spheroidal coordinates used in antenna theory

A NEAR-ZONE TO FAR-ZONE TRANSFORMATION PROCESS UTILIZING A FORMULATED EIGENFUNCTION EXPANSION OF SPHEROIDAL WAVE-HARMONICS Gerald F. Ricciardi Dissertation submitted to the Faculty of the Virginia Polytechnic Institute and State University in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical Engineering Dr. Warren L. Stutzman, Chair Dr. William A. Davis Dr. Sedki M. Riad Dr. John F. Rossi Dr. Ahmad Safaai-Jazi September 21, 2000 Blacksburg, Virginia Keywords: computational electromagnetics, eigenfunctions, FDTD, antennas Copyright © 2000, Gerald F. Ricciardi A NEAR-ZONE TO FAR-ZONE TRANSFORMATION PROCESS UTILIZING A FORMULATED EIGENFUNCTION EXPANSION OF SPHEROIDAL WAVE-HARMONICS Gerald F. Ricciardi (ABSTRACT) In the field of antenna design and analysis, often the need arises to numerically extrapolate the far-zone performance of a radiating structure from its known (or assumedknown) near-zone electromagnetic field. Mathematical processes developed toaccomplish such a task are known in the literature as near-zone to far-zonetransformations (NZ-FZTs) as well as near-field far-field (NF-FF) transformations. Theseprocesses make use of sampled near-zone field quantities along some virtual surface, viz.,the transformation surface, that surrounds the radiating structure of interest. Dependingupon the application, samples of the required near-zone field quantities are supplied viaanalytical, empirical, or computational means. Over the years, a number of NZ-FZT processes have been developed to meet the demands of many applications. In short, their differences include, but are not limited to,the following: (1) the size and shape of the transformation surface, (2) the required near-zone field quantities and how they are sampled, (3) the computational methodology used,and (4) the imbedding of various application-driven features. Each process has its prosand cons depending upon its specific application as well as the type of radiation structureunder consideration. In this dissertation we put forth a new and original NZ-FZT process that allows the transformation surface along which the near-zone is sampled to be spheroidal inshape: namely a prolate or oblate spheroid. Naturally, there are benefits gained in doingso. Our approach uses a formulated eigenfunction expansion of spheroidal wave-harmonics to develop two distinct, yet closely related, NZ-FZT algorithms for each typeof spheroidal transformation surface. The process only requires knowledge of the E-fieldalong the transformation surface and does not need the corresponding H-field. Given is a systematic exposition of the formulation, implementation, and verification of the newly developed NZ-FZT process. Accordingly, computer software isdeveloped to implement both NZ-FZT algorithms. In the validation process, analyticaland empirical radiation structures serve as computational benchmarks. Numerical modelsof both benchmark structures are created by integrating the software with a field solver,viz., a finite-difference time-domain (FDTD) code. Results of these computer models arecompared with theoretical and empirical data to provide additional validation. iiiThe highest result of education is tolerance. Helen Keller (1880-1968) ACKNOWLEDGMENTS First and foremost, I need to thank God; for without divine inspiration and guidance, none of this work would have been possible. Special thanks to my Ph.D. dissertation advisor, Professor Warren L. Stutzman, for allowing me to conduct this endeavor under his tutelage. It was through his patience andunderstanding that this research was able to be completed. I have truly benefited from hisvaluable technical advice throughout my studies here at Virginia Tech. I would like to thank all of my Ph.D. committee members: Professor William A. Davis, Professor Sedki M. Riad, Professor John F. Rossi, and Professor Ahmad Safaai-Jazi. I appreciatethe time and effort they put forth in serving on my committee. Over the years, all of them havecontributed greatly to my growth academically and as a researcher; for that, I am truly grateful. I wish to thank Naval Research Laboratory for furnishing the necessary formal reports that were key to the development of the spheroidal special function routines. With regard to the microstrip patch antenna used as our empirical benchmark, I would be remiss if I did not acknowledge the following institutions and people that were instrumental to itsdevelopment: Rogers Corporation, for donating the costly Duroid substrate material (theirsupport of pure research should be commended); Randall Nealy, for fabricating and testing themicrostrip antenna at Virginia Tech; and NASA Langley and Dr. Dave Shively, for providing thesecond set of pattern data. I am deeply indebted to Dr. Boris Davidson and Matt Monkevich, who have consistently supported me during the most trying of times. At certain points, it was their friendship andsincerity that enabled me to continue with this work. In addition, I must also thank Boris fortechnically reviewing Chapter 2; his comments and suggestions are much appreciated. Further, I must express my gratitude to Bernie Michaels for his much-needed assistance in preparing this document for electronic submission. On a personal note, I would like to thank my parents and my family for always being there. And finally, I owe a heartfelt thank-you to my love, Suzanne, for her years of patience,sacrifice, support, and understanding throughout this long and arduous journey. ivTABLE OF CONTENTS ABSTRACT ....................................................................................................................... ........................... ii ACKNOWLEDGMENTS................................................................................................................ ........... iii CHAPTER 1. INTRODUCTION......................................................................................................... ....1 1.1 BRIEF OVERVIEW OF NZ-FZT RESEARCH AND DEVELOPMENT.........................3 1.2 MOTIVATION FOR THE NEWLY DEVELOPED NZ-FZT PROCESS .........................51.3 ORGANIZATION OF DISSERTATION...........................................................................7 CHAPTER 2. FORMULATION OF THE THEORY AND ALGORITHMIC IMPLEMENTATION OF THE NEWLY DEVELOPED NEAR-ZONE TO FAR-ZONE TRANSFORMATION PROCESS ........................................10 2.1 DETAILED SYSTEM OVERVIEW OF THE NZ-FZT PROCESS ................................11 2.1.1 NZ-FZT Process Input (Transformation Surfaces and Required Near-Zone Fields)........................................................................11 2.1.2 NZ-FZT Process Output (Far-Zone Response)...................................................19 2.2 FORMULATION OF THE NZ-FZT BOUNDARY VALUE PROBLEMS AND CONSTRUCTION OF THE SOLUTIONS .....................................22 2.2.1 Governing Differential Equation (Three-Dimensional Vector Helmholtz Equation)..............................................22 2.2.2 Reduction of the Vector Helmholtz Equation Using a Hybrid of Coordinate Systems (Scalar Spheroidal/Vector Cartesian)............................23 2.2.3 Break Down of the Resulting Scalar Helmholtz Equations into a Set of Ordinary Differential Equations Using the Separation of Variables Method.........................................................................27 2.2.4 Review of Applicable Sturm-Liouville Theory and Self-Adjoint Boundary Value Problems.......................................................32 2.2.5 General Solutions of the Scalar Helmholtz Equation in Spheroidal Coordinates..................................................................................40 2.2.6 Statement of Boundary Conditions Imposed by the NZ-FZT Process.......................................................................................57 2.2.7 Development of Spheroidal Wave-Harmonics....................................................59 v2.2.8 Construction of the Particular Solutions Using Expansions of Spheroidal Wave-Harmonics ......................................................63 2.3 THE NZ-FZT ALGORITHMS FOR THE PROLATE AND OBLATE SPHEROIDAL CASES ..........................................................................69 2.3.1 Algorithmic Implementation of the Constructed Particular Solutions............................................................................................69 2.3.2 Extension of the Algorithmic Implementation for the Constructed Particular Solutions to the Required Vector Solutions.......................................83 CHAPTER 3. NUMERICAL EVALUATION OF REQUIRED SPECIAL FUNCTIONS.................................................................................................87 3.1 OVERVIEW OF SPECIAL FUNCTION NUMERICAL ROUTINES............................88 3.1.1 Description and Organization of Top-Level Numerical Routines ......................883.1.2 Numerical Precision and Accuracy ....................................................................95 3.2 PROLATE AND OBLATE SPHEROIDAL EIGENVALUES ........................................97 3.2.1 PROAPPOXEIG (P-1) and OBLAPPROXEIG (O-1) Numerical Routines ............................................................................................97 3.2.2 JACOBI (B-a) Numerical Routine ....................................................................101 3.3 STURM-LIOUVILLE NORMALIZATION CONSTANTS FOR THE PROLATE AND OBLATE SPHEROIDAL ANGLE FUNCTIONS ............................103 3.3.1 PRONORMCNSTS (P-2) and OBLNORMCNSTS (O-2) Numerical Routines ..........................................................................................103 3.3.2 PRODNCNSTS (P-d) and OBLD NCNSTS (O-d) Numerical Routines ..........................................................................................105 3.4 PROLATE SPHEROIDAL RADIAL FUNCTIONS OF THE FIRST AND SECOND KIND .........................................................................111 3.4.1 PRORADIAL (P-3) Numerical Routine ............................................................1113.4.2 SJBESARRAY (B-b) and SYBESARRAY (B-c) Numerical Routines ..........................................................................................120 3.4.3 PRONEGDNCNSTS (P-e) Numerical Routine .................................................1233.4.4 CALCPQ (P-f) Numerical Routine ...................................................................128 3.5 OBLATE SPHEROIDAL RADIAL FUNCTIONS OF THE FIRST AND SECOND KIND .........................................................................137 vi3.5.1 OBLRADIAL (O-3) Numerical Routine............................................................137 3.5.2 BABHASCNSTS (O-e) Numerical Routine .......................................................1433.5.3 CALCQIM (O-f) Numerical Routine.................................................................146 3.6 PROLATE AND OBLATE SPHEROIDAL ANGLE FUNCTIONS .............................154 3.6.1 PROANG (P-4) and OBLANG (O-4) Numerical Routines ...............................154 3.7 INTEGRATION OF PROLATE AND OBLATE SPHEROIDAL ANGLE FUNCTIONS ..........................................................................157 3.7.1 INTPROANG (P-5) and INTOBLANG (O-5) Numerical Routines ..........................................................................................157 CHAPTER 4. COMPUTER SOFTWARE BASED UPON THE NEWLY DEVELOPED NEAR-ZONE TO FAR-ZONE TRANSFORMATION PROCESS..........................162 4.1 OVERVIEW OF NZ-FZT COMPUTER SOFTWARE..................................................163 4.1.1 Organization of Software..................................................................................1634.1.2 Numerical Precision .........................................................................................1654.1.3 Computer Platform and Programming Language............................................166 4.2 DESCRIPTION OF PHASE I MODULES ....................................................................167 4.2.1 Module (M-1): Input of User Specified Parameters .........................................1674.2.2 Module (M-2): Allocation and Initialization of Data Structures......................1684.2.3 Module (M-3): Read File of Near-Zone Field Samples....................................1684.2.4 Module (M-4): Eigenvalue Computation (First Approximation)......................1694.2.5 Module (M-5): Computation of Sturm-Liouville Normalization Constants and Eigenvalue Refinement .............................................................169 4.2.6 Module (M-6): Setup Spheroidal Sample Grid.................................................1704.2.7 Module (M-7): Computation of Expansion Quantities .....................................1704.2.8 Module (M-8): Computation of Integration Matrices ......................................171 4.3 DESCRIPTION OF PHASE II MODULES ...................................................................172 4.3.1 Module (M-9): Advance to Next Far-Zone Observation Angle ........................1724.3.2 Module (M-10): Far-Zone E-Field Computation via the Spheroidal Wave-Harmonic Expansion ...........................................................172 vii 4.4 DESCRIPTION OF PHASE III MODULE....................................................................174 4.4.1 Module (M-11): Write Far-Zone Fields to Output File....................................174 4.5 IMPROVEMENT OF COMPUTATIONAL EFFICIENCY ..........................................174 4.5.1 Implementation of Memory Caching Technique...............................................174 CHAPTER 5. PROOF-OF-CONCEPT VIA AN ANALYTICAL BENCHMARK ........................178 5.1 EXACT E-FIELD SURROUNDING A FILAMENT DIPOLE .....................................179 5.1.1 Near-Zone E-Field............................................................................................179 5.2 ASYMPTOTIC REDUCTION OF THE EXACT E-FIELD TO OBTAIN THE FAR-ZONE APPROXIMATION....................................................181 5.2.1 Far-Zone E-Field..............................................................................................181 5.3 DEVELOPMENT OF AN ANALYTICAL BENCHMARK SOFTWARE DRIVER ...................................................................................................181 5.3.1 Overview of Software Driver ............................................................................181 5.4 COMPARISON OF ALGORITHMIC RESULTS WITH THEORETICAL RESULTS...........................................................................................182 5.4.1 Centered Dipole Numerical Tests.....................................................................1825.4.2 Offset Dipole Numerical Tests..........................................................................188 CHAPTER 6. DEVELOPMENT OF A COMPUTATIONAL ELECTROMAGNETIC FIELD SOLVER ................................................................202 6.1 FDTD FUNDAMENTALS.............................................................................................203 6.1.1 Overview of Computational Technique ............................................................2036.1.2 Free-Space (or Non-Lossy Dielectric) E-Field Formulation............................2106.1.3 Perfect Electric Conductor E-Field Formulation.............................................2126.1.4 Lossy Dielectric E-Field Formulation..............................................................2126.1.5 Non-Magnetic H-Field Formulation ................................................................2146.1.6 Second-Order Mur Absorbing Boundary Condition ........................................2166.1.7 Modeling Guidelines.........................................................................................221 6.2 NZ-FZT/FDTD SOFTWARE DEVELOPMENT...........................................................224 viii6.2.1 Overview of Software........................................................................................224 6.2.2 Extraction of Near-Zone E-Field Samples Along the Prolate/Oblate Spheroidal Surface...................................................................229 6.3 NZ-FZT/FDTD END-TO-END TEST EMPLOYING THE ANALYTICAL BENCHMARK............................................................................244 6.3.1 NZ-FZT/FDTD Model of the Analytical Benchmark ........................................2446.3.2 Comparison of NZ-FZT/FDTD Results with Theoretical Results.....................251 CHAPTER 7. FURTHER VALIDATION VIA AN EMPIRICAL BENCHMARK .......................262 7.1 MICROSTRIP PATCH ANTENNA ..............................................................................263 7.1.1 Design Layout and Fabrication........................................................................2637.1.2 NASA Langley and Virginia Tech Measurements of Far-Zone Radiation Patterns............................................................................268 7.2 EMPIRICAL VALIDATION OF THE NZ-FZT PROCESS..........................................269 7.2.1 NZ-FZT/FDTD Model of the Empirical Benchmark.........................................2697.2.2 Comparison of NZ-FZT/FDTD Model Results with Measured Results......................................................................................278 CHAPTER 8. SUMMARY AND CONCLUSIONS ...........................................................................283 APPENDIX A. PROLATE AND OBLATE SPHEROIDAL COORDINATE SYSTEMS..........................................................................................286 APPENDIX B. SCALAR LAPLACIAN IN SPHEROIDAL COORDINATES.................................292APPENDIX C. SEPARATION OF VARIABLES OF THE SCALAR HELMHOLTZ EQUATION IN SPHEROIDAL COORDINATES....................................................294 APPENDIX D. LEGENDRE FUNCTIONS..........................................................................................298APPENDIX E. DOUBLE- TO SINGLE-SIDED SPHEROIDAL WAVE-HARMONIC IDENTITY ...............................................................................309 REFERENCES ..................................................................................................................... .....................311 VITA........................................................................................................................... ................................317 1CHAPTER 1. INTRODUCTION In the field of antenna design and analysis, often the need arises to numerically extrapolate the far-zone performance of a radiating structure from its known (or assumedknown) near-zone electromagnetic field. Mathematical processes developed toaccomplish such a task are most popularly known in the literature as near-zone to far-zone transformations (NZ-FZTs) as well as near-field far-field (NF-FF) transformations.In general, most transformations make use of sampled near-zone field quantities alongsome particular closed surface that completely encloses the radiating structure of interest.Samples of the required near-zone field quantities are then supplied via analytical,empirical, or computational means. Yet, other transformations only use near-zone fieldsamples along certain portions of a closed surface where the source of radiation isconsidered most concentrated; consequently, the near-zone quantities along the other less-contributory portions of the closed surface are approximated as zero. Either way, the far-zone response of the radiating structure is then computed from the sampled near-zonedata using a numerical process that is well-rooted in electromagnetic theory. Throughoutour discussion we shall refer to this virtual surface along which the near-zone fieldquantities are sampled as the transformation surface. Over the years, a number of NZ-FZT processes have been developed to meet the demands of many different applications. For the most part, the differences in theseprocesses include, but are not limited to, the following: (1) the size and shape of thetransformation surface, (2) the required near-zone field quantities and the manner inwhich they are sampled, (3) the computational methodology used, and (4) the imbeddingof various application-driven features into the NZ-FZT process. Each of the developedprocesses has its advantages and disadvantages depending upon its specific application aswell as the type of radiation structure under consideration. The thrust of this dissertation is to put forth a new and original NZ-FZT process that allows the transformation surface along which the near-zone is sampled to bespheroidal in shape: namely a prolate or oblate spheroid. Naturally, there are benefits 2gained in doing so. Our approach makes use of a formulated eigenfunction expansion of spheroidal wave-harmonics to develop two distinct, yet closely related, NZ-FZTalgorithms for each type of spheroidal transformation surface. Throughout thisdissertation, we provide a systematic exposition of the formulation, implementation, andverification of our newly developed NZ-FZT process, as well as a demonstration of anovel application. Implementation of both algorithms involves the development ofcomputer software able to carry out the entire NZ-FZT process from start-to-finish in acomputationally efficient manner. Not only must this software include the calculation ofthe formulated eigenfunction expansion, but also the much difficult numerical evaluationof all required special functions. Remarkably, the process is quite practical from anumerical viewpoint in that it is able to be hosted in its entirety on an IBM compatiblepersonal computer. Verification and performance assessment of the new algorithms require the development of both an analytical and an empirical radiation structure to use ascomputational benchmarks. First, a proof-of-concept of the NZ-FZT process is achievedby seeing how well the algorithms are able to compute the far-zone of the analyticalbenchmark. The developed computer software is then integrated with a computationalelectromagnetic field solver (an additional piece of developed software) in order to createnumerical models of both benchmark structures. Ultimately, results from these originalcomputer models are compared with respective theoretical and empirical data to provideadditional validation of our NZ-FZT process. Lastly, we would like to point out thatalthough the joining of the newly developed NZ-FZT process with a computationalelectromagnetic field solver demonstrates one particular application of this work, thesignificance of this research is much more far-reaching and has yet to be explored. 31.1 BRIEF OVERVIEW OF NZ-FZT RESEARCH AND DEVELOPMENT First, we would like to make it clear that it is not our intention to provide another exhaustive account of NZ-FZT research and development: for that, we refer the reader tothe comprehensive survey articles by Johnson et al. [1] and Yaghjian [2]. Both of these works provide an excellent compendium of the many NZ-FZT methods with respect totheir chronological development. Included in these seminal papers are the underlyingtheories behind each of the techniques as well as their various advantages and limitations.However, we shall delve briefly into the history of NZ-FZT methods so as to gain somecontextual understanding of our newly developed NZ-FZT process. Much of the research in this area is driven by the need to determine far-zone antenna patterns from measurements made in the near-zone. Often it is impractical orimpossible to measure radiation patterns of an antenna on a conventional far-field range.Reasons for this difficulty can include, but are not limited to, the following: the distanceto the far-zone may be too long for a given far-field range; or it may be just downrightimpractical to move the antenna from its operating environment to an antenna range [1].For these and other reasons, it is this particular application that spawned initialdevelopment of NZ-FZT techniques and subsequently led to development of near-fieldantenna ranges. According to [2], Richmond and Tice [3], [4] in 1955 put forth the earliest papers to research computation of far-zone radiation patterns from near-zone measurements.Their investigation used air and dielectric-filled open-ended rectangular waveguideprobes for measuring the near-zone of microwave antennas; the calculated far-zone fieldswere then compared with those measured directly. Later, Kerns [5] used plane-waveanalysis to predict the far-zone radiation pattern from near-zone data for a planartransformation surface and included probe compensation to account for the presence ofthe probe. Although sampling of fields along a planar transformation surface is usuallyconducted over a rectangular grid, Rahmat-Samii, Galindo-Israel, and Mittra [6] describe 4an alternative NZ-FZT process that allows the sampling along the planar surface to occur using a plane-polar grid; the transformation is formulated in terms of a Jacobi-Besselseries expansion. Additionally, Williams and Rahmat-Samii [7] developed another NZ-FZT process that uses a bi-polar grid to sample the planar transformation surface. In1961, Brown and Jull [8] put forth a two-dimensional NZ-FZT for a cylindricaltransformation surface (no dependence on the z-coordinate) in which the radiating field isexpanded in terms of cylindrical wave functions. However, it was not until Leach andParis [9] of the Georgia Institute of Technology that the NZ-FZT theory for sampling thenear-zone along a cylindrical surface was fully extended to three dimensions. Jensen [10]of the Technical University of Denmark developed a NZ-FZT for a sphericaltransformation surface that expands the radiating field in terms of spherical wavefunctions. Similarly, James and Longdon [11] describe an alternative method that obtainsthe expansion of spherical wave functions from measurements of the radial component ofthe E- and H-fields over the spherical transformation surface. Beyond just the theory,several institutions have also been extensively involved with the actual design andimplementation of near-field ranges (including practical considerations): the NationalInstitute of Standards and Technology (formerly the National Bureau of Standards) [12];Georgia Institute of Technology [13]; the Technical University of Denmark [14]; andUniversity of California, Los Angeles [15]. Although initial NZ-FZT research stemmed from the development of near-field ranges, these methods have continued to find their way into a host of other applicationsthat are just too numerous to mention. However, we shall cite a few of these applicationsso as to illustrate the widespread use of NZ-FZT methods. While developing designtechniques for Cassegrainian-fed paraboloids, Ludwig [16] formulated an inverse NZ-FZT for a spherical transformation surface that uses an expansion of spherical wavefunctions to transform the far-zone pattern of a given feed antenna to its respective near-zone pattern. In the same vein, Lee et al. used an inverse NZ-FZT for a planar transformation surface to locate defective elements in an array antenna [17]. Rahmat-Samii and Lemanczyk [18] made use of a NZ-FZT process to determine surface distortion 5profiles of a parabolic reflector antenna. Moreover, NZ-FZT methods have also found their way into the realm of computational electromagnetics. Specifically, Taflove andUmashankar [19], [20] used the finite-difference time-domain (FDTD) method coupledwith a NZ-FZT method based on field equivalence theory to analyze electromagneticscattering from complex objects. Later, Luebbers et al. [21] expanded this particular approach to carry out the NZ-FZT in the time-domain. As one can see, NZ-FZT methodshave and continue to play a significant role in the field of antenna design and analysis. 1.2 MOTIVATION FOR THE NEWLY DEVELOPED NZ-FZT PROCESS One advantage that NZ-FZT methods with a spherical transformation surface have over those that are cylindrical and planar based is the ability to provide full coverage ofthe far-zone radiation pattern (i.e., in all directions). In general, methods that employ aplanar transformation surface are confined to determining the fields within the forwardsolid angular region subtended by the edges of the radiating structure and the finitesampling area; consequently, there is limited sidelobe information and no backlobeinformation [2]. Similarly, NZ-FZT methods that use a cylindrical transformation surfacemust exclude the biconical angular region formed by the outer edges of the radiatingstructure and the cylindrical sample area of finite height; accordingly, sidelobeinformation is not available in the far-zone region that corresponds to the top and bottomof the cylindrical surface [2]. On the flip side, the spherical transformation surface must have a radius that is large enough to enclose the radiating structure of interest. For structures that are roughlythe same size in all three Cartesian dimensions, the spherical transformation surface is, bynature, quite appropriate. In these instances, the near-zone is sampled (along the sphere)at distances from the structure that are more or less on the same order. Unfortunately, incases where the radiating structure is planar or linear in shape, sampling of the near-zonemust occur at some points (along the sphere) that are relatively distant from the radiating 6structure. Depending upon the application, not being able to sample the fields close to the radiating structure may pose some difficulties. This lack in capability is one of thereasons why NZ-FZT methods based on planar and cylindrical transformation surfacesare often better suited for planar and linear radiating structures, respectively, than theirspherical based counterparts. All told, we desire a NZ-FZT process for linear and planar structures that is able to deliver the best of both worlds: full coverage of the far-zone radiation pattern whileallowing all of the near-zone samples to remain close to the radiating structure. Towardthis end, we consider geometrically stretching and squashing the spherical transformation surface into prolate and oblate spheroids, respectively. The advantage of doing so is thatthese particular transformation surfaces are capable of closely conforming to both linearand planar types of radiating structures: the prolate spheroid is better suited for linearstructures (e.g., a dipole antenna), and the oblate spheroid is better suited for planarstructures (e.g., a microstrip patch antenna). In essence, the spheroidal surface affords anextra degree of freedom over its spherical counterpart, thus providing the user with addedcontrol over the shape of the transformation surface. (Note that a sphere requires onlyone parameter, i.e., its radius, to describe its geometry; both the prolate and oblatespheroid require two parameters. This difference in specification follows because thesphere is actually a degenerate case of the spheroid.) As we can see, developing a NZ-FZT process that employs a prolate and oblate spheroidal transformation surface can be extremely beneficial. However, Yaghjian [2]states that it is impractical to find the necessary dyadic Green’s function fortransformation surfaces that do not support orthogonal vector wave functions. Morse andFeshbach [22] show that there are only six coordinate systems that support orthogonalvector wave solutions: Cartesian, circular cylinder, spherical, elliptic cylinder, paraboliccylinder, and conical. Much to our chagrin, the two coordinate systems that are bestsuited to describe both spheroidal transformation surfaces, namely prolate spheroidal andoblate spheroidal, are not among these six special coordinate systems. Nevertheless, bothspheroids (prolate and oblate) can still be made to successfully serve as transformation 7surfaces despite not being able to support orthogonal vector wave functions. As we shall see in Chapter 2, our newly developed NZ-FZT process gets around this roadblock byemploying a differential equation approach that solves the three-dimensional vectorHelmholtz equation using a hybrid of coordinate systems. Finally, we address the methodology behind the formulation of our newly developed NZ-FZT process. Although a much simpler NZ-FZT process for a spheroidaltransformation surface could have been developed based on field equivalence theory, amajor drawback of this approach is that it would require knowledge of both E- and H-fields along the transformation surface. The advantage of formulating our NZ-FZTprocess in terms of an eigenfunction expansion of spheroidal wave-harmonics is that onlyone of the two fields along the transformation surface has to be known, not both.Naturally, this reduction in informational content of the input to the NZ-FZT process canprove to be quite indispensable depending upon the application. 1.3 ORGANIZATION OF DISSERTATION Essentially, the dissertation is divided into two parts. Chapters 2, 3, and 4 address the formulation and computational implementation of the newly developed NZ-FZTprocess. Thereafter, Chapters 5, 6, and 7 form the second part of this work that focuseson providing analytical and empirical validation of our spheroidal based NZ-FZT process.Lastly, Chapter 8 provides summary and conclusions of the research. In detail, Chapter 2 discusses the formulation of the theory and algorithmic implementation of the newly developed NZ-FZT process. The first part of the chaptergives a detailed system overview of the NZ-FZT process and explicitly describes what theprocess actuality does. The remainder of the chapter presents a full discourse on thedevelopment of the computational algorithms along with all of the supportingelectromagnetic theory. In addition, the appendices at the end of the document providethe necessary mathematical background that is key to the formulation of our NZ-FZTprocess. 8Chapter 3 is devoted to the numerical evaluation of all required special functions associated with the eigenfunction expansion of spheroidal wave-harmonics. It addressesthe details and issues regarding the development of numerical routines able to calculatethese much involved special functions (i.e., while operating in the context of a workingcomputational algorithm). Because this chapter can be read, for the most part,independently of the others, it is our hope that it provides the interested reader with someof the best ways to go about evaluating these complicated special functions (with regardto both accuracy and computational efficiency). Chapter 4 addresses the formation of computer software based on our newly developed NZ-FZT process. Our software package employs the algorithms developed inChapter 2 along with the special function routines developed in Chapter 3. In detail weexamine the top-down design of the coded software and discuss the organization andparticulars of its various constituent modules. In Chapter 5, we provide a proof-of-concept of our NZ-FZT process by making use of an analytical benchmark. The selected radiating structure has electromagneticfields that are analytically known in both the near- and far-zone regions. Accordingly, a software driver is written to sample the near-zone of our analytical benchmark along auser specified spheroidal transformation surface. Afterwards, the set of sampled near-zone data is able to be processed by the NZ-FZT software developed in Chapter 4.Ultimately, validation of the NZ-FZT process (and the software) is achieved byconducting a series of numerical tests that compares the algorithmic results withanalytical (theoretical) results for an assortment of spheroidal transformation surfaces. Chapters 6 and 7 have been included to provide further validation of our NZ-FZT process together with a successful demonstration of its application. First, Chapter 6concentrates on developing a computational electromagnetic field solver, viz., a FDTDcomputer code, that is able to model the near-zone of a given radiating structure.Naturally, the field solver is designed to interface with the NZ-FZT software packagedeveloped in Chapter 4. Subsequently, an end-to-end test of the combined software isconducted using a computational model of our analytical benchmark (of Chapter 5) in 9order to validate the overall coupled process. In Chapter 7, we use the combined software to model an empirical benchmark radiating structure that has been fabricated and thenpattern tested at both NASA Langley and Virginia Tech. Additional validation of ourNZ-FZT process is achieved by comparing the results from our computer simulation withthe far-zone radiation patterns measured at both test facilities. 10CHAPTER 2. FORMULATION OF THE THEORY AND ALGORITHMIC IMPLEMENTATION OFTHE NEWLY DEVELOPED NEAR-ZONE TOFAR-ZONE TRANSFORMATION PROCESS As previously stated, the aim of our developed NZ-FZT process is to extrapolate the far-zone of a radiating structure from near-zone field samples positioned along aspheroidal transformation surface. Specifically, the input to our process only requires thenear-zone field samples to be of the time-harmonic E-field (that is, knowledge of itscorresponding H-field is not required). Once supplied with this information, the NZ-FZTprocess is then able to determine the desired time-harmonic far-zone response. Becausethe process operates in the frequency-domain, all field quantities throughout the ensuing formulation are in time-harmonic form and thus suppress the tjeω time dependence. A differential equation approach is taken in the development of our NZ-FZT process. The approach first requires us to pose boundary value problems that describe thephysical process. The particular solution to each of these boundary value problems isthen constructed using a suitable expansion of eigenfunctions. Ultimately, the formulatedprolate and oblate spheroidal NZ-FZT algorithms use these eigenfunction expansions tocompute the far-zone response. It is the objective of this chapter to present the theory andthe algorithmic details behind the development of this new NZ-FZT process. Throughout this development we shall observe that the formulations for both the prolate and oblate transformation surfaces closely parallel one another. Accordingly, itstands to reason to develop both formulations concurrently throughout this chapter. Thisis done in a step-by-step fashion by first stating equations and figures for the prolate caseand then following them with their respective oblate counterpart. Both parallel sets ofequations and figures share the same numbering with the only difference being anappended suffix, viz., an “a” or “b,” to distinguish between the prolate and oblate cases,respectively. For the sake of brevity, at some points in the development we choose todiscuss only the prolate case since both spheroidal cases closely resemble one another. 11Consequently, some of the prolate equations are not paired with their oblate counterpart. If need be, these omitted expressions pertaining to the oblate case can be easily derivedfrom their disclosed prolate counterparts. Nevertheless, in these particular instances, weshall maintain the same numbering system and apply the appended suffix “a” to these prolate equations and figures even though their oblate counterparts have been excluded. Lastly, it must be stated that the findings presented in this dissertation are application driven in nature. These results are directed toward applied physicists andengineers practicing computational electromagnetics, and not pure mathematicians.Accordingly, in the formulation that follows, the existence of solutions, completeness ofsets of functions, and permissibility of interchanging order of integration operators andinfinite summations (i.e., termwise integration) are all presumed, not demonstrated. 2.1 DETAILED SYSTEM OVERVIEW OF THE NZ-FZT PROCESS 2.1.1 NZ-FZT Process Input (Transformation Surfaces and Required Near-Zone Fields) The first step in the development of any computational algorithm that approximates a given physical system is to understand the details regarding the systemexcitation (input) and the system response (output). Here, we examine the inputinformation required to implement this new NZ-FZT process. Understanding the input tothe process calls for descriptions of the transformation surfaces and the required near-zone fields. Regarding the desired output response, the section that immediately followsthis one addresses its specific details. For reasons beforehand mentioned in Chapter 1, there are advantages to choosing the transformation surface as either a prolate or oblate spheroid. In doing so, thesesurfaces must first be defined in their respective curvilinear coordinate systems alongwith their relation to Cartesian coordinates. In short, we shall briefly describe the detailsof both the prolate and oblate spheroidal coordinates that are immediately required. 12However, the reader is referred to Appendix A for additional details regarding both of these closely related coordinate systems. It must be stated that unlike the Cartesian and spherical coordinate systems, the pair of spheroidal coordinate systems does not universally conform to a conventionaldefinition and notation. Consequently, one must be careful to avoid the pitfallsassociated with inconsistent definitions and notations when using formulated results frommultiple sources. Generally speaking, both the prolate and oblate spheroidal coordinatesystems are generated by revolving the two-dimensional elliptic curvilinear coordinatesystem (where the orthogonal coordinates are comprised of a family of confocal ellipsesand hyperbolas) about either the major axis (prolate case) or the minor axis (oblate case)of the confocal ellipses. Unfortunately, the lack of a single definition for the spheroidalcoordinate system occurs because multiple versions exist that are able to satisfy thisrequirement. Upon review of the various versions of spheroidal coordinate systems, we found that there are essentially two degrees of freedom responsible for creating the differences:the orientation of the family of ellipses (which determines the axis of revolution) and thereference axis (or plane) of the angular coordinate. Both Flammer [23], Morse andFeshbach [22], and Tai [24] define the z-axis as the axis of revolution for both the prolateand oblate spheroidal coordinate systems. As for AMS-55 [25], the axis of revolution is defined as the x-axis in the prolate case and the y-axis in the oblate case. Althoughsources [22]-[24] remain consistent and designate the z-axis as the reference of theangular coordinate for the prolate spheroidal coordinate system, this is not the case for itsoblate counterpart. Specifically, Tai [24] designates the x-y plane (effectively the ρ- axis) as the reference for the angular coordinate for the oblate spheroidal coordinate system; meanwhile, Flammer [23] and Morse and Feshbach [22] both retain the z-axis asthe angular coordinate reference. As we have shown, there are multiple definitions of the prolate and oblate spheroidal coordinate systems. In order to continue, we must select a pair of spheroidalcoordinate systems that is best suited for our particular application: namely, the pair of 13spheroidal coordinate systems defined by Flammer [23] and Morse and Feshbach [22]. First, since we are dealing with antenna radiation, it is beneficial to remain consistentwith the practices of antenna theory and choose the z-axis as both the reference axis ofthe angular coordinate and the preferred axis of revolution. Second, this choice ofreference axis (for the angular coordinate) remains consistent with spherical coordinates,which also defines its polar angle with respect to the z-axis. As we shall see later on inour discussion, having the spheroidal and the spherical coordinate systems share the z-axis as the preferred axis will prove to be convenient when relating these coordinatesystems. Finally, most of the existing work on spheroidal wave functions is defined interms of the z-axis preferred spheroidal coordinate system. Although this existing workcan be made to account for another preferred axis, doing so has no apparent benefits. The prolate and oblate spheroidal coordinate systems, shown in Figures 2.1a and 2.1b, respectively, are related to Cartesian coordinates by the following transformations[23]: Prolate: xa a=− − =− ξη ϕ ξ ϑ ϕ22 211 1 cos sin cos (2-1a) ya a=− − =− ξη ϕ ξ ϑ ϕ22 211 1 sin sin sin (2-2a) za a==ξη ξ ϑ cos (2-3a) and −≤ ≤11η ()0≤≤ϑπ , 1≤< ∞ξ , 0 2≤≤ϕπ ; (2-4a) Oblate: xa a=+ − =+ ξη ϕ ξ ϑ ϕ22 211 1 cos sin cos (2-1b) ya a=+ − =+ ξη ϕ ξ ϑ ϕ22 211 1 sin sin sin (2-2b) za a==ξη ξ ϑ cos (2-3b) and −≤ ≤11η ()0≤≤ϑπ , 0≤< ∞ξ , 0 2≤≤ϕπ ; (2-4b) where ηϑ=cos in both cases. (2-5) 14z η= η= η=η= η= η= η= η= η= η= η= η=1cos cos cos cos cos -cos-cos-cos-cos0 -1π/12 π/6 π/4 π/3 5π/12 5π/12 π/3 π/4 π/6 η=-cosπ/12aξ^ ^ x,y (or ) ρϕ^ υ1.01.021.077 ξ= ξ=ξ= Figure 2.1a Prolate spheroidal coordinate system [23]. 15az ϕ^ η=η= η= η= η= η= η= η= η= η=η= η=1cos cos cos cos cos 0 -cos -cos -cos -cos -1π/12 π/6 π/4 5π/12 5π/12 π/3 π/4 η=-cosπ/12π/6π/3 x,y (or ) ρξ=ξ= ξ= 00.20.4ξ ^^ υ Figure 2.1b Oblate spheroidal coordinate system [23]. 16Both detailed spheroidal-to-Cartesian transformations, i.e., equations (2-1)-(2-5), indicate that the following parameters must be considered: ξ, η (or ϑ), ϕ, and a. The parameter a represents the distance from the focal point of the confocal ellipses and hyperbolas to the origin of the selected spheroidal coordinate system. Regarding theremaining parameters, they form an ordered triple that represents the coordinates for either spheroidal coordinate system: namely ( , , ) ξηϕ or ( , , ) ξϑϕ (the choice of which depends upon the angular parameter employed). Specifically, the parameter ξ serves as the radial coordinate for both spheroidal coordinate systems. However, unlike the spherical coordinate system, the radial coordinate of the spheroidal coordinate systems does not directly indicate the distance from the origin to the point-of-interest. Instead, ξ is a unitless quantity that implicitly represents the radial distance by indicating whichconfocal ellipsoid of revolution (i.e., ξξ==oconstant surface) intersects with the point- of-interest. In the case of the prolate spheroidal coordinate system, this ellipsoid of revolution (viz., a prolate spheroid) has a major axis of length 2 aξ and a minor axis of length 212aξ−. As for its oblate counterpart, its ellipsoid of revolution (viz., an oblate spheroid) has major and minor axes of lengths 212aξ+ and 2 aξ, respectively. Regarding the parameters η and ϑ, either one can serve as the angular coordinate and is implicitly related to the angle between the point-of-interest and the z-axis. In the case of the prolate spheroidal coordinate system, the surface ηη==oconstant <1 is a hyperboloid of revolution of two sheets with an asymptotic cone whose generating line passes through the origin and is inclined at the angle ϑη=−cos1 to the z-axis [23]. As for its oblate counterpart, the surface ηη==oconstant <1 is in this case a hyperboloid of revolution of one sheet with an asymptotic cone whose generating line also passes through the origin and is inclined at the angle ϑη=−cos1 to the z-axis [23]. Since both parameters ϑ and η are dependent on one another and remain homeomorphic over their limited domains, i.e., maintain a one-to-one mapping, we are able to employ either one as 17the angular coordinate during the formulation of our NZ-FZT process. Still, some care must be exercised whenever making this particular change-of-variable. Lastly, thecoordinate ϕ represents the azimuthal angle between the x-z plane and the plane that passes through both the point-of-interest and the entire z-axis, i.e., ϕϕ==oconstant surface. The spheroidal coordinate ϕ is for all practical purposes the same as the azimuthal coordinate ϕ of both the cylindrical and spherical coordinate systems (or of any other rotational coordinate system). In addition to describing both spheroidal coordinate systems in terms of their scalar coordinates, a review of their respective unit vectors is required. As illustrated inFigures 2.1a and 2.1b, the body-of-revolution confocal quadric surfaces (viz., spheroidsand hyperboloids) intersect each other at right angles. In other words, the tangent planes of all three coordinate surfaces (viz., ξξ=o, ϑϑ=o, and ϕϕ=o constant parameter surfaces) passing through any point-of-interest ( , , ) ξϑϕooo are mutually orthogonal [23]. Consequently, both the prolate and oblate spheroidal coordinate systems are classified as systems of orthogonal curvilinear coordinates. For both spheroidal coordinate systems, the unit vectors /G04ξ, /G04ϑ, and /G04ϕ form a right-handed system that is defined by the following cross products (see Figures 2.1a and 2.1b): /G04 /G04 /G04 ξϑϕ ×= (2-6) /G04 /G04 /G04ϕξϑ ×= (2-7) /G04 /G04 /G04ϑϕξ ×= (2-8) where /G04 /G04ηϑ=− . Having selected and defined an appropriate pair of spheroidal coordinate systems, we are now in a position to mathematically describe our spheroidal transformationsurfaces. From our previous discussion of spheroidal coordinates, the prolate and oblate 18spheroids are easily defined in their respective coordinate systems due to the fact that these surfaces are intrinsically the ξξ=o constant parameter surfaces. Accordingly, these spheroidal transformation surfaces can be formally described as follows: Prolate Surface ΩP:/G26r o(, , )ξηϕξξ== constant= /G26r(, )ηϕ (2-9a) or defined parametrically as /G26rx y z( ( ,) ,(,) , (,) )ηϕ ηϕ ηϕ (2-10a) with xao (,) c o sηϕ ξ η ϕ =− −2 211 yao (,) s i nηϕ ξ η ϕ =− −2 211 zao (, )ηϕ ξη = ; Oblate Surface Ωo:/G26r o(, , )ξηϕξξ== constant= /G26r(, )ηϕ (2-9b) or defined parametrically as /G26rx y z( ( ,) ,(,) , (,) )ηϕ ηϕ ηϕ (2-10b) with xao (,) c o sηϕ ξ η ϕ =+ −2 211 yao (,) s i nηϕ ξ η ϕ =+ −2 211 zao (, )ηϕ ξη = . In both of these descriptions of the prolate and oblate spheroidal transformation surfaces, the vector quantity /G26r represents the position vector in the respective coordinate system. With the transformation surfaces for the NZ-FZT process being mathematically described, our attention is now turned to the required near-zone field quantities. 19Recalling from before that only the time-harmonic E-field along the spheroidal transformation surface is required, we describe the input to our NZ-FZT process as Prolate Process Input:/G26 E P(, , )ξηϕ Ω= /G26 Eo(, , )ξηϕ; (2-11a) Oblate Process Input:/G26 E O(, , )ξηϕ Ω= /G26 Eo(, , )ξηϕ. (2-11b) It is important to understand that the transformation surface is virtual in nature and does not actually alter the operation of the electromagnetic system. That is, the fieldsemanating from the radiating structure are the same with or without the existence of theenclosing transformation surface. Accordingly, the electromagnetic fields sampled alongthis virtual boundary inherently must satisfy Maxwell ’s equations. The fact that all points on the transformation surface lie in free-space implies that the ∇⋅ =/G26 E0 for the specified NZ-FZT process input. 2.1.2 NZ-FZT Process Output (Far-Zone Response) In order to characterize the performance of an antenna, many figures-of-merit have been created and have subsequently evolved through the years to satisfy the needs ofan ever-changing technology. Albeit an all inclusive list would be too large to mention,some of the basic parameters that characterize the performance of an antenna are asfollows: radiation pattern, radiation intensity, directivity, gain, antenna efficiency,bandwidth, polarization, and input impedance. The reader is referred to any text onantenna theory, e.g., [26] or [27], for a more comprehensive review of these basic antennaperformance parameters. Due to the fact that some of these parameters are interrelated,not all of them must be specified in order to give a complete characterization of antennaperformance [27]. As a result, we shall use our NZ-FZT process to determine a quantitythat is so essential that it serves as the foundation for many antenna performanceparameters: the far-zone E-field. This fundamental quantity is so important because it 20can subsequently be used to derive many of the aforementioned antenna performance parameters. Despite having identified the far-zone time-harmonic E-field as the NZ-FZT process output, we now must consider the most suitable form for this quantity to assume.The first issue that must be addressed is the choice of coordinate system to express theprocess output. Upon first consideration, it logically appears that the NZ-FZT processinput and output should share the same coordinate system: namely, the spheroidalcoordinate system. Unfortunately, antenna researchers and designers do not operate interms of this unusual coordinate system. Bearing this in mind, it is desirable to put theNZ-FZT process output, i.e., the far-zone E-field, in the more palatable sphericalcoordinate system and remain consistent with the conventions of antenna theory. Thischoice of using the spherical coordinate system to express the far-zone E-field will notonly apply to the independent coordinates of the scalar functions, but also to its respectivevector quantities. In the far-zone of any radiating structure (in terms of spherical coordinates), it can be shown that the radial dependence, i.e., r, becomes separable from the angular dependence, i.e., θ and ϕ, and can be expressed as follows [27]: l i m (, , ) (, , ) (, )rjkr Er E r Ee rfz fz →∞− ==/G26/G26 /G26 θϕ θϕ θϕ (2-12) where /G26 EE Efz fz fz(, ) (, )/G04(, ) /G04 θϕ θϕθθ ϕϕ θ ϕ=+ . (2-13) It must be pointed out that these expressions are asymptotic approximations of the exact far-zone E-field representations. Even though higher-order terms of 1 / rn for n>1 do exist, in the far-zone they can be safely neglected because they vanish in the limit as 21r→∞ . Once again the reader is referred to any text on basic antenna theory regarding this matter. Note that in expressions (2-12) and (2-13) a bar is placed over the E-field quantities in order to indicate within the notation the suppression of the radial factor ()erjkr−. There are advantages to representing the far-zone E-field with a suppressed radial dependence in much of the same way that the tjeω time dependence is suppressed during the use of phasor notation. With regard to this particular notation, it allows us to avoid having to tote the cumbersome expression throughout the ensuing formulation, i.e.,its presence is implied. Moreover, far-zone parameters most often characterize antennaperformance with respect to only angular variations due to the fact that the form of theradial dependence is known. This is exemplified by the expression for radiation intensitywhich demonstrates that the radial dependence ultimately drops out of the definition [26],[27]: UrEr ofz(, ) (,, )θϕηθϕ =2 2 2/G26 (2-14) and subsequently, UE ofz(, ) (, )θϕηθϕ =1 22 /G26 =+  1 22 2 ηθϕ θϕ θ ϕoEEfz fz( , ) ( , ) (2-15) where ηo is the intrinsic impedance of free-space. Accordingly, we designate the far-zone E-field expressed in conventional spherical coordinates with a suppressed radial dependence, viz., equation (2-13), as the NZ-FZTprocess output. Note that the Sommerfeld radiation condition (see 2.2.6) implies that ∇⋅ =/G26 E0 in the limit as r→∞ for the specified NZ-FZT process output. 222.2 FORMULATION OF THE NZ-FZT BOUNDARY VALUE PROBLEMS AND CONSTRUCTION OF THE SOLUTIONS 2.2.1 Governing Differential Equation (Three-Dimensional Vector Helmholtz Equation) The first step in the formulation of the NZ-FZT process as a boundary value problem is to understand the differential equation that governs electromagnetic radiationin free-space. Our assumption of free-space radiation is justified because the region ofinterest that lies on and outside our closed boundary (i.e., the transformation surface) isconsidered to have no sources and/or any additional media other than free-space (i.e., o oεµ,) . In order to derive the governing differential equation, we must begin with the time-harmonic, differential form of Maxwell ’s equations for free-space: ∇× =−/G26/G26 Ej Hoωµ (2-16) ∇× =/G26/G26 Hj Eoωε (2-17) ∇⋅ =/G26 E0 (2-18) ∇⋅ =/G26 H 0. (2-19) Note that equations (2-16)-(2-19) state Maxwell ’s equations in terms of only E- and H- fields and make use of the free-space constitutive relations /G26/G26 DEo=ε and /G26/G26 BHo=µ (where free-space is obviously taken as both linear and isotropic). We begin decoupling equation (2-16) from (2-17) by applying the curl operator to both sides of (2-16): ∇×∇× =− ∇×/G26/G26 Ej Hoωµ . (2-20) Equation (2-17) can then be substituted into (2-20) to obtain ∇×∇× =/G26/G26 EEoo ωµε2. (2-21) 23Equation (2-22) states a needed vector identity: ∇×∇× =∇ ∇⋅ −∇/G26/G26 /G26 EE E ()2. (2-22) Applying the vector identity of (2-22) to expression (2-21) and assuming no charge density, i.e., ∇⋅ =/G26 E0, we obtain the following [28]: ∇+ =220/G26/G26 Ek E (2-23) where koo =ωµ ε . (2-24) The second-order, linear, homogeneous partial differential equation expressed in (2-23) is often interchangeably referred to as either the vector wave equation [28] or the vectorHelmholtz equation [22]. In the infinite domain exterior to the radiating structure, it isthis differential equation in three dimensions that governs electromagnetic radiation intofree-space. 2.2.2 Reduction of the Vector Helmholtz Equation Using a Hybrid of Coordinate Systems (Scalar Spheroidal/Vector Cartesian) Now that we have resolved that the governing differential equation is the vector Helmholtz equation in three dimensions, our next step is to decide how to best apply thisgeneralized differential equation to meet the system requirements of the NZ-FZT process,viz., its input and output. In expressions (2-11a) and (2-11b), we define the process inputas being the near-zone time-harmonic E-field along either a prolate or oblate spheroidaltransformation surface. Because these particular transformation surfaces form a boundarythat is spheroidal in shape, the obvious approach is to solve the vector Helmholtzequation in spheroidal coordinates. However, as mentioned earlier (see 2.1.2), theprocess output (i.e., the far-zone time-harmonic E-field) is defined in terms ofconventional spherical coordinates. Fortunately, this does not present a problem because 24both spheroidal coordinate systems asymptotically approach that of the conventional spherical coordinate system in the far-zone (see Appendix A). Although we have identified spheroidal coordinates as the coordinate system of choice for our particular process, an obstacle presents itself when trying to solve thevector Helmholtz equation in three dimensions. In coordinate systems other thanCartesian, the vector solution of the three-dimensional Helmholtz equation is not astraightforward extension of its respective scalar solution. This difficulty arises becausethe orientation of the curvilinear unit vector varies as a function of curvilinear coordinatessuch that, in general, the vector Laplacian is not easily expressed in terms of its threerespective scalar Laplacians [28]: ∇ 2 123/G26 Ex x x(, ,) ≠∇2 12311Ex x x xx(, , ) /G04 +∇2 12322Ex x x xx(, ,) /G04 +∇2 12333Ex x x xx(, ,) /G04 (2-25) where in the special case of Cartesian coordinates (2-25) is an equality. Unfortunately, the vector components of the vector Laplacian are mutually coupled (with the exception of Cartesian coordinates) and can be expressed in generalized three-dimensional curvilinear coordinates as follows: ∇ 2 123/G26 Ex x x(, ,) =f EEExxxx(, ,) /G04 123 1 +g EEExxxx(, ,) /G04 123 2 +hE E E xxxx(, ,) /G04 123 3. (2-26) 25Applying the generalized vector Laplacian (2-26) to the three-dimensional vector Helmholtz equation (2-23) yields the following system of scalar equations: fE E E kExxx x(, ,) 123 120 += gE E E k Exxx x(, ,) 123 220 += hE E E k Exxx x(, ,) 123 320 += . (2-27) Upon inspection of the system of scalar equations (2-27), one can see that, in general, the vector components are not easily separated from one another. That is, it is difficult todecouple these equations and formulate them as a related system that has each of its threescalar equations expressed in terms of only one of the respective vector components. Morse and Feshbach [22] indicate that only six of the eleven curvilinear coordinate systems which allow separation of the scalar Helmholtz equation in threedimensions are also capable of presenting separable solutions to the vector Helmholtzequation in three dimensions. Table 2.1 summarizes these eleven coordinate systems andnotes their separable properties with regard to both the scalar and vector Helmholtzequations. These six special coordinate systems are vector-separable because one of theirscale factors is unity, and the ratio of the other two scale factors is independent of thecoordinate corresponding to the unity scale factor [22]. Even though there exists vector-separable solutions for each of these six special coordinates systems, constructing someof them can be quite formidable. In the case of spherical coordinates, variouscombinations of the solutions to the scalar wave equation (viz., spherical harmonics) canbe used to construct a basis set of solution vectors in spherical coordinates (viz., vectorspherical harmonics) [22], [29]. It must be mentioned that although these constructedbasis sets of solution vectors maintain orthogonality, they are not unique. Arfken [29]and Morse and Feshbach [22] offer two different basis sets that are driven by a particularapplication. To digress, the vector spherical harmonics provided by Arfken [29] are usedprimarily in quantum mechanics in which angular momentum is a significant parameter; 26Table 2.1 Summary of Three-Dimensional Curvilinear Coordinate Systems and Their Separable Properties Pertaining to Boththe Scalar and Vector Helmholtz Equations Coordinate System NumberThree-Dimensional Curvilinear Coordinate SystemProvides Scalar- Separable SolutionProvides Vector- Separable Solution 1 Cartesian (rectangular) yes yes 2 circular cylinder yes yes 3 elliptic cylinder yes yes 4 parabolic cylinder yes yes 5 spherical yes yes* 6 conical yes yes* 7 parabolic yes no 8 prolate spheroidal yes no 9 oblate spheroidal yes no 10 ellipsoidal yes no 11 paraboloidal yes no * - Denotes that the coordinate system provides a separable solution to the vector Helmholtz equation only if the preferred coordinate is theradius. meanwhile, Morse and Feshbach [22] describe another set of vector spherical harmonics that offers advantages with regard to separating wave propagation into longitudinal andtransverse components. As we have shown, solving the three-dimensional vector Helmholtz equation is significantly more involved than solving its scalar counterpart. And to further complicatematters, the coordinate systems of choice for our particular NZ-FZT process, viz., prolateand oblate spheroidal coordinates, are not among the six special curvilinear coordinatesystems capable of providing a vector-separable solution. In order to circumvent thisobstacle, we take an approach suggested by Morse and Feshbach [22] when attempting tosolve the vector Helmholtz equation with one of the other five coordinate systems (viz.,those that are scalar-separable, but not vector-separable): Formulate the overall solution 27by taking three solutions of the scalar Helmholtz equation (in the scalar-separable/vector- not-separable coordinate system) as the three Cartesian components of the vector. Yetregarding this approach, Morse and Feshbach [22] express their reservations and warnthat “ ... the fitting of boundary conditions is well-nigh impossible of attainment. ” However, it must be pointed out that this comment more appropriately applies whensolving boundary value problems in analytical form. Fortunately, because our particularNZ-FZT process is numerically based, the difficulties typically associated with the fittingof boundary conditions prove not to be an issue. At this point in our formulation, weimplement this hybrid approach by reducing the vector Helmholtz equation into a systemof three scalar Helmholtz equations in spheroidal coordinates with each representing avector component in Cartesian coordinates: ∇+ = 220 Ek Exx(,, ) (, , )ξηϕ ξηϕ ∇+ =220 Ek Eyy(, , ) (, , )ξηϕ ξηϕ ∇+ =220 Ek Ezz(, , ) (,, )ξηϕ ξηϕ (2-28) where the scalar E-fields are in either spheroidal coordinate system. 2.2.3 Break Down of the Resulting Scalar Helmholtz Equations into a Set of Ordinary Differential Equations Using the Separation of Variables Method Upon inspection of the system of scalar equations presented in (2-28), it is obvious that all three equations are of identical mathematical form. In essence, the vectorproblem has been reduced to three identical scalar problems of the following form: Prolate: ∇+ = 220 ψ ξηϕ ψ ξηϕPPk ( , , ) ( , , ) ; (2-29a) Oblate: ∇+ =220 ψ ξηϕ ψ ξηϕOOk ( , , ) ( , , ) . (2-29b) 28Equations (2-29a) and (2-29b) are the scalar Helmholtz equation in prolate and oblate spheroidal coordinates, respectively. Fortunately, the scalar Helmholtz equation isseparable in both of these coordinate systems (see Table 2.1). Since all three scalarequations of (2-28) are of the same mathematical form (namely, the scalar Helmholtzequation), we need to analyze their respective general solutions for both the prolate and oblate spheroidal coordinate systems, viz., ψξηϕP( , , ) and ψξηϕO( , , ) . These general solutions can then be applied afterwards to each of the three scalar equations of (2-28) along with pertinent boundary conditions to formulate the respective particular solution ofthe system. The first step in determining the general solutions of the scalar Helmholtz equation in both spheroidal coordinate systems is to expand each respective partialdifferential equation and express it in terms of the coordinate specific differentialoperators. This procedure requires that we determine the scalar Laplacians of equations(2-29a) and (2-29b) in their respective coordinate systems: ∇= 2ψξηϕP(, , ) () ()() () ()111 1122 22222 222 2aPP P ()ξ η∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξ η ξ η∂ψ ∂ϕ −−  +−  +− −−    ; (2-30a) ∇=2ψξηϕO(, , ) () ()() () ()111 1122 22222 222 2aOO O ()ξ η∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξ η ξ η∂ψ ∂ϕ ++  +−  ++ +−    . (2-30b) The scalar Laplacians in prolate and oblate spheroidal coordinates are found by employing their respective scale factors (provided in Appendix A) together with thegeneralized expression for the Laplacian in orthogonal curvilinear coordinates, i.e., (B-1). 29The reader is referred to Appendix B for the details behind the derivation of expressions (2-30a) and (2-30b). Having determined the necessary scalar Laplacians, we now apply expressions (2- 30a) and (2-30b), respectively, to equations (2-29a) and (2-29b) to obtain the scalarHelmholtz partial differential equations in terms of coordinate specific differentialoperators [23]: Prolate: () ()() () ()∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξ η ξ η∂ψ ∂ϕξ ηψ2222 222 222 211 110 −  +−  +− −−+− =PP P Ph(); (2-31a) Oblate: () ()() () ()∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξ η ξ η∂ψ ∂ϕξ ηψ2222 222 222 211 110 +  +−  ++ +−++ =OO O Oh(); (2-31b) where hk a a ao ≡≡ ≡ ()22πλ πλ. (2-32) Within the literature, the independent variable of both partial differential equations has been designated by some with the variable h [22] and by others with the variable c [23]. Because of our particular application (i.e., the radiation of electromagnetic waves), we have opted to use the variable h instead of the variable c in order to avoid any confusion with the notation for the speed of light. Moreover, othernotations also exist within the literature for this particular independent variable; for these,the interested reader should consult Table 21.11 of AMS-55 [25] for a summary of the existing notations along with an itemized account of their originators. Finally, equation 30(2-32) indicates that the independent variable h is directly related to the wavelength normalized focal point distance, i.e., aao λ λ= . The general solutions of the scalar Helmholtz equation in spheroidal coordinates may be found by applying one of the oldest systematic methods for solving partialdifferential equations: separation of variables. This method allows a single partialdifferential equation to be reduced to a set of ordinary differential equations that are onlycoupled by separation constants. Although the separation of variables method is notalways successful at solving a given partial differential equation, it will work in our casebecause we have a priori knowledge that the scalar Helmholtz equation is separable in both prolate and oblate spheroidal coordinates (see Table 2.1). When applying the separation of variables method, one assumes the solution of the partial differential equation to be the product of distinct functions with each beingonly dependent on its respective single variable. In our case, we shall assume that thesolutions of the partial differential equations of interest, viz., (2-31a) and (2-31b), are ofthe following form [23]: Prolate: ψξ η ϕ ξ η ϕP PP PRS (, , ) () () () =Φ ; (2-33a) Oblate: ψξ η ϕ ξ η ϕO OO ORS (, , ) () () () =Φ ; (2-33b) where RP()ξ and RO()ξ are radial solutions, SP()η and SO()η are angular solutions, ΦP()ϕ and ΦO()ϕ are azimuthal solutions. The assumed solutions presented in (2-33a) and (2-33b) are then substituted into their respective partial differential equations (2-31a) and (2-31b) to obtain the resultingcorresponding systems of three ordinary differential equations [23], [25]: 31Prolate: ()()d dd dRhmRPPξξξξ λ ξ ξξ22 22 21 10 −  −− + −    = () () w h e r e 1 ≤< ∞ξ , (2-34a) ()()d dd dShmSPPηηηηλ η ηη 1 1022 22 2−  +− − −    = ( ) ( ) where −≤ ≤11η , (2-35a) d dmPP2 220ϕϕϕΦΦ() ()+= where 0 2 ≤≤ϕπ ; (2-36a) Oblate: ()()d dd dRhmROOξξξξ λ ξ ξξ22 22 21 10 +  −− − +    = () () w h e r e 0 ≤< ∞ξ , (2-34b) ()()d dd dShmSOOηηηηλ η ηη 1 1022 22 2−  ++ − −    = () () where −≤ ≤11η , (2-35b) d dmOO2 220ϕϕϕΦΦ() ()+= where 0 2 ≤≤ϕπ . (2-36b) During the separation of variables process, the azimuthal ordinary differential equations, i.e., (2-36a) and (2-36b), are the first to be separated from their respective partial differential equations with −m2 designated as the first separation constant. Subsequently, the remaining respective radial and angular ordinary differential equations are separated from one another using the second separation constant λ (not to be confused with free- space wavelength). Although theory allows the separation constants to be arbitrarily chosen [30], these separation constants have been customarily selected as such becausethey best apply to the resulting solutions. The reason for using these specific separation 32constants will become clearer as we explore the ensuing general solutions. We refer the interested reader to Appendix C for the explicit details behind applying the separation ofvariables method to derive both sets of ordinary differential equations presented in (2-34a)-(2-36a) and (2-34b)-(2-36b). 2.2.4 Review of Applicable Sturm-Liouville Theory and Self-Adjoint Boundary Value Problems So that we may proceed with construction of the general solutions to the scalar Helmholtz equation in spheroidal coordinates, we must begin with some applicableSturm-Liouville theory that is key to the understanding of the differential equations underconsideration, viz., (2-34a)-(2-36a) and (2-34b)-(2-36b). It will become apparent thatthese differential equations are already subject to boundary conditions just based uponexisting physical requirements. Consequently, it is necessary that we gain some insight ofsecond-order self-adjoint boundary value problems and how they apply to the resultingdifferential equations coupled with their respective boundary conditions. This sectionprovides a basic understanding of this type of boundary value problem in addition to theirpertinent special properties. As we shall see, the special properties of self-adjointboundary value problems play a crucial role when constructing the general solutions ofthe scalar Helmholtz equation in spheroidal coordinates. Self-adjoint differential equations form a class of differential equations that fit a particular mathematical form. Sagan [31] provides an excellent treatment of the theorybehind the origin of self-adjoint differential equations and their respective self-adjointoperators. It is here that we refer the reader to obtain the details regarding the origin ofthe definitions that are to follow. A differential operator of second-order, L , is considered self-adjoint if and only if it is equivalent to its adjoint operator, L , and has the following form [29], [31]: )()()()( )(L)(L xuxqdxxduxpdxdxu xu + = = (2-37) 33where the functions px( ) and qx( ) are real functions of x over the region of interest axb≤≤ ; furthermore, qx( ) is continuous and px( ) is twice differentiable (i.e., continuous first and second derivatives). Since the zeros of px( ) will become singularities of the resulting differential equation, we must choose the interval []ab, s o that no singularities are contained in the interior of the interval. However, theory allows for the existence of singular points on either one or both boundaries of the interval [29],[30]. Having defined the differential operator of second-order, we can now express the generalized form of the differential equation (known as the Sturm-Liouville equation)[29], [31]: 0)()( )(L = + xuxw xu λ , (2-38) or more explicitly as (applying (2-37) to (2-38) ) d dxpxdu x dxqxux wxux ()()()() ()()  ++ = λ 0 . (2-39) In these expressions, λ serves as a constant and wx( ) is a known function of x, referred to as a density or weighting function [29]. It is required that wx()>0, except possibly at isolated points at which wx()=0 [29]. If boundary conditions are imposed at both endpoints of the interval, solutions to (2-38) will be limited to a discrete set of ux() functions. Each one of these functions is referred to as an eigenfunction and corresponds to a specific value of λ, known as an eigenvalue. It is important to note that only the discrete set of eigenfunctions (each paired with its respective eigenvalue) is able to satisfy the self-adjoint differential equation of (2-38). Now that we have an understanding of self-adjoint differential equations of second-order, our attention shall focus on some of the underlying boundary conditionsthat are possible for second-order, homogeneous, self-adjoint boundary value problems. 34The boundary conditions are specified at the endpoints of the interval []ab, and shall be classified by us for easy reference as one of the following for homogeneous boundary value problems [30], [31]: Type I boundary conditions: cu a cu a11 12 0 () ()+ ′= cu b cub21 22 0 () ()+ ′=; (2-40) Type II boundary conditions: cu a cu a du b d ub11 12 11 12 0 () () () ()+ ′++ ′= cu a cua du b d ub21 22 21 22 0 () () () ()+ ′++ ′=; (2-41) Type III boundary conditions: cu a cu a11 12 0 () ()+ ′= ub(), ′ub() bounded as xb→; (2-42) Type IV boundary conditions: ua() , ′ua( ) bounded as xa→ ub(), ′ub() bounded as xb→. (2-43) The task of solving the Sturm-Liouville equation, i.e., (2-38), under one of these general types of imposed boundary conditions, viz., (2-40)-(2-43), is referred to as ahomogeneous self-adjoint boundary value problem of second-order [30], [31]. However,these generalized boundary conditions, with the exception of Type I, cannot guaranteethat the resulting boundary value problem is self-adjoint; that is, there is still an additionalcondition that must be satisfied. 35Let ux1( ) and ux2( ) represent every pair of sufficiently differentiable solutions to the self-adjoint differential equation (2-38) that satisfies the given type of boundary condition such that we can formulate the following condition [29]-[31]: () () )(L),( )(),(L2 1 2 1 xuxu xuxu = . (2-44) Note that nomenclature ()fg, represents the complex inner product over the interval []ab, defined as follows [30]: ()fg fx g x d x ab ,( ) * ( )=∫(2-45) where * denotes complex conjugate of the function. A boundary value problem specified by the Sturm-Liouville equation and any one of the defined boundary condition types is considered to be self-adjoint (i.e., on a case-by-casebasis) if the crucial condition (2-44) can be shown to be valid [29]-[31]. Now that we have a basic understanding of what constitutes a homogeneous self- adjoint boundary value problem of second-order, we are now somewhat able to classifythem. First of all, self-adjoint boundary value problems can be divided into one of two types: regular, where no singularities exist anywhere in the interval []ab, including the boundaries; and singular, where singularities exist at either one or both of the boundaries [30]. Type I and II boundary conditions shall apply to regular self-adjoint boundary valueproblems, while the less restrictive Type III and IV boundary conditions are necessary to solve the singular self-adjoint boundary value problem. We first consider the second-order self-adjoint boundary value problem that is regular with separated boundary conditions, viz., Type I (2-40). These linearhomogeneous boundary conditions are said to be separated because each conditioninvolves only one of the endpoints of the interval [30]. Note that these self-adjointboundary value problems are considered extraordinary because sweeping generalizationscan be made about them without having to explicitly investigate each particular one on a 36case-by-case basis. An example being that the identity specified in (2-44) is always true in the particular case that the problem is regular with separated boundary conditions [30],[31]. Unfortunately, this type of self-adjoint boundary value problem does not apply tothe differential equations under consideration. However, the regular self-adjointboundary value problem with the unseparated linear homogeneous boundary conditions,i.e., Type II (2-41), is of importance to us. As we shall see, it is this type of boundaryconditions that enables us to address the periodic boundary conditions that result whenformulating the azimuthal solutions of the scalar Helmholtz equation in spheroidalcoordinates. Finally, we must consider singular, homogeneous self-adjoint boundary value problems with the less restrictive Type III or IV boundary conditions, i.e., (2-42) and (2- 43). In general, Type I and II boundary conditions are not able to be applied in thesingular case because they will only yield the trivial solution [30]. When the endpointsare singular, the boundary condition must be somewhat relaxed and only require that thepossible solutions and their first derivatives remain finite (or bounded). Regarding theregular endpoint, Type III makes use of a separated boundary condition. It must be emphasized that the singular problem must also satisfy the condition put forth in (2-44). According to Boyce and DiPrima [30], a striking difference between regular and singular self-adjoint boundary value problems is that the singular problem may not yielddiscrete eigenvalues; that is, the problem might yield nontrivial solutions for continuousvalues of λ in some particular interval. Moreover, the problem might have a mixture of discrete and continuous eigenvalues. In cases where a singular self-adjoint boundary value problem has only a discrete set of eigenvalues and functions, the expansion of agiven function in terms of a series of eigenfunctions is possible [30]. As we shall see, itis the case where a singular self-adjoint boundary value problem has a singularity at bothendpoints, employs Type IV boundary conditions, and yields only discrete eigenvaluesthat is of utmost importance to us. It is this particular situation that arises whendetermining the angular solutions of the scalar Helmholtz equation in spheroidalcoordinates. 37Now that we have reviewed applicable types of self-adjoint boundary value problems of second-order, we are now ready to investigate their relevant specialproperties. These properties are of extreme importance in physics, both classical andquantum, and will be critical to the formulation of our NZ-FZT process [29]: 1. The eigenvalues of self-adjoint boundary value problems are real. 2. The eigenfunctions of self-adjoint boundary value problems are orthogonal. 3. The eigenfunctions of self-adjoint boundary value problems of second-order form a complete set. An important caveat must be imposed on the validity of these properties. They always hold in the case of the regular self-adjoint problem. To the contrary, these properties arenot guaranteed to remain true in the case of the singular self-adjoint problem. However,Boyce and DiPrima [30] indicate that it can be shown that these properties do remainvalid in the special case that the singular self-adjoint problem has only discreteeigenvalues. The first property is important because one needs to search for only real quantities in the process of finding the eigenvalues. As we shall see, this property is critical in caseswhere the eigenvalues must be numerically determined. The reader is referred to [29] and[30] for the proof of this property. The second property regarding the orthogonality of eigenfunctions implies that the following inner product is true [29]: ux u x w x d x ij ab () *() () = ∫0 where ij≠ and λλij≠ . (2-46) (Notice how this inner product differs from the one previously defined in (2-45); namely, it includes the weighting function wx( ) .) Essentially, the eigenfunctions ux1() and 38ux2( ) are orthogonal with respect to the weighting function wx( ) over the interval []ab,. Orthogonality is the fundamental property behind Sturm-Liouville theory because it makes possible the isolation of the eigenfunction expansion coefficients. The inner product defined in (2-46) is predicated on the fact that the eigenvalues be simple; that is, each eigenvalue is distinct, i.e., λλij≠ , and corresponds to only one linearly independent eigenfunction. As a result, expression (2-46) guarantees orthogonality in this case. However, the situation exists where there are multiple linearlyindependent eigenfunctions that correspond to the same eigenvalue; such a case is labeleddegenerate [29]. Multiple eigenfunctions that correspond to a degenerate eigenvalue mayor may not be orthogonal. In the degenerate case where the eigenfunctions happen not tobe orthogonal, the Gram-Schmidt orthogonalization method can be used to construct anorthogonal basis set [29]. As we shall see, the self-adjoint boundary value problems thatare under consideration do exhibit degenerate eigenvalues. Fortunately, construction ofthe general solutions to the scalar Helmholtz equation in spheroidal coordinates revealsthat the pertinent eigenfunctions remain orthogonal. Consequently, we shall hereafterlimit our discussion to the case where the eigenfunctions are orthogonal and the specifiedcondition (2-46) remains true. Finally, this leads us to the third property regarding the completeness of the set of eigenfunctions. In our context, this completeness implies that any well-behaved function,i.e., at least piecewise continuous, can be approximated by a series expansion oforthogonal eigenfunctions to any degree of accuracy [29]: Fx au x ii i() ()= =∞ ∑ 0. (2-47) In mathematical terms, the set of eigenfunctions is called complete if the limit of the mean square error vanishes [29]: 39l i m () () () Iii iI ab Fx a u x wxd x →∞=−=∑ ∫ 02 0 . (2-48) Notice that as the number of terms increases, (2-48) only requires that the integral of the error squared go to zero, not the magnitude of the error in []ab, [29]. This convergence in the mean is less restrictive than uniform convergence. (Note that uniform convergence implies convergence in the mean; however, the converse is not true.) According toArfken [29], (2-48) is not upset with piecewise continuous functions, which by definitionallows a finite number of finite discontinuities. In order to evaluate the expansion coefficients of (2-47), we first consider normalization of eigenfunctions. The inner product expressed in (2-46) only addressesthe case where the eigenfunctions are different, i.e., ij≠. However, in the special case that the eigenfunctions are the same, i.e., ij=, the inner product can be expressed as follows: u x u x w x dx u x w x dx Nii i i ab ab () *() () () () ==∫ ∫2. (2-49) Note that the square root of the resulting Sturm-Liouville normalization constant Ni is often referred to as the norm of the eigenfunction. Moreover, the value of this normalization constant can be arbitrarily chosen. This degree of freedom exists becauseeach eigenfunction is a solution to a given linear, homogenous differential equation, viz.,the Sturm-Liouville equation; that is, basic differential equation theory indicates that wemay multiply a solution by any constant, and it will still be a solution. A common normalization scheme requires that each eigenfunction be multiplied by 1Ni, i.e., the reciprocal of its norm, so that the inner product expressed in (2-49) yields unity; these rescaled orthogonal eigenfunctions are then said to be orthonormal. In addition to the justdescribed orthonormal scheme, we shall also employ other normalization schemes during 40the formulation of the NZ-FZT process where the Sturm-Liouville normalization constant is not taken as unity. Now that we have addressed the issue of normalization, the individual cases expressed in (2-46) and (2-49) can be combined to yield the single orthogonality relation: u x u x w x dx Nij i ab ij () *() (), = ∫δ (2-50) where δij, represents a Kronecker delta. Finally, we can evaluate the expansion coefficients of (2-47) by using the following expression: aNFxu xwxd xj jj ab =∫1() *( ) ( ) . (2-51) This expression follows by multiplying both sides of (2-47) by [ *() () ]ux w xj and integrating over the interval []ab, . We then proceed to make use of (2-50) and employ the discrete version of the sifting property (due to the Kronecker delta) in order to isolate the j th term. At this point in the discussion, it is quite obvious that it is orthogonality that enables the eigenfunction expansion method to be such an effective analytical approach. 2.2.5 General Solutions of the Scalar Helmholtz Equation in Spheroidal Coordinates We are now ready to begin construction of the general solutions to the scalar Helmholtz equation in spheroidal coordinates by applying Sturm-Liouville theory to theseparated differential equations, i.e., (2-34a)-(2-36a) and (2-34b)-(2-36b). Physicalrequirements impose two sets of boundary conditions that limit the general solutions to adiscrete set of eigenfunctions with corresponding eigenvalues: single-valuedness of theazimuthal solutions and finiteness of the angular solutions (specifically at the poles of thespheroid) [23]. Consequently, self-adjoint boundary value problems must be formulatedthat address both the azimuthal and the angular solutions of the separated differential 41Table 2.2 Overview of the Azimuthal and Angular Self-Adjoint Boundary Value Problems Differential Equationpx() qx() w(x) [,]ab Boundary Conditionsλ (eigenvalues) Prolate Angular (2-35a)12−η−−−mh2 222 1ηη1[ , ]−11 (singular)Type IV (2-43) (finite at both boundaries)λmlh,() Oblate Angular (2-35b)12−η−−+mh2 222 1ηη1[ , ]−11 (singular)Type IV (2-43) (finite at both boundaries)λml jh,()− Prolate/Oblate Azimuthal (SHO) (2-36a)/(2-36b)10 1 [ , ] 02π (regular)Type II (2-41) (periodic)m2 Note: px() ,qx( ) , and w(x) are with regard to the Sturm-Liouville equation (2-38). equations. An overview of these self-adjoint boundary value problems is provided in Table 2.2. Regarding the radial solution, its behavior is driven by the results of theazimuthal and angular self-adjoint boundary value problems; that is, the allowableazimuthal and angular solutions (i.e., the eigenfunctions) dictate the correspondingpermissible separation constants (which directly correspond to the eigenvalues).Accordingly, the radial solution can be understood using conventional differentialequation theory only after analyzing the azimuthal and angular self-adjoint boundaryvalue problems. Let us first address the separated differential equations that pertain to the azimuthal solution, namely, (2-36a) and (2-36b). Since both of these equations aremathematically identical, it is only necessary to analyze their form. The form of theseazimuthal differential equations is quite common and has been traditionally referred to asthe simple harmonic oscillator (SHO) [29]. As we can see, the SHO is easily put intoself-adjoint form, i.e., satisfies the Sturm-Liouville equation (2-38). The interval of theazimuthal coordinate ϕ is designated as [ , ] 02π by definition (see 2.1.1). As we have 42previously stated, physical requirements demand that the azimuthal solution be single- valued, i.e., ΦΦ() ( )ϕπ ϕ+=2 . (2-52) The constraint imposed by (2-52) is equivalent to requiring the azimuthal solution to have a period of 2 π or some integral multiple of it [29]. This physical requirement can be represented as the following periodic boundary conditions for the interval [ , ] 02π [30]: ΦΦ() ( )02 0−=π and ′−′= ΦΦ() ( )02 0π . (2-53) Notice that the boundary conditions of (2-53) actually are of Type II, i.e., (2-41). Furthermore, it is easy to show that the SHO under these particular Type II boundaryconditions satisfies the condition required for all second-order self-adjoint boundaryvalue problems, viz., (2-44). Since there are no singularities at the endpoints, ourazimuthal self-adjoint boundary value problem is considered regular. The eigenvalues of the azimuthal self-adjoint boundary value problem can be shown from basic differential equation theory to be [30]: λmm=2 where /G15,2,1,0=m . (2-54) This result is consistent with Sturm-Liouville theory. Differential equation theory can also be used to determine the allowable azimuthal solutions (i.e., the eigenfunctions)[29]: sin cosm mϕ ϕ  . (2-55) In addition to (2-55), differential equation theory also allows us to express the azimuthal eigenfunctions as the following linearly independent complex exponentials [29]: 43e ejm jm+ −    ϕ ϕ. (2-56) As we shall see, these complex exponentials will be the form of choice during our construction of the general solutions. Upon review of (2-55) and (2-56), one mayrecognize the linearly independent solutions as the basis sets for both the trigonometricFourier series and its corresponding exponential Fourier series, respectively. It is important to note that there are two linearly independent eigenfunctions corresponding to each nonzero eigenvalue; that is, with the exception of λ00=, the remaining eigenvalues are two-fold degenerate. Although Sturm-Liouville theory cannot guarantee orthogonality of eigenfunctions that correspond to a particular degenerateeigenvalue, direct evaluation of (2-50) with the exponential form of the eigenfunctionsdoes confirm that all solutions are orthogonal; this leads us to the following neededorthogonality relation [29]: () ee djm jm mm++ ′ ′ ∫=ϕϕπ ϕπ δ 02 2* , (2-57) where Nm=2π and wx()=1. We shall now turn our attention to the separated differential equations that pertain to the angular solutions, namely, (2-35a) and (2-35b). Gaining insight into the behaviorof both of these angular differential equations under their imposed boundary conditions ismuch more involved than their azimuthal counterparts. At this point in our discussion,we shall predominately rely on the monograph put forth by Flammer [23] (which isconsidered one of the most comprehensive works on the subject) and the text by Morseand Feshbach [22] to develop the required understanding of these angular solutions.Furthermore, we shall subsequently refer to both of these works regarding thecorresponding radial solutions. Upon review of the prolate and oblate separated differential equations for the radial and angular solutions, i.e., (2-34a)/(2-34b) and (2-35a)/(2-35b), it is important to 44note that the oblate differential equations can be obtained from their prolate counterparts using the following transformation [23]: hj h→− (or +jh); and ξξ→+ j (or −jξ). (2-58) Since these sets of differential equations are closely related through this simple transformation, much of the oblate solution development closely parallels that of itsprolate counterpart. Consequently, for the sake of brevity, throughout the ensuingdevelopment there will be instances where the results of the oblate case are obtained fromthe prolate case using the aforementioned transformation. However, as always, one mustexercise caution and not apply this transformation blindly; there will be points (especiallyin the computational process) where the approaches between the two cases differ.Because the separated differential equations are so closely related, many investigatorshave included these transformations within the notation of the solutions and theirrespective eigenvalues to distinguish between the prolate and oblate cases. The separated angular differential equations, i.e., (2-35a) and (2-35b), can be put into self-adjoint form such that both satisfy the Sturm-Liouville equation. By definition,the interval of the angular coordinate η for both the prolate and oblate cases is designated as [ , ] −1 1 (see 2.1.1). Upon inspection of the angular differential equations, we are able to see that singularities exist at both endpoints of the interval: η=±1. Since physical requirements demand that the angular solutions of both the prolate and oblate case remain finite over the specified interval in the presence of these endpointsingularities, we must appropriately apply Type IV boundary conditions. These boundaryconditions coupled with their respective differential equations enable us to formulate theneeded angular self-adjoint boundary value problems (which happen to be singular inboth spheroidal cases). Determining the eigenvalues of the angular self-adjoint boundary value problems is much more complicated than finding those of their azimuthal counterparts (i.e., theSHO). As a matter of fact, computation of the exact eigenvalues is not straightforward 45and requires the use of a sophisticated numerical process that is subsequently addressed in Chapter 3. However, some relevant points regarding these eigenvalues and how theyrelate to the angular solutions can be made at this point in our discussion. In the limiting case that h=0, both angular differential equations (2-35a) and (2- 35b) approach the form of the associated Legendre equation [29]: ()()/G2Cd dd dSmS llSηηηη ηηλ η 1 11022 2−  − −+ ≡+= () () ()( ) (2-59) where −≤ ≤11η . The associated Legendre equation is the ordinary differential equation that results when solving the scalar Helmholtz equation in spherical coordinates (after applying theseparation of variables method). Essentially, the associated Legendre equation is aspecial case of both angular differential equations; this intuitively makes sense because asphere is a special case of a spheroid. We shall use the insight gained from this specialcase as a springboard to extend our understanding to the more general angular self-adjointboundary value problems for both the prolate and oblate cases. The eigenvalues of the self-adjoint boundary value problem associated with this special case (which also requires the solution to remain bounded over the interval) areknown to be [29]: λmlhl l,() ( )==+0 1 where −≤ ≤ +lm l and /G15,2,1,0=l . (2-60) Equation (2-60) indicates that each real eigenvalue is referenced by a two-dimensional index where the index m is considered the order (based upon the azimuthal separation constant, −m2) and the index l is referred to as the degree (based upon the angular separation constant, λ=+ll()1). Table 2.3 depicts a portion of an infinite two- dimensional matrix where the elements are the eigenvalues with indices that correspond to the order and degree, respectively. Inspection of both (2-60) and Table 2.3 reveals that 46Table 2.3 Portion of Infinite Two-Dimensional Eigenvalue Matrix for the Special Case where λmlh,()=0 Index l = 0 123 m = - 3 --- 1 2 -2 - - 6 12 -1 - 2 6 12 0 026 1 2 1- 2 6 1 2 2- - 6 1 2 3 --- 1 2 Note: A hyphen denotes no corresponding eigenvalue. for the special case where h=0, the eigenvalues are independent of the order m and thus are degenerate; namely, the eigenfunctions of (2-59) having the same order m happen to share the same eigenvalue. In addition, equation (2-60) implies that all of the eigenvaluesform a monotonic increasing sequence for a given order m: λλml mlhh,,() ()=< =+ 001 . (2-61) The eigenfunctions of the self-adjoint boundary value problem associated with this special case (which also requires the solution to remain bounded over the interval)are well known and are referred to as the associated Legendre functions of the first kind [29]: P lm()η, where m is the order and l is the degree. Both indices are the same as those used to reference the corresponding eigenvalue. Throughout this dissertation, we shall always employ Ferrer’s definition of the associated Legendre functions of the first kindwhen the argument is on the −< <11x branch cut, namely, 47Pxlm()=−() ( )122xd dxPxmm m l (Ferrer ’s definition), (2-62) and Hobson ’s definition when the argument lies elsewhere in the finite complex plane, namely, Pzlm()=−() ( )zd dzPzmm m l221 (Hobson ’s definition). (2-63) Notice how both definitions approach the same values at the ±1 branch points. It is at this point in our discussion that we refer the reader to Appendix D for a morecomprehensive understanding of Legendre functions. It is important for us to understand that for negative orders of m, these eigenfunctions are not linearly independent from their positive order counterparts. Thisstatement remains consistent with the associated Legendre equation (2-59) because the solutions are symmetric with respect to the order m (due to the m 2 term). Accordingly, the associated Legendre functions of the first kind with negative orders are proportionalto their positive order counterparts as follows [29]: ()Plm lmPlmm lm −=−− +() ( )! () !() ηη 1 . (2-64) Note that the presence of a proportionality constant between the positive and negative order solutions is justified based upon differential equation theory. This is so because alinear combination of solutions of a linear homogeneous differential equation is also asolution of the equation [30]. One advantage of using this particular constant ofproportionality is that it enables the recurrence relations that are valid for lm≤≤0 to remain valid for 0<≤− ml [29]. Once again Sturm-Liouville theory cannot guarantee orthogonality of eigenfunctions corresponding to a particular degenerate eigenvalue. However, theeigenvalues are distinct with respect to order because they form a monotonic increasingsequence for any given order m, i.e., (2-61). The fact that all of the eigenvalues for a 48given order m differ from one another implies that the corresponding eigenfunctions for a given order m are mutually orthogonal. This assertion is demonstrated by Arfken [29] when he shows that the associated Legendre functions of the first kind for a given order m are indeed mutually orthogonal and satisfy the following orthogonality relation: () ()PP dllm lmlm lm ll () ()! !, ηη η δ −′′∫=++ −112 21(2-65) where () ()Nllm lmml,! !≡++ −2 21 and wx()=1. Since the eigenfunctions and the eigenvalues are referenced by the indices m and l, it is logical that we do the same for the corresponding Sturm-Liouville normalizationconstants. Moreover, we would like to point out that the normalization given in (2-65) isthe one that has been customarily used for associated Legendre functions of the first kind.Even though it is possible to develop an orthogonality relation for these associatedLegendre functions for a given degree l (with differing orders m), its usefulness is limited because in physical problems orthogonality of the azimuthal dependence ties together thepossible orders m and thus leads to (2-65) [29]. If we lift the condition imposed on the associated Legendre self-adjoint boundary value problem that requires the solutions to remain finite, a second set of linearlyindependent solutions becomes acceptable [29]. These solutions are logarithmic innature, i.e., become infinite at the boundaries of the interval, and are referred to as the associated Legendre functions of the second kind, Q lm()η. However, these solutions are seldom used in the angular context, i.e., over the interval [ , ] −1 1 , because of the imposed physical requirements. Having explored the behavior of the solutions for the special case where h=0, we are now in a better position to understand the more general solutions to the angularself-adjoint boundary value problems. In the general case where h≠0, the eigenvalues are now a function of both indices, m and l, in addition to the independent variable h [23]. 49Because of these dependencies, the eigenvalues can be represented for the prolate and oblate cases within the notation as λmlh,( ) and λml jh,()− , respectively [23]. Unlike the special case, the eigenvalues need to be evaluated numerically because they do not have closed form representations. However, both angular differential equations, i.e., (2-35a)and (2-35b), indicate that corresponding positive and negative order eigenfunctions sharethe same eigenvalue for a given order m (just like the special case where h=0) [23]: Prolate: λλ−=ml mlhh,,( ) ( ) ; (2-66a) Oblate: λλ−−= −ml ml jh jh,,( ) ( ) . (2-66b) Once again, this occurs because the solutions to both self-adjoint boundary value problems are symmetric with respect to the order m due to the m2 term (see Table 2.2). Another important similarity between the general and special cases is that eigenvalues forthe general cases also form a monotonic increasing sequence for a given order m [22]: Prolate: λλml mlhh,,() ()<+1; (2-67a) Oblate: λλml ml jh jh,,() ()−< −+1 ; (2-67b) where −≤ ≤ +lm l and /G15,2,1,0=l . Making use of the symmetry of the eigenvalues with respect to the order m (i.e., 2-66a and 2-66b), we are able to display in Table 2.4 a portion of the infinite two-dimensionaleigenvalue matrix for both the prolate and oblate cases. As for the eigenfunctions of the self-adjoint boundary value problems associated with the general cases (i.e., where h≠0), they are not so well known and have been referred to as prolate and oblate spheroidal angle functions of the first kind: Shml,()(, )1η and Sj hml,()(, )1−η, respectively [23]. Once again, indices m and l are still the same as those used to reference the corresponding eigenvalue. However, unlike the 50Table 2.4 Portion of Infinite Two-dimensional Eigenvalue Matrix for the Prolate and Oblate Cases ( h≠0) Index l =0 1 2 m = -2 - - λ22,()h; λ22,()−jh -1 - λ11,()h; λ11,()−jhλ12,()h; λ12,()−jh 0λ00,()h; λ00,()−jhλ01,()h; λ01,()−jhλ02,()h; λ02,()−jh 1- λ11,()h; λ11,()−jhλ12,()h; λ12,()−jh 2- - λ22,()h; λ22,()−jh Note: A hyphen denotes no corresponding eigenvalue. eigenfunctions for the special case, i.e., Plm()η, the spheroidal angle functions of the first kind not only depend on the angular parameter η, but also on the independent variable h. Flammer [23] states that both spheroidal angle functions of the first kind can be expressed as an infinite expansion of associated Legendre functions of the first kind: Prolate: Sh d h Pml rml mrm r,() , ,(, ) () ()1 01ηη=′ + =∞ ∑ where 0≤≤ml ; (2-68a) Oblate: Sj h d j h Pml rml mrm r,() , ,(, ) () ( )1 01−=′−+ =∞ ∑ηη where 0≤≤ml . (2-68b) In these expressions and from this point forward, the prime over the summation sign indicates that the summation is over only even values of r when ( l - m ) is even, and over only odd values of r when ( l - m ) is odd. Consequently, spheroidal angle functions are themselves even functions when ( l - m ) is even, and odd functions when ( l - m ) is odd. 51Later, we shall exploit this property to improve the computational efficiency of the special function routines and the overall NZ-FZT computer software. Like the special case where h=0, the eigenfunctions of the general case, i.e., the spheroidal angle functions, that correspond to negative orders of m are also not linearly independent from their positive order counterparts. Once again, this is so because thesolutions to the prolate and oblate self-adjoint boundary value problems are symmetricwith respect to the order m (see Table 2.2). Similarly, the eigenfunctions that correspond to negative orders of m are also directly proportional to their positive order counterparts. Regarding the proportionality constant between the positive and negative ordereigenfunctions, Flammer [23] recommends using a constant that is identical to the oneused for the associated Legendre functions of the first kind, i.e., contained in (2-64). Thisparticular proportionality constant would allow the infinite expansions provided in (2-68a) and (2-68b) to remain valid for negative orders of m. However, for our particular application, it will facilitate our development if we assume a proportionally constant ofunity: Prolate: Sh S h ml ml−≡,() ,()(, ) (, )11ηη ; (2-69a) Oblate: Sj hS j hml ml−−≡ −,() ,()(, ) (, )11ηη . (2-69b) As before, we are justified in doing so based upon fundamental differential equation theory. The advantage of assuming a unity proportionality constant is that it prevents usfrom having to unnecessarily tote around an additional factor for negative values of m; this factor for negative orders of m would make no difference in our final formulation because it would ultimately end up being absorbed into the eigenfunction expansioncoefficients. Accordingly, we have appropriately applied the restriction 0≤≤ml to the expansion representation of the spheroidal angle functions presented in (2-68a) and (2-68b). Akin to the special case where h=0, each of the angular self-adjoint boundary value problems also possesses a second set of linearly independent solutions for thegeneral case. These respective sets of linearly independent solutions are referred to as the 52prolate and oblate spheroidal angle functions of the second kind: Shml,()(, )2η and Sj hml,()(, )2−η, respectively [23]. These solutions are also logarithmic in nature and can be expressed as an infinite expansion of associated Legendre functions of the second kind, Qlm()η [23]. However, the physically imposed boundary condition requiring the solutions to remain finite precludes both Shml,()(, )2η and Sj hml,()(, )2−η as acceptable solutions to their respective angular self-adjoint boundary value problem. Consequently, we shall refrain from any further discussion of spheroidal angle functions of the secondkind. Because we will not be dealing with spheroidal angle functions of the second kind,we are now able to drop the superscript in the notation and subsequently avoid confusion when referring to the spheroidal angle functions of the first kind: Sh Sh ml ml,,()(, ) (, )ηη≡1 and Sj h Sj hml ml,,()(, ) (, )−≡ −ηη1. We shall now turn our attention to orthogonality of the spheroidal angle functions of the first kind. Upon review of numerical results for selected values by Hanish andKing [32] and the power-series expansions provided by Flammer [23], the degenerateeigenvalues for the special case (i.e., h=0) appear to become distinct and simple for both the prolate and oblate general cases. Furthermore, expressions (2-67a) and (2-67b)indicate that in both general cases the eigenvalues are distinct with respect to orderbecause they too form a monotonic increasing sequence for any given order m. By way of Sturm-Liouville theory, this fact implies that the eigenfunctions corresponding to a givenorder m are mutually orthogonal. Flammer [23] supports this assertion by stating that spheroidal angle functions of the first kind for a given order m remain mutually orthogonal such that they satisfy the following orthogonality relations: Prolate: ShS hd Nh ml ml ml ll,, , ,(, ) (, ) ()ηη η δ −′′∫= 11 ; (2-70a) Oblate: S jh S jh d N jhml ml ml ll,, , ,(, )(, ) ()−− = − −′′ ∫ηη η δ 11 . (2-70b) 53This leads us to the last issue pertaining to the angular solutions that must be addressed: normalization. Since we are dealing with linear homogeneous differentialequations in the angular self-adjoint boundary value problems, the possible solutions, i.e.,the eigenfunctions, can be scaled by any arbitrary constant. In the special case where h=0, the solutions (i.e., the associated Legendre functions of the first kind) already have a universally accepted normalization scheme. However, in both general cases of theangular self-adjoint boundary value problems, we are not so fortunate. Several differentnormalization schemes have been presented within the literature and are tabulated byFlammer [23]. The one thing that is common among almost all of the tabulatednormalization schemes (with the exception of one) is that they somehow scale thespheroidal angle function to behave like the associated Legendre function of the first kind(i.e., the special case solution where h=0) with regard to a certain parameter. For the sake of brevity, we shall only discuss the popular normalization schemes put forth by Flammer [23] and Morse and Feshbach [22]. Flammer [23] presents anormalization scheme that dictates the following as η→0 : all even spheroidal angle functions behave like their corresponding associated Legendre functions, and all first derivatives of odd spheroidal angle functions behave as the first derivatives of theirrespective associated Legendre functions. In mathematical terms, Prolate: Sh P ml lm ,(,) ()00= for ()lm− even, (2-71a) ′=′ Sh Pml lm ,(,) ()0 0 for ()lm− odd; (2-72a) Oblate: Sj h Pml lm ,(, ) ( )−=0 0 for ()lm− even, (2-71b) ′−= ′ Sj h Pml lm ,(, ) ( ) 0 0 for ()lm− odd. (2-72b) As for Morse and Feshbach [22], their normalization scheme stipulates that the spheroidal angle functions behave like their corresponding associated Legendre functions as η→1: 54Prolate: () () 122 1,22)( 1 ),( 1 =− =−  −= − η ηη η η ηm lm lmm P h S ; (2-73a) Oblate: () () 122 1,22)( 1 ), ( 1 =− =−  −= − − η ηη η η ηm lm lmm P jh S . (2-73b) In general, conversion factors between any pair of normalization schemes can be developed. Furthermore, the choice of the normalization scheme is immaterial becausethe eigenfunction expansion coefficients will ultimately assume their proper values basedon the normalization of the eigenfunctions. An important point that may not be quite soobvious is that the Sturm-Liouville normalization constants, i.e., Nh ml,( ) and Nj hml,()− , are also dependent on the normalization scheme chosen. In the ensuing theoretical development of our NZ-FZT process, we shall maintain generality and assume anarbitrary normalization scheme. However, in Chapters 3 and 4 when we addresscomputational details of the NZ-FZT process, we shall opt to use the normalizationscheme of Morse and Feshbach for the computation of the spheroidal angle functions(including their respective Sturm-Liouville normalization constants). Equipped with an understanding of the solutions to the azimuthal and angular self-adjoint boundary value problems, we are now able to address the radial solutions tothe scalar Helmholtz equation in spheroidal coordinates. As we mentioned earlier in thissection, the allowable radial solutions are driven by the possible separation constants(which directly correspond to the eigenvalues of the azimuthal and angular self-adjointboundary value problems; see Table 2.2). Two sets of linearly independent solutions exist that satisfy the radial differential equations presented in (2-34a) and (2-34b). These linearly independent solutions sets arereferred to as the prolate and oblate spheroidal radial functions of the first and second kind: { Rh ml,()(,)1ξ, Rhml,()(,)2ξ} and { Rj h jml,()(, )1−ξ, Rj h jml,()(, )2−ξ}, respectively. These functions appear to be universally defined so that in the special case where h=0, both sets of spheroidal radial functions approach the spherical Bessel and Neumann functions 55{jl()ξ, nl()ξ} [22], [23]. Based upon differential equation theory, both sets of linearly independent radial solutions can be combined to produce a complex form of these solutions referred to as the prolate and oblate spheroidal radial functions of the third and fourth kind: { Rhml,()(,)3ξ, Rhml,()(,)4ξ} and { Rj h jml,()(, )3−ξ, Rj h jml,()(, )4−ξ}, respectively. The spheroidal radial functions of the third and fourth kind are related to those of the first and second kind by the following expressions [23]: Prolate: Rh R h j R hml ml ml,() , () ,() ,()(,) (,) (,)34 1 2ξ ξ ξ =± ; (2-74a) Oblate: R jh j R jh j jR jh jml ml ml,() , () ,() ,()(, ) (, ) (, )34 1 2−= −± −ξ ξ ξ. (2-74b) It must be noted that the spheroidal radial functions of the third and fourth kind are somewhat analogous to complex exponentials. As a matter of fact, both of these spheroidal radial functions asymptotically approach complex exponentials as ξ→∞ [23]: Prolate: () lim ( , ) exp,() , () ξξξξ π →∞=± − +  Rhhjh lml34 11 21 ; (2-75a) Oblate: () lim ( , ) exp,() , () ξξξξ π →∞−=± − +  Rj h jhjh lml34 11 21 . (2-75b) As we shall see, both of these asymptotic expressions will prove useful later on in our development. Furthermore, these functions also appear to be universally defined so thatin the special case that h=0, both complex sets of spheroidal radial functions approach the spherical Hankel functions of the first and second kind { h l()()1ξ, hl()()2ξ} [23]. Just like the spheroidal angle functions, the spheroidal radial functions that correspond to negative orders of m are not linearly independent from their positive order counterparts. This is so because the solutions to the both the prolate and oblate radialdifferential equations, i.e., (2-34a) and (2-34b), are also symmetric with respect to the 56order m due to the m2 term. As for the proportionality constant between the positive and negative order solutions, our development is simplified if we define it to be unity: Prolate: RhRhml ml− ≡,() , () ,() , ()(,) (,)34 34ξξ ; (2-76a) Oblate: Rj h j Rj h jml ml−−≡−,() , () ,() , ()(, ) (, )34 34ξξ . (2-76b) Once again, we are at liberty to do so because a linear combination of solutions of a linear homogeneous differential equation is also a solution of the equation [30]. The spheroidal radial functions of the third and fourth kind have significant meaning in the study of wave phenomena when using the spheroidal coordinate systems. Both of these linearly independent solutions, i.e., Rml,()3 and Rml,()4, represent incoming and outgoing traveling waves, respectively. Note that the mathematical form of the complex exponentials in the asymptotic relations of (2-75a) and (2-75b) is consistent with thisphysical interpretation of the spheroidal radial functions. Now that the separated solutions for the scalar Helmholtz equation in spheroidal coordinates have been understood, we are at the point where the overall general solutionscan be constructed. Referring back to (2-33a) and (2-33b), the overall solution isobtained by taking the product of the separated solutions. However, our investigation hasrevealed that the possible solutions are limited to a discrete set of eigenfunctions basedupon the order m and the degree l. Therefore, the general solutions must be expressed as a linear combination of these resulting eigenfunctions [23]: Prolate: () ()ψ ξηϕ ψ ξηϕP ml mlmlPa ,, ,,, ,, =∑ (2-77a) where() {} ψξ η ϕξ ξηϕ ϕ mlP ml mlmljm jmRh RhShe e,,() ,() , ,,(,) (,)(, ) =        + −3 4; (2-78a) 57Oblate: () ()ψξηϕ ψξηϕO ml mlmlOa ,, ,,, ,, =∑ (2-77b) where () {} ψξηϕξ ξηϕ ϕ mlO ml mlmljm jmRj h j Rj h jSj he e,,() ,() , ,,(, ) (, )(, ) =− −    −  + −3 4. (2-78b) Note that we have opted to use the exponential form of the solutions in the expressions for the general solutions. By evoking Sturm-Liouville theory, we are allowed to approximate the solution to the scalar Helmholtz equation in spheroidal coordinates to any degree of accuracy with anexpansion of its respective eigenfunctions. In expressions (2-78a) and (2-78b), thefunctions encased in the brackets are understood to be the basis sets of the eigenfunctionexpansions. Based upon differential equation theory, a particular solution can beformulated from the general solutions only after imposing boundary conditions. Furtherin our development, we shall apply the specific boundary conditions to our generalsolutions to formulate the particular eigenfunction expansion needed for our NZ-FZTprocess. Lastly, having presumed the completeness of these developed basis sets, we canexpect convergence in the mean for an eigenfunction expansion of a well-behavedparticular solution. 2.2.6 Statement of Boundary Conditions Imposed by the NZ-FZT Process In the previous section, we applied the boundary conditions imposed by physical requirements when constructing both general solutions of the scalar Helmholtz equationin spheroidal coordinates (viz., single-valuedness of the azimuthal solutions andfiniteness of the angular solutions). One can recall that it was these boundary conditionsthat limited the general solutions to a discrete set of eigenfunctions. However, in order toconstruct the particular solutions to our overall prolate and oblate spheroidal NZ-FZTboundary value problems, we must further impose additional boundary conditions. It ishere that we state these additional boundary conditions, as well as express their impact onnarrowing the general solutions to the desired particular solutions. 58As stated at the outset of this chapter, the purpose of our NZ-FZT process is to use near-zone E-field samples along a spheroidal transformation surface (which surrounds theradiating structure) to extrapolate the far-zone response. The first significant boundarycondition is implicitly contained within this objective. Specifically, the fact that thespheroidal transformation surface fully encompasses the radiating structure of interestimplies spatial causality with respect to wave and energy propagation; that is, at all pointsoutside the closed spheroidal transformation surface (i.e., the source region), onlyoutgoing traveling waves must exist. Enforcing this boundary condition on a surface atinfinity was first introduced by Sommerfeld and is referred to as the “radiation condition”[33]. As indicated by Felsen and Marcuvitz [33], the Sommerfeld radiation conditionalso implies that the transverse components of a spherically diverging wave propagating in a homogeneous medium behave as ()erjkr−. By virtue of the Sommerfeld radiation condition, the particular solutions to our overall spheroidal NZ-FZT boundary value problems must only contain the outward traveling spheroidal radial functions of the fourth kind, viz., Rhml,()(,)4ξ and Rj h jml,()(, )4−ξ. Furthermore, these radial functions must also remain consistent with the radiation condition and behave as ()erjkr− in the far-zone; this is demonstrated by the following asymptotic expressions for spheroidal radial functions of the fourth kind: Prolate: () lim ( , ),() ξξ →∞=+ −Rhj kermll jkr 41 ; (2-79a) Oblate: () lim ( , ),() ξξ →∞−=+ −Rj h jj kermll jkr 41 . (2-79b) (Note that these equations can be found from the asymptotic expressions of (2-75a) and (2-75b) together with the fact that hk rξ→ in the far-zone.) It should be emphasized that these expressions prove to be very important in the development of our NZ-FZT process. They essentially mean that the radial dependence of the field in the far-zone will 59approach that of a spherically diverging wave (expressed in spherical coordinates) regardless of the independent variable h. Moreover, the far-zone behavior of the radial dependence is identical for both the prolate and oblate cases. Having discussed the boundary condition imposed on the spheroidal surface at infinity, we now must address the boundary condition that exists on the spheroidaltransformation surface. The NZ-FZT process input specifies the E-field on this boundary.As a result, the particular field solutions for both the prolate and oblate cases must beconstructed to match the near-zone along their respective transformation surface. Usingthe mathematical description of the spheroidal transformation surfaces provided earlier(see 2.1.1), we express these Dirichlet boundary conditions as follows: Prolate: ψξηϕP P(, , ) Ω≡ψξηϕP o(, , ) = known ; (2-80a) Oblate: ψξηϕO O(,, ) Ω≡ψξηϕO o(, , ) = known . (2-80b) The boundary conditions imposed by (2-80a) and (2-80b) are classified as Dirichlet boundary conditions because they specify the values of a function on a given boundary[29]. 2.2.7 Development of Spheroidal Wave-Harmonics In order to facilitate construction of the particular solutions to our overall NZ-FZT boundary value problems, we must first develop an understanding of spheroidal wave-harmonics for both the prolate and oblate cases. Since no reference appears to addressthem, we must take some time to develop and define our own. Fortunately, we have alogical starting point to base our development: conventional spherical harmonics, i.e., thespecial case where h=0. Furthermore, we shall define our spheroidal wave-harmonics with a notation that somewhat resembles that of conventional spherical harmonics. In the section that follows, the particular solutions to our overall NZ-FZT boundary value problems have to be expanded on the spheroidal transformation surface in 60terms of their azimuthal and angular eigenfunctions. Because of this requirement, it is logical to develop a two-dimensional basis set (dependent on azimuthal and angularcoordinates) for both the prolate and oblate cases that remains orthonormal over a givenspheroidal surface. This spheroidal wave-harmonic for both spheroidal cases can beobtained by taking the product of their respective normalized azimuthal and angulareigenfunctions. We can express the azimuthal eigenfunctions presented in (2-56) using the following orthonormal scheme: Φmjme ()ϕ πϕ=+ 1 2 (2-81) ΦΦmm m m d () *(), ϕϕ ϕ δπ ′′ = ∫ 02 (2-82) where −≤ ≤ +lm l . Here we have used the overbar to indicate that the eigenfunction has been normalized. Additionally, we are able to include both sets of complex exponentials by extending theindex m over positive and negative orders. Regarding the angular eigenfunctions, we would like to replace the angular parameter η with ϑ. As mentioned beforehand, the two angular parameters are mutually related by the transformation given in equation (2-5), i.e., ηϑ=cos . Employing this change of variable together with the orthonormal scheme, we can express both angular eigenfunctions and their orthogonality relations as follows: Prolate: Sh NhShml mlml , ,, (, c o s ) ()(, c o s ) ϑϑ=1(2-83a) Sh S h dml ml ll,, ,(, c o s ) (, c o s ) s i nϑϑ ϑ ϑ δπ 0∫ ′′ = ; (2-84a) 61Oblate: Sj h Nj hSj hml mlml , ,, (, c o s ) ()(, c o s ) −= −− ϑϑ1(2-83b) Sj h S j h dml ml ll,, ,(, c o s )(, c o s ) s i n−− =∫ ′′ϑϑ ϑ ϑ δπ 0. (2-84b) Once again we have used the overbar to denote normalization of the eigenfunction. In the orthogonality relations presented in (2-84a) and (2-84b), it is important to understand thatthe sinϑ integration factor that arises from the change of variable is consistent with Sturm-Liouville theory. One can show by performing a change of independent variable, viz., ηϑ→ , on the differential equations for both angular self-adjoint boundary value problems (prolate and oblate) leads to a non-unity weighting function that is equal to the sinϑ integration factor. As previously stated in the definitions of (2-69a) and (2-69b), we have assumed a unity proportionality constant relating positive and negative orders of spheroidal anglefunctions. This implies that the Sturm-Liouville normalization constants that correspondto the positive orders of spheroidal angle functions must be equal to their negative ordercounterparts: Prolate: Nh N h ml ml,,() ()≡− ; (2-85a) Oblate: Nj h N j hml ml,,() ()−≡ −− . (2-85b) Therefore, the normalized angular eigenfunctions expressed in (2-83a) and (2-83b) must also possess a unity proportionality constant between the positive and negative orders m. We can now define our spheroidal wave-harmonics for both the prolate and oblate cases by taking the product of their corresponding normalized azimuthal and angulareigenfunctions: 62Prolate: Yh NhSh eml mlmljm , ,, (, , ) ()(, c o s ) ϑϕ πϑϕ=+ 1 2 (2-86a) where −≤ ≤ +lm l , Yh Y h dml m l P mm ll P,, , ,(, , ) *(, , ) ϑϕ ϑϕ δδ′′ ′ ′ ∫∫=Ω Ω(2-87a) where { ΩP: 0≤≤ϑπ , 0 2≤≤ϕπ, and dd dPΩ= sinϑϑϕ}, Yh Y hml ml,,*(, , ) (, , )ϑϕ ϑϕ =− ; (2-88a) Oblate: Yj h Nj hSj h eml mlmljm , ,, (, , ) ()(, c o s ) −= −−+ϑϕ πϑϕ 1 2(2-86b) where −≤ ≤ +lm l , Yj h Y j h dml m l O mm ll O,, , ,(, , ) *(, , ) −− =′′ ′ ′ ∫∫ϑϕ ϑϕ δδΩ Ω(2-87b) where { ΩO: 0≤≤ϑπ , 0 2≤≤ϕπ, and dd dOΩ= sinϑϑϕ}, Yj h Y j hml ml,,*( ,,) ( ,,)−= −− ϑϕ ϑϕ. (2-88b) As we can see, the spheroidal wave-harmonics defined in (2-86a) and (2-86b) are valid over the orders where −≤ ≤ +lm l ; it is in this instance that our unity proportionality constant relating positive and negative orders of the angular solutions proves to beconvenient. The two-dimensional orthogonality relations provided in (2-87a) and (2-87b)are derived by taking the product of the azimuthal orthogonality relation, (2-82), and theirrespective angular orthogonality relations, (2-84a) and (2-84b). These resultingorthogonality relations demonstrate that the defined spheroidal wave-harmonics areorthonormal over their respective spheroidal surfaces. The fact that the complex 63conjugate of a spheroidal wave-harmonic is equal to its respective negative order, i.e., (2- 88a) and (2-88b), follows by inspection. Finally, something must be said about why we choose to refer to the product of the angular and azimuthal spheroidal solutions for the prolate and oblate cases as“spheroidal wave-harmonics. ” Traditionally, the term “harmonic ” referred to solutions of the Laplace equation (i.e., the static equation where k=0 ) because they were called harmonic functions [29]. When solving both the Helmholtz and Laplace equations inspherical coordinates, the angular and azimuthal Helmholtz solutions happen to exactlymatch those of the Laplace solutions. It is the product of these angular and azimuthalsolutions that are often referred to as “spherical harmonics ” [29]. Thus, in this particular case, the solutions to the Helmholtz equation happen to be referred to as “harmonics. ” However, when solving the Helmholtz and Laplace equations in the prolate and oblatespheroidal coordinate systems, the angular part of the solution is no longer the same forthe two partial differential equations. This is why the angular solutions (as well as theradial solutions) of the scalar Helmholtz equation (i.e., the wave equation) in bothspheroidal coordinate systems are referred to as “spheroidal wave functions ” [23]. In short, we label the product of these angular and azimuthal solutions as “spheroidal wave- harmonics. ” We do so in order to maintain some sort of kinship with conventional spherical harmonics, yet still convey that the two-dimensional basis sets are solutions tothe scalar Helmholtz equation (i.e., the wave equation). 2.2.8 Construction of the Particular Solutions Using Expansions of Spheroidal Wave- Harmonics Equipped with the developed spheroidal wave-harmonics and their related properties, we are now ready to continue with the construction of the particular solutionsto our overall NZ-FZT boundary value problems (i.e., for the prolate and oblate cases).Construction of the particular solutions begins with applying the Sommerfeld radiationcondition to both sets of general solutions. Application of this boundary condition leadsto the appropriate expansions of spheroidal wave-harmonics. Both of these expansions 64have coefficients that can be determined only after applying their respective Dirichlet boundary conditions. To do so, each particular solution, i.e., the expansion of spheroidalwave-harmonics, must be forced to match the known field quantities (specified by theNZ-FZT process input) along its respective spheroidal boundary. Lastly, we approximateeach of these particular solutions in the far-zone region by using the appropriateasymptotic expression, viz., (2-79a) or (2-79b). We recall that the Sommerfeld radiation condition has limited our radial solutions to only outward traveling waves, viz., Rh ml,()(,)4ξ and Rj h jml,()(, )4−ξ; this eliminates the spheroidal radial functions of the third kind, i.e., Rhml,()(,)3ξ and Rj h jml,()(, )3−ξ, from both sets of possible solutions. Applying this boundary condition to the general solutions expressed in (2-77a)-(2-78a) and (2-77b)-(2-78b), we can construct both particularsolutions in terms of the following spheroidal wave-harmonic expansions: Prolate: ψξϑϕ ξ ϑϕP ml ml mll lml bh RhYh (, , ) () (,) (, , ),,() , = =−+ =∞ ∑∑4 0; (2-89a) Oblate: ψξϑϕ ξ ϑϕO ml ml mll lml b jh R jh j Y jh ( , , ) ()(, ) (, , ),,() , =− −− =−+ =∞ ∑∑4 0. (2-89b) Note that both expansions represent the linear combination of all possible linearly independent solutions to their respective boundary value problem. The limits of theindices m and l are based upon the possible eigenvalues and the definition of the developed spheroidal wave-harmonics. Our next step is to determine the expansion coefficients for both the prolate and oblate cases. In order to accomplish this task, we must first enforce each respectiveDirichlet boundary condition over its corresponding spheroidal boundary. Thisessentially means that both particular solutions must match the known field quantities along their respective spheroidal transformation surface (i.e., the ξξ=o constant parameter surface): 65Prolate: ψξϑϕ ξ ϑϕP om l m l mll lom l known b h R h Y h ( , , ) () (, ) (, , ),,() , == =−+ =∞ ∑∑4 0; (2-90a) Oblate: ψξϑϕ ξ ϑϕO om l m l mll lom l known b jh R jh j Y jh (, , ) ( ) ( , ) ( , , ),,() , == − − − =−+ =∞ ∑∑4 0. (2-90b) We can now multiply each of these equations by its respective conjugated spheroidal wave-harmonic, Yhml′′,*(, , )ϑϕ or Yj hml′′−,*(, , )ϑϕ, and then integrate over its corresponding spheroidal surface: Prolate: ψξϑϕ ϑϕP om l P Yh d P(, , ) *(, , ),′′ ∫∫Ω Ω = bh Rh Yh Y h dml ml mll lom l m l P P,,() ,, () (, ) (, , ) *(, , )4 0=−+ =∞ ′′ ∑∑∫∫ξ ϑϕ ϑϕΩ Ω; (2-91a) Oblate: ψξϑϕ ϑϕO om l O Yj h d O(, , ) *(, , ),′′− ∫∫Ω Ω = b jh R jh j Y jh Y jh dml ml mll lom l m l O O,,() ,, ()(,) (, , ) *(, , ) −− − − =−+ =∞ ′′ ∑∑∫∫4 0ξ ϑϕ ϑϕΩ Ω. (2-91b) Assuming the permissibility of termwise integration, we first interchange the integration operator and the summation on the right-hand side of the equation. Having performed 66this operation, we then make use of the orthogonality relation for prolate spheroidal wave-harmonics, i.e., (2-87a), such that we are able to employ the discrete version of thesifting property to isolate each expansion coefficient: Prolate: ψξϑϕ ϑϕP om l P Yh d P(, , ) *(, , ),′′ ∫∫Ω Ω = bh Rh Yh Y h dml ml mll lom l m l P P,,() ,, () (, ) (, , ) *(, , )4 0=−+ =∞ ′′ ∑∑ ∫∫ξ ϑϕ ϑϕΩ Ω = bh Rhml ml mll lom m l l ,,() ,, () (, )4 0=−+ =∞ ′′ ∑∑ ξδδ =bh Rhml ml o′′ ′′,,()() (, )4ξ. (2-92a) For the oblate case, similar mathematical operations can be performed to yield the following: Oblate: ψξϑϕ ϑϕO om l O Yj h d O(, , ) *(, , ),′′− ∫∫Ω Ω =bj h Rj h jml ml o′′ ′′−−,,()() (,)4ξ. (2-92b) We are now able to drop the prime from the indices and express the coefficients for both expansions of spheroidal wave-harmonics, i.e., (2-89a) and (2-89b), as follows: Prolate: bhRhYh dml ml oP om l P P, ,() , ()(, )(, , ) *(, , ) = ∫∫1 4ξψξϑϕ ϑϕΩ Ω; (2-93a) Oblate: bj hRj h jYj h dml ml oO om l O O, ,() , ()(,)(, , ) *(, , ) −=−− ∫∫1 4ξψξϑϕ ϑϕΩ Ω. (2-93b) Since the objective of our NZ-FZT process is to compute the far-zone response of a radiating structure, we shall approximate the developed spheroidal wave-harmonic 67expansions in the far-zone region by using asymptotic expressions for the spheroidal radial functions of the fourth kind, i.e., Rhml,()(,)4ξ and Rj h jml,()(, )4−ξ. The far-zone behavior of these radial functions has already been examined and is described in expressions (2-79a) and (2-79b). The asymptotic expression for the prolate spheroidalradial function of the fourth kind can then be applied to (2-89a) to yield the following far-zone spheroidal wave-harmonic expansion for the prolate case: Prolate: ψfzP = lim ( , , ) ξψξϑϕ →∞P = bh Yh Rhml mll lml ml ,, ,()() (, , ) l i m (,) =−+ =∞ ∑∑→∞   04ϑϕ ξξ = () bh Yhj kerml mll lmll jkr ,,() (, , ) =−+ =∞ + −∑∑   01 ϑϕ = ()j kbh ahYh e rl ml mlmll lmljkr+ =−+ =∞ −    ≡∑∑1 0, ,, () ()(, , ) /G09/G0A/G0C/G0C/G0B/G0C/G0Cϑϕ . (2-94a) For the sake of simplicity, we have elected to incorporate the complex factor []jkl+1 into our new expansion coefficient ahml,( ) . Furthermore, notice how the presence of the ()erjkr− radial dependence is consistent with the Sommerfeld radiation condition. For reasons expressed earlier (see 2.1.2), we shall once again suppress the ()erjkr− radial factor. In short, by using a bar over the scalar field quantity to indicate suppression of the radial dependence, we express the far-zone prolate spheroidal wave-harmonic expansionwith its expansion coefficients as follows: 68 Prolate: ψϑϕfzP(,) = ah Yhml mll lml ,,() (, , ) =−+ =∞ ∑∑ 0ϑϕ (2-95a) where ahj kR hYh dmll ml oP om l P P, ,() , ()(, )(, , ) *(, , ) =+ ∫∫1 4ξψξϑϕ ϑϕ Ω Ω. (2-96a) Likewise, following similar mathematical operations and rationale, we are also able to express the far-zone oblate spheroidal wave-harmonic expansion with its expansioncoefficients: Oblate: ψϑϕfzO(,) = aj h Yj hml mll lml ,,() (, , )−− =−+ =∞ ∑∑ 0ϑϕ (2-95b) where aj hj kR j h jYj h dmll ml oO om l O O, ,() , ()(,)(, , ) *(, , ) −=−−+ ∫∫1 4ξψξϑϕ ϑϕ Ω Ω. (2-96b) Having satisfied all of the imposed boundary conditions, both of these spheroidal wave- harmonic expansions individually form the particular solution to their respective NZ-FZTboundary value problem. 692.3 THE NZ-FZT ALGORITHMS FOR THE PROLATE AND OBLATE SPHEROIDAL CASES 2.3.1 Algorithmic Implementation of the Constructed Particular Solutions In this section we take the constructed prolate and oblate particular solutions and reduce each of them to a numerical algorithm capable of being implemented by acomputer code. The first step in the formulation of the algorithms requires that eachexpansion of spheroidal wave-harmonics be expressed only in terms of special functionsthat are able to be numerically evaluated. This is followed by addressing the particularsregarding the sampling of the known near-zone E-field along the respective spheroidaltransformation surface, including how these samples are incorporated into ourcomputational process. Finally, numerical algorithms are put forth that enable thecalculation of the prolate and oblate particular solutions for our NZ-FZT process.Recalling that both particular solutions have been constructed for the scalar case, weexplain in the section that immediately follows (i.e., 2.3.2) how these algorithms areextended to the vector case using our hybrid of coordinate systems approach (see 2.2.2). At this point in the development of our NZ-FZT process, it has become evident that the formulations for both the prolate and oblate cases have begun to closely resembleone another. Prior to now, we have elected to explicitly state results for both spheroidalcases throughout our discussion on behalf of providing a theoretical development that isboth thorough and complete. Furthermore, this approach of stating the results for bothspheroidal cases in a side-by-side manner has also allowed us to compare the twoformulations, as well as define their respective nomenclature and special functions.However, we shall now depart from this practice in the following discussion regardingthe development of the NZ-FZT algorithms. For the sake of brevity, the discussion thatensues shall only provide a step-by-step development of the numerical algorithm for theprolate case, not the oblate case. Since these numerical algorithms closely parallel oneanother, details regarding development of the algorithm for the oblate case can easily be 70derived from the discussion of its prolate counterpart. Nonetheless, we shall still disclose both prolate and oblate numerical algorithms at the conclusion of this discussion. Since the Dirichlet boundary condition specifies the field quantities on the prolate spheroidal surface (i.e., the ξξ=o constant parameter surface), we define this boundary condition as a function of only angular and azimuthal parameters: ψξϑϕP P(, , ) Ω≡ψξϑϕP o(, , ) ≡fP(,)ϑϕ. (2-97a) As we shall ultimately see, this two-dimensional function will serve as the NZ-FZT process input for each respective Cartesian vector component of the near-zone E-field.We can now substitute this two-dimensional function into our expression for the wave-harmonic expansion coefficients presented in (2-96a): ahj kR hfY h dmll ml oP ml P P, ,() , ()(, )(,) *(, , ) = ′′ ′′ ′+ ′∫∫1 4ξϑϕ ϑϕ Ω Ω. (2-98a) It is important to note that in (2-98a), the variables of integration, viz., ′ϑ and ′ϕ, have been primed in order to distinguish them as boundary parameters that exist along the prolate spheroidal transformation surface; they are considered in the near-zone. As fortheir unprimed counterparts, i.e., ϑ and ϕ, they represent the far-zone observation parameters in the prolate spheroidal coordinate system. Thus, we can now substitute (2- 98a) into the far-zone prolate wave-harmonic expansion presented in (2-95a) to obtain ψ ϑϕfzP(,) =j kR hfY h Y h dl ml oP ml ml P mll lP+ ′ =−+ =∞ ′′ ′′ ′ ∫∫ ∑∑1 4 0 ,() ,,(, )(,) *(, , ) (, , )ξϑϕ ϑϕ ϑϕΩ Ω. (2-99a) Once again presuming the permissibility of interchanging the summation with the integration operator, we yield 71ψ ϑϕfzP(,) =j kYh Yh Rhfdl ml ml ml o mll lP P P+ =−+ =∞ ′′′    ′′ ′ ∑ ∑∫∫1 4 0,, ,()*(, , ) (, , ) (, )(,)ϑϕ ϑ ϕ ξϑϕ Ω Ω. (2-100a) At this juncture in our development, we make use of the following crucial identity derived in Appendix E: Yh Yh Rhml ml ml o mll ,, ,()*(, , ) (, , ) (, )′′ =−+ ∑ϑϕ ϑ ϕ ξ4 =1 2200 0044 1 πϑϑ ξϑϑ ξϕϕSh Sh Nh R hSh Sh Nh Rhmll ll oml ml ml ml o ml ,, ,,(),, ,,()(, c o s ) (, c o s ) () (, )(, c o s ) (, c o s ) () (, )cos ( )′+′−′   =∑ . (2-101a) It is this double- to single-sided wave-harmonic identity that enables us to reduce our eigenfunction expansion into a mathematical form that is expressed only in terms ofcomputable special functions. We do so by substituting (2-101a) into (2-100a): ψ ϑϕfzP(,) = j kSh Sh Nh Rh Sh Sh Nh Rhmfdlll ll o ml ml ml ml o ml lP P P+ ==∞ ′′ +′−′          ′′ ′ ∑∑∫∫100 004 4 1022πϑϑ ξ ϑϑ ξϕϕϑϕ,, ,,() ,, ,,()(, c o s ) (, c o s ) () (, ) (, c o s ) (, c o s ) () (, )cos ( )(,) Ω Ω. (2-102a) Assuming that we are able to termwise integrate the resulting expression, we reduce it to an eigenfunction expansion in terms of a set of indexed integrals dependent upon the Dirichlet boundary condition function fP(,)′′ϑϕ : 72ψ ϑϕfzP(,) =j kSh Nh RhfS h d Sh Nh RhfS h m dll ll oP lP ml ml ml o ml P ml PlP P+′ = ′=∞′′ ′ ′ + ′′ ′ −′′        ∫∫ ∑ ∫∫∑10 004 0 4 1022πϑ ξϑϕ ϑ ϑ ξϑϕ ϑ ϕ ϕ, ,,() , , ,,() ,(, c o s ) () (, )( , ) ( ,cos ) (, c o s ) () (, )( , ) ( ,cos ) cos ( )Ω ΩΩ Ω =j kSh Nh R hIhSh Nh RhIhl l ll olml ml ml olm ml l+ = =∞ +   ∑ ∑1 0 004 0 4 1 022πϑ ξϑ ξϕ, ,,() ,, ,,() ,(, c o s ) () (, )()(, c o s ) () (, )(, ) (2-103a) where Ih f Sh dlP lP P00,,( ) ( , ) ( ,cos )= ′′ ′ ′ ′∫∫ϑϕ ϑ Ω Ω for l≥0 (2-104a) and Ih f Sh m dmlP ml P P,,(, ) ( , ) (, c o s ) c o s ( )ϕϑ ϕ ϑ ϕ ϕ= ′′ ′ −′′ ′∫∫Ω Ω for 1 ≤≤ml . (2-105a) Now that we have expressed the expansion of prolate spheroidal wave-harmonics in terms of computable special functions, the next order of business is to determine howto best approximate the indexed two-dimensional integrals of (2-104a) and (2-105a) interms of the sampled near-zone E-field along the prolate spheroidal transformationsurface. As illustrated in Figure 2.2a, let us first partition the surface of the prolate spheroid (i.e., the ξξ=o constant parameter surface) into uniform surface elements ∆′Sij, (with respect to ′ϑ and ′ϕ dimensions) that are centered at ( , ) ′′ϑϕij and have respective widths ∆′ϑ and ∆′ϕ. If we allow the subscripts i and j to index the surface elements in the angular and azimuthal directions, respectively, we can define the center of each surface element as follows: 73(υ ,ϕ )ij ∆ϕz ∆υυ y x ϕ Figure 2.2a The i th, j th surface element ∆′Sij, of the partitioned prolate spheroidal transformation surface (i.e., the ξξ=o constant parameter surface). 74Angular: ∆′=ϑπ I (2-106a) ′=− ′ ϑϑii()1 2∆ (2-107a) where I = total number of samples in the angular direction and I i ,,2,1/G15 = ; Azimuthal: ∆′=ϕπ 2J (2-108a) ′=− ′ ϕϕj j()1 2∆ (2-109a) where J = total number of samples in the azimuthal direction and J j ,,2,1/G15 = . As we can see, expressions (2-104a) and (2-105a) both require an integration operation that must be performed over the entire prolate spheroidal transformationsurface. Because we have partitioned the spheroidal surface into ( ) IJ× surface elements, these integration operations can be broken down into a finite summation of individual integration operations performed over each surface element ∆′S ij,: Ih f Sh dlP l ijp Sij00,, ,( ) ( , ) ( ,cos ) ,= ′′ ′ ′ ′∫∫∑ ϑϕ ϑ ∆Ω (2-110a) Ih f Sh m dmlP ml ijp Sij,, ,(, ) ( , ) (, c o s ) c o s ( ) ,ϕϑ ϕ ϑ ϕ ϕ= ′′ ′ −′′ ′∫∫∑ ∆Ω. (2-111a) We shall now presuppose that the two-dimensional function fP(,)′′ϑϕ varies slowly over any particular surface element ∆′Sij, provided that the size of its area is made fine enough. This assumption allows us to approximate the function fP(,)′′ϑϕ over the entire prolate spheroidal transformation surface as a mosaic of constant values with each one corresponding to an individual surface element ∆′Sij,. Furthermore, these values 75shall equal the function fP(,)′′ϑϕ sampled at the center of each respective surface element: in mathematical terms, ffijPP ij , (, ) ≡ ′′ ≡ ϑϕ constant . Using this approximation of the two-dimensional function fP(,)′′ϑϕ , we approximate the finite summation of individual integration operations defined in (2-110a) and (2-111a) in terms of the indexed samples: Ih f Sh dli jP l ijp Sij00,, , ,() (, c o s ) ,≅ ′′ ′∫∫∑ ∆Ω ϑ = ′′ ′∫∫ ∑fS h dijP l ijp Sij,, ,,(, c o s )0 ∆Ω ϑ , (2-112a) p jilmP ji lm d m h Sf hI jiSΩ′ ′− ′ ≅∑∫∫ ′∆) ( cos) cos,( ),( ,, , , ,ϕϕ ϑ ϕ p jilmP ji d m h S f jiSΩ′ ′− ′ =∑ ∫∫ ′∆) ( cos) cos,( ,, , ,ϕϕ ϑ . (2-113a) Note that the indexed samples fijP , can be brought outside both integrals because they remain constant with respect to the variables of integration ′ϑ and ′ϕ over each individual surface element ∆′Sij, (by definition of the approximation for the two- dimensional function fP(,)′′ϑϕ ). So that we may further reduce the approximated integrations of (2-112a) and (2- 113a), each of the individual integration operations (with respect to i and j) must first be explicitly performed over its corresponding surface element ∆′Sij,: 76Sh dlp Sij0, ,(, c o s ) ∆Ω ′∫∫′′ϑ = Sh ddl ii jj 0 22 22 ,(, c o s ) s i n ′′ ′ ′ ′−′′+′ ′−′′+′           ∫ ∫ϑ ϑϑϕ ϑϑϑϑ ϕϕϕϕ ∆∆ ∆∆ = ∆ ∆∆ ′′ ′ ′ ′−′′+′      ∫ϕϑ ϑ ϑ ϑϑϑϑ Sh dl ii 0 22 ,(, c o s ) s i n for l≥0, (2-114a) Sh m dml p Sij, ,( ,cos ) cos ( ) ∆Ω ′∫∫′ −′′ ϑϕ ϕ = Sh m d dml ii jj ,( ,cos ) cos ( ) sin ′ −′′ ′ ′ ′−′′+′ ′−′′+′           ∫ ∫ϑ ϕϕ ϑϑϕ ϑϑϑϑ ϕϕϕϕ ∆∆ ∆∆ 22 22 =2 2 22 mmm S h djm l ii sin cos ( ) ( ,cos ) sin,∆ ∆∆′   ′− ′′ ′ ′−′′+′      ∫ϕϕϕ ϑ ϑ ϑ ϑϑϑϑ for 1 ≤≤ml . (2-115a) Using these explicit results for the individual integration operations and the approximated integrations of (2-112a) and (2-113a), we can now present the overall approximations forthe indexed integrals of (2-104a) and (2-105a): Ih fI S li jP liP ij00,, , , ,()≅ ′∑∆ϕ for l≥0 (2-116a) where IS S h dliP l ii 00 22,, , (, c o s ) s i n ≡ ′′ ′ ′−′′+′      ∫ϑϑ ϑ ϑϑϑϑ ∆∆ , (2-117a) 77Ihmmf mI Sml i jP jm l iP ij,, , , ,(, ) s i n c o s ( )ϕϕϕ ϕ ≅′   ′− ∑2 2∆ for 1≤≤ml (2-118a) where IS S h dmliP ml ii ,, , (, c o s ) s i n ≡ ′′ ′ ′−′′+′      ∫ϑϑ ϑ ϑϑϑϑ ∆∆ 22 . (2-119a) From a computational viewpoint, these approximations require numerical integrations of a function, viz., the prolate spheroidal angle function of the first kind, thatin and of itself must be numerically evaluated. At first glance, these numericalintegrations may appear to be computationally costly. However, as we shall see, thesecomputations are quite plausible and will only need to be calculated once for eachparticular spheroidal transformation surface, regardless of the number of far-zoneobservation angles. As a matter of fact, these numerical integrations do not account for amajority of the computation time for an average computational run. Particulars regardingthese numerical integrations are addressed in Chapter 3, which focuses on computationsof the required special functions. An alternative approach to approximating the indexed integrals of (2-104a) and (2-105a) could have been to use standard Riemann sums and thus avoid the requirednumerical integrations. Unfortunately, a drawback of this approach is that theconvergence of the integration is then coupled to the order and degree, i.e., m and l, of the two-dimensional basis functions; that is, the more oscillatory wave-harmonics (dependenton l and m) would require finer sampling of the Dirichlet boundary condition function f P(,)′′ϑϕ regardless of how slow or fast the function varies over the prolate spheroidal transformation surface. The approach that we have elected to use, i.e., (2-116a)-(2-119a), effectively allows us to more efficiently sample the two-dimensional function fP(,)′′ϑϕ. Now that we have determined how to best approximate the indexed two- dimensional integrals of (2-104a) and (2-105a), we return our attention to our expansionof prolate spheroidal wave-harmonics. As explained earlier, we want our extrapolated 78far-zone field response to be in conventional spherical coordinates (see 2.1.2). Fortunately, the angular coordinate for both the prolate and oblate cases approaches thatof the spherical coordinate system in the far-zone (see Appendix A); this means that in the far-zone, i.e., where ξ→∞ , we can apply the following coordinate transformation: ϑθ→ . (2-120) As for the azimuthal parameter, it obviously remains unchanged by definition of the involved coordinate systems, i.e., ϕϕ≡. Since the far-zone prolate spheroidal wave- harmonic expansion presented in (2-103a) exists in the far-zone by definition, we can apply (2-120) to transform only the variables that represent the angular observationparameter, viz., ϑ (recalling that the angular boundary parameter ′ϑ is not in the far- zone): ψ θϕfzP(, ) = j kSh Nh R hIhSh Nh RhIhl l ll olml ml ml olm ml l+ = =∞ +   ∑ ∑1 0 004 0 4 1 022πθ ξθ ξϕ, ,,() ,, ,,() ,(, c o s) () (, )()(, c o s) () (, )(, ). (2-121a) The far-zone prolate spheroidal wave-harmonic expansion in (2-121a) presents itself as an infinite series expansion. In order for the expansion to be useable from anapplied perspective, we must be able to truncate it at some prescribed order and degree,i.e., m and l, and still obtain valid results; that is, the truncated wave-harmonic expansion needs to converge upon a solution as higher-order terms of m and l are considered. Having presumed the completeness of the developed basis sets (i.e., the spheroidal wave-harmonics) implies that the formulated spheroidal wave-harmonic expansion will renderconvergence in the mean for well-behaved solutions. As a result, we are able to truncateboth series of (2-121a) at some user prescribed degree L (since the range of the order m is dependent on the degree l) and still achieve valid results. 79At this point, we are able to insert our overall approximations for the indexed integrals, viz., (2-116a) and (2-118a), into the truncated version of (2-121a) to obtain thefollowing finite expansion of wave-harmonics: ≅),(ϕθ ψP fz λ πϕθ ξ ϕ θ ξϕϕol l ll oijP liP ij lL l ml ml ml oijP jm l iP ij ml lLSh Nh RhfI S m mSh Nh Rhfm I S()()(, c o s) () (, ) ()sin ( ) (, c o s) () (, )cos ( ), ,,() ,, , , , ,,() ,, , ,21 41221 0 004 0 0 1 4 1 1− ′ +−′′−      + = + = =∑ ∑ ∑ ∑∑∆ ∆. (2-122a) (Note that we replaced the complex factor jl+1 with ()−+11l in order to avoid any confusion with the azimuthal sample index j.) Because the expansion of (2-122a) is finite, we can interchange the order of the summations and then factor out the indexed samples P jif, (which are independent of l and m): ψ θϕfzP(, ) ≅ [] [] [] []λ πϕ ξθ ϕ ξθϕ ϕo ijPl ll oliP l lL l ml ml omliP ml j ml lLijfNh RhIS S h m mN h R hIS S h m()() () (, )(, c o s) () s i n ( ) () (, )( ,cos ) cos ( ),,,() ,, , ,,() ,, ,, 21 41221 004 00 0 1 4 1 1×− ′    +− ′    ′−        + = + = =∑ ∑∑∑∆ ∆. (2-123a) For both spheroidal cases, we are now able to put forth the algorithmic implementations of the constructed particular solutions capable of computing a Cartesianvector component of the desired far-zone E-field. Shown in Tables 2.5a and 2.5b are therequired computations (including the overall finite wave-harmonic expansions for boththe prolate and oblate cases) and the memory storage format of the NZ-FZT process for 80Table 2.5a Prolate Spheroidal NZ-FZT Algorithm for Each Cartesian Vector Component ψ θϕfzP(, ) ≅ [] [ ] [] [] [] []CfCI SS h CI S S h mijPlP liP l lL mlP mliP ml j ml lL jJ iI 12 300 0 1 11 1,,, , ,, , ,(, c o s) ( ,cos ) cos ( )× + ′−     = = == =∑ ∑∑∑∑θ θϕ ϕ (2-124a) Notation Definition Indices Memory Storage Format C1 λ πo ()22  none scalar constant []ClP2 () () (, ),,()− ′     +11 004l ll oNh Rh∆ϕ ξ),,1,0( L l/G15 = complex vector: ()L+1 elements []CmlP3, 4121 4() s i n ( ) () (, ),,()− ′   +l ml ml om mN h R h∆ϕ ξ),,2,1( L l/G15 = ; ),,2,1 ( l m/G15 =complex matrix: LL× elements {upper triangular} []ISliP 0, , Sh dl ii 0 22 ,(, c o s ) s i n ′′ ′ ′−′′+′      ∫ϑϑ ϑ ϑϑϑϑ ∆∆),,1,0( L l/G15 = ; ),,2,1( I i/G15 =real matrix: I L ×+)1( elements []ISmliP ,, Sh dml ii ,(, c o s ) s i n ′′ ′ ′−′′+′      ∫ϑϑ ϑ ϑϑϑϑ ∆∆ 22 ),,2,1( L l/G15 = ; ),,2,1 ( l m/G15 = ; ),,2,1( I i/G15 =real hyper-matrix: ILL ×× elements {triangular prism} fijP , fP ij(, )′′ϑϕ ),,2,1( I i/G15 = ; ),,2,1( J j/G15 =complex matrix: JI× elements 81Table 2.5b Oblate Spheroidal NZ-FZT Algorithm for Each Cartesian Vector Component ψ θϕfzO(, ) ≅ [] [ ] [] [] [] []CfCI SS j h CI S S j h mijOlO liO l lL mlO mliO ml j ml lL jJ iI 12 300 0 1 11 1,,, , ,, , ,(, c o s ) ( ,cos ) cos ( )×− +− ′−     = = == =∑ ∑∑∑∑θ θϕ ϕ (2-124b) Notation Definition Indices Memory Storage Format C1 λ πo ()22  none scalar constant []ClO2 () () (,),,()− ′ −−    +11 004l ll oNj h Rj h j∆ϕ ξ),,1,0( L l/G15 = complex vector: ()L+1 elements []CmlO3, 4121 4() s i n ( ) ()(,),,()− ′ −−  +l ml ml om mN jh R jh j∆ϕ ξ),,2,1( L l/G15 = ; ),,2,1 ( l m/G15 =complex matrix: LL× elements {upper triangular} []ISliO 0, , Sj h dl ii 0 22,(, c o s ) s i n− ′′ ′ ′−′′+′      ∫ϑϑ ϑ ϑϑϑϑ ∆∆),,1,0( L l/G15 = ; ),,2,1( I i/G15 =real matrix: I L ×+)1( elements []ISmliO ,, Sj h dml ii ,(, c o s ) s i n− ′′ ′ ′−′′+′      ∫ϑϑ ϑ ϑϑϑϑ ∆∆ 22 ),,2,1( L l/G15 = ; ),,2,1 ( l m/G15 = ; ),,2,1( I i/G15 =real hyper-matrix: ILL ×× elements {triangular prism} fijO , fO ij(, )′′ϑϕ ),,2,1( I i/G15 = ; ),,2,1( J j/G15 =complex matrix: JI× elements 82the scalar case. In the next section, we show how to extend these algorithms to the vector case using our approach that employs a hybrid of coordinate systems. Upon review of the algorithms disclosed in Tables 2.5a and 2.5b, important observations can be made regarding the implementation of the required computations. For any user selected spheroidal transformation surface, i.e., specified by h and ξo, all of the respective constants, vectors, and matrices only need to be computed once, regardless of the number of desired far-zone observation angles (, )θϕ. From the viewpoint of computational efficiency (with respect to computation time and the total number of floating point operations), this algorithmic feature is quite beneficial because most ofthese quantities require many evaluations of computationally demanding specialfunctions. However, this improvement in computational efficiency is only gained at theexpense of memory storage. Fortunately, present technology does possess the memorystorage capabilities required to implement these algorithms using the outlined storage scheme. It is important to note that some of the matrices, specifically, []CmlP3,, []ISmliP ,,, []CmlO3,, and []ISmliO ,,, could be more efficiently stored than the disclosed matrix format if need be. Further inspection of both algorithms reveals that for each desired far-zone observation angle θ, the only special functions that need to be recalculated are their respective spheroidal angle functions, viz., Shml,(, c o s) θ or Sj hml,(, c o s )− θ, where ),,1,0( L l/G15 = and ) ,,1,0 ( l m/G15 = . Correspondingly, for each user requested far-zone observation angle there are a total of () ( )LL++1 2 2 additional evaluations of the respective spheroidal angle functions. Finally, the prolate or oblate wave-harmonic expansions of (2-124a) and (2-124b) must be summed for each user requested far-zone observation angle (, )θϕ. It is in these summations that temporary storage can be exploited to further improve computational efficiency. 832.3.2 Extension of the Algorithmic Implementation for the Constructed Particular Solutions to the Required Vector Solutions The final step in completing the newly developed NZ-FZT process is to extend the developed algorithms so that they can handle the computation of the required vectorsolutions. We do so by using the hybrid of coordinate systems approach discussed earlierin this chapter (see 2.2.2). The approach requires us to solve the vector Helmholtzequation by using its scalar solution (which corresponds to the scalar Helmholtz equationin spheroidal coordinates) to individually represent all three vector components inCartesian coordinates. The overall vector solution is then obtained by using vectoraddition in Cartesian coordinates to combine all three scalar solutions. The algorithms put forth in Tables 2.5a and 2.5b are capable of computing each Cartesian vector component of the far-zone E-field for both spheroidal transformationsurfaces. Both algorithms individually operate on each Cartesian vector component ofthe Dirichlet boundary condition in one-to-one correspondence in order to extrapolate allthree Cartesian vector components of the far-zone E-field. In essence, the same exact algorithm operates on E xij(, )′′ϑϕ to yield Efzx(, )θϕ , Eyi j(, )′′ϑϕ to yield Efzy(, )θϕ , and Ezi j(, )′′ϑϕ to yield Efzz(, )θϕ ; we mathematically describe this process as follows: Prolate:E E Efzx fzy fzzfzP f ijP(, ) (, ) (, )(, ) ,θϕ θϕ θϕψθ ϕ      = (2-125a) where ffE E EijPP ijP oi jxi j yi j zi j, (, ) (,, )(, ) (, ) (, )= ′′ = ′′ =′′ ′′ ′′    ϑϕ ψ ξϑϕϑϕ ϑϕ ϑϕ; (2-126a) 84Oblate:E E Efzx fzy fzzfzO f ijO(, ) (, ) (, )(, ) ,θϕ θϕ θϕψθϕ      = (2-125b) where ffE E EijOO ijO oi jxi j yi j zi j, (, ) (,, )(, ) (, ) (, )= ′′ = ′′ =′′ ′′ ′′    ϑϕ ψξϑϕϑϕ ϑϕ ϑϕ. (2-126b) It should be reiterated that the bar over each Cartesian component of the far-zone E-field indicates the suppression of the radial dependence (see 2.1.2). Meanwhile, eachCartesian component of the Dirichlet boundary condition is the near-zone E-field sampledon the spheroidal transformation surface in spheroidal coordinates. Since each Cartesian component employs the same exact algorithm, the overall NZ-FZT process can take advantage of eliminating redundant computations. Specifically,we have formulated the required computations (i.e., the constants, vectors, matrices,spheroidal angle function evaluations, and part of the wave-harmonic expansion) to beindependent of the samples of the Dirichlet boundary conditions, i.e., E xij(, )′′ϑϕ , Eyi j(, )′′ϑϕ , and Ezi j(, )′′ϑϕ. However, the overall wave-harmonic summation, i.e., (2- 124a) or (2-124b), still has to be performed individually for each Cartesian component. Nevertheless, elimination of these redundant computations does significantly improve thecomputational efficiency of the overall NZ-FZT process. Specific details regarding thebest way to implement these computations are addressed in the subsequent chapter onNZ-FZT software development (see Chapter 4). Now that each Cartesian vector component of the far-zone E-field has been determined, we are able to put the resulting far-zone E-field in spherical coordinates asspecified by the NZ-FZT process output, i.e., equation (2-13): 85Efzθθϕ(, ) = cos cos ( , ) cos sin ( , ) sin ( , )θ ϕ θϕ θ ϕ θϕ θ θϕ EE Efzx fzy fzz+− (2-127) EE Efz fz x fzy ϕθϕ ϕ θϕ ϕ θϕ(, ) s i n (, ) c o s (, ) =− + . (2-128) This final step concludes the rigorous development of our newly formulated NZ-FZT process. Figure 2.3 summarizes the entire NZ-FZT process for both prolate and oblatespheroidal transformation surfaces by depicting an overall system diagram. 86E E Exi j yi jzi j(, ) (, ) (, )′′ ′′ ′′    ϑϕ ϑϕ ϑϕProlate/Oblate Spheroidal NZ-FZT Process: Table 2.5a/2.5b, (2-124a)/(2-124b), (2-125a)-(2-126a)/ (2-127) and (2-128).E Efz fzθ ϕθϕ θϕ(, ) (, )    Process Input: Process Output: Near-Zone E-Field Transformation SurfaceSamples Along Prolate/Far-Zone E-Field with Suppressed Radial Dependence {Prolate/Oblate Spheroidal Coordinates}{Spherical Coordinates}(2-125b)-(2-126b), Oblate SpheroidalEquations EquationsEquation ξo h (constant and ) Figure 2.3 Overall system diagram of the newly developed NZ- FZT process for both prolate and oblate spheroidaltransformation surfaces. 87CHAPTER 3. NUMERICAL EVALUATION OF REQUIRED SPECIAL FUNCTIONS Successful implementation of the newly developed NZ-FZT algorithms put forth in the preceding chapter requires the evaluation of special functions pertaining to prolateand oblate spheroidal wave functions. Unfortunately, calculating them is far from asimple process. Yet without the ability to evaluate these much needed special functionsin an efficient and accurate manner, the scope of our developed algorithms wouldunfortunately remain confined to only theoretical applications. This chapter addresses thedetails and issues regarding the development of numerical routines capable of efficientlycalculating these special functions while operating in the context of a workingcomputational algorithm. As we shall see, these numerical routines are quite involvedwith respect to their computational procedures, as well as their interactions with oneanother. Since the development of these numerical routines is application driven, ourdiscussion shall be procedurally based, thus leaving the interested reader to examine thesupporting references for the theoretical details. One of the major difficulties that must be avoided in the development of these numerical routines is the numerical roundoff error that often occurs in computationalprocesses. As any individual experienced in the field of scientific computing will attest,theoretical expressions for desired quantities do not always lend themselves well tonumerical techniques. Accordingly, our concern becomes whether or not these numericalroutines for the special functions can furnish results that are precise enough for the NZ-FZT algorithms. Moreover, these routines must be able to provide accurate results for allorders m and degrees l that are necessary to achieve convergence of the spheroidal wave- harmonic expansions contained within our algorithms. Lastly, we do not want to rely onesoteric computer platforms that have more precision than that which is generallyavailable for a typical IEEE-compliant 32-bit machine. For the most part, theaccessibility of such extraordinary machines is quite often limited and costly.Fortunately, we are able with much finesse to implement these routines using double 88precision on a standard IBM compatible personal computer (considered a typical IEEE- compliant 32-bit machine). At the time of writing, the use of this particular platform isconsidered quite advantageous because it is relatively inexpensive and, more often thannot, readily available. 3.1 OVERVIEW OF SPECIAL FUNCTION NUMERICAL ROUTINES 3.1.1 Description and Organization of Top-Level Numerical Routines As mentioned at the outset of this chapter, the numerical routines necessary to accurately evaluate these special functions are extremely involved. This is partly sobecause computation of some of these functions requires the evaluation of otherconstituent special functions that, in and of themselves, are complicated. Furthermore,the level of complexity for some of these special function routines is increased by virtueof having totally different computational approaches for different regions of interest. Inthis section, we describe the required top-level numerical routines and their relationshipto one another; we do so in order to gain an understanding of the overall computationalprocess for the special functions. So as to be concise, the scope of our discussion islimited to the required top-level numerical routines. Toward that end, we haveintentionally excluded details concerning the supporting lower-level routines. Many of the developed special function routines make use of computational approaches (including the resultant numerical experience) described in a series of formalreports published by the Acoustics Division of the Naval Research Laboratory (NRL)[32], [34]-[36]. Additionally, some of the routines are streamlined versions of Fortransubroutines presented in Numerical Recipes in FORTRAN [37]. And finally, others are developed directly from the outcomes of many numerical experiments, which were soconducted in order to determine the best computational approaches. By using thiseclectic mix of methods and routines, an altogether numerical process is developed that 89computes the required special functions in the best possible way, i.e., in terms of accuracy and efficiency, for our specific application. Shown in Figures 3.1a and 3.1b are structure charts depicting the overall organization of the top-level numerical routines that is necessary to compute the prolateand oblate special functions. The name of each routine is chosen to somewhat reflect itsrespective computational objective. Accompanying these figures are Tables 3.1a and3.1b, which furnish an overview of the top-level routines by providing a brief functionaldescription of each routine. To facilitate our discussion, we have chosen to tag eachnumerical routine with a reference index. Although the meaning of the lettered prefixcontained within the reference index may be somewhat obvious, we nevertheless describeits interpretation and bearing on the routine referenced: “P,” only needed in the prolatespecial function computational process; “O,” only needed in the oblate special functioncomputational process; and “B,” needed in both computational processes. Upon review of Figures 3.1a and 3.1b, we recognize that there are essentially five primary numerical routines for each spheroidal case. All of the primary routines havebeen assigned a reference index with a numbered suffix so as to distinguish them fromthe remaining secondary routines. The computational objectives of the primary routinesare as follows: (P-1) and (O-1) compute an initial approximation of all desiredeigenvalues; (P-2) and (O-2) compute the Sturm-Liouville normalization constants for therespective spheroidal angle function; (P-3) and (O-3) evaluate the respective spheroidalradial function of the first and second kind along with both of their first derivatives; (P-4)and (O-4) evaluate the respective spheroidal angle functions; (P-5) and (O-5) compute theintegral of the respective spheroidal angle functions over a given interval. As we can see,these routines provide all of the computations necessary to implement the prolate andoblate spheroidal NZ-FZT algorithms of Chapter 2. All of the secondary numericalroutines, i.e., those that have a reference index that contains a lettered suffix, essentiallysupport the computation of one or more of these primary routines. It is important to note that although the first derivatives of the spheroidal radial functions are not used directly by the NZ-FZT algorithms, they are nevertheless provided 90PRODNCNSTSPRORADIAL PRONEGDNCNSTS CALCPQ SJBESARRAY SYBESARRAYPROAPPROXEIG JACOBIPRONORMCNSTS PRODNCNSTS PRODNCNSTSPRODNCNSTSPROANGINTPROANG(P-1) (P-2) (P-3) (P-4)(P-5) PROANG (P-4)(B-a) (P-d) (P-d) (P-e) (P-f) (B-b) (B-c) (P-d) (P-d) Figure 3.1a Structure chart indicating organization of top-level numerical routines required to compute the prolatespheroidal special functions. 91Table 3.1a Overview of Top-Level Numerical Routines Required to Compute the Prolate Spheroidal Special Functions Reference IndexTop-level Routine NameTop-level Routine Functional Description (P-1) PROAPPROXEIG Provides an initial approximation to all desired prolate spheroidal eigenvalues. (P-2) PRONORMCNSTS * Provides Sturm-Liouville normalization constants for prolate spheroidal angle functions. (P-3) PRORADIAL Evaluates the prolate spheroidal radial functions of the first and second kind along with theirrespective first derivatives. (P-4) PROANG * Evaluates the prolate spheroidal angle function. (P-5) INTPROANG * Computes the integral of the prolate spheroidal angle function over a given interval. (B-a) JACOBI Employs the Jacobi transformation method to solve eigenvalues of a real symmetric matrix. (B-b) SJBESARRAY Evaluates required values of spherical Bessel functions. (B-c) SYBESARRAY Evaluates required values of spherical Neumann functions. (P-d) PRODNCNSTS * Computes dhrml,( ) expansion coefficients. Also, given an initial approximation to the prolate eigenvalue, this routine replaces it with one thathas a higher degree of accuracy. (P-e) PRONEGDNCNSTS * Computes dhrml −,( ) and dhrml ρ/,( ) expansion coefficients. (P-f) CALCPQ Computes the required values of associated Legendre functions of the first and second kindfor real arguments that do not lie on the ()−< < +11x branch cut or the x=±1 singular points. * - Denotes that the routine yields results in terms of the Morse and Feshbach normalization scheme. 92OBLDNCNSTSOBLRADIAL BABHASCNSTS SJBESARRAY SYBESARRAYOBLAPPROXEIG JACOBIOBLNORMCNSTS OBLDNCNSTS OBLDNCNSTSOBLDNCNSTSOBLANGINTOBLANG(O-1) (O-2) (O-3) (O-4)(O-5) OBLANG (O-4)(B-a) (O-d) (O-d) (O-e) (B-b) (B-c) (O-d) (O-d)CALCQIM (O-f) Figure 3.1b Structure chart indicating organization of top-level numerical routines required to compute the oblatespheroidal special functions. 93Table 3.1b Overview of Top-Level Numerical Routines Required to Compute the Oblate Spheroidal Special Functions Reference IndexTop-level Routine NameTop-level Routine Functional Description (O-1) OBLAPPROXEIG Provides an initial approximation to all desired oblate spheroidal eigenvalues. (O-2) OBLNORMCNSTS * Provides Sturm-Liouville normalization constants for oblate spheroidal angle functions. (O-3) OBLRADIAL Evaluates the oblate spheroidal radial functions of the first and second kind along with theirrespective first derivatives. (O-4) OBLANG * Evaluates the oblate spheroidal angle function. (O-5) INTOBLANG * Computes the integral of the oblate spheroidal angle function over a given interval. (B-a) JACOBI Employs the Jacobi transformation method to solve eigenvalues of a real symmetric matrix. (B-b) SJBESARRAY Evaluates required values of spherical Bessel functions. (B-c) SYBESARRAY Evaluates required values of spherical Neumann functions. (O-d) OBLDNCNSTS * Computes dj hrml,()− expansion coefficients. Also, given an initial approximation to the oblate eigenvalue, this routine replaces it with one thathas a higher degree of accuracy. (O-e) BABHASCNSTS Computes Baber and Hasse normalized expansion coefficients ()lm mlm r, , −/G24/G24 . (O-f) CALCQIM Computes the required values of associated Legendre functions of the second kind for purelyimaginary arguments that do not lie on the ()−< < +11x branch cut (i.e., excludes x=0) * - Denotes that the routine yields results in terms of the Morse and Feshbach normalization scheme. 94to the user. As fortune would have it, this feature does not have a negative impact on computational efficiency because determination of these first derivatives is required in theprocess of computing their respective spheroidal radial functions. Since the firstderivatives of the spheroidal radial functions are computational by-products and come atno expense, they are provided in order to make the set of numerical routines moreversatile and complete. Regarding the primary numerical routines that directly or indirectly involve the spheroidal angle functions, namely, (P-2), (P-4), (P-5), (O-2), (O-4), and (O-5), theyrender their results in terms of the Morse and Feshbach normalization scheme (see 2.2.5).Consequently, some of the secondary routines, viz., (P-d), (P-e), and (O-d), also furnishtheir results in terms of Morse and Feshbach normalization. Notwithstanding, it shouldbe made clear that even though our special function routines happen to employ thisparticular normalization scheme, both of our developed NZ-FZT algorithms workproperly regardless of the normalization selected. However, the use of a differentnormalization scheme would ultimately require all of our special functions that aredependent on the Morse and Feshbach normalization to be modified accordingly. Upon further review of the structure charts shown in Figures 3.1a and 3.1b, it is important to note that there is an element of redundancy within the overall computationalprocess for the special functions. For each spheroidal case, four of the five primarynumerical routines share a common secondary routine: namely, PRODNCNSTS (P-d) inthe prolate case, and OBLDNCNSTS (O-d) in the oblate case. Furthermore, in eachspheroidal case one of the primary routines evokes another primary routine. In particular,the routines that compute the integration of the spheroidal angle functions must each usethe routine that calculates the respective spheroidal angle function, viz., INTPROANG(P-5) uses PROANG (P-4), and INTOBLANG (O-5) uses OBLANG (O-4). Finally, we would like to highlight that the computational processes for both prolate and oblate special functions do have some significant differences. Some of theroutines for the oblate special functions can be derived from their respective prolatecounterparts with some minor modifications. However, other routines require totally 95different numerical approaches. The principal difference between both sets of numerical routines occurs when evaluating the respective spheroidal radial functions, viz., (P-3) and(O-3), as well as a few of their supporting secondary routines. Further details concerningthese differences are addressed in the sections of this chapter that discuss each of thespheroidal radial function routines. 3.1.2 Numerical Precision and Accuracy As a numerical algorithm progresses through its computations, the possibility exists that inherent roundoff error will cause the accuracy of each passing stage offloating-point computations to continually degrade until the overall results becomecompletely swamped with erroneous calculations [37]. In these catastrophic cases wherenumerical algorithms fail to converge upon usable results, the computations are said tohave become numerically unstable. More often than not, it is very difficult tocharacterize these numerical errors and predict the performance of a given algorithmwithout conducting numerical experiments. Furthermore, a given algorithm may workwell for one particular region of operating parameters while failing miserably for others. The ultimate goal of our NZ-FZT process is to provide single precision results for the far-zone E-field response (i.e., between six and seven significant digits on a typicalIEEE-compliant 32-bit machine). However, numerical experience has shown thatcomputation of the NZ-FZT algorithms should be conducted in double precision (i.e.,between fifteen and sixteen significant digits on a typical IEEE-compliant 32-bitmachine) in order to achieve the desired single precision results. In particular, the use ofdouble precision is instrumental in minimizing the roundoff error when summing thespheroidal wave-harmonic expansions. Since our NZ-FZT algorithms need to be performed with double precision, logic dictates that our special function routines must also support double precision. Besidesusing double precision to achieve more accurate results for the special functions, its use isalso required in certain instances to avoid underflow/overflow errors. As we shall see, 96some of the numerical routines require expansions of functions that contain values that lie outside the permissible range of magnitude for single precision, viz., ()10 1038 38−+<<x for a typical IEEE-compliant 32-bit machine [37]. By using double precision, the range of magnitude can be extended to ()10 10308 308−+<<x on a typical IEEE-compliant 32- bit machine [37]. During the development of the numerical routines for the special functions, each of the secondary routines is individually tested and validated before being integrated withits respective primary routine. Results from the secondary routines are compared withpublished tabulated data whenever available to help confirm their accuracy. Followingthe validation and integration of the secondary routines, the overall primary routines arethen tested. First, the results of the primary routines are compared with tabulated data fora given set of parameters whenever possible. Unfortunately, most of the published datasets that are available are limited with respect to the number of decimal places providedand the number of test cases presented. In some cases, we make use of computational checks to further corroborate the accuracy of the developed routines. Some of these computational checks consist ofcomparing the results of the developed primary routines with other routines that calculatethe same special functions by way of a different numerical approach. In general, theseother approaches were not considered the method of choice because of their lack ofcomputational efficiency and limited range of validity. Nonetheless, the development ofthese less efficient routines is not in vain because they still contribute to our overallvalidation process. The remaining computational checks consist of comparing the results of both spheroidal radial function routines with their known Wronskian value. Specifically, bothroutines take advantage of the Wronskian check and use it during the computationalprocess to ensure the accuracy of their results. Details regarding this process areaddressed in the sections pertaining to the prolate and oblate spheroidal radial functions. 97As mentioned earlier, these numerical routines must be able to provide accurate results for all orders m and degrees l necessary to achieve convergence of the spheroidal wave-harmonic expansions. Naturally, the number of terms required to achieveconvergence depends on the functions that are to be expanded, viz., the Dirichletboundary conditions that exist on the spheroidal transformation surface. However,computational testing for reasonable values of h (viz., where h<20 ) indicates that a ceiling of 50 should be imposed on all orders m and degrees l (i.e., L≤50 ) so as to ensure that the numerical routines safely provide reasonable results. We do so becausethe special function routines may deliver questionable results for wave-harmonics that liebeyond this imposed ceiling. Fortunately, experience shows that for our particularapplication, the prolate and oblate spheroidal wave-harmonic expansions converge wellbefore reaching the imposed ceiling with a considerable margin of allowable orders anddegrees remaining. 3.2 PROLATE AND OBLATE SPHEROIDAL EIGENVALUES 3.2.1 PROAPPROXEIG (P-1) and OBLAPPROXEIG (O-1) Numerical Routines The first step in evaluating all of the prolate and oblate spheroidal wave functions is the determination of their respective eigenvalues: ) ( ,hlmλ and λml jh,()− . It should be noted that each of the other primary numerical routines for the special functions must first be supplied with its respective eigenvalues before it can proceed with its calculations. Aspreviously pointed out in Chapter 2, computation of these eigenvalues is notstraightforward and involves the use of a complicated numerical process. Thecomputational approach that we have decided to incorporate into our numerical routinesis that which is used and explained in NRL formal reports [32], [34]-[36]. Ourexperience shows that this approach performs the best with regard to computationalefficiency and accuracy. The approach is essentially a two-step process that first requiresthe eigenvalues to be determined approximately and then subjects them to a refining 98process to improve their accuracy. Both numerical routines addressed in this section only carry out the first part of this process (this explains the choice of their names). Thesecond part of the process that refines the eigenvalues for each spheroidal case isperformed by the commonly shared secondary routines PRODNCNSTS (P-d) andOBLDNCNSTS (O-d); details concerning refinement of the eigenvalues are addressed inthe subsequent section that features both of these routines (see 3.3.2). Hanish and King [32] present the detailed theory behind the computational process that provides an initial approximation to the spheroidal eigenvalues. Althoughthis intricate theory lies outside the scope of our discussion, the interested reader isreferred here to gain further insight into the theoretical basis of this computationalprocess. With regard to practicality, the other NRL formal reports, viz., [34]-[36],describe the matrix methods required to compute an initial approximation of both sets ofspheroidal eigenvalues. These matrix methods are able to compute an accurate initial approximation of the eigenvalues for both spheroidal cases by solving a set of linear algebra eigenvalueproblems. In keeping with the nomenclature of fundamental linear algebra, theseeigenvalue problems can be expressed using the following standard matrix equations[35]: Prolate: [] 0)()(-),( =hdh mh/G26 Iλ /G2F ; (3-1a) Oblate:[] 0) () (-),(- =−− jhdjh mjh/G26 Iλ /G2F . (3-1b) In these matrix equations, the parameters are defined as follows: ),(mh/G2F and ), ( mjh−/G2F are infinite square real symmetric matrices dependent on h and m; )(hd/G26 and /G26 dj h()− are possible eigenvectors that represent dhrml,( ) and dj hrml,()− expansion coefficients with respect to normalized associated Legendre functions; λ()h and λ()−jh represent the desired spheroidal eigenvalues for a given order m, i.e., /G15,1 ,+= mml , when arranged in 99a monotonic increasing sequence; and I is the corresponding identity matrix. Accordingly, a separate eigenvalue problem must be solved for each desired order m so that all of the needed eigenvalues can be determined. In order to continue with our discussion of the eigenvalue problems at hand, we must first explore the matrices ),(mh/G2F and ), ( mjh−/G2F in more detail. Although both of these matrices are infinite in size, King et al. [35] have indicated that truncated matrices of modest proportions will yield an initial approximation of the eigenvalues accurateenough to supply the refining part of the computational process. More specifically, theirexperience has found that in the prolate case a truncated matrix of size ( ) 50 50× delivers good approximations of the eigenvalues for 49 ,,1 ,++= m mml /G15 when h<10. However, as h increases in magnitude, the size of the truncated matrix must also increase in order to supply approximations for the lowest 50 eigenvalues that are sufficientlyaccurate. According to [35], truncated matrices of size ( ) 90 90× have been shown to deliver sufficiently accurate approximations for the lowest 50 eigenvalues for the specific case where h=80. Because the size of the truncated matrix required to deliver an acceptable approximation of the eigenvalues is not known exactly, King et al. [35] provide a practical approach to solve this problem. Their approach selects the size of thetruncated matrix ( ) KK× between 50 100 ≤≤K by using a linear function of h that passes through the two numerical benchmarks, namely, ( h=0, K=50) and ( h=80 , K=90 ). Although King et al. [35] cite this practical approach for only the prolate spheroidal case, another NRL formal report, viz., [34], extends the approach (withslightly different numerical benchmarks) to its oblate counterpart. Based upon thesuccess of their experiences for both spheroidal cases, we have chosen to adopt theirapproach using ( h=0, K=51) and ( h=80 , K=91) to determine the size of our truncated ),(mh/G2F and ), ( mjh−/G2F matrices. Now that we have an understanding of how to truncate the matrices ),(mh/G2F and ), ( mjh−/G2F , we proceed to examine the content of their elements. King et al. [34], [35] 100explicitly put forth in their computational procedures the following expressions for the elements of the truncated matrices: Prolate: =),(ji, mh/G2F ij p p hpp m pp ij jih ppm pm pm pm pp=+ ++− − −+   =+ =+ +++ ++ −+ −+ ++          :( )() () () :()( ) () ( ) () () ()121 2 1 21 23 2 2 232121 21 25 22 2 elsewhere: 0 (other elements); (3-2a) Oblate: =− ), (ji, mjh /G2F ij p p hpp m pp ij jih ppm pm pm pm pp=+ −+− − −+   =+ =+− +++ ++ −+ −+ ++          :( )() () () :()( ) () ( ) () () ()121 2 1 21 23 2 2 232121 21 25 22 2 elsewhere: 0 (other elements); (3-2b) where K ji ,,2,1,0 , /G15= and pm i=+ for both cases. Upon inspection of these expressions, we are able to recognize an important property of the matrices ),(mh/G2F and ), ( mjh−/G2F : the only nonzero elements are those situated on the principal diagonal (i.e., ij=), the second subdiagonal (i.e., ij=+ 2 ), and the second 101superdiagonal (i.e., ji=+ 2 ). From a viewpoint of filling these matrices, its symmetry can be used to easily derive elements of the second subdiagonal from those of the second superdiagonal, e.g., 7,5 5,7/G2F /G2F= , or vice-versa. One should also note that the matrix for the oblate case (3-2b) can be easily derived from its prolate counterpart (3-2a) by using the previously mentioned transformation hj h→− . Having discussed the matrix method for approximating the set of eigenvalues for a given order m, we now must address its connection to the eigenvalue matrix described in Table 2.4 (which differs from the matrices ),(mh/G2F and ), ( mjh−/G2F ). Each row of the eigenvalue matrix (i.e., corresponding to a given order m) is approximated by solving a separate linear algebra eigenvalue problem. Disregarding the negative orders of theeigenvalue matrix (due to its symmetry with respect to the order m; see 2.2.5), we are left with solving ( ) L+1 separate linear algebra eigenvalue problems in order to fill the desired eigenvalue matrix with its respective initial approximations (recalling from Chapter 2 that L represents the user prescribed degree as to where the wave-harmonic expansion is truncated). This eigenvalue matrix is then stored in memory for non-negative orders of m (including m = 0) using a ( ) ( ) LL+×+1 1 two-dimensional matrix; note that zero-fill is used for elements that do not have an eigenvalue corresponding to the indices m and l. 3.2.2 JACOBI (B-a) Numerical Routine In the process of computing an initial approximation of the spheroidal eigenvalues, the matrix methods employed in both PROAPPROXEIG (P-1) andOBLAPPROXEIG (O-1) call for a number of linear algebra eigenvalue problems to besolved for numerically. Since both of these primary routines provide the computationalsetup for the resulting eigenvalue problems, a generic numerical process is then needed toindividually solve them. As a result, it is the purpose of this supporting secondary routineto numerically evaluate the eigenvalues for a given ),(mh/G2F or ), ( mjh−/G2F truncated matrix. 102There are many existing numerical techniques and routines available able to compute the eigenvalues of a given matrix. The computational advantage of any one ofthese techniques over another depends on some a priori knowledge of the matrix under consideration. This is so because many of these numerical techniques are speciallydesigned to handle a specific class of matrices possessing certain characteristics. Moreoften than not, these specialized numerical techniques perform much better for theirintended class of matrices than their generalized counterparts. Naturally, it is our intentto use the eigenvalue technique that is the most beneficial for our particular application. As mentioned earlier, the truncated ) ,(mh/G2F and ) , ( mjh−/G2F matrices employed during the matrix method of (P-1) and (O-1) routines are real and symmetric. Accordingly, the Jacobi method proves to be a suitable eigenvalue technique for ourparticular application because it takes advantage of both of these matrix characteristics.The method employs a sequence of orthogonal similarity transformations (i.e., Jacobirotations) with each one being a plane rotation designed to eliminate one of the off-diagonal matrix elements [37]. Although successive transformations undo previously setzeros, the off-diagonal elements do continue to get smaller with each passing iterationuntil the matrix is diagonal to machine precision. The on-the-fly product (i.e., the matrixmultiplication) of these successive transformations leads to the overall transformationmatrix that is comprised of its corresponding eigenvectors. After applying the similaritytransformation to the original matrix, the resulting elements of the final diagonal matrixare the eigenvalues. The interested reader is referred to Numerical Recipes in FORTRAN [37] for further details regarding this robust eigenvalue numerical technique. According to [37], the Jacobi method is considered absolutely foolproof for all real symmetric matrices. Although they do mention that the algorithm can besignificantly slower than others when considering large matrices, they also cite that theJacobi method is much simpler than these other more efficient methods. Because ourapplication requires us to compute the set of eigenvalues only once for a giventransformation surface, the additional computational expense associated with thisnumerical method does not seriously lengthen our overall computation time. 103The JACOBI (B-a) numerical routine is a streamlined version of the one presented in [37]. Since our application does not derive the eigenvectors from this portion of theprocess, we have chosen to eliminate the eigenvector feature from their routine.Furthermore, this routine puts the eigenvalues in monotonic increasing order as requiredby our application. 3.3 STURM-LIOUVILLE NORMALIZATION CONSTANTS FOR THE PROLATE AND OBLATE SPHEROIDALANGLE FUNCTIONS 3.3.1 PRONORMCNSTS (P-2) and OBLNORMCNSTS (O-2) Numerical Routines We now proceed with the development of the two primary routines that compute the Sturm-Liouville normalization constants for the prolate and oblate spheroidal anglefunctions. As mentioned beforehand, both of these numerical routines providenormalization constants that are consistent with the Morse and Feshbach normalizationscheme. In order for these routines to operate properly, they must be supplied with initialapproximations to all eigenvalues corresponding to the desired normalization constants.Furthermore, the Sturm-Liouville normalization constants only have to be computed oncefor a given spheroidal transformation surface. A computational by-product of these primary routines, i.e., (P-2) and (O-2), is that the approximate eigenvalues that are supplied to these routines (computed by the primaryroutines (P-1) and (O-1)) are replaced afterwards with more accurate values. In essence,it is the execution of these routines that serves as the second part of the eigenvaluecomputation process that refines the initial set of approximate eigenvalues. Thisrefinement occurs because the secondary routines PRODNCNSTS (P-d) andOBLDNCNSTS (O-d), which directly perform the eigenvalue refinement portion of theprocess, must be used by these primary routines. It should be noted that if the initialapproximation of any eigenvalue happens to match the refined eigenvalue within the 104precision provided by either of the secondary routines (P-d) and (O-d), then its value virtually remains unchanged by the routine. Our numerical experience indicates that the best way to compute the Sturm- Liouville normalization constants for both spheroidal angle functions is to use thefollowing expansions presented by Flammer [23]: Prolate: []() ()Nh dhmr mr rml rml r,, ,() ()! !=′ + ++=∞ ∑22 22 12 01; (3-3a) Oblate: []() ()Nj h dj hmr mr rml rml r,, ,() ()! !−= −′ + ++=∞ ∑22 22 12 01; (3-3b) where the prime over the summation sign indicates that the summation is over only even values of r when ( ) lm− is even, and over only odd values of r when ( ) lm− is odd. Both of these expressions can be derived from taking the inner product of their respective spheroidal angle function while representing them with their appropriate expansion ofassociated Legendre functions, viz., equations (2-68a) and (2-68b). In derivingexpansions (3-3a) and (3-3b), the factorial expressions are a result of using theorthogonality property of the associated Legendre functions of the first kind, i.e., equation(2-65). The routines PRODNCNSTS (P-d) and OBLDNCNSTS (O-d) are summoned to deliver the dh rml,( ) and dj hrml,()− expansion coefficients. It must be pointed out that during the computational process the ratio of factorial expressions must be evaluated carefully in order to avoid overflow conditions. Furthermore, convergence of these expansions is quite rapid due to the fact that the dhrml,( ) and dj hrml,()− coefficients are squared. Flammer [23] presents tabulated values of the Sturm-Liouville normalization constants for the prolate spheroidal angle functions, yet for some reason does not providevalues of the normalization constants for the oblate case. Consequently, we take a two-pronged approach when validating these primary routines. In the prolate spheroidal case, 105we naturally make use of the available tabulated data (after accounting for the difference in normalization schemes) to validate the PRONORMCNSTS (P-2) routine. In general,results of the prolate routine are fairly consistent with the tabulated values. As for theoblate spheroidal case, we must resort to some sort of computational check. Specifically,our computational check consists of a conducting numerical integration of the square ofthe oblate spheroidal angle function over the [ , ] −1 1 interval; the results are then compared to those obtained from the OBLNORMCNSTS (O-2) routine, i.e., found by using (3-3b). For the cases tested, the two different methods yield results that match oneanother extremely well. 3.3.2 PRODNCNSTS (P-d) and OBLDNCNSTS (O-d) Numerical Routines These secondary routines compute their respective set of dh rml,( ) and dj hrml,()− expansion coefficients using the Morse and Feshbach normalization scheme for a given order m, degree l, and independent parameter h. Because these routines are called by many of the primary routines, they are essentially at the core of all special functioncomputations. In the organization of this chapter, the discussion of these routines couldhave easily been placed in any of the other sections that describe the primary routines forwhich they support. However, we have chosen to include their discussion here becausePRONORMCNSTS (P-2) and OBLNORMCNSTS (O-2) are chronologically the firstprimary routines in our NZ-FZT process to evoke these much shared secondary routines. Both routines make use of a variational method devised by Bouwkamp in [38] and further described in [23], [34], [35]. Also, it must be mentioned that this variationalmethod was developed independently by Blanch [39]. In order for the Bouwkamp/Blanchmethod to function properly, it requires an initial approximation of the eigenvalue, viz., λmlh,()()1 or λml jh,()()1− ; this requirement explains the necessity of both the PROAPPROXEIG (P-1) and OBLAPPROXEIG (O-1) numerical routines. The Bouwkamp/Blanch method is essentially an iterative numerical process that convergesupon the desired eigenvalue within some desired degree of accuracy using an initial 106approximation of the eigenvalue as its starting point. Moreover, in the process of improving the accuracy of the approximate eigenvalue, the method simultaneously renders the much needed dhrml,( ) and dj hrml,()− expansion coefficients. Thus, both of these numerical routines intrinsically carry out the second part of the eigenvalue computational process that refines the eigenvalue. Since the computational procedures to calculate the expansion coefficients for both spheroidal cases, i.e., dhrml,( ) and dj hrml,()− , closely parallel one another, we shall from this point forward (for the remainder of this section) only discuss in detail the process that corresponds to the prolate spheroidal case. Regarding the process thatcomputes the expansion coefficients for the oblate spheroidal case, it can be easilyderived from its prolate counterpart using the ubiquitous hj h→− transformation. Accordingly, both PRODNCNSTS (P-d) and OBLDNCNSTS (O-d) routines essentially use the same procedure with the only exception being the difference in sign betweensome of the terms containing the independent parameter h. The first step in computing the dh rml,( ) expansion coefficients is to compute the correction term δλmlh,( ) to the initial approximation of the eigenvalue λmlh,()()1. After finding this correction term, it can be applied to the approximate eigenvalue as follows to determine a more accurate numerical representation of the eigenvalue [23]: λλδ λml ml mlhh h,,() , () () ()=+1. (3-4a) With the accuracy of the eigenvalue improved, the process can then be applied again to further refine its initial result. According to Flammer [23], one or two applications of thisiterative method will lead to remarkably accurate eigenvalues. Moreover, Flammer [23]indicates that Bouwkamp [38] showed the method to improve the approximate eigenvaluefrom two- to six-decimal accuracy with only one iteration of the process. Recalling that the dh rml,( ) expansion coefficients exist for only even terms when ()lm− is even and only odd terms when ()lm− is odd, we formally state 107 −−=odd. ) (for ,5,3,1even, ) (for ,4,2,0 mlmlr/G15/G15(3-5a) Bearing this property in mind, we now make use of two closely related forms of a three- term recurrence relation (derived from one another using a shift of index) [35]: ΛΛrm rm rm ml rmh=− −++β γλ,()()1 2 for r≥2 , (3-6a) ΛΛrm rm mlrm rmh =− + −−− −γλβ 21 2 2,()( ) for r≥4 , (3-7a) where γrmmr mr hm mr mr=+ + + + −− +− ++   () ( )() ()1141 22 1 22 31 222 for r≥0, (3-8a) βrm r r mr mr mr mr mrh =−++ − +− +− ++() ( ) ( ) () () ()12 2 1 22 1 22 3 22 124 for r≥2 . (3-9a) The difference between these two forms of the recurrence relation is the direction in which they compute the Λrm terms: recurrence relation (3-6a) works its way downward (i.e., backward); as opposed to recurrence relation (3-7a), which works its way upward (i.e., forward). The fact that both recurrence expressions relate a given term to anotherterm that differs in index by two is consistent with (3-5a). Upon review of (3-6a), one can see that the downward recurrence relation can be written as a continued fraction. Since Flammer [23] indicates that lim rrm →∞=Λ 0, the continued fraction can be safely terminated at some reasonable large value for the index r, viz., rrterm= (remembering to select the appropriate even or odd index), by letting Λrtermm +=20. As a result, the starting point for the downward recurrence relation can be taken as [34]: 108Λrtermm rtermm rtermm mlh=− −β γλ,()()1. (3-10a) Using this starting point, the remaining values in the Λrm sequence can then be determined by working downward using the recurrence relation (3-6a). Likewise, the upward recurrence relation (3-7a) can also be written in terms of a continued fraction. In this case, we can exploit the fact that both β0m and β1m equal zero in order to terminate the sequence; thus, we are led to the following starting point [34]: Λ201 mm mlh =− +γλ,()( ) for ()lm− even, Λ311 mm mlh =− +γλ,()( ) for ()lm− odd. (3-11a) Accordingly, the Λrm terms can be computed by working upward from the appropriate starting point using the recurrence relation (3-7a). With these upward and downward recurrence relations, we are now able to compute the correction term δλmlh,( ) to the initial approximation of the eigenvalue λmlh,()()1. First, the initial approximation for the eigenvalue λmlh,()()1 is used by both recurrence relations to compute two numerical values for Λlmm −+ 2: Λdw lmm −+ 2 by (3-6a) and Λuplmm −+ 2 by (3-7a). In addition, the routine must store in memory all of these upward and downward intermediate Λrm terms used in the process of determining both numerical values for Λlmm −+ 2 (beginning with each of their respective starting points). Using all of these numerical results, we are then able to determine the eigenvalue correction term as follows [23], [34], [35]: 109δλmlh,()= ()() ( )()() ()ΛΛ ΛΛ ΛΛΛ Λdw up up up updw dw dwlmm lmm lmm lmmlmm lmm lmm lmmlmm lmmlmm lmm lmm lmm−+ −+ − −−− − −− −−+ −+−+ −+ −+ −+− ++ +    ++ +    22 22 2 2222 222 42 241β ββ β ββ/G16/G16. (3-12a) Each of the series in the denominator for this expression is computed until convergence has been achieved. Lastly, the entire Bouwkamp/Blanch process, i.e., described by (3-4a)-(3-12a), is repeated until λmlh,( ) reaches the desired accuracy. Now that the Bouwkamp/Blanch method has been used to refine the initial approximation of the eigenvalue, we return our attention to the main objective of the PRODNCNSTS (P-d) secondary routine: computing the much needed set of dhrml,() expansion coefficients for a particular order m and degree l. Fortunately, these expansion coefficients are related to the Λrm terms (taken from the respective upward and downward intermediate terms) by the following relation [35]: dh dhmr mr mr mr hrml rml rm, ,() ()() () () ( )−=+− ++ ++ −2222 1 22 1 22 1Λ for r≥2 . (3-13a) It is important to point out that repeated applications of the Bouwkamp/Blanch method will not only refine a particular eigenvalue λmlh,( ) , but also its corresponding Λrm terms. Equation (3-13a) relates the computed Λrm terms to the ratio of consecutive dhrml,() expansion coefficients. The list of the expansion coefficients normalized with respect to its first term in the sequence, i.e., dhml 0,( ) or dhml 1,( ) , can be easily determined from this sequence of ratios by using the appropriate finite product [23]: 110       = −lm rlm r lmlm lmlm lmlm r dd dd dd h dh d , 2, , 2, 4 , 0, 2 , 0, )()(/G16 for ()lm− even and r≥2,        = −lm rlm r lmlm lmlm lmlm r dd dd dd h dh d , 2, , 3, 5 , 1, 3 , 1, )()(/G16 for ( ) lm− odd and r≥3. (3-14a) As we can see, all of the computed expansion coefficients of (3-14a) are normalized with respect to the first term in their sequence. Thus, by applying its respective first term as a scaling constant, i.e., dhml 0,( ) or dhml 1,( ) , to the entire computed sequence, i.e., dh dhrml ml,,() ()0 or dh dhrml ml,,() ()1 , the list of dhrml,( ) expansion coefficients can be found explicitly. However, before being able to do so, we must first determine the first term in the sequence. Up to this point, the process for computing the dhrml,( ) expansion coefficients has been independent of the chosen Sturm-Liouville normalization scheme. However, the selected normalization scheme will have a direct bearing on this next and final step in thiscomputational process. Since we have chosen to implement our NZ-FZT process usingthe normalization scheme of Morse and Feshbach [22], the first term of the sequence canbe found by appropriately using one the following normalizing relations [35]: ()dhlm lmdh dhmr rml rml ml r0 0 01 2,, ,()() ! () !() ()! !=+ −′ +    =∞− ∑ for ()lm− even, ()dhlm lmdh dhmr rml rml ml r1 1 11 2,, ,()() ! () !() ()! !=+ −′ +    =∞− ∑ for ()lm− odd. (3-15a) To digress for a moment, other normalization schemes would require the use of different normalization relations. As an example, Flammer [23] presents a different pair ofnormalization relations to be used with the scheme presented in his monograph (recallingthat his particular normalization scheme for prolate spheroidal angle functions is 111described in equations (2-71a) and (2-72a) of Chapter 2). Once again, care must be exercised when evaluating the ratio of factorial expressions in (3-15a) in order to avoidoverflow conditions. The first step in validating both PRODNCNSTS (P-d) and OBLDNCNSTS (O-d) numerical routines consists of checking the accuracy of the refined prolate and oblatespheroidal eigenvalues. Results of these routines are compared to computed values madeavailable in [35], [36]. Comparisons with these data sets indicate that for the cases testedboth numerical routines refine their respective eigenvalues to better than twelve-decimalaccuracy. The next step in validating both PRODNCNSTS (P-d) and OBLDNCNSTS (O-d) routines is to check the accuracy of the dh rml,( ) and dj hrml,()− expansion coefficients. Fortunately, Flammer [23] provides tabulated data for both of these expansion coefficients that ranges anywhere between four- and eight-decimal accuracy. Resultsfrom both of our routines fare extremely well when compared with this available data(i.e., after accounting for the difference in normalization schemes). Nevertheless, theultimate validation is left to how well these routines support their respective primaryroutines. 3.4 PROLATE SPHEROIDAL RADIAL FUNCTIONS OF THE FIRST AND SECOND KIND 3.4.1 PRORADIAL (P-3) Numerical Routine This numerical routine evaluates the prolate spheroidal radial functions of the first and second kind (both real quantities) together with their respective first derivatives for agiven order m, degree l, and independent parameter h. However, upon review of our prolate spheroidal NZ-FZT algorithm (see Chapter 2), we recognize that its wave-harmonic expansion actually calls for the corresponding prolate spheroidal radial functionof the fourth kind (a complex quantity). Fortunately, this seeming incompatibility doesnot present any real difficulty because the fourth kind solution is actually comprised of its 112corresponding first and second kind solutions by way of (2-74a). Furthermore, because the prolate spheroidal radial function of the second kind is singular at ξ=1, we have chosen to limit both radial functions of this numerical routine to arguments greater than unity, i.e., where ξ>1. In this section, we choose to address only the computation of the spheroidal radial functions for the prolate case and not their oblate counterparts. We do so because the numerical approaches for computing both spheroidal cases are somewhatdifferent. Consequently, our routine that handles computation of the oblate spheroidalradial functions, viz., OBLRADIAL (O-3), is later discussed in another section (see 3.5.1). Since this primary routine also makes use of the shared secondary routine PRODNCNSTS (P-d) to compute the dh rml,( ) expansion coefficients, it too must be provided with either an initial approximation of the respective eigenvalue or an already refined eigenvalue. In the case that an approximation of the eigenvalue is supplied to theroutine, the numerical representation of the eigenvalue is replaced with a value that ismore accurate. Although the numerical process to compute an individual prolate spheroidal radial function is quite involved, its frequency of use is relatively limited in the course ofcomputing the far-zone response for a single transformation surface. In fact, PRORADIAL (P-3) only has to calculate the set of Rh ml,()(,)4ξ functions once for all orders m and degrees l that are necessary to achieve convergence of the corresponding spheroidal wave-harmonic expansion (see Table 2.5a). It has therefore been ourexperience that this set of function evaluations only accounts for a small fraction of thetotal computation time for a given prolate spheroidal transformation surface. This numerical routine is somewhat complicated because it makes use of one of two different approaches to compute the prolate spheroidal radial functions of the secondkind and its first derivatives. The decision making process that determines whichapproach yields the most accurate results for a given region of input arguments is basedupon the computational experience provided in NRL Formal Report 7012 [35]. However, 113in one particular input parameter region the approach that performs the best is not known decisively. Fortunately, we are able to use the known closed form of the Wronskian tovalidate the computed results in an on-the-fly manner when operating in this ambiguousregion [35]. By comparing the numerically evaluated Wronskian (resulting from thecomputed prolate spheroidal radial functions and their first derivatives) from eachnumerical approach with the theoretical Wronskian (resulting from the known closedform), the approach which performs the best in this region of ambiguity is thusdetermined. It is this Wronskian check that lies behind the need for computing both firstderivatives. First, we shall focus on computing the prolate spheroidal radial function of the first kind together with its first derivative. To do so, King et al. [35] recommend using the following expansions of spherical Bessel functions jh n()ξ: Rhml,()(,)1ξ=() ! () !()() ! !(), ,lm lmjd hmr rjhm rml rml rrm− +−  ′ ++− =∞ + ∑ξ ξξ2 22 011 2(3-16) d dRhlm lmmlm ξξξ ξ,()(,)() ! () !12 221=− +−  × hj dhmr rm rmr j h mr j h mjd hmr rjhrml rml rm rm r rml rml rrm+− +− ++ =∞ +− =∞ +′ + +++ −+ +   +−′ +      ∑ ∑, , , ,()() ! !( )() ( ) () ( ) ()()() ! !()2 22 1 1 121 1 01 2 01ξ ξ ξξξ. (3-17) The equation presented in (3-17) is easily derived from (3-16) by using a standard recurrence relation for spherical Bessel functions that relates the first derivative of thespecial function to those of other orders. In addition, one should note that the complex j rml+− term in both expressions can be misleading; that is, it belies the fact that the results of both expansions are always real quantities. This is so because the quantity ()rml+− is always an even integer; note that this property is consistent with (3-5a). As for the 114required set of spherical Bessel function evaluations, it is supplied to this routine by the secondary routine SJBESARRAY (B-b). From a perspective of reducing computations, we are able to take advantage of the fact that both (3-16) and (3-17) share many redundant computations: the foremost beingthat both expressions happen to contain an identical expansion of spherical Besselfunctions. In removing these redundant computations, we are able to significantlyimprove the computational efficiency of this numerical routine. Moreover, all of theexpansions contained within (3-16) and (3-17) are truncated at the point where numericalconvergence has been achieved in order to eliminate non-contributory calculations. Onceagain, care must be taken when handling the ratios of the factorial expressions containedwithin these expansions. As stated earlier, this numerical routine makes use of one of two different approaches to compute the prolate spheroidal radial functions of the second kind. Thefirst approach closely resembles that which is taken to compute the prolate spheroidalradial function of the first kind, i.e., (3-16) and (3-17), with the only difference being thesubstitution of spherical Neumann functions for the spherical Bessel functions [35]: Rh ml,()(,)2ξ=() ! () !()() ! !(), ,lm lmjd hmr rnhm rml rml rrm− +−  ′ ++− =∞ + ∑ξ ξξ2 22 011 2(3-18) d dRhlm lmmlm ξξξ ξ,()(,)() ! () !22 221=− +−  × hj dhmr rm rmr n h mr n h mjd hmr rnhrml rml rm rm r rml rml rrm+− +− ++ =∞ +− =∞ +′ + +++ −+ +   +−′ +      ∑ ∑, , , ,()() ! !( )() ( ) () ( ) ()()() ! !()2 22 1 1 121 1 01 2 01ξ ξ ξξξ. (3-19) Unfortunately, the expansions of Neumann functions that are contained in these expressions do not converge well for small values of hξ [35]. The convergence of these 115expansions is difficult in this region because the Neumann functions possess a singularity at the origin. As a matter of fact, Morse and Feshbach [22] indicate that these expansionsare actually asymptotic series and are not absolutely convergent for any finite values of h ξ. As a result, the usefulness of the spherical Neumann expansion approach is limited to large values of hξ. An advantage of the spherical Neumann expansion approach is that we can exploit the fact that many of the calculations are identical to those required to compute its firstkind counterpart. In fact, all of the overall expansion coefficients of the sphericalNeumann expansion approach and the spherical Bessel expansion approach are exactlythe same. As for the set of spherical Neumann function evaluations, it is provided to thisprimary routine by evoking the secondary routine SYBESARRAY (B-c). Because thespherical Neumann functions can assume extremely large magnitudes as their orderincreases, the resulting set of evaluations must be scaled by some large constant at somepoint in the sequence of functions in order to avert an impending floating-point overflowcondition. This scaling scheme effectively expands the computational dynamic range of the spherical Neumann functions (i.e., allows magnitudes beyond 10 308+) and is critical to the successful implementation of this particular numerical approach. Essentially, eachspherical Neumann function is an intermediate calculation of a particular term in theexpansion. Accordingly, all expansion coefficients that correspond to the scaled sphericalNeumann functions must now be scaled by the reciprocal of the scaling constant. Thisscaling scheme works because the combining of the spherical Neumann functions (withor without scaling) with their corresponding expansion coefficients (with or withoutscaling) yields expansion terms that lie within the range of magnitude afforded by doubleprecision. All expansions contained within expressions (3-18) and (3-19) are alsotruncated at the point where numerical convergence has been achieved in order toeliminate non-contributory calculations. Due to the limitations of the spherical Neumann function expansion approach, an alternative approach is necessary. Flammer [23] has shown that the prolate spheroidal 116radial function of the second kind can be expressed in terms of associated Legendre functions: Rhhdh Q dh Pml mlrml mrm rm mrml rmm rm m,() ,(), ,/, ,(,)()() () () ()2 2 2 211 22 211ξκξ ξρ =′+′     + =− −+∞ −− =+ +∞ ∑∑ (3-20) d dRhhdhd dQ dhd dPml mlrml mrm rm m rml rmm rm mξξκξξ ξξρ,() ,(), , /, ,(,)()() () () ()2 22 21 1 22 211=′ +′     + =− −+∞ −− =+ +∞∑ ∑; (3-21) where κmlh,()()2= )( )( !22!22)! 2()1( )! ()! ( !)!2( )1 2()1(, 21 0,2 12) ( h d h drm rrm mlml mm h mlm m rlm r rr mml −− ∞ =−−      +  + ′−   −+   −−∑ for (l - m) even,   −+   −−− −+− )! ()! ( !)!2( )3 2)(1 2()1( 22)1 ( mlml mm h m mmml )( )( !21 2!212)!1 2( )1(, 121 1,2)1( h d h drm rrmlm m rlm r rr +−− ∞ =−      ++  −++ ′−×∑ for (l - m) odd. (3-22) 117Note that the expansions of (3-20) and (3-21) not only include the coefficients dhrml,() , but also contain the additional expansion coefficients dhrml ρ/,( ) . Moreover, both expressions also require the evaluation of dhrml,( ) expansion coefficients for negative subscripts. Values of these additional expansion coefficients are supplied to this routine by the secondary routine PRONEGDNCNSTS (P-e). Because both prolate spheroidal radial and angle functions of the second kind are integral functions and actually satisfy the same differential equation, i.e., (2-34a) and (2-35a), the two functions must be proportional to one another [23]. This constant of proportionally is referred to as the joining factor of the second kind κmlh,()()2 and can be computed by the expression given in (3-22). Since this joining factor involves the prolate spheroidal angle function of the second kind, it must be dependent upon the selectedSturm-Liouville normalization scheme. Accordingly, the expression presented in (3-22)computes the joining factor for the desired Morse and Feshbach normalization scheme.One should note that expression (3-22) is actually derived from the one given in Flammer[23] by using a normalization conversion factor; the interested reader is referred here forfurther details and theory behind this particular joining factor. Both expressions (3-20) and (3-21) involve expansions of associated Legendre functions of the first and second kind. However, caution must be exercised whenworking with these associated Legendre functions because their arguments are positive,real, and do not lie on the ( ) −< < +11x branch cut (as defined by Erdelyi et al. [40]; see also Appendix D). As mentioned earlier, because the prolate spheroidal radial function of the second kind is singular at ξ=1, the region of validity for these computations shall be limited to real arguments with values greater than unity, i.e., ξ>1. Thus, evaluation of these associated Legendre functions of the first and second kind for this particular region requires the use of the specialized routine CALCPQ (P-e). It is this supporting secondaryroutine that provides the necessary set of associated Legendre functions for theexpansions contained in (3-20) and (3-21). 118Upon further review of (3-21), we recognize that the associated Legendre function expansion approach not only requires the evaluation of associated Legendre functions, butalso their corresponding first derivatives. These derivatives are computed from theassociated Legendre functions (i.e., provided by CALCPQ (P-e)) by way of the followingrecurrence formulas that are valid for real arguments that do not lie on the branch cut: d dPrmm ξξ−− 1() =−+ −−− −−+ mPPrmm rmm ξ ξξ ξξ2 1211 11 1() () ( 3 - 2 3 ) d dQmrm ξξ+() =−+ −+++ mQQmrm mrm ξ ξξ ξξ221 11 1() () =++ −−−+− +() ( )() ()21 1 121 2mr rQmQmrm mrm ξξξ ξξ. (3-24) These expressions can be derived from the generalized complex variable form of the recurrence relations that are valid off the branch cut provided by Erdelyi et al. [40] (see equation (D-36) of Appendix D). Both recurrence formulas relate the derivative of theassociated Legendre function for a given order m to the function itself and one of an adjacent order (all of which share the same degree). Having discussed both approaches capable of computing the prolate spheroidal radial function of the second kind and its first derivative, we now need to determinewhich method is the most applicable (with respect to accuracy) for a given region of inputparameters. As mentioned earlier, we shall take advantage of the computationalexperience of King et al. [35] when implementing this algorithmic decision making process. Supplied in Table 3.2 is a summary of their recommendations regarding which method to use when given a particular set of values for ξ and h. As indicated by Table 3.2, the recommended method for one of the input parameter regions requires the selection of the best result from both numerical approachesusing the Wronskian. Earlier in this section, we discussed that the numerically evaluated 119Table 3.2 Best Method to Compute the Prolate Spheroidal Radial Function of the Second Kind Rhml,()(,)2ξ for a Given Input Parameter Region [35] Input Parameter Region Best Method 11 0 5<≤ξ . Associated Legendre Function Expansion Approach 105 13..<≤ξ or h<01. Both (select better result using Wronskian) 13.<ξ and 10≤h Spherical Neumann Function Expansion Approach 2<ξ and 1≤h Spherical Neumann Function Expansion Approach 5<ξ and 01.≤h Spherical Neumann Function Expansion Approach ξ and houtside the above stated rangesAssociated Legendre Function Expansion Approach Wronskian computed from each approach can be compared with the known theoretical Wronskian in order to determine which approach provides the better result; both of theneeded formulations for the Wronskian check are expressed as follows [23]: WR h R h numerical m l m l ( (,) , (,) ),() ,() 12ξξ= Rhd dRh Rhd dRhml ml ml ml,() ,() ,() ,()(,) (,) (,) (,)12 21ξξξ ξξξ − (3-25) WR h R hhtheory m l m l( (,) , (,) )(),() ,() 12 21 1ξξξ=−. (3-26) Validation of this numerical routine consists of comparing its results with known published data provided by King et al. [35]. On the whole, this routine performs well (i.e., at least six-decimal accuracy) for reasonable values of h and ξ (viz., where h<20 and ξ>101. ). However, some difficulties do arise when the routine attempts to evaluate the prolate radial functions and their derivatives for some extreme regions of ξ and h that are not of interest for our particular application. In the unlikely case that the user attempts to push the routine past its numerical capabilities, the on-the-fly Wronskiancheck will warn the user of numerical results that fail to meet some specified level of 120accuracy. An advantage of using the Wronskian check during the computational process of the routine is that it provides the user with an added level of confidence regarding thevalidity of the numerical results. 3.4.2 SJBESARRAY (B-b) and SYBESARRAY (B-c) Numerical Routines These secondary routines provide the necessary evaluations of spherical Bessel and spherical Neumann functions. It is important to understand that both of thesefunctions play an important role when computing the prolate and oblate spheroidal radialfunctions. As such, both of these secondary routines together must support eachspheroidal radial function routine, viz., PRORADIAL (P-3) and OBLRADIAL (O-3).We choose to address the routines SJBESARRAY (B-b) and SYBESARRAY (B-c) herebecause PRORADIAL (P-3) is sequentially the first of the two primary routines in ourdiscussion that utilizes these much needed secondary routines. Both SJBESARRAY (B-b) and SYBESARRAY (B-c) must provide a truncated list of evaluations from a single family of their respective functions to the routines (P-3)and (O-3) (where a family is defined as the set of all orders of the function possessing thesame argument). Furthermore, both of these secondary routines must also provideenough orders (i.e., from the zeroth order up to the point of truncation) from theirrespective families to enable convergence of the relevant expansions contained withinboth spheroidal radial function routines. Because the expansions of spherical Neumannfunctions are actually asymptotic expansions (see 3.4.1), they numerically converge muchslower than their spherical Bessel function counterparts. Consequently, the sphericalNeumann function routine SYBESARRAY (B-c) must furnish (to the spheroidal radialfunction routines) more orders from its family than its counterpart spherical Besselfunction routine SJBESARRAY (B-b). The routine SJBESARRAY (B-b) computes the spherical Bessel function using a bifurcated approach that is dependent upon the value of its argument. For arguments that 121are relatively small (viz., x < 0.4), the following ascending series is used to compute the family of spherical Bessel functions [25]: jxx nx nx nnnn ()( )!! !( )() !( )( )=+−++++−     21112 3 22 32 51 22 1 222 /G15 (3-27) where the double factorial notation denotes the product of only odd integers. For larger arguments (viz., 0 4 100 .≤<x ), an approach suggested by Arfken [29] is employed that works downward (i.e., backward) from two starting values using the following recurrencerelation: jxn xjxjxnn n() () ()=+−++23 12 . (3-28) The two consecutive starting values needed to implement this downward recurrence relation are chosen as some arbitrary constant and zero. As for the starting order of thedownward recurrence relation, it should be somewhat greater than the highest ordernecessary. This procedural requirement is imposed because the initial values that resultfrom the downward recurrence relation are not accurate and are needed to prime thedownward computational pipeline. After this initial start-up procedure, the values for theremaining lower orders become extremely accurate. Since we choose an arbitraryconstant as one of the starting values, the process yields a family of results that differs bya single scaling constant from that of the conventionally defined spherical Besselfunctions. This situation leaves us with having to normalize the entire sequence of resultsas the last step in the spherical Bessel function computation process. Normalization isperformed by comparing the sequence with either one of the known forms for jx 0() and jx1( ) to determine the scaling constant [29]: jx x xo() s i n= (3-29) jx x x x x12() s i n c o s=− . (3-30) 122In order to maintain the greatest accuracy possible for this approach, King et al. [35] recommend determining the scaling constant with the jx1() form when the function jx0( ) is in the neighborhood of a zero. Regarding computation of the spherical Neumann functions, the routine SYBESARRAY (B-c) only makes use of the recurrence relation approach. However, thistime around, the evaluation of the family of functions is determined by working upward(i.e., forward) using a recurrence relation similar to (3-28) [29]: nxn xnxnxnn n() () ()=−−−−21 12 . (3-31) For obvious reasons, both nx0( ) and nx1( ) serve as the starting values and are computed from their respective closed form representations [29]: nx x xo() c o s=− (3-32) nx x x x x12() c o s s i n=− − . (3-33) During our discussion of the routine PRORADIAL (P-3) (see 3.4.1), we mention that the spherical Neumann functions belonging to a given family have to be scaled atsome point as their order increases so as to avoid an overflow condition.Correspondingly, the spherical Neumann function routine SYBESARRAY (B-c) at somepoint during the process of working upward with (3-31) must scale its results.Consequently, the spheroidal radial function routines not only must receive the family ofspherical Neumann function evaluations, but also the index of the leading order in thesequence at which scaling had to be imposed. For obvious reasons, this additional pieceof information is necessary for the spheroidal radial function routines to properly interpretthe family of function evaluations. These spherical Bessel and Neumann function routines are initially validated using the tabulated data provided in AMS-55 [25]. Although the routines perform 123extremely well with the tabulated data, we need to validate our routines for orders of the functions that are higher than those available in this particular data set. Toward that end,we compare the results of our routines with those computed by an alternative numericalroutine for spherical Bessel and Neumann functions (namely, the one made available in Numerical Recipes in FORTRAN [37]). In terms of accuracy, our developed routines perform well when compared to the results of this alternative routine. It must be pointedout that the alternative routine provided in [37] was not selected for our particularapplication because its numerical approach is not the most computationally efficient wayto evaluate a large family of spherical Bessel and spherical Neumann functions. 3.4.3 PRONEGDNCNSTS (P-e) Numerical Routine This secondary routine computes the dh rml,( ) expansion coefficients with negative subscripts and the expansion coefficients dhrml ρ/,( ) that are necessary to compute the prolate spheroidal radial function of the second kind when employing the associated Legendre function expansion approach. Accordingly, this routine only has to support thePRORADIAL (P-3) primary routine. The inputs of this routine require two criticalparameters for a given order m and degree l that are computed by the routine PRODNCNSTS (P-d): either dh ml 0,( ) or dhml 1,( ) , and the refined eigenvalue λmlh,(). The dhrml,( ) expansion coefficients with negative subscripts are computed in a manner that is somewhat similar to its non-negative counterparts: the difference being that the procedure need not employ an iterative process because the eigenvalue is alreadyassumed to be an accurate representation, not an initial approximation. Maintainingconsistency with the expansions of associated Legendre functions of the second kind within (3-20) and (3-21), the range of the index r for the dh rml,( ) expansion coefficients with negative subscripts can be formally stated as  − +−−−− −−−=odd. ) (for 1 2,,3,1even, ) (for 2,,4,2 ml mml mr/G15/G15(3-34) 124We now make use of another three-term recurrence relation using the already refined eigenvalue λmlh,( ) to determine the ratios of the dhrml,( ) expansion coefficients with negative subscripts [23]: dh dhA BCdh dhrml rmlrm rml rm rml rml, , ,, ,() () () ()++ −−=− +  22 22(3-35) where Amr mr mr mrrm=++ − +− ++() ( ) () ()22 1 22 1 22 1(3-36) Bmr mr m mr mrrml, () ( ) () ()=++ + −− +− ++21 2 1 22 1 22 32 +++ + −() ( ) ( ), mr mr h hml 1 2λ(3-37) Crr mr mrrm=++ ++ ++() ( ) () ()12 22 1 22 3. (3-38) (Note that our recurrence constants differ slightly from those of Flammer [23] by a factor of h−2. Since this alternative definition does not make a difference with respect to the recurrence relation of (3-35), we define the constants as such to minimize floating-point operations.) The fact that both recurrence constants Amm −2 and Amm −+21 vanish (by inspection of (3-36)) implies that both expansion coefficient ratios dh dhmml mml−− −22 2, ,() () and dh dhmml mml−− −+21 21, ,() () also must vanish. Using a zero value for either of these expansion coefficient ratios along with the recurrence relation of (3-35) leads us to the following starting points: 125dh dhA Bmml mmlmm mml− −+−+ −=−2 2222 2, ,,() ()for ( ) lm− even, dh dhA Bmml mmlmm mml−+ −+−+ −+=−21 2323 21, ,,() () for ( ) lm− odd. (3-39) We are now able to compute the sequence of dhrml,( ) expansion coefficients ratios with negative subscripts by working upward (i.e., forward) with recurrence relation (3-35) using the appropriate starting point of (3-39) for all required indices of r stated in (3-34). Now that we have determined the dhrml,( ) expansion coefficient ratios with negative subscripts, the list of expansion coefficients normalized with respect to either dhml 0,( ) or dhml 1,( ) can be determined by using one of the following appropriate expressions [23]:        = + −− − lm rlm r lmlm lmlm lmlm r dd dd dd h dh d , 2, , 2, 4 , 0, 2 , 0, )()(/G16 for ()lm− even and −≤≤ −22mr ,         = + −− − lm rlm r lmlm lmlm lmlm r dd dd dd h dh d , 2, , 1, 3 , 1, 1 , 1, )()(/G16 for ( ) lm− odd and −+ ≤≤ −21 1mr . (3-40) We now appropriately apply either dhml 0,( ) or dhml 1,( ) , i.e., calculated by PRODNCNSTS (P-d), to properly scale the list and explicitly provide the desired sequence of dhrml,() expansion coefficients with negative subscripts. In our particular case, the values provided for dhml 0,( ) or dhml 1,( ) correspond to those required for the Morse and Feshbach normalization scheme. Having addressed the dhrml,( ) expansion coefficients with negative subscripts, we now turn our attention to the numerical procedure put forth by Flammer [23] that is necessary to compute the dhrml ρ/,( ) expansion coefficients. It is important to understand 126that the dhrml ρ/,( ) expansion coefficients are related to the dhrml,( ) expansion coefficients, albeit for a different range of subscripts, through the following relation [23]: dh drml rml ρ ρ ρρ/,,( ) lim=→− +0 (3-41) where  − ++− ++=odd. ) (for ,3 2,1 2even, ) (for ,4 2,2 2 ml m mml m mr/G15/G15(3-42) Note that the range of indices specified in (3-42) remains consistent with the expansions of associated Legendre functions of the first kind included in expressions (3-20) and (3-21). We can now apply (3-41) to the three-term recurrence relation of (3-35) to obtain an expression for the first element in the sequence of dh rml ρ/,( ) ratios: []    +−+−= ++ −− −−− −+ )()()21)(21( )()( , 2 2/, 4 2/ 4 2, 2 21 , 2, 2 2/ h dh dC Bm m h dh d lm mlm m m mlm mlm mlm m ρρρfor ()lm− even, []    +−−−= ++ −−−−− +−+ )()()23)(21( )()( , 1 2/, 3 2/ 3 2, 121 , 1 2, 1 2/ h dh dC Bm m h dh d lm mlm m m mlm mlm mlm m ρρρ for ( ) lm− odd; (3-43) where the recurrence constants are computed using equations (3-36)-(3-38). Moreover, the numerator in each expression is derived from evaluating the respective limiting formulation: limρ ρρ→− +02Amm or limρ ρρ→− + +02 1Amm. Note that both expressions of (3- 43) can be written as continued fractions and may be computed to any degree of accuracy by truncating the fractions after a sufficient number of terms [23]. We now make use ofthe following three-term recurrence relation to work downward (i.e., backward) to the second element in the sequence of dh rml ρ/,( ) ratios: 127dh dhA BCdh dhrml rmlrm rml rm rml rmlρ ρ ρ ρ/, /, , /, /,() () () ()−−+ −− −+=− +  22 22for rm≥+2 3 . (3-44) The first element in the sequence, i.e., dh dhmml mmlρ/, ,() ()22 2+ − or dh dhmml mmlρ/, ,() ()21 21+ −+, can then be computed from the second element by using the appropriate expression of (3-43). The fact that Flammer [23] indicates that the continued fraction may be safely truncated implies that the sequence of dhrml ρ/,( ) ratios continues to get smaller in magnitude as the index r increases, i.e., lim() ()/, /, rrml rmldh dh→∞+=ρ ρ20 . By terminating the continued fraction at some reasonable large value for the index r, viz., rrterm= (remembering to choose the appropriate even or odd index), and using the approximation dh dhrtermml rtermmlρ ρ/, /,() ()+=20 , we create the following starting point for the downward recurrence relation of (3-44): dh dhA Brtermml rtermmlrtermm rtermmlρ ρ/, /,,() ()−−+ −=− 22. (3-45) From this calculated list of dhrml ρ/,( ) expansion coefficient ratios, the sequence of expansion coefficients normalized with respect to either dhmml −2,( ) or dhmml −+21,( ) can be found by appropriately applying one of the following expressions (where the index r still conforms to the range defined in (3-42)) [23]: 128       = − ++ −+ − )()( )()( , 2/, / , 2 2/, 4 2/ , 2, 2 2/ , 2, / h dh d dd dd h dh d lm rlm r lm mlm m lm mlm m lm mlm r ρρ ρρ ρ ρ/G16 for ()lm− even and rm≥+22 ,        = − ++ +−+ +− )()( )()( , 2/, / , 12/, 3 2/ , 1 2, 1 2/ , 1 2, / h dh d dd dd h dh d lm rlm r lm mlm m lm mlm m lm mlm r ρρ ρρ ρ ρ/G16 for ()lm− odd and rm≥+21 . (3-46) At this point, we scale the entire computed sequence by either dhmml −2,( ) or dhmml −+21,() t o explicitly determine the desired dhrml ρ/,( ) expansion coefficients. Fortunately, the necessary scaling constant has already been computed by the first part of this routine that calculates the dhrml,( ) expansion coefficients with negative subscripts. Consequently, the values of the dhrml ρ/,( ) expansion coefficients also must correspond to the Morse and Feshbach normalization scheme. Validation of this routine consists of comparing both the computed dhrml,() expansion coefficients with negative subscripts and the computed expansion coefficients dhrml ρ/,( ) to tabulated data provided by Flammer [23]. Results for both computed sets of expansion coefficients do extremely well when compared with this available data (i.e., after accounting for the difference in normalization schemes). However, the ultimatevalidation is once again left to how well this routine supports the expansions ofassociated Legendre function expansions contained within expressions (3-20) and (3-21). 3.4.4 CALCPQ (P-f) Numerical Routine This secondary routine provides the necessary evaluations of the associated Legendre functions of the first and second kind to the PRORADIAL (P-3) routine. These 129computations are only necessary when invoking the associated Legendre function expansion approach of (3-20) and (3-21) to compute the prolate spheroidal radial functionof the second kind. An essential feature of this routine is that both associated Legendrefunctions must be evaluated for positive real arguments (in terms of the radial parameter)that do not lie on the ( ) −< < +11x branch cut or at the x=±1 singular points, viz., where ξ>1. Expressions (3-20) and (3-21) coupled with the recurrence formulas of (3-23) and (3-24) indicate that the associated Legendre function expansion approach requires theevaluation of the m, m+1, and m−1 order families for the associated Legendre functions. (By definition, a family is considered to be a set of associated Legendrefunctions that correspond to a given order m.) Specifically, the approach only requires evaluations of the m and m+1 families in the case of the associated Legendre functions of the first kind, and evaluations of all three families in the case of the associatedLegendre functions of the second kind. Even though the expansions of (3-20) and (3-21)only call for every other member of a particular associated Legendre function family (i.e.,either even or odd degrees), this routine shall nevertheless provide consecutive membersto the PRORADIAL (P-3) routine in the interest of being more versatile and complete.Since this secondary routine only provides the necessary computations for a given orderm, PRORADIAL (P-3) must again evoke this routine in order to compute the m+1 and m−1 order families. Evaluation of the associated Legendre function of the first kind for real arguments that do not lie on the branch cut can be performed in a variety of ways. These approachesinclude the use of closed form expressions, viz., found by using (D-20), as well as otherexplicit series expansions. Unfortunately, these techniques do not lend themselves wellto the numerical evaluation of this function from a viewpoint of coding efficiency andnumerical accuracy. Instead, we make use of the ingenious computational approachrecommended by Numerical Recipes in FORTRAN [37]. Although the computational approach presented in [37] is made to address real arguments that are either on the branch cut or at the singular points, with slight 130modifications this method can be augmented to also handle real arguments that are off the branch cut. Numerical Recipes in FORTRAN [37] recommends using the following numerically stable recurrence relation to compute a family of associated Legendrefunctions of the first kind for a given order m by working upward (i.e., forward) with respect to the degree l: Pl lmPlm lmPlm lm lm()() ()()() ()() ξξ ξ ξ =− −−+− −−−21 1 12 . (3-47) As fortune would have it, in this instance this particular recurrence relation not only applies to real arguments that lie on the branch cut, but also to those that exist off thebranch cut. For that reason, recurrence relation (3-47) can be directly incorporated intoour numerical routine. The fact that (3-47) remains valid off the branch cut for realarguments can be shown using the generalized complex variable form of the recurrencerelation, i.e., (D-34), provided by Erdelyi et al. [40]. It must be noted that having a recurrence relation hold for real arguments on and off the branch cut is generally not truefor associated Legendre functions (see Appendix D). We now employ a closed form expression that is valid for real arguments off the branch cut as one of the two starting values needed for the recurrence relation (3-47): Pmmmm() ( ) ! ! ( )ξ ξ =− −21 122. (3-48) This expression can be derived from a finite power series representation of the associated Legendre function of the first kind (found in MTP-1945 [41]) which terminates quickly when lm=. Using (3-47) and setting Pmm −=1 0 ()ξ , we can determine the second starting value [37]: Pm Pmm mm +=+1 21 () ( ) ()ξ ξξ. (3-49) 131Working upward with (3-47) from these two starting values, we are able to generate an entire family of associated Legendre functions of the first kind for a given order m, i.e., /G15,1 ,+= mml , with a single pass. As we can see, this method is remarkably efficient for our particular application. We now turn our attention to evaluating the associated Legendre functions of the second kind for positive real arguments that do not lie on the branch cut or at the singular points, i.e., ξ>1. These functions must be computed for both positive and negative degrees l, i.e., /G15/G15 ,1,0,,1 ,+−−= mm l , when the given order m is greater than zero (as opposed to just non-negative degrees l when m=0) . Let us first address the computation of the required set of associated Legendre functions of the second kind with non-negative degrees (i.e., l≥0). Due to the associated Legendre function of the second kind being singular at x=1, computation of the necessary set of these functions becomes progressively more difficult as the argument ξ gets closer to unity. As a result, we found that using a two- pronged approach that is based upon the value of the argument is the best way to compute the set of associated Legendre functions of the second kind with non-negative degrees fora given order m, i.e., where /G15,1,0=l . For the region 1 11 << ξ . where the positive real argument is close to the singular point, the set of associated Legendre functions of the second kind with non-negative degrees is individually computed for all respective degrees of l using the following general explicit expression (based upon the example case given in MTP-1945 [41]): 132Qlm()ξ= 1 21 1Plm() l nξξ ξ+ −   −−−+ −−+− =∑ 2 2 1 )1 ()1()1()()! ()1( 2! ss s sm lm ss Psmsm ξξ ξξ −−− +−=−   −− ∑() () ( )()() 24 1 2101 21 21lq ql qP qInt l lqmξ for l≥1; (3-50) where Int x[ ] is considered the greatest integer function. Note that expression (3-50) reduces to the following when l=0 and m=0 (convention drops the superscript for the zeroth order): Q0()ξ= 1 21 1lnξ ξ+ −. (3-51) The expression (3-50) is derived from the definitive expression for the zeroth-order associated Legendre function of the second kind Ql()ξ, viz., (D-4), and the differentiation formula that relates Ql()ξ for a given degree l to its higher-order counterparts, viz., (D-22). One makes use of Leibnitz’s formula for the nth derivative of a product, i.e., (D-40), to remove the resulting derivatives and reduce the explicitexpression for the associated Legendre functions of the second kind to the form presentedin (3-50). When using this explicit approach to compute the set of functions with non- negative degrees, the necessary evaluations of the associated Legendre functions of thefirst kind are provided by using the approach outlined in (3-47)-(3-49). In addition, itmust be pointed out that this explicit approach is not very efficient because it requireseach function to be individually evaluated using a computationally intensive formula.Unfortunately, we are limited to this approach when operating so close to the singularity. 133Regarding the region where ξ≥11. , the argument has been numerically determined to be far enough from the singularity to successfully implement the following more computationally efficient approach. In order to compute the set of associated Legendre functions of the second kind with non-negative degrees for ξ≥11., w e s h a l l take advantage of the following downward (i.e., backward) traveling recurrence relation (with respect to the degree l ) that is valid off the branch cut for a given order m [40]: Ql lmQlm lmQlm lm lm()() ()()() ()() ξξ ξ ξ =+ ++−−+ ++++23 12 112 . (3-52) Note that (3-52) is of similar form to the upward traveling recurrence relation of (3-47) with the exception of a shift of indices. Numerical experience has shown that therecurrence relation of (3-52) is numerically stable when traveling in the downwarddirection for this particular function. In order to begin working downward to Q m 0()ξ using the recurrence relation (3- 52), we need to have two consecutive starting values. These two starting values must be situated at some point far enough from l=0 to provide enough members of the family to ensure convergence of the associated Legendre functions of the second kind expansionscontained within (3-20) and (3-21). Furthermore, we can take advantage of some knownexpressions for associated Legendre functions of the second kind to compute both ofthese starting values. These expressions are in open form and can be represented in termsof the Gauss hypergeometric function (defined for a complex argument z) [25]: ! )!1 ()!1( )!1()!1 ( )!1()!1 ();;,( 0 kz kcc bkb akazcbaFk k   −+−    −−+    −−+=∑∞ =. (3-53) The region of convergence for this function without exception includes all values of the complex argument z enclosed within the unit circle, i.e., | | z<1, and excludes all values which lie outside it. Regarding complex arguments situated on the unit circle boundary, convergence of the hypergeometric function is dictated by a set of conditional rules 134dependent upon the parameters a, b, and c. Fortunately, having knowledge of whether or not a particular hypergeometric function theoretically converges on the unit circle isbeyond our computational purposes given the fact that it would be numerically imprudentto operate the expansion at this boundary. Consequently, the interested reader is referredto AMS-55 [25] for the details regarding convergence of the hypergeometric function for complex arguments situated on the unit circle. According to MTP-1945 [41], the following hypergeometric expression is the one to use when computing the associated Legendre function of the second kind with anargument whose modulus is less than or equal to three: Q lm()ξ= () Bml F m ml t(, ,) , ; ;ξ1 21 23 2 +−+ − (3-54) where ()  + −+− +−= + )!2()! (! 1 )1 (2) (4)1(),,( 21 2 21 2 21lmll llmB ll m ξξ ξξ t=−− −ξξ ξ2 21 21. However, it can be shown that the hypergeometric expression of (3-54) possesses a convergence boundary at ξ=≈3 421 0 6 . (found by setting the modulus of −t to unity). Since (3-54) is valid for positive real arguments off the branch cut that are greater than this convergence boundary, our routine makes use of this hypergeometric expression to compute both starting values when operating in the 11 3 .≤≤ξ region. As for arguments that have a modulus greater than three, MTP-1945 [41] recommends using the following less complicated, but slower converging, 135Table 3.3 Summary of Approach Used to Compute the Set of Associated Legendre Functions of the Second Kindwith Non-Negative Degrees for a Given Order m Region of Argument ξApproach Used to Compute Set of Qlm()ξfunctions for l≥0Formula Used to Compute Starting Values 11 1<<ξ .Explicit Expression Approach (use (3-50) and (3-51) to individuallycompute all members of set)NA 11 3.≤≤ξRecurrence Relation Approach (use (3-52) to work downward fromstarting values)Hypergeometric Expression (3-54) ξ>3Recurrence Relation Approach (use (3-52) to work downward fromstarting values)Hypergeometric Expression (3-55) hypergeometric expression when computing the associated Legendre functions of the second kind: Qlm()ξ= () Aml F l m l m l(, ,) ( ) ,( ) ; ;ξ ξ1 21 23 2221−+ −+ +−(3-55) where   + −+−=+−− )!2()! (! )1 )( (2)1(),,(1 2 2 211 lmll llmAml ml m ξ ξξ . Accordingly, (3-55) is employed by this routine to compute both starting values when the argument is in the ξ>3 region. Although these hypergeometric expressions can be used to individually evaluate each function of the set, the recurrence relation approach is much more computationally efficient. Having completed the discussion of our approach forcomputing the set of associated Legendre functions of the second kind with non-negativedegrees, we present a summary of its details in Table 3.3. 136We now turn our attention to computing the set of associated Legendre functions of the second kind with negative degrees l, viz., 1,,1 , −+−−= /G15 mm l , for a given order m (i.e., when m>0) . These computations are done by using the recurrence relation (3- 52) to work downward from the starting values Qm 0()ξ and Qm 1()ξ until Qmm −()ξ is reached. Note that both starting values have already been determined from the part of the routine that computes the set of associated Legendre functions of the second kind withnon-negative degrees (see Table 3.3). The results of this routine are first compared to tabulated data available in MTP- 1945 [41] (i.e., six-decimal accuracy) for both sets of associated Legendre functions. For all of the test cases, the routine performs extremely well and yields results that agree withthe published data. In order to further validate the performance of our routine, additionalcomputational checks are made. First, the associated Legendre functions of the first kindare computed by the CALCPQ (P-f) routine and then compared to results obtained from aselect set of closed form representations: both sets of results match one another extremelywell. As for the associated Legendre functions of the second kind, we continue ourvalidation process by conducting consistency checks between the computational methods. For arguments situated in the regions where ξ≥11. , results obtained from the recurrence relation approach are compared to those computed directly with the hypergeometric expression appropriate for its region, viz., (3-54) and (3-55). For both of these regions, i.e., 11 3 .≤≤ξ and ξ>3 , the two computational approaches are in excellent agreement (i.e., match to better than twelve decimal places) for the cases that are tested. As a result, it is these additional tests that allow us to gain additional confidence in the developedCALCPQ (P-f) numerical routine. 1373.5 OBLATE SPHEROIDAL RADIAL FUNCTIONS OF THE FIRST AND SECOND KIND 3.5.1 OBLRADIAL (O-3) Numerical Routine With similarity to its prolate counterpart, this numerical routine computes the oblate spheroidal radial functions of the first and second kind with their first derivativesfor a given order m, degree l, and independent parameter h. The oblate spheroidal radial function of the fourth kind, which is required by the NZ-FZT algorithm for the oblatecase, is then determined from both of these computed radial solutions by way of (2-74b).Because this primary routine makes use of the shared secondary routine OBLDNCNSTS (O-d) to compute the dj h rml,()− expansion coefficients, it too must be provided with either an initial approximation of the respective eigenvalue or an already refined eigenvalue. In the case that an approximation of the eigenvalue is supplied to the routine,the numerical representation of the eigenvalue is replaced with a value that is moreaccurate. As we shall see, the principal difference between the prolate and oblateapproaches presents itself when computing the spheroidal radial function of the secondkind and its first derivative. Even though both oblate spheroidal radial functions are defined (i.e., not singular) at ξ=0 , we have elected to confine this routine to those arguments that are greater than zero, i.e., ξ>0 . In the limiting case where ξ=0 , the oblate spheroidal surface becomes a circular disc of infinitesimal thickness (refer to Figure 2.1b). From an applied perspective, the usefulness of this particular transformation surface is somewhat limitedbecause of the difficulty it has with enclosing a realistic radiating structure possessingfinite dimensions. Furthermore, evaluating both oblate spheroidal radial functions and their first derivatives at ξ=0 requires the use of special formulas [36]. In fact, the two radial solutions no longer remain linearly independent in this limiting case [36]. Taking into account these reasons, we decided that being able to evaluate these functions at ξ=0 is beyond what is really necessary for our particular application. 138Computation of the oblate spheroidal radial function of the first kind and its first derivative uses essentially the same procedure as its prolate counterpart with theexception of some sign reversals in the expressions containing the spherical Besselfunction expansions [36]: Rj h j ml,()(, )1−=ξ() ! () !()() ! !(), ,lm lmjdj hmr rjhm rml rml rrm− ++  ′−++− =∞ + ∑ξ ξξ2 22 011 2 (3-56) d dRj h jlm lmmlm ξξξ ξ,()(, )() ! () !12 221−=− ++  × hj d j hmr rm rmr j h mr j h mjdj hmr rjhrml rml rm rm r rml rml rrm+− +− ++ =∞ +− =∞ +′−+ +++ −+ +   −+′−+      ∑ ∑, , , ,()() ! !( )() ( ) () ( ) ()()() ! !()2 22 1 1 121 1 01 2 01ξ ξ ξξξ. (3-57) Evaluation of these expressions is performed in the same manner as that which is done for the prolate case, i.e., using (3-16) and (3-17), and need not be discussed any further. This numerical routine makes use of two different approaches to compute the oblate spheroidal radial function of the second kind and its first derivative much in thesame way as its prolate counterpart. This time around, the decision making process thatdetermines the best approach to use is based upon the computational experience providedin NRL Report 6959 [36]. Once again, the Wronskian check (using the appropriate oblate form) plays a significant role. The first of these two approaches uses essentially the same procedure as its prolate counterpart with the exception of some sign reversals in the expressions containing thespherical Neumann function expansions [36]: 139Rj h jml,()(, )2−=ξ() ! () !()() ! !(), ,lm lmjdj hmr rnhm rml rml rrm− ++  ′−++− =∞ + ∑ξ ξξ2 22 011 2 (3-58) d dRj h jlm lmmlm ξξξ ξ,()(, )() ! () !22 221−=− ++  × hj d j hmr rm rmr n h mr n h mjdj hmr rnhrml rml rm rm r rml rml rrm+− +− ++ =∞ +− =∞ +′−+ +++ −+ +   −+′−+      ∑ ∑, , , ,()() ! !( )() ( ) () ( ) ()()() ! !()2 22 1 1 121 1 01 2 01ξ ξ ξξξ. (3-59) Because the evaluation of these expressions is performed in the same exact manner as that which is done for the prolate case, i.e., using (3-18) and (3-19), it too will not bediscussed in any further depth. Just like its prolate counterpart, the usefulness of the spherical Neumann function expansion approach of (3-58) and (3-59) is limited to large values of h ξ because of the spherical Neumann functions. Accordingly, Van Buren et al. [36] put forth another approach capable of evaluating the oblate spheroidal radial function of the second kind and its first derivative when given small values of hξ. Their approach fundamentally differs from its prolate counterpart, i.e., equations (3-20)-(3-24), and requires the use of an expression presented by Flammer [23] that expands the oblate spheroidal radialfunction of the third kind only in terms of associated Legendre functions of the secondkind [36]: /G0C/G0C/G0C/G0C/G0B/G0C/G0C/G0C/G0C /G0A /G09/G0C/G0C/G0B/G0C/G0C/G0A/G09 δγξ βαξπ ξ jj Qhmj je jjh R mrm rm lm mlm rm l hj lm +≡  × +≡=− ∑∞ −=+ −+ +−)(!), (,, 12 ))2)(1( ( )3( ,/G24/G24(3-60) where /G15,1 ,+−−= mm r . 140As a point of information, this expression is developed from a special expansion for the oblate spheroidal angle function of the first kind discovered by Baber and Hasse; theinterested reader is referred to [23], [36] regarding the specific details. Contrary to theother expansions explored within this chapter, the summation in this case is over eachconsecutive value of r as opposed to every other value of r. The Baber and Hasse normalized expansion coefficients ()lm mlm r, , −/G24/G24 are computed using another three-term recurrence relation that is employed in the secondary routine BABHASCNSTS (O-e); these computations are detailed in the next section (see 3.5.2). Upon inspection of (3-60), one can see that this expression can be represented as the product of two complex quantities: ()αβ+j and ()γδ+j. The oblate spheroidal radial function of the second kind can now be related to its first kind counterpart using the following expression derived from (2-74b) and (3-60) [36]: [] R j hj R j hjml ml,() ,()(, ) (, )21−= + −+ξαδβ αξβδ. (3-61) This expression allows us to compute the oblate spheroidal radial function of the second kind from the results of (3-56), i.e., Rj h jml,()(, )1−ξ, after determining the parameters α, β, and δ from (3-60). While the computation of the parameters α and β is straightforward, this is not the case for the parameter δ. Evaluation of the quantity ()γδ+j requires the complex summation of an expansion of associated Legendre functions of the second kind. Although the ()lm mlm r, , −/G24/G24 normalized expansion coefficients are real quantities, the summation is complex because the associated Legendre functions of the second kind Qjmrm +()ξ can be either a real or purely imaginary quantity. The calculation of this complex set of associated Legendre functions of the second kind is provided by the CALCQIM (O-f) secondary routine and is discussed later(see 3.5.3). 141Regarding the first derivative of the oblate radial function of the second kind, it can be computed from its first kind counterpart (determined from (3-57)) using thefollowing expressions [36]: d dRj h jd dRj h jml mlξξανβ αξξβν,() ,()(, ) (, )21−= + −+  (3-62) where ∑∞ −= −++ −     +++ + ++   =+ mr m rmm rm lm mlm r j Qrmjjh rm j Q j )( 1) 2(1) ()( 1 22 ,, ξ ξξξξ νµ /G24/G24. (3-63) By taking the imaginary part of the complex expansion (3-63), expression (3-62) is provided with the necessary parameter ν. Both of these expressions can be found from (3-60) and (3-61) using the following recurrence relation involving the derivative of the associated Legendre function of the second kind that is valid for arguments that are offthe branch cut and are purely imaginary: d dQj m r Qj jmrQjmrm mrm mrm ξξξ ξξξξ++ + −=++++ +()( ) ()()()22 112 1. (3-64) This recurrence relation can be derived from its generalized complex variable form, i.e., (D-35), provided by Erdelyi et al. [40]. One should take note that (3-64) relates the first derivative of this radial function to only associated Legendre functions of the second kindthat correspond to the same family of a given order m. Practically speaking, this feature means that the evaluation of the derivative only requires associated Legendre functions ofthe second kind from a single family when using formulations (3-62) and (3-63).Moreover, this family of functions is the same as the one used when computing itscorresponding oblate spheroidal radial function of the second kind with (3-60) and (3-61).As a result, the routine CALCQIM (O-f) only needs to be evoked once when using the 142Table 3.4 Best Method to Compute the Oblate Spheroidal Radial Function of the Second Kind Rj h jml,()(, )2−ξ for a Given Input Parameter Region Input Parameter Region Best Method 01<≤ξ or hξ≤10 Baber and Hasse Expansion Approach ξ and houtside the above stated rangesBoth Spherical Neumann Function Expansion Approach and Baber and Hasse Expansion Approach (select betterresult using the Wronskian) Baber and Hasse expansion approach to compute this particular radial function and its first derivative. Now that we have discussed both approaches capable of computing the oblate spheroidal radial function of the second kind and its first derivative, our attention mustfocus on which method is the best with respect to accuracy for a given region of inputparameters. Taking advantage of the computational experience of Van Buren et al. [36], Table 3.4 summarizes their recommendations regarding the best method to use when given a particular set of values for ξ and h. Upon inspection of Table 3.4, we notice that the second input parameter region requires the selection of the best result from both numerical approaches by comparingtheir numerical Wronskians. This computational check, which is initially described in thesection pertaining to the PRORADIAL (P-3) numerical routine, consists of comparing thenumerical Wronskian of each approach with the known theoretical Wronskian todetermine which approach yields the better result. Using the Wronskian formulation thatcorresponds to the oblate case, we obtain the necessary expressions for the Wronskiancheck [23]: 143W R jh j R jh jnumerical m l m l ( ( ,) , ( ,) ),() ,() 12−− =ξξ Rj h jd dRj h j Rj h jd dRj h jml ml ml ml,() ,() ,() ,()(, ) (, ) (, ) (, )12 21−− − −−ξξξξξξ (3-65) W R jh j R jh jht h e o r y ml ml( ( ,) , ( ,) )(),() ,() 12 21 1−− =+ξξξ. (3-66) Validation of the OBLRADIAL (O-3) routine consists of comparing its results with published data provided by Van Buren et al. [36]. The routine performs extremely well (i.e., at least twelve-decimal accuracy) when compared to this data set. Once again,we choose to implement the Wronskian check in an on-the-fly manner such that theroutine can warn the user of results that fail to meet some specified level of accuracy. Wedo so in order to provide the user with an added level of confidence regarding the validityof the computed results. 3.5.2 BABHASCNSTS (O-e) Numerical Routine This routine computes the Baber and Hasse normalized expansion coefficients ()lm mlm r, , −/G24/G24 required to implement the approach outlined in equations (3-60)-(3-63). In order to perform these computations, our routine makes use of the technique put forth by Van Buren et al. [36]. Like the other special function expansion coefficients, this technique also evokes the use of a three-term recurrence relation that relates consecutiveexpansion coefficients. We begin computing these normalized expansion coefficients byusing the following downward (i.e., backward) traveling form of the three-termrecurrence relation [36]: 144m rlm mlm r lm r lm mlm r m r lm mlm r WV U 1,, 1 , 1 ,, 2 1 ,, +−+ + −+ + −  +   −=   /G24/G24 /G24/G24 /G24/G24(3-67) where Urm r m rmrm=++ + + ++21 2 1 22 3() ( ) ()(3-68) Vj h rm rm hhrml ml , ,() ( ) ( )=−− + + ++λ 1(3-69) Wrr m rmrm=−+ +−2 22 1() (). (3-70) Note that (3-69) requires this routine to be provided with a refined version of the respective oblate spheroidal eigenvalue. In order to begin working downward with therecurrence relation of (3-67), we must generate two consecutive starting values. Just likewe do when computing the spherical Bessel functions, these consecutive starting valuesare arbitrarily chosen as some constant and zero. The index of the starting values shouldthen be chosen somewhat greater than the highest index required to achieve convergenceof the expansions in (3-60) and (3-63). As before, this procedural requirement must beimposed because the initial values that result from the downward recurrence relation arenot accurate and are needed to prime the downward computational pipeline. However,the values of the lower indices do become extremely accurate after the initial start-upprocedure. Because we have chosen an arbitrary constant as one of the starting values,this procedure yields expansion coefficients that are only accurate relative to one another;that is, they still need to be scaled by a correction constant to achieve the proper results. Unfortunately, this particular three-term recurrence relation may not provide accurate results for the entire pass in the downward direction. From a computationalviewpoint, one would prefer to work a recurrence relation in the direction of increasingmagnitude (of the related terms) in order to minimize the loss of precision. However in 145this particular case, the set of normalized expansion coefficients ()lm mlm r, , −/G24/G24 is not always guaranteed to be continually increasing in the downward direction. To overcome this difficulty, Van Buren et al. [36] suggest using the downward traveling recurrence relation to compute the coefficients only as long as the they continue to increase inmagnitude. Accordingly, it is possible that the entire set of normalized coefficients couldbe computed using a single downward pass. However, if the normalized coefficientsshould begin to decrease in magnitude during the downward traveling computations, thenthose remaining coefficients need to be computed differently. This second part of theapproach requires that the remaining lower indexed portion of the normalized coefficientsbe evaluated from rm=− up to this resulting cross-over index (i.e., the point where the terms begin to get smaller with respect to the downward direction) using the followingupward traveling form of the three-term recurrence relation [36]: m rlm mlm r m r lm mlm r lm r lm mlm r UW V 1,, 2 1 ,, 1 , 1 ,, −−− − −− − −  +   −=   /G24/G24 /G24/G24 /G24/G24. (3-71) The two consecutive starting values necessary to begin working upward with (3-71) are given as follows: 1,, =   −− lm mlm m /G24/G24 and m mlm m lm mlm m UV −− −+−−=  , ,, 1 /G24/G24. The first starting value is unity for obvious reasons, and the second starting value can be found directly from (3-71) using the fact that Wmm −=0 by way of (3-70). Since both of these starting values are correct in absolute terms, the normalized coefficients that are computed using the upward recurrence relation need not be scaled by a correction constant. Having determined the normalized expansion coefficients using both forms of the three-term recurrence relation, we must now address the fact that the upper portion of thecoefficients computed using the downward form (3-67) must be scaled by a correctionconstant. Since the normalized coefficients that are computed by the upward form (3-71)are accurate in absolute terms (as opposed to relative terms), the necessary correction 146constant can be found by reconciling the difference between the two differently computed portions at the resulting cross-over index. In the special case that all of the normalizedcoefficients are computed using the downward recurrence relation, the correction constantis then determined using the fact that the first normalized coefficient in the set is unity. We are not able to directly validate this routine because of the lack of available published data for these obscure normalized expansion coefficients. Once again, theultimate validation is left to how well this routine supports its respective primary routine,i.e., OBLRADIAL (O-3). 3.5.3 CALCQIM (O-f) Numerical Routine This secondary routine provides the necessary evaluations of the associated Legendre functions of the second kind to the OBLRADIAL (O-3) routine. In particular,CALCQIM (O-f) evaluates a family of these functions for a given order m with degrees corresponding to /G15,1,0,1−=l . This supporting routine is always called by (O-3) because the overall approach to computing the oblate spheroidal radial function of the second kind requires the evaluation of the Baber and Hasse expansions of (3-60) and (3-63) (see Table 3.4). Furthermore, CALCQIM (O-f) must provide enough members of thefamily to ensure convergence of both of these expansions. A unique feature of thisroutine is that it evaluates the associated Legendre functions of the second kind for a purely imaginary argument, viz., Qj lm()ξ. Correspondingly, the formulations used in this routine are derived from generalized complex variable expressions for the associated Legendre functions of the second kind that are valid off the ( ) −< < +11x branch cut. In keeping with the requirements of OBLRADIAL (O-3), this supporting routine is confined to purely imaginary arguments that lie on the positive half of the imaginary axis, viz., where jξ>0. Initially, we discuss the how and when of the three different methods used by this routine to directly evaluate the associated Legendre functions of the second kind forpurely imaginary arguments with non-negative degrees (i.e., l≥0 ). As we will see, the 147methods for computing the associated Legendre functions of the second kind for purely imaginary arguments somewhat resemble those methods that address real argumentsgreater than unity (see 3.4.4). These similarities exist because the methods arefundamentally derived from the same generalized complex variable expressions.Although using these direct evaluation methods to calculate the entire set of complex-valued associated Legendre functions of the second kind is very feasible, doing so is stillcomputationally costly. For that reason, we proceed to explain how to improve thecomputational efficiency of this routine by fusing the direct evaluation methods with arecurrence relation approach. Finally, we address computation of the lone function with a negative degree, viz., Qj m −1()ξ. When directly evaluating the associated Legendre functions of the second kind with non-negative degrees for purely imaginary arguments, the routine selects one of the three following methods depending upon the modulus of the argument jξ. For arguments in the region where 0 0 01 <<ξ . , our numerical experience shows that the following explicit expression performs well: Qjlm()ξ= )2 (tan)(1πξξ −−j jPm l             ++       ++ ×−+− =∑ odd, 11Re2even, 11Im2 )()! ( 2! 22 1 sjsjj j Psmsj m ss sm lm ss ξξξξ ξ )() )(1 2()1 42( 12)1( 21 0ξj Pql qqlm qll Int q−− − =∑−+−−− for l≥1; (3-72) where Int x[ ] is considered the greatest integer function. 148Note that (3-72) reduces to the following when l=0 and m=0 (convention drops the superscript for the zeroth order): Qj0()ξ= )2 (tan1πξ−−j . (3-73) The explicit expression of (3-72) can be found directly (with some reduction) from its prolate counterpart (3-50) by way of the ξξ→j transformation. Although the Baber and Hasse expansion approach does not explicitly use the associated Legendre functions of the first kind, the evaluation of these functions for apurely imaginary argument is necessary when using (3-72) to directly compute theassociated Legendre function of the second kind. Fortunately, we are able to expand thepreviously outlined approach that computes the associated Legendre functions of the firstkind for real arguments off the branch cut to handle purely imaginary arguments (whichalso happen to be off the branch cut). These modifications consist of applying the ξξ→j transformation to equations (3-47)-(3-49) to yield the following complex equations: Pj jl lmPjlm lmPjlm lm lm()() ()()() ()() ξξ ξ ξ=− −−+− −−−21 1 12 (3-74) Pj j mmmm m() ( ) ! ! ( )ξξ=−+ 21 122(3-75) Pj j m P jmm mm +=+1 21 () ( ) ()ξξ ξ . (3-76) Using these modified equations in a manner identical to that which is performed for real arguments off the branch cut, we begin working upward with the complex recurrencerelation of (3-74) from the starting values provided by (3-75) and (3-76). This processallows us to evaluate the required associated Legendre functions of the first kind for a purely imaginary argument j ξ. It is important to recognize that the evaluation of this 149particular function for a purely imaginary argument yields either a real or purely imaginary result whose phase behaves as jl. We now consider direct computation of the associated Legendre functions of the second kind for purely imaginary arguments jξ where ξ≥00 1. . Our numerical experience shows that the hypergeometric approaches recommended by MTP-1945 [41] perform extremely well in the ξ≥00 1. region when modified to handle purely imaginary arguments. For the case where 0 01 3 .≤≤ξ , we make use of the following complex hypergeometric expression (obtained from (3-54) using the ξξ→j transformation): Qjlm()ξ= () Bml j F m m l t(, , ) , ; ;ξ1 21 23 2 +−+ − (3-77) where ()Bml jj lll m llm l l(, , ) () ( )!( )! () !() ξ ξξξ= ++ + ++  −+ + +21 1 221 2 21 24 21 12 t=−+ +ξξ ξ2 21 21. We are able to employ (3-77) for purely imaginary arguments jξ in this particular region because its respective hypergeometric function converges for all positive values of ξ. Regarding the region where ξ>3 , we make use of the less complicated, but slower converging, complex hypergeometric expression to directly compute the associated Legendre function of the second kind (obtained from (3-55) using the ξξ→j transformation): 150Qjlm()ξ= () Aml j F l m l m l(, , ) ( ) ,( ) ; ;ξ ξ1 21 23 2221−+ −+ +−−(3-78) where Aml jj lll m llm l ml m(, , )() ( )!( )! () !() ξξξ=+++  −+ + − −+21 1 1 222 12 1 2. Because expression (3-78) converges for all positive values of ξ greater than unity, it is capable of directly evaluating the associated Legendre function of the second kind for purely imaginary arguments jξ in the region where ξ>3. Having discussed the three methods used to directly evaluate the associated Legendre functions of the second kind for purely imaginary arguments, we are now ableto improve the computational efficiency of this routine by incorporating the followingrecurrence relation approach. We begin computing these complex-valued functions bytaking advantage of the following downward (i.e., backward) traveling complexrecurrence relation that is valid for purely imaginary arguments (obtained from (3-52) using the ξξ→j transformation): Qj jl lmQjlm lmQjlm lm lm()() ()()() ()() ξξ ξ ξ =+ ++−−+ ++++23 12 112 . (3-79) Yet before we can begin working downward with this complex recurrence relation, we must first have two consecutive starting values. In a manner that is similar to theprocedure used when computing the spherical Bessel functions, the moduli of theseconsecutive starting values are chosen as some arbitrary constant and zero. However, thistime around, the nonzero starting value is either real or purely imaginary. We know thisis the case because the phase of the associated Legendre function of the second kind for purely imaginary arguments behaves as j lm−+ +()21. As a result, we are able to assign the correct phase to the nonzero starting value such that the resulting correction constant is a scalar quantity, not a complex quantity. (Recall from before that a correction constant is 151needed because the nonzero starting value is arbitrarily chosen.) Once again, the index l of the nonzero starting value, i.e., the degree of the associated Legendre function, should be chosen somewhat greater than the highest index required to achieve convergence ofthe expansions in (3-60) and (3-63). Like the process for computing the spherical Besselfunctions, this procedural requirement must be imposed because the initial values thatresult from the downward recurrence relation are not considered usable and are necessaryto prime the downward computational pipeline. Regarding the complex recurrence relation of (3-79), numerical experiments indicate that it may not always yield accurate results for the entire pass in the downwarddirection. This difficulty occurs because a given family of associated Legendre functionsof the second kind with a purely imaginary argument may not provide a sequence ofmoduli that is always continually increasing in the downward direction. As previouslymentioned, one would prefer to work a recurrence relation in the direction of increasingmagnitude (of the related terms) in order to minimize the loss of precision. In order tosidestep this difficulty, we make use of a procedure that somewhat resembles theapproach used when computing the Baber and Hasse normalized expansion coefficients(see 3.5.2). We begin by using the downward traveling complex recurrence relation toevaluate the family of functions only as long as the moduli of the functions continue toincrease. Once again, it is conceivable that the entire family of associated Legendrefunctions of the second kind is computed using a single downward pass. However, if themoduli of these functions should begin to decrease during the downward travelingcomputations, then those remaining functions of the family (viz., those with the lowerdegrees) need to be computed differently. Our numerical experience shows that the bestway to evaluate this lower portion of the family from l=0 to this cross-over index is by individually calculating each function using direct evaluation; in essence, the remaininglower indexed portion of the family is forward filled by appropriately using one of thethree direct evaluation approaches that depends upon the modulus of the argument. Now that the family of associated Legendre functions of the second kind has been computed using either the recurrence relation or the direct evaluation approach, we turn 152our attention to applying the proper correction constant to the upper indexed portion of the family that is computed using the downward recurrence relation of (3-79). Since thelower indexed portion of the family (computed by using direct evaluation) is correct inabsolute terms (as opposed to relative terms), the proper correction constant is easilyfound by reconciling the difference between the two differently computed portions at theresulting cross-over index (this procedural step closely parallels that which is used in theBABHASCNSTS (O-e) routine). In the special case that all of the functions arecomputed using the downward recurrence relation, the second element in the sequence(which corresponds to l=1) is reconciled with its direct evaluation counterpart in order to determine the proper correction constant. In order to obtain the proper set of results,we then apply the computed correction constant to the portion of the family that iscomputed using the downward recurrence relation (3-79). Numerical testing reveals that the use of this particular recurrence relation approach can provide a significant savings in computation time for these much neededevaluations of the associated Legendre functions of the second kind. In the cases where amajority of the functions are able to be computed using the recurrence relation approach,a better than ten-fold savings in computation time has been observed when compared toonly using the direct evaluation approach. However, for obvious reasons there is little orno improvement in computation time for the cases where most of the associated Legendrefunctions must be found using direct evaluation. Since the computation time of thefunctions is reduced on average with the recurrence relation approach, we have prudentlychosen to incorporate this time-saving technique into our routine. Unfortunately, furthertesting of the recurrence relation approach indicates that its effectiveness is limited to arguments where ξ≤5 due to computational difficulties that occur when operating beyond this region (viz., numerical overflow conditions); consequently, we exclusively employ the direct evaluation approach to compute the family of functions when ξ>5. Table 3.5 summarizes our overall numerical approach to evaluating the required family of 153Table 3.5 Summary of Overall Approach to Compute the Set of Associated Legendre Functions of the Second Kind withNon-Negative Degrees for a Given Order m and Purely Imaginary Argument j ξ Region of Argument jξApproach Used to Compute the Set of Qjlm()ξfunctions for l≥0Appropriate Direct Evaluation Approach 00 0 1<<ξ .Recurrence Relation Approach (3-79) Using the Appropriate Direct EvaluationApproach to Forward FillExplicit Expression (3-72) and (3-73) 00 1 3.≤≤ξRecurrence Relation Approach (3-79) Using the Appropriate Direct EvaluationApproach to Forward FillHypergeometric Expression (3-77) 35<≤ξRecurrence Relation Approach (3-79) Using the Appropriate Direct EvaluationApproach to Forward FillHypergeometric Expression (3-78) ξ>5Direct Evaluation Approach to Compute All Members of the Required SetHypergeometric Expression (3-78) associated Legendre functions of the second kind with non-negative degrees for a purely imaginary argument jξ. Finally, we must address the computation of the lone associated Legendre function of the second kind possessing a negative degree for a purely imaginary argument, i.e., Qjm −1()ξ. Evaluation of this particular function is obtained by using the complex relation of (3-79) to work downward from the just computed Qjm 0()ξ and Qjm 1()ξ (viz., the first two members of the set of function evaluations computed via the procedure outlined in Table 3.5). In the special case where m=0 , the downward recurrence relation of (3-79) becomes undefined for l=−1 and cannot compute Qj−10()ξ. Fortunately, the expansion of (3-63) does not need this particular function because its respective expansion coefficient happens to conveniently vanish in this special case. 154Validation of this routine includes the comparison of its results with tabulated data available in MTP-1945 [41] (i.e., six-decimal accuracy tables of associated Legendre functions of the second kind with purely imaginary arguments). For all of our test cases,the routine performs extremely well and has results that match exactly with the publisheddata. The routine is further validated by comparing the results obtained using therecurrence relation approach with those computed using only the direct evaluationapproach (i.e., computed outside the normal operation of the routine). In general, theresults provided by the two approaches match within twelve decimal places. Althoughthe results provided by both methods are not as consistent when operating close to theorigin, numerical tests indicate that in this region they still match one another withinseven decimal places. In the midst of establishing the proper combination of the two methods to use for the different regions of the argument, we are presented with an engineering tradeoff ofaccuracy (provided by direct evaluation) versus efficiency (provided by the recurrencerelation). In short, accuracy of the routine could be improved in some regions at theexpense of computation time by exclusively using the direct evaluation approach.Ultimately, our overall numerical approach strikes a practical balance between accuracyand computational efficiency for our particular application. 3.6 PROLATE AND OBLATE SPHEROIDAL ANGLE FUNCTIONS 3.6.1 PROANG (P-4) and OBLANG (O-4) Numerical Routines Each of these primary routines evaluates its respective spheroidal angle function in terms of the Morse and Feshbach normalization scheme for a given order m, degree l, independent parameter h, and real argument η. In keeping with the definition of η put forth in Chapter 2, both numerical routines evaluate the spheroidal angle functions for arguments where −≤ ≤ +11η . Furthermore, these primary routines must be supplied with either an initial approximation of the eigenvalue or a previously refined eigenvalue 155for the particular function under consideration. Once again, this requirement stems from the fact that each of these primary routines must evoke the secondary routine which computes its respective expansion coefficients, viz., dhrml,( ) or dj hrml,()− . With respect to the discussion that follows, we have chosen to address both routines concurrently throughout this section on account of how closely they parallel one another. From a computational standpoint, it is important to understand that both spheroidal NZ-FZT algorithms need to recalculate their respective set of spheroidal anglefunctions for each far-zone observation angle under consideration (see Tables 2.5a and2.5b). In addition, each of these primary routines must also provide support to itsrespective integration routine, i.e., INTPROANG (P-5) or INTOBLANG (O-5). By andlarge, the routines PROANG (P-4) and OBLANG (O-4) get an intensive workout in thecourse of computing the NZ-FZT process for their respective spheroidal transformationsurface. Consequently, repeated execution of these routines accounts for a large share ofthe total computation time for the overall NZ-FZT process. It is for this reason that bothroutines must be able to evaluate their respective spheroidal angle functions in the bestpossible way with respect to computational efficiency. In order to compute the spheroidal angle functions, we make use of the following expansions of associated Legendre functions of the first kind initially presented in (2-68a)and (2-68b): Prolate: Sh d h P ml rml mrm r,, ,(, ) () ()ηη=′ + =∞ ∑ 01 where 0≤≤ml ; (3-80a) Oblate: Sj h d j h Pml rml mrm r,, ,(, ) () ( )−=′−+ =∞ ∑ηη 01 where 0≤≤ml . (3-80b) Once more, the expansion coefficients dhrml,( ) and dj hrml,()− are provided by the secondary routines PRODNCNSTS (P-d) and OBLDNCNSTS (O-d). As a result, we are now left with having to compute the associated Legendre functions of the first kind that 156are part of these expansions. However, this time around these functions must be evaluated for arguments that are either on the ( ) −< < +11x branch cut or at the x=±1 branch points. Fortunately, we are able to use the computational approach put forth in [37] without having to make any modifications. As before, we begin by using the upward(i.e., forward) traveling recurrence relation that relates associated Legendre functions ofthe first kind for a given order m [37]: Pl lmPlm lmPlm lm lm()() ()()() ()() ηη η η=− −−+− −−−21 1 12 . (3-81) Note that this recurrence relation is identical in form to the one presented in (3-47) by virtue of being valid on and off the branch cut. We now employ the following closedform expressions that are valid for real arguments on the branch cut and at the branchpoints to determine the two necessary starting values [37]: Pmmmm() ( ) ! ! ( )ηη=− −21 122(3-82) Pm Pmm mm +=+1 21 () ( ) ()ηη η . (3-83) Accordingly, the associated Legendre functions required to evaluate (3-80a) and (3-80b) are computed by working upward with the recurrence relation of (3-81) from these twostarting values. With regard to computational efficiency, this numerical approach lendsitself quite well to the evaluation of both spheroidal angle functions. This is so becausethe associated Legendre function of the first kind for each expansion term can becomputed on an as-needed basis until convergence of the respective expansion, i.e., (3-80a) or (3-80b), is achieved. Namely, family members for a given order m are progressively computed on-the-fly using a single upward pass of the recurrence relation(3-81) until the expansion has converged; at this point, evaluation of the family ofassociated Legendre functions is then halted in order to eliminate any unnecessary 157computations. Much to our satisfaction, both routines PROANG (P-4) and OBLANG (O- 4) continually prove to be remarkably efficient. Validation of the PROANG (P-4) routine consists of comparing its results with those computed by an alternative numerical method. We start by considering a numericaltechnique put forth in [37] capable of calculating the spheroidal eigenvalues. Theapproach consists of numerically solving the prolate angular self-adjoint boundary valueproblem of Chapter 2 (see Table 2.2) through the use of an ordinary differential equation(ODE) integrator. In our particular case, we happen to use the fifth-order Cash-KarpRunge-Kutta method with adaptive stepsize (also presented in [37]) as the ODEintegrator. At length, we are then able to expand their numerical approach to a pointwhere it can evaluate prolate spheroidal angle functions. Although this alternativemethod proved to be too slow for our particular algorithm, it still provides us with anexcellent computational check. For the cases tested, results from PROANG (P-4) arequite consistent with those computed using this alternative numerical method: they matchone another anywhere between seven and fourteen decimal places. In general, weconsider the expansion approach employed in (P-4) to be more accurate than itsalternative counterpart due to the limitations generally associated with ODE integrators. In the case of OBLANG (O-4), validation of the routine consists of comparing its calculated results with tabulated data (i.e., four-decimal accuracy) provided by Flammer[23]. For the cases tested, the routine performs extremely well when compared to thispublished data (i.e., after accounting for the difference in normalization schemes). 3.7 INTEGRATION OF PROLATE AND OBLATE SPHEROIDAL ANGLE FUNCTIONS 3.7.1 INTPROANG (P-5) and INTOBLANG (O-5) Numerical Routines Both of these primary routines integrate their respective spheroidal angle function over a specified interval for a given order m, degree l, independent parameter h, and lower/upper boundaries ϑlw and ϑup. This computational objective is met by conducting 158a numerical integration of the specified function. In order to perform this operation, these routines must be able to sample the given spheroidal angle function at many differentpoints over the specified interval. These samples are acquired through many repeatedcalls to their respective spheroidal angle function routine, viz., PROANG (P-4) orOBLANG (O-4). Upon review of Tables 2.5a and 2.5b, we recognize that the amount of use each integration routine receives is based upon the NZ-FZT process input. For a singlespheroidal transformation surface, the number of times the integration routine is calleddepends on the number of surface partitions in the angular direction and the total numberof spheroidal wave-harmonics necessary for the expansion to achieve convergence.Fortunately, this set of integral evaluations only has to be computed once for a givenspheroidal transformation surface despite the number of far-zone observation anglesunder consideration. In developing these routines, our aim is to put together an overall approach capable of conducting these numerical integrations in the best possible way with respectto accuracy and computational efficiency. Consequently, both of these objectives promptus to investigate already existing integration techniques to arrive at one which is well-suited for integrating the spheroidal angle functions. After conducting many numericalexperiments, we are ultimately led to using Romberg integration (also referred to asRomberg quadrature). According to [37], this integration technique is very powerful forsufficiently smooth integrands which contain no singularities anywhere on the intervalunder consideration (including the endpoints). In our particular case, the integrand iseither the prolate or oblate spheroidal angle function with the interval of interest beingany segment of the −≤ ≤ +11 η (0≤≤ϑπ ) domain. Since the behavior of the spheroidal angle functions satisfies both of these requirements, the Romberg integration technique is a natural for our specific application. In particular, the routines INTPROANG (P-5) and INTOBLANG (O-5) make use of the Romberg integration routine provided in [37] (i.e., with some slight modifications).This integration routine employs Romberg’s method of order 2 K and is viewed as the 159natural generalization of Simpson ’s rule (i.e., where K=2 ). For a more comprehensive review of this standard integration technique, the interested reader is referred to [37]and/or any text on numerical analysis, e.g., [42]. Our investigation also reveals that Romberg integration performs extremely well (in terms of accuracy and convergence rate) for both spheroidal angle functions only afterperforming the ηϑ=cos change of variable. Numerical testing demonstrates that convergence of these integrations can be considerably slower without carrying out this particular change of variable. As a result, both INTPROANG (P-5) and INTOBLANG(O-5) routines integrate their corresponding spheroidal angle functions with respect to theangular parameter ϑ. In mathematical terms, the numerical integrations implemented by these routines can be expressed as follows: Prolate: Ih m llw up (, ,, , )ϑϑ=Sh dml lwup ,(, c o s ) s i n′′ ′∫ϑϑ ϑ ϑϑ ; (3-84a) Oblate: Ij h m llw up (, , ,,)−=ϑϑ Sj h dml lwup ,(, c o s ) s i n−′′ ′ ∫ϑϑ ϑ ϑϑ ; (3-84b) where 0≤≤ϑπlw and 0≤≤ϑπup . One may recall from Chapter 2 that the sin′ϑ integration factor found in both integrals is introduced by the aforementioned change of variable, while the angular parameter has been primed to identify it as a variable of integration. Moreover, it is important torecognize that both of these integral representations are conveniently in the formnecessary to implement our prolate and oblate spheroidal NZ-FZT algorithms (see Tables2.5a and 2.5b). Additional testing indicates that the Romberg integration technique does experience some difficulties when integrating the spheroidal angle functions over certainintervals in the defined domain. As we already know, the spheroidal angle functions 160exhibit even or odd symmetry over the −≤ ≤ +11η domain based on whether ( ) lm− is even or odd. Furthermore, these symmetries remain intact after performing the ηϑ=cos change of variable in spite of the domain being mapped to 0≤≤ϑπ . Consequently, this change of variable causes the spheroidal angle functions to now become symmetric about the point (, )π2 0 instead of the origin ( , ) 0 0 . This point of symmetry plays an important role in the implementation of these routines because numerical testing shows that Romberg integration has difficulties with integrating the spheroidal angle functionsthrough this particular point, viz., where ϑπ=2. That is, the integration technique appears to breakdown when evaluating the integrals of (3-84a) and (3-84b) for cases where the boundaries ϑlw and ϑup lie in different halves of the 0≤≤ϑπ domain: namely [, ]02π and [, ]ππ2 (it is important to recognize that the ϑπ=2 symmetry point is shared by both halves). We overcome this computational difficulty for these special cases by dividing the troubled numerical integration into two separate integrals that share the ϑπ=2 symmetry point as a common boundary. Because Romberg integration has no problems integrating to this symmetry point (as opposed to integrating through it), both of theseseparate integrals can be numerically evaluated without a glitch. In essence, thisapproach allows the routines to piecewise integrate the spheroidal angle function over anygiven interval without ever having to pass through this menacing point. For obviousreasons, only a single integral must be evaluated for cases where both boundaries lie inthe same half of the 0≤≤ϑπ domain. When considering the special class of intervals where the boundaries ϑlw and ϑup are equidistant from the ϑπ=2 symmetry point and lie in different halves of the 0≤≤ϑπ domain, both routines exploit the symmetries of the spheroidal angle functions to reduce the computation time of the overall integration. If the spheroidal angle function happens to be even, only one of the two separate integrals (i.e., for the piecewiseintegration) needs to be computed because the overall integration is then found by 161doubling the results. When the spheroidal angle function is odd, no calculations are necessary since the overall integration clearly vanishes for this special class of intervals. As anticipated, no integral tables of the prolate and oblate spheroidal angle functions are able to be found. It is this lack of available data that prompts us to devise acomputational check that is able to provide some validation of the INTPROANG (P-5)and INTOBLANG (O-5) routines. We choose to termwise integrate the expansions of (3-80a) and (3-80b) to procure an alternative, albeit much slower, approach to computing theintegrals of (3-84a) and (3-84b). Essentially, Romberg integration is used to individuallyintegrate the associated Legendre function identified with each term until convergence ofthe expansion is achieved. For the cases tested, results from the INTPROANG (P-5) andINTOBLANG (O-5) routines compare extremely well with those obtained using thisalternative method; that is, they match one another within eleven decimal places (thislevel of accuracy is consistent with the fractional accuracy imposed on the Rombergintegration routine). 162CHAPTER 4. COMPUTER SOFTWARE BASED UPON THE NEWLY DEVELOPED NEAR-ZONE TOFAR-ZONE TRANSFORMATION PROCESS Using the algorithms developed in Chapter 2 along with the special function routines developed in Chapter 3, we are now able to put together a software packagecapable of realizing the entire NZ-FZT process for both spheroidal cases. It is this stageof the project that allows our theoretical concept to emerge from the drawing board andbe put into actual practice. Later on in our investigation, i.e., in the chapters that follow,we shall use the developed software to theoretically validate the overall NZ-FZT processas well as assess its computational performance. Because the software is constructed in amanner that is both generalized and computationally efficient, it has the capacity tohandle a wide variety of radiation problems. This chapter addresses the top-down design of the developed software by examining the organization and the particulars of its various constituent modules. Inorder to gain a broad understanding of the software without getting lost in the details, weelect to keep the discussion on a top-level by describing the entire process usingpseudocode, i.e., a series of English-like statements. An additional benefit of such anapproach is that it allows the computational process to be freely implemented using anystructured, high-level computer language. Nevertheless, a section addressing thelanguage and computer platform used during our investigation is included in order toprovide a complete description of the software package. Finally, a memory storagetechnique used by the developed software to significantly improve its computationalefficiency is discussed in the last section of this chapter. 1634.1 OVERVIEW OF NZ-FZT COMPUTER SOFTWARE 4.1.1 Organization of Software During the software development process, we found that using two separate computer codes to individually implement the NZ-FZT algorithms (for the prolate andoblate cases) is more practical than using one very large, all-encompassing code. Uponinspection of both algorithms (see Chapter 2), one can see that the top-level processes forboth spheroidal cases are identical to each other with regard to structure, viz., memorystorage formats, expansion formats, etc. As a result, both of the developed codes are verymuch the same with only one major difference: each code evokes a separate library ofnumerical routines when evaluating its respective special functions (one may recall fromChapter 3 that these numerical routines can vary significantly between the prolate andoblate cases). It is this common structure of both codes that prompts us to continue ourdiscussion of the software package in a generalized manner; in particular, we describeboth codes in terms of a generalized top-level procedure that applies to either spheroidalcase. However, specific references to the pertinent algorithmic quantities, numericalroutines, and special functions for each spheroidal case are included whenever applicable(using the nomenclature defined in Chapters 2 and 3). At this point in our discussion, we feel that it is necessary to digress for a moment and reiterate that the NZ-FZT process from start-to-finish is designed to compute far-zonefield quantities for a single spheroidal transformation surface. Consequently, the entirecomputational process must be performed each time a different transformation surface isconsidered. The developed software mirrors this process and handles only one spheroidaltransformation surface per execution; thus, additional transformation surfaces requireadditional computer runs. We now turn our attention to Figure 4.1 which uses a flowchart to depict the top- level modular design of the NZ-FZT computer codes for the prolate and oblate spheroidaltransformation surfaces. Included in the flowchart for each module is a functionaldescription along with a numerical tag. One can plainly see that the numbering of the 164 Start Input of User Specified Parameters (M-1)Allocation & Initialization of Data Structures (M-2) Read File of Near-Zone Field Samples (M-3) Eigenvalue Computation (First Approximation) (M-4)Eigenvalue Refinement (M-5)Strm-Lvlle Norm Cnsts &Computation of Spheroidal Sample GridSetup (M-6) Quantities (M-7)Computation of Expansion Are all Far-Zone Fields Computed? Observation Angle (M-9)Advance to Next Far- Zone Write Far-Zone Fields to Output File (M-11)Yes No Spheroidal Wave- (M-10)Far-Zone E-Field Computation via the Harmonic ExpansionStopPhase III Phase IIPhase I Matrices (M-8)Computation of Integration Figure 4.1 Flowchart of the NZ-FZT computer codes for the prolate and oblate spheroidal transformation surfaces. 165modules corresponds to the sequential order in which they are initially encountered during the computational process. In addition, the functional descriptions of the modulesare stated in pseudocode and are applicable to either spheroidal case. The top-level modular design of both NZ-FZT computer codes is partitioned into three sequential phases so as to give us a better understanding of the developed software.As shown in Figure 4.1, Phase I is made up of the first eight modules, i.e., (M-1)-(M-8),and is executed only once for a given spheroidal transformation surface. The purpose ofthis phase is to take the information provided by the user and conduct the many one-timecomputations associated with the NZ-FZT process. Since the number of far-zoneobservation angles has no bearing on these one-time computations, we view thecalculations of Phase I as computational overhead. In essence, the computational expenseassociated with these calculations is essentially fixed because it remains the same despitethe number of desired far-zone field quantities. However, the computational expensecontributed by Phase II does depend on the number of far-zone observation angles. Thesecond phase consists of two modules, viz., (M-9) and (M-10), and forms a loop thatmust be repeated each time that a different far-zone field quantity is desired. It is thisphase that accounts for most of the computation time when addressing a large number offar-zone field quantities. Lastly, Phase III is simply made up of the single m odule (M-11) and handles the output only after all of the desired far-zone field quantities have beencomputed. The computational expense contributed by this phase is relatively minor whencompared to that of the previous two phases. 4.1.2 Numerical Precision The developed software is designed to accept input data and to furnish output results with single precision. Despite this requirement, computations of the NZ-FZTprocess are nevertheless conducted with double precision in order to minimize theroundoff error associated with computing the spheroidal wave-harmonic expansion.Consequently, all of the software modules perform their calculations using double 166precision. To meet this demand, the special function routines (developed in Chapter 3) must in turn provide double precision results to the modules. Ultimately, the NZ-FZTsoftware rounds the final results to single precision at the conclusion of the entire process. 4.1.3 Computer Platform and Programming Language One of our objectives is to take the developed NZ-FZT algorithms and map them to an IBM compatible personal computer. There are several distinct advantages in doingso. Just recently, personal computers have become powerful enough to hostcomputationally intensive numerical algorithms. At present, both speed and memorystorage of these machines have been steadily increasing while the base cost of thesemachines has continued to remain more or less the same. Accordingly, CPU time forpersonal computers is relatively inexpensive when compared to that of other alternativeplatforms, e.g., mainframes, workstations, etc. From a standpoint of availability, thesemachines are now much more accessible since becoming so pervasive throughout thescientific and industrial community. All things considered, the personal computer iseasily the platform of choice for our specific application. As for the programming language, we choose to implement the NZ-FZT computer software using Fortran albeit any structured, high-level language could be used (e.g.,Pascal, C, etc.). The foremost reason for selecting Fortran stems from the fact that overtime it has become the most commonly used programming language in the field ofcomputational electromagnetics. Bearing this in mind, we opted to have our softwareconform to this industry standard. Maintaining portability of our software also ranks high on our list of priorities. Portability between compilers can be improved by limiting the use of compiler-specificstatements throughout the software. (Of course, achieving 100 percent portability is nearimpossible for a software project of this magnitude.) In addition, our concern extendsbeyond just portability of the code between compilers and addresses its portability toother structured, high-level languages. By limiting statements, control structures, and 167data types that are only intrinsic to Fortran, we are able to develop a software package capable of being translated with minimal work to other structured, high-level languages.This approach gives our software the flexibility that it may need for future applications. 4.2 DESCRIPTION OF PHASE I MODULES 4.2.1 Module (M-1): Input of User Specified Parameters By way of the computer terminal and an ASCII input file, this input module queries the user for information that is key to both ends of the NZ-FZT process (seeFigure 2.3). Specifications requested through the terminal allow the software toconfigure the process output to that desired by the user. Meanwhile, data contained in theASCII input file supplies the code with the necessary process input. The advantage ofsuch an approach is that it allows the user to request a variety of output arrangements fora single input file. In addition, the information gathered by this module is also needed tointernally configure the data structures necessary to implement the NZ-FZT process. At the start of the module, the user must indicate the location of the ASCII input file by specifying its filename via the computer keyboard. Included in this input file is aheader that contains parameters describing the geometry of the transformation surface aswell as the configuration of the sampling scheme. The module then proceeds to open thefile and read the following input parameters contained in the header: the free-spacewavelength, λo; the focal point of the spheroidal transformation surface, a (related to independent parameter h by equation (2-32)); the radial coordinate of the spheroidal transformation surface, ξo; the total number of samples in the angular and azimuthal directions, I and J; and the widths of the partitioned surface elements, ∆′ϑ and ∆′ϕ. The remainder of the input file contains samples of the near-zone E-field that are to be read in by a subsequent module, viz., (M-3). After reading the required input parameters, the code proceeds to query the user by way of the terminal for details regarding the computational process and its 168corresponding output. One of the parameters that the user must provide is the degree at which he or she wants the spheroidal wave-harmonic expansion to be truncated duringthe computational process, viz., the parameter L of the NZ-FZT algorithms (see Tables 2.5a and 2.5b). In addition, the user must specify the desired pattern cuts of the far-zoneE-field together with the preferred type of pattern normalization. 4.2.2 Module (M-2): Allocation and Initialization of Data Structures The purpose of this housekeeping module is to first configure and then zero- initialize all data structures required to implement the NZ-FZT process. Referring toTables 2.5a and 2.5b, one can see that the memory storage format of the data structures isdependent on the input parameters I, J, and L. However, it is the amount of available memory that dictates the maximum allowable size of these data structures (that is,assuming no software limitations are imposed by the compiler or the operating system).Thus, it is the memory capacity of the host platform that is the limiting factor whenconsidering extremely fine sampling of the spheroidal transformation surface or a largenumber of orders and degrees for the spheroidal wave-harmonic expansion. At the start of each run, this module employs dynamic memory allocation to configure all of the data structures to the exact size needed. The main advantage of usingdynamic allocation over fixed allocation is that it enables the software to automaticallymanage the available memory. As a result, the code need not be recompiled every timethe problem under consideration is larger than the present configuration or if the softwareis ported to a machine with a different memory capacity. Furthermore, by customizingthe data structures to each problem under consideration, the software is able to make themost of the available memory resources. 4.2.3 Module (M-3): Read File of Near-Zone Field Samples Having configured and initialized all of the necessary data structures, the code is now ready to read the remainder of the ASCII input file. This input module proceeds to 169read in the near-zone E-field samples along the spheroidal transformation surface, i.e., Exij(, )′′ϑϕ , Eyi j(, )′′ϑϕ , and Ezi j(, )′′ϑϕ , and then to place them in their respective data structures. We recall from Chapter 2 that each of these samples represents a Cartesian vector component in complex phasor form. The input samples are converted from singleto double precision by zero-filling the additional significant figures. 4.2.4 Module (M-4): Eigenvalue Computation (First Approximation) As indicated in Chapter 3, the first step in evaluating the prolate and oblate spheroidal wave functions is to compute their respective eigenvalues λmlh,() a n d λml jh,()− . Recalling from before, calculating these eigenvalues is a two-step process that initially requires a first approximation of the eigenvalue to be later refined using the Bouwkamp/Blanch method. This computational module only handles the first part of this process and produces the first approximation of all necessary eigenvalues, i.e., λmlh,()()1 or λml jh,()()1− where ) ,,1,0( L l/G15 = and ) ,,1,0 ( l m/G15 = , by appropriately calling either the PROAPPROXEIG (P-1) or OBLAPPROXEIG (O-1) primary routine. The approximated eigenvalues are then stored in a ( ) ( ) LL+× +1 1 two-dimensional matrix that is subsequently refined during the execution of the next module. 4.2.5 Module (M-5): Computation of Sturm-Liouville Normalization Constants and Eigenvalue Refinement The purpose of this computation module is essentially twofold. First, the module proceeds to compute all of the necessary Sturm-Liouville normalization constants for agiven spheroidal transformation surface, i.e., Nh ml,( ) or Nj hml,()− where ) ,,1,0( L l/G15 = and ) ,,1,0 ( l m/G15 = , by appropriately evoking either the PRONORMCNSTS (P-2) or OBLNORMCNSTS (O-2) primary routine. These computed normalization constants are then stored in a ( ) ( ) LL+× +1 1 two-dimensional matrix that uses zero-fill for elements not having a corresponding normalization constant. Second, by way of performing these 170calculations, the module proceeds to also refine the eigenvalue matrix (i.e., initially computed by the previous module), thus obtaining the necessary values of λmlh,() o r λml jh,()− (see Chapter 3). These refined eigenvalues are then used for the remainder of the NZ-FZT process. 4.2.6 Module (M-6): Setup Spheroidal Sample Grid This computation module uses information provided in the header of the ASCII input file to calculate the samples points ( , ) ′′ϑϕij along the spheroidal transformation surface in radian measure. The total number of samples in the angular and azimuthal directions (i.e., I and J, respectively) and the widths of the partitioned surface elements (i.e., ∆′ϑ and ∆′ϕ) are used with (2-107a) and (2-109a) to determine the spheroidal sample grid. These sample points coincide with the near-zone E-field samples provided by the ASCII input file. 4.2.7 Module (M-7): Computation of Expansion Quantities The next step in the NZ-FZT process is to compute the following quantities associated with the spheroidal wave-harmonic expansion: C1, []ClP2, and []CmlP3, for the prolate case; and C1, []ClO2, and []CmlO3, for the oblate case. Tables 2.5a and 2.5b summarize the specifics regarding the computation and memory storage format of these quantities. In order to carry out these calculations, this module utilizes the respectiveSturm-Liouville normalization constants together with the corresponding spheroidalradial functions of the fourth kind. At this point in the process, the normalizationconstants have already been computed and stored by the (M-5) software module. As forthe required spheroidal radial functions, either PRORADIAL (P-3) or OBLRADIAL (O-3) supplies the necessary function evaluations. 1714.2.8 Module (M-8): Computation of Integration Matrices The final step of Phase I is to compute the following integration matrices of the spheroidal wave-harmonic expansion: []ISliP 0, , and []ISmliP ,, for the prolate case; and []ISliO 0, , and []ISmliO ,, for the oblate case. Details regarding the definition and memory storage format of these quantities are summarized in Tables 2.5a and 2.5b. We recall from Chapter 2 that each element of these matrices is related to the integration of aparticular spheroidal angle function (indexed by m and l) over a given interval. In turn, these intervals correspond to a partition of the spheroidal transformation surface in theangular dimension (indexed by i). Accordingly, the integration matrices are computed (and filled) by individually evoking the appropriate integration routine, i.e.,INTPROANG (P-5) or INTOBLANG (O-5), for all values of m, l, and i. As discussed in Chapter 3 (see 3.7.1), both spheroidal angle functions exhibit even or odd symmetry over the 0≤≤ϑπ domain (i.e., about the ϑπ=2 symmetry point) based on whether ( ) lm− is even or odd. In addition, the spheroidal transformation surface is uniformly partitioned in the angular dimension so that the resulting intervals in the upper-half of the spheroid (i.e., [, ]02π ) mirror those in the lower-half (i.e., [, ]ππ2 ). Under these conditions, each of the integration matrices exhibits some form of combined symmetry. In the cases where the spheroidal angle function is even (i.e., when ()lm− even), corresponding upper- and lower-half matrix elements are equivalent, viz., IS ISmliP mlI iP ,, ,,=+−1 where ]2)1([,,2,1 + = IInt i/G15 . Likewise, when the spheroidal angle function is odd (i.e., when ()lm− odd), corresponding upper- and lower-half matrix elements are the same magnitude with a sign reversal, viz., IS ISmliP ml I iP ,, ,,=−+−1 where ]2)1([,,2,1 + = IInt i/G15 . By taking advantage of these symmetries, this module is able to cut the overall computation time roughly in half. Note that during the instances when the total number of partitions I is odd, the reduction 172in the number of computations is a little less than a half because the middle partition, i.e., iI=+()12 , does not have a symmetrical counterpart. 4.3 DESCRIPTION OF PHASE II MODULES 4.3.1 Module (M-9): Advance to Next Far-Zone Observation Angle The purpose of Phase II is to calculate the desired far-zone E-field response using the near-zone sample data together with the quantities computed during the first phase.As the first part of the Phase II loop, this module determines the sequence of observationangles ( , ) θϕ required to obtain the desired far-zone E-field pattern cuts. For each iteration of the loop, the next observation angle (, )θϕ in the sequence is fed to the second part of the loop that actually computes the far-zone E-field, viz., module (M-10). Module (M-9) also supplies the trigonometric constants associated with the coordinatetransformation of (2-127) and (2-128) for each observation angle ( , ) θϕ . 4.3.2 Module (M-10): Far-Zone E-Field Computation via the Spheroidal Wave- Harmonic Expansion This second and final part of the Phase II loop computes the far-zone E-field for a particular observation angle (, )θϕ by employing the appropriate spheroidal wave- harmonic expansion (i.e., either prolate or oblate). We recall from Chapter 2 that the NZ- FZT process requires the overall expansion, i.e., either (2-124a) or (2-124b), to beindividually evaluated for each Cartesian component of the E-field. Correspondingly,this computational module must evaluate three distinct expansions by individuallyemploying each component of the near-zone field samples, viz., E xij(, )′′ϑϕ , Eyi j(, )′′ϑϕ , and Ezi j(, )′′ϑϕ , as either fijP , or fijO ,; see (2-126a) and (2-126b). The results of these expansions yield all three Cartesian components of the far-zone E-field, viz., Efzx(, )θϕ , Efzy(, )θϕ , and Efzz(, )θϕ ; see (2-125a) and (2-125b). 173From a computational viewpoint, the manner in which these three expansions are evaluated can have a serious impact on the total computation time. As always, the issueat hand once again becomes an engineering tradeoff of computational efficiency versusmemory storage. For a fixed amount of memory, the computational speed of the NZ-FZTprocess can be increased only at the expense of decreasing the maximum allowableproblem size. In short, the computational efficiency is improved by storing intermediateresults in order to minimize redundant calculations. However, storage of theseintermediate results can be quite memory intensive depending upon the size of theproblem under consideration. Because our goal is to put together a software package ableto address as many realistic radiation problems as possible, we opt to implement all threeexpansions in a fashion that generally favors maximum allowable problem size overcomputational speed. On the whole, the three expansions (corresponding to each Cartesian component) are implemented as they are written in (2-124a) and (2-124b). Yet, to reduce redundantcalculations, we do take advantage of the fact that all three expansions share the sameexpansion coefficients, viz., the bracketed terms of (2-124a) and (2-124b). To minimizememory storage, this module computes the corresponding bracketed term on-the-fly andseparately applies it to each of the samples E xij(, )′′ϑϕ , Eyi j(, )′′ϑϕ , and Ezi j(, )′′ϑϕ ; the results each contribute to their respective running summation (over the indices i and j). Since the one-dimensional summation included in the bracketed term (over the degrees L l ,,1,0/G15 = ) is only dependent on the index i, we are able to eliminate other redundant computations by using a simple storage scheme. By looping over all values of j for a given index i, the module buffers on-the-fly the part of the bracketed term that corresponds to the one-dimensional summation. Thus, all of the bracketed terms for agiven index i can be computed without continually having to recompute this one- dimensional summation. The overall process is repeated over all values of j until contributions from all near-zone E-field samples have been included. When computingthe bracketed term, the module evaluates the necessary prolate or oblate spheroidal anglefunctions by appropriately evoking either PROANG (P-4) or OBLANG (O-4). After 174computing the far-zone E-field in Cartesian coordinates, module (M-10) then applies the coordinate transformation of (2-127) and (2-128) to yield the desired end-product for agiven observation angle: the far-zone E-field in spherical coordinates with suppressed radial dependence, viz., Efzθθϕ( , ) and Efzϕθϕ(, ) . 4.4 DESCRIPTION OF PHASE III MODULE 4.4.1 Module (M-11): Write Far-Zone Fields to Output File After Phase II computes the desired far-zone E-field response, this single module phase applies the proper normalization (if any) to the computed results, i.e., Efzθθϕ(, ) and Efzϕθϕ( , ) , and writes them to an ASCII output file. The choice of pattern normalization is specified by the user during the (M-1) input module and includes the following: (1) no normalization; (2) normalization with respect to the θ-component, Efzθ; (3) normalization with respect to the ϕ-component, Efzϕ; and (4) normalization with respect to the total magnitude, EEfz fzθ ϕ2 2 + . Since the far-zone E-field components are complex quantities, the ASCII output file expresses them in terms of both polar and rectangular formats. 4.5 IMPROVEMENT OF COMPUTATIONAL EFFICIENCY 4.5.1 Implementation of Memory Caching Technique As we have just seen, the implementation of the overall NZ-FZT process is quite involved. The developed software package must go through a multitude of floating-pointoperations in order to arrive at the final results. Accordingly, we searched for additionalways to reduce the number of computations without seriously impacting the available 175memory. After examining the manner in which the modules interface with the developed special function routines of Chapter 3, we are able to devise a memory storage techniquethat significantly improves the computational efficiency of both spheroidal codes. The developed technique makes use of the fact that certain special functions utilize redundant sets of computed data at some points during the evaluation process. Bybuffering this common data, the software is able to eliminate a significant amount offloating-point operations. However, storage of this large amount of data will naturallyhave a negative impact on the overall memory capacity. In order to circumvent thisdifficulty, both codes have been designed to take advantage of the sequence in whichtheir special function routines are called. The approach requires only one set ofredundant data to be buffered at any point in the computational process. Using thisbuffered data, the software then proceeds to compute all of the connected specialfunctions before computing and storing the next set of redundant data. With thistechnique, we are able to reduce the number of floating-point operations without seriouslyimpacting the amount of available memory. However, the modules now have to be smartabout the order in which they call these particular special function routines. Inserted in these specific primary routines is a memory caching mechanism that stores the relevant data upon exit of the routine. Computation of this data is intrinsicallybased on some of the arguments passed to the numerical routine. Upon a subsequent call,each of these routines proceeds to compare the relevant arguments with those from thelast call: in the instances where these arguments differ, the routine recomputes thenecessary data; in the instances where these arguments match, the routine skips theselengthy computations and uses the data buffered from the previous call. As one can see,the technique is not very memory intensive because it only requires the code to buffer therelevant data one-step back. The specific primary routines able to take advantage of thistechnique consist of the following: PRORADIAL (P-3), OBLRADIAL (O-3), PROANG(P-4), and OBLANG (O-4). In the case of PRORADIAL (P-3), the data from both families of spherical Bessel and spherical Neumann functions is buffered to speed up the computations of equations 176(3-16)-(3-19). Since both of these families are evaluated for the single argument hoξ and the software only addresses a single transformation surface at a time, these calculations are only performed once for a given computational run. In addition, this routine alsobuffers the associated Legendre functions of the first and second kind together with theirfirst derivatives in order to reduce the number of computations associated with (3-20) and(3-21). It is important to note that all of the associated Legendre function data is based ona given order m and independent parameter h. Accordingly, module (M-7) is able to employ the memory caching technique to speed up the evaluation of expansion quantities that include the prolate spheroidal radial function: []ClP2 and []CmlP3, (see Table 2.5a). However, this is where the software module must be smart about the sequential order in which it evokes PRORADIAL (P-3). The associated Legendre function data has to berecalculated every time a call to PRORADIAL (P-3) uses a given order m that differs from the one used during the previous call. In order to take advantage of the memory caching technique, module (M-7) must compute []ClP2 and []CmlP3, by evaluating row- by-row the elements that correspond to all degrees of l for a given order m (this is done by using a nested loop with the degree l as the index for the internal loop). As for OBLRADIAL (O-3), it employs the memory caching technique in much of the same way as its prolate counterpart. It too buffers the data from both families ofspherical Bessel and spherical Neumann functions to speed up its respectivecomputations, i.e., equations (3-56)-(3-59). However, this routine only has to buffer theassociated Legendre functions of the second kind in order to reduce the number ofcomputations associated with the Baber and Hasse approach, i.e., equations (3-60)-(3-63).In a manner that is identical to the prolate case, module (M-7) is also able to employ thememory caching technique to speed up the evaluation of expansion quantities that include the oblate spheroidal radial function: []ClO2 and []CmlO3, (see Table 2.5b). The primary routines PROANG (P-4) and OBLANG (O-4) make use of the memory caching technique to buffer their respective expansion coefficients dhrml,() a n d 177dj hrml,()− for a given order m, degree l, and independent parameter h. By storing this data at the end of these routines, the software is able to significantly reduce the number of floating-point operations associated with the computation of the integration matrices: []ISliP 0, , and []ISmliP ,, for the prolate case; []ISliO 0, , and []ISmliO ,, for the oblate case. Module (M-8) computes these quantities by evoking the appropriate integration routine, i.e., INTPROANG (P-5) or INTOBLANG (O-5), for all values of m, l, and i. Each time a different pair of m and l is considered, these routines repeatedly evaluate the same spheroidal angle function in order to carry out the numerical integration. Although theangular parameter differs for each of these successive function evaluations, the expansion coefficients, i.e., dh rml,( ) and dj hrml,()− , of (3-80a) and (3-80b) remain the same. As a result, the technique considerably speeds up the integration processes of INTPROANG (P-5) and INTOBLANG (O-5). However, in order to take full advantage of the memorycaching technique (in this regard), module (M-8) should compute group-by-group theelements of the integration matrices that correspond to all values of i for a given order m and degree l. 178CHAPTER 5. PROOF-OF-CONCEPT VIA AN ANALYTICAL BENCHMARK In the preceding chapters, we fully disclose the details of the newly developed NZ-FZT process as well as the particulars necessary to put together a software packagecapable of carrying out its implementation. In keeping with the scientific method, thenext step in the development process is to demonstrate the validity and viability of thesenew computational algorithms. This chapter addresses the first step in this validationprocess and consists of using a well-characterized radiation structure as an analyticalbenchmark. By applying the algorithms to a radiating structure with known near- and far-zone E-fields, we are able to setup and conduct multiple numerical tests that serve as aproof-of-concept for the newly developed NZ-FZT process. Our first task at hand involves the choice of radiation structure to use as the analytical benchmark. Although the far-zone is often well-characterized for manyrealistic radiation structures, the near-zone is ordinarily only approximately understood.For our purposes, we want an analytical benchmark where the field quantities are exactlyknown in both the near- and far-zone regions. After searching, we are able to find such aradiating structure: a filament dipole with sinusoidal current distribution. Fortunately,Balanis [27] derives and presents an exact closed form representation for the surroundingE-field of this radiating structure (z-directed) that is valid everywhere including the near-zone. It is these exact field equations (after some manipulation) that are used tomathematically represent the analytical benchmark during our proof-of-conceptexperiments. Finally, in order to conduct these numerical experiments, we must create asoftware driver capable of sampling the near-zone of the analytic benchmark and thengenerating the corresponding ASCII input file required by the NZ-FZT software package.Comparisons of the algorithmic results are then made with their theoretically computedcounterparts to assess the overall performance of the newly developed NZ-FZT process. 1795.1 EXACT E-FIELD SURROUNDING A FILAMENT DIPOLE 5.1.1 Near-Zone E-Field In defining the exact E-field for a filament dipole of length l, we choose to present the corresponding field equations in Cartesian form because it is the most suitable for ourparticular application. As described in Chapter 2, the NZ-FZT process input requires thenear-zone E-field samples (situated along the spheroidal transformation surface) to be invector Cartesian form, i.e., Exij(, )′′ϑϕ , Eyi j(, )′′ϑϕ , and Ezi j(, )′′ϑϕ. Although the E- field must be sampled at specific points in the respective spheroidal coordinate system, viz., ( , ) ′′ϑϕij, defining the equations in terms of Cartesian parameters fortunately does not present any difficulties. In short, the spheroidal sample grid is easily converted to Cartesian coordinates by using the respective prolate or oblate spheroidal coordinatetransformation, i.e., equations (2-1)-(2-5). After mathematically manipulating the exactfield equations that are provided by Balanis [27], we obtain the following closed formrepresentation for the near-zone E-field of a z-directed filament dipole with sinusoidalcurrent distribution: Exyz x(,,)λλ λ =jI x xyoo oηπλλ λλ422+  ×()(,,) ()(,,) cos( )(,,)(,, ) (,, ) (,, )zle Rx y z zle Rx y z zle rx yzjR x y z jR x y z jrx y zλλπλ λλλ λ λλπλ λλλ λ λλπλ λλλ λλλ λ λλ λ λλ λ π− ++ −        − − −21 22 221 22 2(5-1) 180 Exyzy(,,)λλ λ =jI y xyoo oηπλλ λλ422+  ×()(,,) ()(,,) cos( )(,,)(,, ) (,, ) (,, )zle Rx y z zle Rx y z zle rx yzjR x y z jR x y z jrx y zλλπλ λλλ λ λλπλ λλλ λ λλπλ λλλ λλλ λ λλ λ λλ λ π− ++ −        − − −21 22 221 22 2(5-2) Exyzz(,,)λλ λ =−×jI oo oηπλ4e Rx y z e Rx y z le rx yzjR x y z jR x y z jrx y z− − −+ −        21 22 21 2 2πλ λλλ λ πλ λλλ λ λπλ λλλ λλλ λ λλ λ λλ λ π(,, ) (,, ) (,, )(,,) (,,) cos( )(,,)(5-3) where Rx y z x y zl1222 2 λλλ λ λ λ λ λ(,,) ( ) =+ + − (5-4) Rx y z xy zl2222 2 λλλ λ λ λ λ λ(,,) ( ) =+ + + (5-5) rx yz x y zλλλ λ λ λ λ(,,) =+ +22 2. (5-6) It is important to note that all spatial parameters of these equations have been normalized with respect to free-space wavelength (e.g., xxo λ λ= ). Normalizing these equations in this fashion enables our findings to be much more far-reaching because they now apply to spatial parameters that are defined relative to wavelength. In addition, the parameter ηo refers to the intrinsic impedance of free-space. 1815.2 ASYMPTOTIC REDUCTION OF THE EXACT E-FIELD TO OBTAIN THE FAR-ZONE APPROXIMATION 5.2.1 Far-Zone E-Field Although the field equations presented in (5-1)-(5-6) are valid for all regions surrounding the z-directed filament dipole, we would like to express the far-zone E-fieldin a form that is consistent with the NZ-FZT process output in order to facilitate ournumerical studies. As indicated in Chapter 2 (see 2.1.2), the desired form of the far-zoneE-field consists of expressing the field quantities in spherical coordinates (with respect toboth independent coordinates and vector quantities) together with the suppression of the radial factor ()erjkr−. By taking the limit as r→∞ , we are able to asymptotically approximate the field equations of (5-1)-(5-6) to obtain the following representation of the far-zone E-field [27]: EjI ll fz oo θθϕ ηππθ π θλλ(, )cos( cos ) cos( ) sin=−   2(5-7) Efzϕθϕ(, ) =0 . (5-8) It is this far-zone E-field representation that serves as the theoretical yardstick for which the results of the NZ-FZT process are subsequently compared. 5.3 DEVELOPMENT OF AN ANALYTICAL BENCHMARK SOFTWARE DRIVER 5.3.1 Overview of Software Driver The developed software driver samples the near-zone E-field of the z-directed filament dipole using equations (5-1)-(5-6) and then assembles the corresponding ASCIIinput file necessary to run the NZ-FZT software package (see Chapter 4). By way of thecomputer terminal, the user specifies the spheroidal transformation surface, 182corresponding sampling scheme, free-space wavelength, and length of the filament dipole. Also included in the specification of the spheroidal transformation surface is itsplacement relative to the center of the filament dipole; that is, the center of the spheroidalsurface can be offset from that of the dipole. As we shall subsequently see, this addedfeature allows us to further examine the effectiveness of the developed algorithms.Lastly, it must be mentioned that for all of our numerical studies we have arbitrarily assumed an oI of unity when defining the E-field of the filament dipole. 5.4 COMPARISON OF ALGORITHMIC RESULTS WITH THEORETICAL RESULTS 5.4.1 Centered Dipole Numerical Tests Our first set of numerical tests consists of using a z-directed filament dipole that is centered with respect to the spheroidal transformation surface. Both spheroidalalgorithms are individually examined by repeatedly applying the developed NZ-FZTprocess to the analytical benchmark using a family of spheroidal transformation surfaces.By comparing the computed far-zone results with those that are theoretically known, i.e.,computed using equations (5-7) and (5-8), we are able to quantify the performance ofboth the prolate and oblate spheroidal NZ-FZT algorithms. Throughout all of the numerical studies the length of the dipole remains the same and is always taken as a tenth of a wavelength, i.e., l o =01.λ. As for the focal point of the spheroidal coordinate system, it is taken as ao =01.λ for the prolate tests and ao =10.λ for the oblate tests. For the sake of clarity, we have depicted both of these test configurations in Figures 5.1a and 5.1b. Regarding the two families of spheroidal transformation surfaces, they range in our numerical studies from ξo=12. t o ξo=50. for the prolate case and from ξo=015. t o ξo=080. for the oblate case. In setting up these test configurations, we must pay close attention to the position of the radiating structure relative to that of the transformation surface so as to ensure that the two do not 183l = 0.1 λa = 0.1 λ xyz ξ = ξo Figure 5.1a Configuration of centered dipole numerical tests for the prolate spheroidal case. l = 0.1 λ a = 1.0 λxyz ξ = ξo Figure 5.1b Configuration of centered dipole numerical tests for the oblate spheroidal case. 184intersect. All of the numerical tests employ a uniform sample grid with 1o intervals in both the ϑand ϕ dimensions (i.e., IJ×= × 180 360 ). Lastly, we choose to compute the far-zone radiation patterns using 10o increments in the θ-direction for two planar cuts: ϕ=0o and ϕ=45o. Having defined the configurations of the centered dipole numerical tests, we now turn our attention to assessing the performance of the developed algorithms implementedby the NZ-FZT software package. This performance evaluation is accomplished byconducting the following numerical studies: (1) an error analysis of the computed results;(2) an analysis of how well the algorithms approximate theoretical values of zero (i.e.,establishing a numerical noise floor); and (3) a convergence study of the computedresults. The error analysis consists of comparing the computed results with the knownfar-zone E-field of the centered dipole, i.e., specified by equations (5-7) and (5-8). Sincethe far-zone E-field is a complex quantity, it is logical to conduct the error analysis inpolar form individually with respect to magnitude and phase. Because theory indicatesthat only the θ-component of the far-zone E-field is nonzero, we limit our error analysis to only Efzθθϕ( , ) . Nevertheless, we still address its cross-polarized counterpart Efzϕθϕ( , ) by assessing how well both algorithms approximate its theoretical value of zero. In this regard, we must use this approach to evaluate the performance of the algorithms because conventional error analysis is rendered meaningless when itscorresponding theoretical value vanishes. As for the convergence study, it consists offinding the value of L (i.e., the degree at which the spheroidal wave-harmonic expansion is truncated; see 2.3.1) where all nonzero results of Efzθθϕ( , ) have converged within single precision (i.e., between six and seven significant figures). Note that the convergence study is individually performed for each prolate and oblate spheroidaltransformation surface under consideration. 185For all of the error analyses, we use the following standard formula to compute the relative (or fractional) error in magnitude of Efzθθϕ(, ) : Relative Error of |( , ) |Efzθθϕ (ppm)=computed theoretical theoretical−×106. (5-9) As we shall see, we choose to express the relative error in parts per million (ppm) as opposed to percent (%) because it more fittingly describes the results of our numerical tests. Regarding the phase error of Efzθθϕ( , ) , it is computed in terms of degrees as follows: Phase Error of ∠Efzθθϕ(, ) ( d e g ) = computed theoretical − . (5-10) Using formulas (5-9) and (5-10) together with the results of the centered dipole numerical tests, we conduct the error analysis for both spheroidal cases. Presented inTables 5.1a and 5.1b is a compilation of the findings. It is at this point that we would liketo emphasize that our error analysis makes use of computed and theoretical data that areunnormalized with respect to both magnitude and phase; more precisely, unnormalizedfigures computed by the NZ-FZT algorithms are compared directly to those found usingthe theoretical expressions of (5-7) and (5-8). Upon review of the compiled error analysis data, one can see that each algorithm delivers the same performance (with respect to itself) when predicting the E-field in theupper hemisphere of the far-zone as compared to that of the lower hemisphere.Furthermore, the performance of each algorithm is also the same when computing its respective far-zone E-field for both the ϕ=0o and ϕ=45o planar cuts. Both of these observations are exactly what should be expected for such a problem setup that has angular and azimuthal (rotational) symmetry. From a quantitative perspective, the worst 186Table 5.1a Comparison of Prolate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the Filament Dipole of lλ=01. (Centered with Respect to Transformation Surface): ϕ=0o and 45o lλ=01.; aλ=01.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=<0 45o o ProlateξoConvrgce Lθ=10o θ=170oθ=20o θ=160oθ=30o θ=150oθ=40o θ=140oθ=50o θ=130oθ=60o θ=120oθ=70o θ=110oθ=80o θ=100oθ=90o 1.2 6 7 ppm; 0.0012 deg7 ppm; 0.0012 deg8 ppm; 0.0012 deg9 ppm; 0.0012 deg9 ppm; 0.0012 deg10 ppm; 0.0012 deg11 ppm; 0.0012 deg11 ppm; 0.0012 deg11 ppm; 0.0012 deg 1.5 6 8 ppm; 0.0012 deg7 ppm; 0.0012 deg6 ppm; 0.0012 deg5 ppm; 0.0012 deg3 ppm; 0.0012 deg2 ppm; 0.0012 deg< 1 ppm; 0.0012 deg1 ppm; 0.0013 deg1 ppm; 0.0013 deg 2.0 6 22 ppm; 0.0010 deg21 ppm; 0.0010 deg19 ppm; 0.0010 deg17 ppm; 0.0010 deg15 ppm; 0.0010 deg13 ppm; 0.0010 deg11 ppm; 0.0011 deg10 ppm; 0.0011 deg9 ppm; 0.0011 deg 3.0 6 32 ppm; 0.0008 deg31 ppm; 0.0008 deg29 ppm; 0.0008 deg26 ppm; 0.0008 deg24 ppm; 0.0008 deg21 ppm; 0.0007 deg19 ppm; 0.0007 deg18 ppm; 0.0007 deg17 ppm; 0.0007 deg 4.0 6 35 ppm; 0.0006 deg34 ppm; 0.0006 deg32 ppm; 0.0006 deg30 ppm; 0.0006 deg27 ppm; 0.0006 deg24 ppm; 0.0006 deg22 ppm; 0.0006 deg21 ppm; 0.0005 deg21 ppm; 0.0005 deg 5.0 6 37 ppm; 0.0005 deg36 ppm; 0.0005 deg34 ppm; 0.0005 deg31 ppm; 0.0005 deg29 ppm; 0.0005 deg26 ppm; 0.0005 deg24 ppm; 0.0004 deg23 ppm; 0.0004 deg22 ppm; 0.0004 deg 187Table 5.1b Comparison of Oblate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the Filament Dipole of lλ=01. (Centered with Respect to Transformation Surface): ϕ=0o and 45o lλ=01.; aλ=10.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=<0 45o o OblateξoConvrgce Lθ=10o θ=170oθ=20o θ=160oθ=30o θ=150oθ=40o θ=140oθ=50o θ=130oθ=60o θ=120oθ=70o θ=110oθ=80o θ=100oθ=90o 0.15 14 781 ppm; 0.1105 deg826 ppm; 0.1002 deg805 ppm; 0.0849 deg677 ppm; 0.0679 deg464 ppm; 0.0525 deg225 ppm; 0.0406 deg18 ppm; 0.0327 deg119 ppm; 0.0283 deg167 ppm; 0.0270 deg 0.20 14 494 ppm; 0.0527 deg521 ppm; 0.0452 deg505 ppm; 0.0358 deg427 ppm; 0.0272 deg314 ppm; 0.0209 deg203 ppm; 0.0171 deg118 ppm; 0.0151 deg67 ppm; 0.0142 deg51 ppm; 0.0139 deg 0.30 14 289 ppm; 0.0184 deg307 ppm; 0.0138 deg297 ppm; 0.0087 deg254 ppm; 0.0052 deg201 ppm; 0.0041 deg164 ppm; 0.0047 deg149 ppm; 0.0059 deg147 ppm; 0.0070 deg148 ppm; 0.0074 deg 0.40 14 207 ppm; 0.0072 deg221 ppm; 0.0042 deg213 ppm; 0.0009 deg183 ppm; 0.0009 deg151 ppm; 0.0006 deg136 ppm; 0.0013 deg140 ppm; 0.0037 deg152 ppm; 0.0054 deg157 ppm; 0.0060 deg 0.60 14 125 ppm; 0.0002 deg135 ppm; 0.0018 deg134 ppm; 0.0037 deg121 ppm; 0.0044 deg105 ppm; 0.0032 deg102 ppm; 0.0006 deg112 ppm; 0.0025 deg126 ppm; 0.0048 deg132 ppm; 0.0056 deg 0.80 14 71 ppm; 0.0027 deg84 ppm; 0.0036 deg94 ppm; 0.0047 deg94 ppm; 0.0048 deg87 ppm; 0.0036 deg84 ppm; 0.0010 deg89 ppm; 0.0021 deg98 ppm; 0.0044 deg102 ppm; 0.0053 deg 188case magnitude and phase error for our centered dipole numerical tests is as follows: (a) prolate, 37 ppm and 0.0013 deg; (b) oblate, 826 ppm and 0.1105 deg. Moreover, theprolate algorithm converges more rapidly than the oblate algorithm for this particularradiating structure, i.e., L=6 for the prolate numerical tests as compared to L=14 for the oblate numerical tests. Further inspection of Tables 5.1a and 5.1b reveals that the error analysis does not include performance of the algorithms at the far-zone observation angles θ=0o and θ=180o. These particular observation angles have been omitted because the far-zone E- field pattern possesses a null at these points; as a result, the relative error in magnitude becomes a meaningless quantity at these observation angles (i.e., equation (5-9) becomesundefined when the value of the theoretical result vanishes). However, we do examinehow well both algorithms approximate the null at these observation angles. For all of thetest cases, both algorithms compute the null as values that are significantly below -120 dB from the main beam peak (i.e., which exists at θ=90o). Regarding the cross-polarized far-zone E-field, both algorithms compute the theoretical zero of Efzϕθϕ( , ) as values that are also well below -120 dB from the main beam peak. In all of the instances where we examine how well the algorithms approximate the theoretical value of zero, thecorresponding phase is ignored because it essentially represents some form of numericalphase noise. 5.4.2 Offset Dipole Numerical Tests Having investigated how well the developed NZ-FZT algorithms address a problem setup that has both angular and azimuthal (rotational) symmetry, we would nowlike to assess their performance when considering an asymmetric problem. Fortunately,we can use the same analytical benchmark to generate such a test problem by simplyoffsetting the dipole with respect to the spheroidal transformation surface. By doing so,both algorithms must now address a near-zone E-field that is no longer symmetrically 189l = 0.1 λa = 0.1 λ xyz ξ = ξo r' r'= ( 0.01 , 0.03 , 0.04 ) λ λ λ Figure 5.2a Configuration of offset dipole numerical tests for the prolate spheroidal case. l = 0.1 λ a = 1.0 λxyz ξ = ξo r'r'= ( 0.1 , 0.3 , 0.05 ) λλ λ Figure 5.2b Configuration of offset dipole numerical tests for the oblate spheroidal case. 190distributed over the spheroidal transformation surface. Consequently, it is specifically for this purpose that the dipole offset feature has been added to our analytical benchmarksoftware driver (see 5.3.1). In order to maintain a basis for comparison, we shall keep the parameters and test cases of the offset dipole numerical tests the same as those that are used for the centereddipole numerical tests. Essentially, the offset tests are identical to the centered tests withthe only exception being the placement of the dipole. For these offset dipole numericaltests, the center of the dipole is translated from the center of the spheroidal transformationsurface by the following offset vector: (a) /G26′=r(′=x o 00 1.λ, ′=yo 00 3.λ, ′=zo 00 4.λ) for the prolate tests; (b) /G26′=r(′=xo01.λ, ′=yo 03.λ, ′=zo 00 5.λ) for the oblate tests. Portrayed in Figures 5.2a and 5.2b are the configurations of the offset dipole numerical tests for both spheroidal cases. Note that the offset in the x- and y-directions removes theazimuthal symmetry while the offset in the z-direction eliminates the angular symmetry.Once again we have ensured that the dipole and the spheroidal transformation surface donot intersect for any of the test cases. Evaluating the performance of the NZ-FZT algorithms for the offset dipole numerical tests is done in the same way as that which is done for the centered dipolenumerical tests. This time the error analysis consists of comparing the computed resultswith the known far-zone E-field of the offset dipole. Basic radiation theory indicates thatthe magnitude of the far-zone E-field remains unchanged for the offset dipoleconfiguration when compared to that of the centered dipole configuration. As a result,the magnitude of equation (5-7) is still used directly to determine the theoretical datanecessary to conduct the error analysis for the offset dipole numerical tests. However, thesame basic radiation theory also indicates that the corresponding phase of the far-zone E-field differs between the offset and centered dipole configurations. Using the parallel rayapproximation that is so often evoked in basic antenna theory (see Stutzman and Thiele[26]), one can show that the phase difference between the two dipole configurations is 191Table 5.2a Comparison of Prolate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=0o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 00 100 3 00 4 lλ=01.; aλ=01.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=0o ProlateξoConvrgce Lθ=10oθ=20oθ=30oθ=40oθ=50oθ=60oθ=70oθ=80oθ=90o 1.2 7 70 ppm; 0.0081 deg50 ppm; 0.0044 deg42 ppm; 0.0032 deg38 ppm; 0.0026 deg35 ppm; 0.0023 deg32 ppm; 0.0021 deg30 ppm; 0.0020 deg28 ppm; 0.0020 deg26 ppm; 0.0020 deg 1.5 7 9 ppm; 0.0063 deg9 ppm; 0.0039 deg10 ppm; 0.0031 deg11 ppm; 0.0026 deg12 ppm; 0.0023 deg12 ppm; 0.0021 deg13 ppm; 0.0019 deg12 ppm; 0.0018 deg11 ppm; 0.0017 deg 2.0 7 27 ppm; 0.0035 deg20 ppm; 0.0024 deg17 ppm; 0.0020 deg13 ppm; 0.0017 deg11 ppm; 0.0015 deg8 ppm; 0.0014 deg7 ppm; 0.0013 deg5 ppm; 0.0013 deg5 ppm; 0.0012 deg 3.0 7 41 ppm; 0.0016 deg35 ppm; 0.0011 deg31 ppm; 0.0009 deg28 ppm; 0.0008 deg24 ppm; 0.0007 deg22 ppm; 0.0007 deg19 ppm; 0.0007 deg18 ppm; 0.0007 deg17 ppm; 0.0007 deg 4.0 7 43 ppm; 0.0009 deg38 ppm; 0.0006 deg35 ppm; 0.0005 deg32 ppm; 0.0004 deg28 ppm; 0.0004 deg26 ppm; 0.0004 deg23 ppm; 0.0004 deg22 ppm; 0.0004 deg21 ppm; 0.0005 deg 5.0 7 43 ppm; 0.0006 deg39 ppm; 0.0003 deg36 ppm; 0.0003 deg33 ppm; 0.0002 deg30 ppm; 0.0002 deg27 ppm; 0.0002 deg25 ppm; 0.0003 deg24 ppm; 0.0003 deg23 ppm; 0.0004 deg 192Table 5.2a Comparison of Prolate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the (cont.) Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=0o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 00 100 3 00 4 lλ=01.; aλ=01.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=0o ProlateξoConvrgce Lθ=90oθ=100oθ=110oθ=120oθ=130oθ=140oθ=150oθ=160oθ=170o 1.2 7 26 ppm; 0.0020 deg23 ppm; 0.0020 deg21 ppm; 0.0021 deg19 ppm; 0.0022 deg17 ppm; 0.0024 deg14 ppm; 0.0027 deg11 ppm; 0.0030 deg4 ppm; 0.0037 deg16 ppm; 0.0058 deg 1.5 7 11 ppm; 0.0017 deg10 ppm; 0.0016 deg9 ppm; 0.0016 deg7 ppm; 0.0016 deg5 ppm; 0.0017 deg3 ppm; 0.0018 deg< 1 ppm; 0.0020 deg 2 ppm; 0.0024 deg7 ppm; 0.0037 deg 2.0 7 5 ppm; 0.0012 deg6 ppm; 0.0012 deg7 ppm; 0.0012 deg9 ppm; 0.0012 deg11 ppm; 0.0013 deg13 ppm; 0.0013 deg15 ppm; 0.0015 deg17 ppm; 0.0017 deg19 ppm; 0.0024 deg 3.0 7 17 ppm; 0.0007 deg18 ppm; 0.0008 deg18 ppm; 0.0008 deg21 ppm; 0.0009 deg23 ppm; 0.0010 deg25 ppm; 0.0011 deg27 ppm; 0.0012 deg30 ppm; 0.0014 deg32 ppm; 0.0018 deg 4.0 7 21 ppm; 0.0005 deg21 ppm; 0.0006 deg22 ppm; 0.0007 deg24 ppm; 0.0008 deg27 ppm; 0.0009 deg29 ppm; 0.0010 deg32 ppm; 0.0011 deg34 ppm; 0.0012 deg37 ppm; 0.0016 deg 5.0 7 23 ppm; 0.0004 deg23 ppm; 0.0004 deg25 ppm; 0.0005 deg26 ppm; 0.0007 deg28 ppm; 0.0008 deg31 ppm; 0.0009 deg34 ppm; 0.0010 deg36 ppm; 0.0012 deg38 ppm; 0.0014 deg 193Table 5.3a Comparison of Prolate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=45o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 00 100 3 00 4 lλ=01.; aλ=01.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=45o ProlateξoConvrgce Lθ=10oθ=20oθ=30oθ=40oθ=50oθ=60oθ=70oθ=80oθ=90o 1.2 7 143 ppm; 0.0215 deg86 ppm; 0.0111 deg65 ppm; 0.0075 deg55 ppm; 0.0057 deg46 ppm; 0.0046 deg40 ppm; 0.0039 deg34 ppm; 0.0034 deg29 ppm; 0.0031 deg24 ppm; 0.0030 deg 1.5 7 13 ppm; 0.0146 deg13 ppm; 0.0079 deg14 ppm; 0.0056 deg15 ppm; 0.0043 deg15 ppm; 0.0035 deg15 ppm; 0.0029 deg15 ppm; 0.0025 deg13 ppm; 0.0022 deg11 ppm; 0.0021 deg 2.0 7 44 ppm; 0.0076 deg27 ppm; 0.0043 deg19 ppm; 0.0031 deg15 ppm; 0.0025 deg11 ppm; 0.0021 deg8 ppm; 0.0017 deg6 ppm; 0.0015 deg5 ppm; 0.0013 deg5 ppm; 0.0013 deg 3.0 7 57 ppm; 0.0033 deg42 ppm; 0.0019 deg35 ppm; 0.0014 deg29 ppm; 0.0011 deg25 ppm; 0.0009 deg22 ppm; 0.0008 deg19 ppm; 0.0007 deg17 ppm; 0.0006 deg17 ppm; 0.0006 deg 4.0 7 55 ppm; 0.0019 deg44 ppm; 0.0011 deg38 ppm; 0.0008 deg33 ppm; 0.0006 deg29 ppm; 0.0005 deg26 ppm; 0.0004 deg23 ppm; 0.0003 deg22 ppm; 0.0003 deg21 ppm; 0.0004 deg 5.0 7 53 ppm; 0.0014 deg43 ppm; 0.0007 deg39 ppm; 0.0005 deg34 ppm; 0.0003 deg30 ppm; 0.0002 deg27 ppm; 0.0002 deg25 ppm; 0.0002 deg23 ppm; 0.0002 deg23 ppm; 0.0002 deg 194Table 5.3a Comparison of Prolate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the (cont.) Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=45o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 00 100 3 00 4 lλ=01.; aλ=01.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=45o ProlateξoConvrgce Lθ=90oθ=100oθ=110oθ=120oθ=130oθ=140oθ=150oθ=160oθ=170o 1.2 7 24 ppm; 0.0030 deg19 ppm; 0.0029 deg14 ppm; 0.0030 deg9 ppm; 0.0032 deg3 ppm; 0.0036 deg4 ppm; 0.0041 deg15 ppm; 0.0051 deg34 ppm; 0.0071 deg88 ppm; 0.0129 deg 1.5 7 11 ppm; 0.0021 deg 9 ppm; 0.0020 deg6 ppm; 0.0020 deg4 ppm; 0.0021 deg< 1 ppm; 0.0023 deg3 ppm; 0.0027 deg 6 ppm; 0.0033 deg 12 ppm; 0.0046 deg25 ppm; 0.0083 deg 2.0 7 5 ppm; 0.0013 deg6 ppm; 0.0012 deg7 ppm; 0.0013 deg10 ppm; 0.0014 deg12 ppm; 0.0016 deg15 ppm; 0.0018 deg17 ppm; 0.0022 deg19 ppm; 0.0029 deg22 ppm; 0.0049 deg 3.0 7 17 ppm; 0.0006 deg17 ppm; 0.0007 deg19 ppm; 0.0008 deg21 ppm; 0.0009 deg23 ppm; 0.0011 deg26 ppm; 0.0013 deg29 ppm; 0.0016 deg32 ppm; 0.0020 deg37 ppm; 0.0031 deg 4.0 7 21 ppm; 0.0004 deg22 ppm; 0.0004 deg23 ppm; 0.0006 deg25 ppm; 0.0007 deg27 ppm; 0.0009 deg30 ppm; 0.0011 deg33 ppm; 0.0013 deg37 ppm; 0.0017 deg42 ppm; 0.0025 deg 5.0 7 23 ppm; 0.0002 deg23 ppm; 0.0003 deg25 ppm; 0.0004 deg27 ppm; 0.0006 deg29 ppm; 0.0008 deg32 ppm; 0.0010 deg35 ppm; 0.0012 deg38 ppm; 0.0015 deg44 ppm; 0.0022 deg 195Table 5.2b Comparison of Oblate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=0o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 0103 00 5 lλ=01.; aλ=10.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=0o OblateξoConvrgce Lθ=10oθ=20oθ=30oθ=40oθ=50oθ=60oθ=70oθ=80oθ=90o 0.15 15 154 ppm; 0.0035 deg74 ppm; 0.0009 deg 8 ppm; 0.0014 deg64 ppm; 0.0007 deg102 ppm; 0.0005 deg133 ppm; 0.0020 deg159 ppm; 0.0032 deg176 ppm; 0.0039 deg181 ppm; 0.0041 deg 0.20 15 91 ppm; 0.0080 deg56 ppm; 0.0017 deg8 ppm; 0.0004 deg58 ppm; 0.0009 deg95 ppm; 0.0003 deg129 ppm; 0.0009 deg158 ppm; 0.0022 deg178 ppm; 0.0031 deg184 ppm; 0.0035 deg 0.30 15 57 ppm; 0.0092 deg12 ppm; 0.0037 deg34 ppm; 0.0008 deg62 ppm; 0.0008 deg89 ppm; 0.0009 deg120 ppm; 0.0001 deg150 ppm; 0.0014 deg173 ppm; 0.0025 deg181 ppm; 0.0031 deg 0.40 15 133 ppm; 0.0058 deg60 ppm; 0.0026 deg58 ppm; 0.0002 deg70 ppm; 0.0013 deg89 ppm; 0.0013 deg114 ppm; 0.0003 deg142 ppm; 0.0012 deg164 ppm; 0.0024 deg172 ppm; 0.0030 deg 0.60 15 137 ppm; 0.0002 deg78 ppm; 0.0007 deg72 ppm; 0.0019 deg78 ppm; 0.0025 deg87 ppm; 0.0021 deg103 ppm; 0.0007 deg124 ppm; 0.0009 deg143 ppm; 0.0023 deg150 ppm; 0.0031 deg 0.80 15 100 ppm; 0.0024 deg69 ppm; 0.0026 deg70 ppm; 0.0030 deg77 ppm; 0.0029 deg83 ppm; 0.0023 deg94 ppm; 0.0010 deg110 ppm; 0.0007 deg126 ppm; 0.0021 deg131 ppm; 0.0029 deg 196Table 5.2b Comparison of Oblate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the (cont.) Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=0o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 0103 00 5 lλ=01.; aλ=10.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=0o OblateξoConvrgce Lθ=90oθ=100oθ=110oθ=120oθ=130oθ=140oθ=150oθ=160oθ=170o 0.15 15 181 ppm; 0.0041 deg171 ppm; 0.0036 deg146 ppm; 0.0027 deg111 ppm; 0.0014 deg76 ppm; 0.0002 deg43 ppm; 0.0013 deg10 ppm; 0.0011 deg20 ppm; 0.0016 deg3 ppm; 0.0090 deg 0.20 15 184 ppm; 0.0035 deg173 ppm; 0.0033 deg146 ppm; 0.0025 deg109 ppm; 0.0012 deg74 ppm; 0.0001 deg45 ppm; 0.0011 deg24 ppm; 0.0007 deg 12 ppm; 0.0020 deg63 ppm; 0.0080 deg 0.30 15 181 ppm; 0.0031 deg169 ppm; 0.0030 deg141 ppm; 0.0023 deg104 ppm; 0.0011 deg70 ppm; 0.0003 deg50 ppm; 0.0011 deg45 ppm; 0.0009 deg55 ppm; 0.0008 deg126 ppm; 0.0038 deg 0.40 15 172 ppm; 0.0030 deg161 ppm; 0.0030 deg133 ppm; 0.0022 deg97 ppm; 0.0009 deg66 ppm; 0.0006 deg50 ppm; 0.0016 deg52 ppm; 0.0017 deg71 ppm; 0.0008 deg141 ppm; 0.0002 deg 0.60 15 150 ppm; 0.0031 deg139 ppm; 0.0030 deg114 ppm; 0.0022 deg83 ppm; 0.0007 deg60 ppm; 0.0011 deg49 ppm; 0.0025 deg52 ppm; 0.0029 deg68 ppm; 0.0026 deg120 ppm; 0.0026 deg 0.80 15 131 ppm; 0.0029 deg120 ppm; 0.0029 deg97 ppm; 0.0020 deg73 ppm; 0.0004 deg57 ppm; 0.0013 deg51 ppm; 0.0026 deg51 ppm; 0.0033 deg57 ppm; 0.0032 deg85 ppm; 0.0032 deg 197Table 5.3b Comparison of Oblate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=45o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 0103 00 5 lλ=01.; aλ=10.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=45o OblateξoConvrgce Lθ=10oθ=20oθ=30oθ=40oθ=50oθ=60oθ=70oθ=80oθ=90o 0.15 15 119 ppm; 0.0127 deg179 ppm; 0.0013 deg 236 ppm; 0.0052 deg265 ppm; 0.0057 deg279 ppm; 0.0041 deg286 ppm; 0.0013 deg291 ppm; 0.0018 deg290 ppm; 0.0041 deg281 ppm; 0.0053 deg 0.20 15 186 ppm; 0.0206 deg131 ppm; 0.0030 deg160 ppm; 0.0031 deg189 ppm; 0.0050 deg214 ppm; 0.0041 deg234 ppm; 0.0016 deg247 ppm; 0.0013 deg252 ppm; 0.0036 deg247 ppm; 0.0047 deg 0.30 15 368 ppm; 0.0207 deg133 ppm; 0.0062 deg94 ppm; 0.0006 deg106 ppm; 0.0037 deg133 ppm; 0.0039 deg162 ppm; 0.0020 deg186 ppm; 0.0005 deg199 ppm; 0.0026 deg198 ppm; 0.0037 deg 0.40 15 429 ppm; 0.0125 deg146 ppm; 0.0044 deg79 ppm; 0.0006 deg74 ppm; 0.0033 deg95 ppm; 0.0036 deg123 ppm; 0.0021 deg148 ppm; 0.0001 deg163 ppm; 0.0021 deg166 ppm; 0.0031 deg 0.60 15 338 ppm; 0.0012 deg126 ppm; 0.0002 deg66 ppm; 0.0022 deg53 ppm; 0.0034 deg64 ppm; 0.0034 deg86 ppm; 0.0021 deg107 ppm; 0.0002 deg119 ppm; 0.0015 deg122 ppm; 0.0025 deg 0.80 15 236 ppm; 0.0024 deg100 ppm; 0.0021 deg56 ppm; 0.0029 deg44 ppm; 0.0034 deg52 ppm; 0.0032 deg68 ppm; 0.0020 deg84 ppm; 0.0004 deg93 ppm; 0.0011 deg93 ppm; 0.0019 deg 198Table 5.3b Comparison of Oblate Spheroidal NZ-FZT Algorithmic Results with Theoretical Results for the (cont.) Filament Dipole of lλ=01. (Offset with Respect to Transformation Surface): ϕ=45o Offset: ( , , ) ( . , . , . ) ′′′ = xyzλλ λ 0103 00 5 lλ=01.; aλ=10.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (ppm) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=45o OblateξoConvrgce Lθ=90oθ=100oθ=110oθ=120oθ=130oθ=140oθ=150oθ=160oθ=170o 0.15 15 281 ppm; 0.0053 deg260 ppm; 0.0051 deg226 ppm; 0.0036 deg182 ppm; 0.0011 deg136 ppm; 0.0014 deg100 ppm; 0.0030 deg84 ppm; 0.0023 deg98 ppm; 0.0027 deg257 ppm; 0.0187 deg 0.20 15 247 ppm; 0.0047 deg227 ppm; 0.0045 deg194 ppm; 0.0030 deg149 ppm; 0.0007 deg102 ppm; 0.0016 deg69 ppm; 0.0026 deg65 ppm; 0.0014 deg 111 ppm; 0.0034 deg337 ppm; 0.0162 deg 0.30 15 198 ppm; 0.0037 deg183 ppm; 0.0035 deg152 ppm; 0.0022 deg109 ppm; 0.0001 deg66 ppm; 0.0017 deg40 ppm; 0.0024 deg51 ppm; 0.0011 deg127 ppm; 0.0022 deg391 ppm; 0.0080 deg 0.40 15 166 ppm; 0.0031 deg152 ppm; 0.0030 deg125 ppm; 0.0018 deg87 ppm; 0.0001 deg50 ppm; 0.0018 deg28 ppm; 0.0025 deg44 ppm; 0.0016 deg120 ppm; 0.0003 deg361 ppm; 0.0016 deg 0.60 15 122 ppm; 0.0025 deg112 ppm; 0.0024 deg91 ppm; 0.0013 deg62 ppm; 0.0003 deg35 ppm; 0.0019 deg21 ppm; 0.0027 deg33 ppm; 0.0026 deg88 ppm; 0.0021 deg255 ppm; 0.0031 deg 0.80 15 93 ppm; 0.0019 deg86 ppm; 0.0018 deg71 ppm; 0.0009 deg49 ppm; 0.0005 deg29 ppm; 0.0018 deg19 ppm; 0.0026 deg31 ppm; 0.0028 deg70 ppm; 0.0027 deg178 ppm; 0.0034 deg 199related to the offset vector /G26′=′′′ rx y z ( , , ) and the observation angle ( , ) θϕ by the following expression: ∆Φ(radians) = []2π λθϕ θϕ θ oxy z′ +′ +′ sin cos sin sin cos . (5-11) By adding this phase difference to the phase of equation (5-7), we compute the theoretical phase data necessary to conduct the phase error analysis for the offset dipole numericaltests. Presented in Tables 5.2a, 5.2b, 5.3a, and 5.3b is a compilation of the overall erroranalysis for both spheroidal cases of this second set of numerical tests. Both Tables 5.2aand 5.3a provide the overall error analysis of the prolate spheroidal algorithm for the ϕ=0o and ϕ=45o planar cuts, respectively. Likewise, Tables 5.2b and 5.3b respectively address the overall error analysis for the oblate spheroidal algorithm for the ϕ=0o and ϕ=45o planar cuts. Upon review of these findings, we see that both algorithms respond differently to the offset dipole numerical tests when compared to that of the centered dipole numericaltests. Included in these differences is the fact that significantly more analysis data ispresented in the case of the offset dipole numerical tests. This increase of data occursbecause both algorithms now deliver varying performances for each observation angle (, ) θϕ under consideration. As a result, the performance of the algorithms is no longer the same when predicting the far-zone E-field in the upper and lower hemisphere. Moreover, both algorithms now deliver performances that differ when comparing their results for the ϕ=0o planar cut with those for the ϕ=45o planar cut. It must be pointed out that this asymmetric performance of the algorithms is what should be expected when addressing a problem setup that has no angular and azimuthal (rotational) symmetry. Further inspection of the error analysis data reveals that the worst case magnitudeand phase error for our offset dipole numerical tests (i.e., for both planar cuts) is asfollows: (a) prolate, 143 ppm and 0.0215 deg; (b) oblate, 429 ppm and 0.0207 deg. Theperformance of the prolate spheroidal algorithm appears to slightly degrade when 200compared to its centered dipole counterpart; however, the opposite is true for the performance of the oblate spheroidal algorithm. Furthermore, both algorithms converge at a slightly slower rate for the offset numerical tests, viz., L=7 for prolate cases and L=15 for the oblate cases, as compared to their centered dipole counterparts. In a manner identical to that conducted for the centered dipole numerical tests, the next step in the analysis of the offset dipole numerical tests is to determine how well thealgorithms approximate theoretical values of zero. Once again, this analysis not only applies to the nulls of the pattern Efzθθϕ( , ) at the observation angles θ=0o and θ=180o, but also to the cross-polarized far-zone E-field Efzϕθϕ( , ) . The worst case null computed with respect to the main beam peak is as follows for all of the offset dipole test cases: (a) prolate, -83.74 dB; (b) oblate, -78.79 dB. As for the cross-polarized far- zone E-field Efzϕθϕ( , ) , its worst case magnitude (considering all offset dipole test cases) computed with respect to the main beam peak is as follows: (a) prolate, -83.23 dB; (b) oblate, -78.28 dB. Note that for each spheroidal algorithm the worst caseapproximation of the theoretical value of zero is roughly the same when computing either the far-zone E-field nulls or the cross-polarized E-field Efzϕθϕ( , ) . Moreover, the results of the offset dipole numerical tests show that the numerical noise floor is significantly higher than that of the corresponding centered dipole numerical tests. Inspite of this, the increased level of the numerical noise floor still remains quite acceptablefor both spheroidal algorithms. Having completed the analysis for both the offset and centered dipole numerical tests, we are now ready to summarize the overall performance of the NZ-FZT algorithms.We do so by combining the results of both sets of analysis data to establish an overallworst case performance for all of the numerical test cases. The outcome of thisperformance assessment is compiled and presented in Table 5.4. 201Table 5.4 Overall Worst Case Performance of the Spheroidal NZ-FZT Algorithms for the Offset and Centered Dipole Numerical Tests NZ-FZT AlgorithmSlowest Convrgce LWorst Case Relative Error of |( , ) |Efzθθϕ (ppm)Worst Case Phase Error of ∠Efzθθϕ(, ) ( d e g )Highest Level of Zero Approximation (dB) Prolate 7 143 ppm 0.0215 deg -83.23 dB Oblate 15 826 ppm 0.1105 deg -78.28 dB From this performance assessment we are able to draw several important conclusions about the newly developed NZ-FZT algorithms. Based on these results, itappears that the prolate spheroidal algorithm performs a little better than its oblatecounterpart. On average, the prolate algorithm delivers more accurate results for bothsets of numerical tests while maintaining a numerical noise floor that is a few decibels(dB) lower. Regardless, both algorithms perform remarkably well: both algorithmsdeliver far-zone E-field computations that are within ±01. % (or ±1000 ppm) in magnitude and ±012. deg in phase with at least 78 dB of numerical dynamic range. As for the difference in the rate of convergence, it is most likely attributed to the fact that the prolate spheroidal transformation surface geometrically conforms to the filament dipolebetter than its oblate counterpart. Most importantly, the results of these numerical studiesdemonstrate that our overall NZ-FZT concept is sound both in theory and inimplementation. 202CHAPTER 6. DEVELOPMENT OF A COMPUTATIONAL ELECTROMAGNETIC FIELD SOLVER This chapter addresses the development of an electromagnetic field solver capable of computing the surrounding near-zone fields for a variety of radiation structures. Thepurpose of this software development is essentially twofold. First, by combining acomputational field solver with our NZ-FZT computer software, we are able to providefurther validation of our newly developed NZ-FZT process. (Note that this additionalvalidation is completed in the chapter that follows using the combined software to modelan empirical benchmark.) Second, the coupling of this software allows us to illustrate arealistic example of an application that makes excellent use of our NZ-FZT process. Although many different computational techniques exist that are able to calculate the surrounding near-zone fields for a given radiation structure, we have chosen toimplement our electromagnetic field solver using a specific form of the finite-differencetime-domain (FDTD) method. Since our prime objective is to further validate the newlydeveloped NZ-FZT process, only one electromagnetic field solver needs to be developed;that is, the examination of other field solvers would prove nothing additional with respectto the verification of our algorithms. In particular, the FDTD method is chosen to fulfillthis function in our validation process because it is an extremely versatile technique thathas become quite popular in most recent years. However, it is important to note that anyone of the many other well-abled computational techniques, e.g., the finite elementmethod (FEM), could have equally been chosen to serve as our electromagnetic fieldsolver. Initially, we begin our discussion by examining the relevant basics of the FDTD method and how they pertain to our particular application. We follow this by discussingthe development of a FDTD field solver that is designed to interface with our NZ-FZTsoftware. Finally, we conclude this chapter by conducting an end-to-end test of thecombined NZ-FZT/FDTD software using a computational model of our analyticalbenchmark (initially described and used in Chapter 5). 2036.1 FDTD FUNDAMENTALS 6.1.1 Overview of Computational Technique In the field of computational electromagnetics, the term FDTD applies to all techniques that employ a finite-difference scheme both in time and space to numericallyapproximate the time-dependent differential form of Maxwell’s equations. The techniqueof finite-differencing is not new to numerical analysis and is often considered the methodof choice when computing a numerical solution for a system of partial differentialequations. However, the use of the finite-difference method to treat partial differentialequations is in itself quite an expansive subject: one that is too vast to even beginconsidering here. Consequently, the reader is referred to texts [37], [43], [44] for themany schemes and intricate details associated with this technique of numerical analysis.To further complicate matters, many finite-difference schemes and mesh configurations(used to approximate Maxwell’s equations) exist that come under the umbrella of theFDTD method. These various approaches range from higher-order techniques to non-uniform grid configurations (all of which are too numerous to mention here). In general,this ever-growing diversity of FDTD stems from the continual desire to improve one ormore of the following performance factors: accuracy, dispersion, stability, memorystorage, and total computation time. For a comprehensive survey of the many variationsof this computational technique and their advantages, we refer the interested reader toreferences [45], [46] (both presently considered the foremost texts on FDTD) and theliterature survey article [47] (which points to the myriad of various FDTD techniquesavailable in the literature). It is because of the sheer volume of these methods that ourattention remains focused on the specific variation of FDTD utilized by ourelectromagnetic field solver. In developing the field solver, we elect to make use of the most basic form of FDTD originally pioneered by Yee [48]. As we shall see, this approach combined withthe second-order Mur absorbing boundary condition (ABC) [49] proves to be quitesufficient for our purposes of demonstrating the performance and effectiveness of the 204newly developed NZ-FZT process. Yee ’s approach makes use of a two-step finite- difference explicit scheme with central differencing both in time and space (often referredto as a leapfrog scheme [45]) to represent both E- and H-fields in Cartesian coordinates.As a result, this scheme leads to a mesh configuration that represents a given solutionvolume with samples of E- and H-fields that are interleaved both spatially and temporally.Since this particular finite-differencing scheme only retains first-order terms, it providessecond-order accuracy with respect to spatial and temporal sample periods. Furthermore,because the samples are uniformly distributed with respect to each Cartesian dimension,we are able to partition the solution volume into a set of uniform cells (also known as Yeecells). Each of these Yee cells consists of a set of E- and H-field samples with a spatialarrangement of that shown in Figure 6.1 (which results from central differencing inspace). Upon inspection of the Yee cell geometry, one should notice that the E-fieldsamples in each Cartesian plane surround the corresponding orthogonal H-field samplesand vice-versa. Intuitively this arrangement makes sense because it remains consistentwith both curl equations contained within the differential form of Maxwell ’s equations. Alternatively, one can recognize by inspection that the arrangement of the fields is alsoconsistent with the integral form of Maxwell ’s equations (as should be expected). Likewise, it is also important to realize that the arrangements for each Cartesiancomponent of the E- and H-fields do not coincide and are all spatially offset with respectto one another. Referring to Figure 6.1, we summarize the spatial locations of theindexed field samples as follows: Ei j kx(, , )+1 2 is at (( ) , , )ix j y k z+1 2∆∆∆ ; Ei j ky(, , )+1 2 is at (, ( ) , )ix j y kz∆∆ ∆ +1 2 ; Ei j kz(, , ) +1 2 is at (, , ( ) )ix jyk z∆∆ ∆ +1 2 ; Hi j kx(, , )++1 21 2 is at (, ( ) , ( ) )ix j yk z∆∆∆ ++1 21 2 ; Hi j ky(, , )++1 21 2 is at (( ) , ,( ) )ix j y kz++1 21 2 ∆∆ ∆ ; Hi j kz(, , )++1 21 2 is at (( ) ,( ) , )ix jy k z++1 21 2 ∆∆ ∆ . 205 E (i+1/2,j,k)xE (i,j+1/2,k)yE (i,j,k+1/2)z E (i,j+1,k+1/2)z E (i+1/2,j+1,k)x E (i+1,j+1/2,k)yH (i,j+1/2,k+1/2)x H (i+1/2,j,k+1/2)y H (i+1/2,j+1/2,k)z xyz ∆y∆z∆x Figure 6.1 Yee cell geometry with index ( i, j, k). 206As we can see, each Yee cell is referenced in the solution volume by the spatial index (, , )ij k and is actually made up of twelve E-field samples and six H-field samples. By duality, the three-dimensional lattice can be viewed as a set of magnetic cells that instead contain twelve H-field samples and six E-field samples. Nonetheless, partitioningthe solution volume into electrically defined Yee cells is much more suitable for ourparticular application; that is, the electrical approach is intrinsically better able to model aradiation structure solely made of non-magnetic materials (i.e., materials possessing apermeability of µo). The modeling of a non-magnetic (electrical) structure using the FDTD method requires that each E-field sample be computed using a distinct formulation thatcorresponds to the medium present: (1) free-space (or non-lossy dielectric materials), (2)perfect electric conductor, and (3) lossy dielectric materials. Yet because all mediarequired for our particular application are considered non-magnetic, computation of therespective H-field samples only requires a single formulation. Each of these resultingFDTD formulations is detailed in the sections that follow. Nevertheless, it is byspecifying different materials at E-field sample points and subsequently using the properformulations that effectively allow the construction of a FDTD electrical structure. Now that we have a basic understanding of how FDTD functions spatially, we need to focus on its operation with respect to time. The technique is considered a time-marching algorithm that starts at some point in time (i.e., t=0 in our case) with an initially defined solution volume and continues calculating the fields until somedesignated stop-point is reached. For our purposes of modeling radiation structures, thesolution volume is considered at rest (i.e., all fields at zero) prior to the start of the FDTDsimulation (i.e., t<0 ). By receiving some form of excitation during the simulation, the FDTD field solver is then able to compute the resulting E- and H-field samples in thetime-domain as the fields propagate throughout the entire three-dimensional lattice. In a manner similar to the partitioning of the solution volume, the overall simulation time is first divided into uniform time-steps of duration ∆t. At each of these time-steps there exists a distinct set of E- and H-field samples that represents the entire 207solution volume. Moreover, these E- and H-field samples are staggered in time with respect to one another by a half time-step (i.e., ∆t2). For a given time-step (indexed with respect to n), the set of E-field samples is computed using the set of H-field samples from a half time-step back and the set of E-field samples from a whole time-step back.Likewise, the immediately following set of H-field samples is then found by using therecently computed set of E-field samples (now a half time-step back) and the set of H-fields samples from a whole time-step back. This two-step process is more easilyunderstood by examining Figure 6.2. By repeating this multistep process, the field solverprogresses through all of the time-steps until the designated stop-point index has beenreached. The finite-difference scheme employed (i.e., leapfrog) is considered explicitbecause any given field sample is computed using only prior-time field samples [45]. Having reviewed both the spatial and temporal operation of this computational method, we now appropriately include a time-step index n in addition to the spatial index (, , )ij k when describing both E- and H-field samples (see Figure 6.2). It is important to note that within the literature there are a number of other notations that describe this multistep process. For the most part, the differences occur in the way that the indicesrefer to a particular sample both spatially and temporally. Nevertheless, all of the othernotations are essentially describing the same process. We now turn our attention to the fact that our three-dimensional lattice of E- and H-field samples is not infinite in extent. Although this limitation has no bearing on themodeling of closed electromagnetic problems (e.g., cavity type problems), it does posesome difficulties when modeling open electromagnetic problems (e.g., free-spaceradiation problems). In short, truncation of the three-dimensional lattice gives rise tounwanted reflections at the boundaries of the solution volume that will ultimately corruptthe results of a FDTD simulation. Because our particular application requires us tocompute the near-zone fields of a radiation structure situated in free-space, theseunwanted reflections can seriously impact the validity of our results. In order to attenuatethese boundary reflections, we apply what is referred to as an absorbing boundary 208t = (n-1/2) t ∆ t = n t ∆ t = (n+1/2) t ∆ Needed to compute entire solution space (all values of i,j,k). (i,j+1/2,k+1/2)xn+1/2 H (i+1/2,j,k+1/2)yn+1/2 H (i+1/2,j+1/2,k)zn+1/2 H(i,j+1/2,k+1/2)xn-1/2 H (i+1/2,j,k+1/2)yn-1/2 H (i+1/2,j+1/2,k)zn-1/2 H(i+1/2,j,k)xn+1 E (i,j+1/2,k)yn+1 E (i,j,k+1/2)zn+1 E(i+1/2,j,k)xn E (i,j+1/2,k)yn E (i,j,k+1/2)zn E Needed to compute entire solution space (all values of i,j,k).t = (n+1) t ∆ Needed to compute entire solution space (all values of i,j,k). Figure 6.2 Temporal operation of FDTD. 209condition (ABC) at the perimeter of the solution volume. Essentially, the ABC mathematically absorbs electromagnetic waves that are propagating outward with respectto the solution volume. Since there are so many accepted ABC techniques (includingtheir respective variations), we will not even attempt to review them here. For acomprehensive examination of these existing techniques, the reader is referred to theextensive survey provided by Taflove [46] that details the theory and performance levelfor each respective ABC. However, in addition to these techniques, new ABCs arecontinually under development by the research community; for the most part, all of theseefforts are directed toward finding the ideal method able to provide the quietest boundaryreflections using the fewest computations. As mentioned earlier, our electromagneticfield solver makes use of the second-order ABC put forth by Mur [49]: experience showsthat it provides a level of attenuation (of the boundary reflections) that is quite acceptablefor our particular application. Modern-day FDTD simulations typically make use of formulations based upon scattered-fields only, total-fields only, or some combination of both [45], [46]. Thescattered-field formulation is based upon separating all E- and H-fields within thesolution volume into incident and scattered components (this decomposition is justifiedby the linearity of Maxwell ’s equations [46]). In doing so, the incident field (serving as the excitation) can be specified analytically throughout the solution volume such that thefield solver only has to compute the scattered fields. Moreover, an added benefit of thescattered-field formulation is that only the scattered fields need to be absorbed at theperimeter of the solution volume. According to Kunz and Luebbers [45], this feature isimportant because the scattered fields can be more readily absorbed by the ABC than thecorresponding total fields; thus, the scattered-field formulation provides more accurateresults in cases where the amplitude of the scattered fields is much less than that of thetotal fields. It is for these reasons that the scattered-field approach is better suited tomodel the scattering of incident waves impinging upon material objects. To the contrary,our application requires us to model a given radiation structure that is considered the solesource of any field disturbances (i.e., no incident fields). By internally exciting the 210radiation structure, the total-field formulation allows the field solver to directly compute the total radiated fields. The total-field formulation is a special case of the scattered-fieldformulation where the incident fields are set to zero; that is, the scattered-fieldformulation inherently becomes the total-field formulation in the absence of incidentfields. Accordingly, the total-field approach is the most logical choice for our purposes ofvalidating the developed NZ-FZT algorithms. It is important to note that each of theFDTD formulations detailed in the sections that follow is particular to the total-fieldformulation and reflects the fact that the incident E- and H-fields are assumed to be zero.Finally, we dispense with the combination approach because it too, like the scattered-fieldformulation, is better suited for the modeling of scattering problems. 6.1.2 Free-Space (or Non-Lossy Dielectric) E-Field Formulation In applications where FDTD is used to model electromagnetic radiation, a portion of the solution volume must be reserved for a free-space buffer that completely surroundsthe entire radiation structure. This spatial buffer must be present in order to properlysimulate the interaction of the radiation structure with free-space; without it, the modeledradiation structure improperly reacts with the ABC (situated at the perimeter of thesolution volume) to yield inaccurate numerical results. Hence, all E-field samples that arepart of this spatial buffer must be computed using a FDTD formulation that correspondsto free-space. In addition, this free-space formulation not only applies to E-field samplessituated in the spatial buffer, but also to E-field samples that represent the absence ofmaterials internal to a radiating structure (e.g., the internal space of a horn antenna). In order to get a basic understanding of the rationale behind the free-space E-field formulation, we begin by considering the time-dependent differential form of theMaxwell-Ampere equation for free-space (assuming σ=0) : ()∂ ∂εtEH/G26/G26 =∇ ×1 0. (6-1) 211As we can see, the vector equation contains derivatives with respect to both time and space (with the curl operator giving rise to the spatial derivatives). Operating in Cartesiancoordinates, we are able to separate the vector equation of (6-1) into its three scalarcomponents. By applying finite-differencing to approximate the temporal and spatialderivatives of each of the resulting scalar equations, we obtain the following FDTDupdate equations for free-space (using the total-field formulation) [45]: E i jk E i jkxn xn ++= +1 1 21 2 (, , ) (, , )     ∆− + −+ +−∆− + − + + ∆++ ++ + zkj i H kj i Hyk j i Hk j i H t n yn yn zn z o ) ,, ( ) ,, (), , ( ), , ( 21 21 21 21 21 2121 21 21 21 21 21 ε(6-2) E i jk E i jkyn yn ++= +1 1 21 2 (, , ) (, , )     ∆+ − − + +−∆− + −+ + ∆++ ++ + xk j i Hk j i Hzk ji H k ji H t n zn zn xn x o ), , ( ), , () , ,( ) , ,( 21 21 21 21 21 2121 21 21 21 21 21 ε(6-3) E i jk E i jkzn zn ++= +1 1 21 2 (, , ) (, , )     ∆+ − −+ +−∆+ − −+ + ∆++ ++ + yk ji H k ji Hxkj i H kj i H t n xn xn yn y o ) , ,( ) , ,() ,, ( ) ,, ( 21 21 21 21 21 2121 21 21 21 21 21 ε. (6-4) Our electromagnetic field solver makes use of these update equations to compute the E- field for all samples that are situated in free-space. A given sample for each Cartesiancomponent of the E-field is computed by using its previous sample (a whole time-stepback) together with several neighboring orthogonal H-field samples (a half time-step 212back). Note how these update equations conform to the spatial and temporal format of FDTD examined earlier in Figures 6.1 and 6.2. Furthermore, this free-space formulationcan be easily made to address other non-lossy dielectric media by replacing the free-spacepermittivity with that of the desired dielectric material (i.e., εεo→). 6.1.3 Perfect Electric Conductor E-Field Formulation The parts of a radiation structure that are made of metal are typically modeled in FDTD simulations using a perfect electric conductor (i.e., σ→∞ ). For most applications, this approximation is quite valid and often yields extremely good numerical results. In which case, we need a FDTD formulation able to compute E-field samplescorresponding to this ideal material. Fortunately, we are able to make use of the fact thatthe total E-field in a perfect electric conductor must vanish; this fundamental propertythus allows us to obtain the following update equations (using the total-field formulation)[45]: Ei j kxn++=1 1 2 0 ( , , ) (6-5) Ei j kyn++=1 1 2 0 ( , , ) (6-6) Ei j kzn++=1 1 20 ( , , ) . (6-7) At each time step, our electromagnetic field solver computes the E-field samples corresponding to a perfect electric conductor by simply setting them equal to zero. 6.1.4 Lossy Dielectric E-Field Formulation The third and final medium that our electromagnetic field solver must be able to address is a lossy dielectric possessing a finite conductivity of σ and a permittivity of ε. As we shall see, simulating this particular type of material is critical in the modeling of our empirical benchmark (which is used in Chapter 7 to further validate our newlydeveloped NZ-FZT process). Consequently, we need an additional FDTD formulation 213able to compute E-field samples for a variety of lossy dielectrics. Once again, we begin to understand the rationale behind this E-field formulation by evoking the time-dependentdifferential form of the Maxwell-Ampere equation; however, this time assuming anonzero finite conductivity (i.e., 0 << ∞ σ ): ()∂ ∂σ εε tEE H/G26/G26 /G26 =− + ∇×1. (6-8) Accordingly, the vector equation now contains an additional term when compared to equation (6-1) that is proportional to the conduction current density, i.e., /G26/G26 JE=σ. By separating the vector equation of (6-8) into its three scalar equations (in Cartesian coordinates) and then applying finite-differencing to approximate the temporal and spatialderivatives, we obtain the following FDTD update equations for a lossy dielectric (usingthe total-field formulation) [45]: Ei j ktEi j kxn xn ++=+   +1 1 21 2 (, , ) (, , )ε εσ ∆     ∆− + −+ +−∆− + − + + ∆+∆++ ++ + zkj i H kj i Hyk j i Hk j i H tt n yn yn zn z ) ,, ( ) ,, (), , ( ), , ( 21 21 21 21 21 2121 21 21 21 21 21 σε(6-9) Ei j ktEi j kyn yn ++=+   +1 1 21 2 (, , ) (, , )ε εσ ∆     ∆+ − − + +−∆− + −+ + ∆+∆++ ++ + xk j i Hk j i Hzk ji H k ji H tt n zn zn xn x ), , ( ), , () , ,( ) , ,( 21 21 21 21 21 2121 21 21 21 21 21 σε(6-10) 214Ei j ktEi j kzn zn ++=+   +1 1 21 2 (, , ) (, , )ε εσ ∆     ∆+ − −+ +−∆+ − −+ + ∆+∆++ ++ + yk ji H k ji Hxkj i H kj i H tt n xn xn yn y ) , ,( ) , ,() ,, ( ) ,, ( 21 21 21 21 21 2121 21 21 21 21 21 σε. (6-11) During the formulation of these update equations, it is important to use the most recent samples of the E-field, viz., Ei j kxn++1 1 2(, , ) , Ei j kyn++1 1 2 ( , , ) , or Ei j kzn++1 1 2 (, , ), t o determine each respective Cartesian component of the term ( −σ ε/G26 E); Kunz and Luebbers [45] stress that this approach is key to obtaining numerical stability for large conductivity values. Note how in the limiting cases of σ→0 and σ→∞ , these FDTD update equations reduce to those for free-space (or non-lossy dielectric), i.e., (6-2)-(6-4), and those for the perfect electric conductor, i.e., (6-5)-(6-7), respectively. Our electromagnetic field solver makes use of update equations (6-9)-(6-11) to compute the E-field for all samples corresponding to a lossy dielectric. Different valuesof σ and ε are then used to represent each lossy dielectric that is part of a given FDTD simulation. In a manner that is quite similar to the formulation for free-space, each individual Cartesian component of the E-field is computed by using its previous sample(a whole time-step back) together with several neighboring orthogonal H-field samples (ahalf time-step back). 6.1.5 Non-Magnetic H-Field Formulation Having addressed how to compute E-field samples corresponding to a variety of non-magnetic media, we now concentrate on computation of their respective H-fieldsamples. As stated earlier, our electromagnetic field solver only needs one FDTDformulation to compute all H-field samples for the three types of non-magnetic media 215under consideration. We begin by considering the time-dependent differential form of the Maxwell-Faraday equation with a non-magnetic permeability (assuming µµ=o): ()∂ ∂ µ tHE/G26/G26 =− ∇×1 0. (6-12) Once again, we are able to separate the vector equation into its three Cartesian scalar equations. Applying finite-differencing to approximate the temporal and spatialderivatives of each of the resulting scalar equations, we obtain the following FDTDupdate equations to compute all H-field samples corresponding to non-magnetic media(using the total-field formulation) [45]: Hi jk Hi jkxn xn +−++ = ++12 1 21 212 1 21 2 (, , ) (, , )     ∆+ −+ +−∆+ −+ + ∆+ ykjiE k jiEzk jiE k jiE tn zn zn yn y o ) ,,( ) ,1,(), ,( )1, ,( 21 2121 21 µ(6-13) Hi j k Hi j kyn yn +−++ = ++12 1 21 212 1 21 2 (, , ) (, , )     ∆+ −+ +−∆+ −+ + ∆+ zkj iE kj iExkjiE kj iE tn xn xn zn z o ),, ( )1,, () ,,( ) ,,1( 21 2121 21 µ(6-14) 216Hi jk Hi jkzn zn +−++ = ++12 1 21 212 1 21 2 (, , ) (, , )     ∆+ − + +−∆+ − + + ∆+ xk jiEk j iEykj iEk j iE tn yn yn xn x o ), ,( ), ,1(),, ( ),1, ( 21 2121 21 µ. (6-15) Since our particular application only requires non-magnetic media, our electromagnetic field solver uses these update equations to compute the H-field for all samples regardlessof the media type. A given sample for each Cartesian component of the H-field iscomputed by using its previous sample (a whole time-step back) together with severalneighboring orthogonal E-field samples (a half time-step back). As we can see, theseupdate equations closely resemble the form of the free-space E-field formulation; thesimilarity of the two formulations should be expected by virtue of duality. 6.1.6 Second-Order Mur Absorbing Boundary Condition The second-order Mur ABC used by our electromagnetic field solver is based upon applying an approximation of a one-way wave equation at the planar boundaries ofthe FDTD solution volume to numerically absorb impinging radiating waves. Theapproach can be traced back to a theory of one-way equations originally developed byEngquist and Majda [50] and can be understood in terms of the factoring of partialdifferential operators. Citing this theory in his seminal paper, Mur [49] devises an ABCfor the FDTD method that is both practical and effective. We begin our understanding of the second-order Mur ABC by considering the time-dependent three-dimensional wave equation [46]: ∂ ∂∂ ∂∂ ∂∂ ∂2 22 22 222 210U xU yU zcU t++− = (6-16) 217where U is a scalar field quantity and c is the free-space phase velocity (i.e., speed of light). Based on this equation, we are able to define the corresponding partial differentialoperator [46]: Lxyz c tDDDcDxyz t ≡ + + − ≡++−∂ ∂∂ ∂∂ ∂∂ ∂2 22 22 222 2222 22 11. (6-17) Thus, the three-dimensional wave equation can be concisely represented as follows [46]: LU=0. (6-18) According to Taflove, the partial differential operator L can then be factored in the following manner [46]: LU L L U==+−0 (6-19) where LDD cSxt −≡− − 12(6-20) LDD cSxt +≡+ − 12(6-21) and SD DcD Dcy tz t≡  +  22 . (6-22) At this point, we now make use of an important finding by Engquist and Majda: Application of L− to the field quantity U exactly absorbs a plane wave propagating toward the x=0 planar boundary at any incident angle [50]. In mathematical terms, LU−=0. (6-23) As a result, the L− operator applied at x=0 acts as an exact analytical ABC for plane waves impinging on this particular planar boundary of the solution volume. Likewise, the 218L+ operator proceeds to absorb plane waves propagating at an arbitrary angle toward the opposite planar boundary of the solution volume. The radical contained within equations (6-20) and (6-21) classifies both L− and L+ as pseudodifferential operators that are nonlocal in both the space and time variables [46]. Unfortunately, this particular feature prevents the direct numerical implementationof both operators as an ABC. However, the square root function can be approximated bycreating a set of conventional partial differential equations able to be numericallyevaluated using finite-differences. Since this approach is not exact, the resulting ABCdoes allow a small amount of reflection as the waves reach each planar boundary of thesolution volume. Nonetheless, Taflove [46] reports that the second-order Mur ABC isable to truncate the FDTD lattice with an overall level of spurious reflections in the rangeof 1% to 5%. This level of performance proves to be sufficient for our purposes ofproviding further validation of the newly developed NZ-FZT algorithms. For the sake of clarity and brevity, we limit our discussion to the planar boundary of the solution volume situated at x=0 . As we shall see, the equations for the first- and second-order Mur ABCs at this particular boundary are easily extended to formulate thoserequired by the five other planar boundaries of the solution volume. Note that the first-order version of the ABC has been included in our discussion because it is still requiredby a FDTD simulation even if the second-order Mur ABC is being applied [45]. We begin by considering the one- and two-term Taylor series approximations for the square root function contained in equation (6-20) [46]: 112−≅S (first-order) (6-24) 111 222−≅ −SS (second-order). (6-25) Applying these one- and two-term expansions to (6-20), we obtain the following approximations for the L− operator [46]: 219LDD cxt −≡− (first-order) (6-26) LDD ccD DcD Dxt y tz t−≡−+ +2 2 22(second-order). (6-27) These equations are then used to approximate the L− operator of equation (6-23) to yield the following first- and second-order ABCs at the x=0 planar boundary [46]: ∂ ∂∂ ∂U xcU t−=10 (first-order) (6-28) ∂ ∂∂∂ ∂∂ ∂∂ ∂22 22 22 21 220U xt cU tcU ycU z−++= (second-order). (6-29) In order to apply these mathematical ABCs, they must first be put into a numerical form that is compatible with the FDTD method. Fortunately, Mur [49] is able to do just this byfinding a stable finite-difference scheme to discretize both of these partial differentialequations. Partitioning the solution volume into electrically defined Yee cells as described earlier causes E y and Ez to be the only electrical field components at the x=0 planar boundary (see Figure 6.1). Having knowledge of these two field components along this planar boundary is sufficient to compute the Hx-field samples located at x=0 as well as the Ex-, Hy-, and Hz-field samples located at xx=∆ 2. Unfortunately, both Ey and Ez components at the x=0 planar boundary cannot be calculated using the standard FDTD update equations because the needed H-field samples at xx=−∆ 2 lie outside the solution volume. As a result, both of these field components are computed at the x=0 planar boundary using Mur ’s finite-difference expressions to numerically approximate the mathematical ABCs of (6-28) and (6-29). During the development of the finite-difference expressions, Mur chooses to approximate the partial differential equations of (6-28) and (6-29) using central 220differencing both in time and space to expand the partial derivatives about an auxiliary sample point. Although the auxiliary sample points are spatially the same for both thefirst- and second-order Mur expressions, they do differ with respect to time: (,, )1 21 212jkn++ for the first-order ABC (6-28); and (,, )1 21 2 jkn+ for the second-order ABC (6-29). Utilizing this approach, we obtain the first- and second-order Mur finite- difference expressions for the Ez samples situated at the x=0 planar boundary [45]: Ej kzn++=1 1 2 0(,, ) Ej kzn(, , )11 2+ [] +− ++− ++ ct x ct xEj k E j kzn zn ∆∆ ∆∆1 1 21 2 10( , , ) ( , , ) (first-order) (6-30) Ej kzn++=1 1 2 0(,, ) −+−Ej kzn1 1 2 1(, , ) [] +− +++ ++− ct x ct xEj k Ej kzn zn ∆∆ ∆∆1 1 21 1 2 10( , ,) ( , ,) [] ++++ +2011 21 2∆ ∆∆x ct xEj k E j kzn zn( , ,) ( , ,) ++++ − + +− + ++ + −+ + − +      ∆∆ ∆∆ ∆xc t yc t xEj k Ej k Ej k Ej k Ej k Ejkzn zn zn zn zn zn() () ( )(, , ) (,, ) ( , ,) ( , ,) (, , ) (, , )2 21 21 2 1 21 2 1 21 2201 20 01 11 21 11 +++− + +− ++ −+ + −      ∆∆ ∆∆ ∆xc t zc t xEj k Ej k Ej k E j k Ej k Ej kzn zn zn zn zn zn() () ( )(,, ) (,, ) (,, ) ( ,, ) (, , ) (, , )2 23 21 2 1 23 2 1 21 2202 0 01 21 1 (second-order). (6-31) Regarding the corresponding Mur expressions for the Ey samples situated at the x=0 planar boundary, both can be found by applying the same central differencing approach. 221Generally speaking, we can now repeat the entire procedure (with the proper modifications) to determine the corresponding first- and second-order Mur expressionsfor the remaining five planar boundaries of the solution volume. As a matter of fact, thefirst- and second-order Mur expressions required by these other planar boundaries can besubsequently found by properly modifying expressions (6-30) and (6-31) using coordinatesymmetry arguments to permute all subscripts and spatial indices [46]. Upon inspection of (6-30) and (6-31), we point out that the first-order Mur expression only requires samples from a single time-step back while the second-orderMur expression needs samples from two time-steps back. Moreover, the second-orderexpression also requires samples from adjacent Yee cells (in both directions along theplanar boundary to approximate the second-order spatial derivatives). Consequently, it isfor this reason that the second-order ABC cannot be applied at the intersection of any twoplanar boundaries. To sidestep this difficulty, we make use of the approachrecommended by Kunz and Luebbers [45]: The first-order Mur expressions update allfield samples situated at the edges of the rectangular solution volume; meanwhile, thesecond-order Mur expressions update the other remaining samples along the faces of theplanar boundaries. 6.1.7 Modeling Guidelines Equipped with a fundamental understanding of FDTD, we now review key modeling guidelines that need to be observed when modeling a radiation structure usingthis computational method. Although there is always room for exceptions dependingupon the application, compliance with these guidelines, more often than not, will allow usto escape the many numerical pitfalls that often crop up during a FDTD simulation.Development of these modeling guidelines is the product of many years of intense workconducted by the research community. As a result, we have prudently chosen to takeadvantage of this research and construct all of our FDTD models (in this chapter and thenext) in a manner that carefully adheres to these established guidelines. 222We begin by understanding the choice of Yee cell size for a given FDTD simulation. Fundamentally, the size of the cell must be significantly smaller that that ofthe shortest wavelength under consideration in order to obtain accurate results [45]. Ofcourse basic sampling theory does impose a limit on the cell size based on the Nyquistsampling theorem (to avoid any aliasing): ∆∆ ∆xy z, ,and ≤λ 2. (6-32) Nevertheless, it is important to understand that this limit is only a theoretically maximum for size of the cell. Instead, Kunz and Luebbers [45] advise using a cell size thatcorresponds to ten or more samples (cells) per wavelength at the highest frequency (orshortest wavelength) of interest in order to achieve accurate and stable results: ∆∆ ∆xy z, ,and ≤λ 10. (6-33) Because FDTD is a volumetric computational method, the wavelength used in the modeling guideline of (6-33) must correspond to that which exists in the mostelectrically-dense penetrable medium that is part of a given simulation [45]. Accordingly,models containing dielectric materials will need to have smaller cells than thosecontaining only free-space and perfect electrical conductors. Failure to observe thisparticular modeling guideline can result in grid dispersion errors that can severely impactthe accuracy and stability of a given simulation (depending on the structure beingmodeled). Another factor that must be considered when choosing the cell size is the geometry of the structure under consideration. In cases where the structures have minutegeometrical features, the cell size must be made small enough to properly model theintricate details. When modeling such electrically detailed structures, one mustunderstand that all cells in the solution volume are forced to be relatively small; 223consequently, the total number of cells required to partition the solution volume may be significantly increased depending upon the structure being modeled. Having understood the factors behind choosing the cell size for a particular FDTD simulation, we now turn our attention to determining the maximum duration for eachtime-step ∆t. From a finite-difference perspective, the choice of time-step duration is crucial to establishing numerical stability for a given simulation. In short, a plane wave propagating through the FDTD lattice (of samples) must not pass through more than 1cell during a single time-step [45]. Consequently, the length of each time-step ∆t must comply with the stability criterion for the full three-dimensional Yee algorithm (rigorously derived by Taflove and Brodwin [51]): () () ()∆ ∆∆∆t c xyz≤ ++1 111 222(6-34) where c represents the maximum phase velocity for a given FDTD simulation. For our application of modeling radiating structures situated in free-space, the maximum phasevelocity c is taken as the speed of light. Kunz and Luebbers [45] indicate that their experience has shown that for actual computations, the ∆t value given by the equality of (6-34) does yield accurate results. Ironically, in most circumstances a smaller time-step does not necessarily provide results that are more accurate. As a matter of fact, they state that when the equality holds,simulated wave propagation most closely approximates actual wave propagation such thatgrid dispersion errors are minimized [45]. Nonetheless, it must be mentioned that thereare always certain circumstances where numerical stability requires time-steps that aresmaller than this limiting value. The interested reader is referred to [45] for detailsregarding these special cases. Fortunately, these anomalies have no direct bearing on ourparticular application; thus, we use the equality of (6-34) to compute the time-stepduration ∆t for all of our FDTD simulations (in this chapter and the next). 224The final modeling guideline needed to be addressed is the distance the radiation structure must be from the boundaries of the solution volume. The more distant themodeled structure is from the ABC (situated at the boundaries of the solution volume) thegreater the absorption of outward traveling waves [45]. This improvement inperformance occurs because outward traveling waves become much more like planewaves as they get farther from the radiation structure. However, increasing the size of thefree-space buffer does not come without the expense of more computation time andgreater memory storage. Keeping this engineering tradeoff in mind, Kunz and Luebbers[45] quote a minimum of 10 cells between the radiation structure and all boundaries ofthe solution volume. Failure to comply with this modeling guideline can result ininstabilities of the ABC implementation. 6.2 NZ-FZT/FDTD SOFTWARE DEVELOPMENT 6.2.1 Overview of Software Using the FDTD approach described in the preceding sections along with the second-order Mur ABC, we are now able to put together a computational electromagneticfield solver capable of meeting the demands of our validation objective. We start thisdevelopment by building upon an already existing single precision FDTD scattering codeput forth by Kunz and Luebbers [45]. It is through modifications and additions to thiscode that we are able to construct a field solver capable of being coupled with our NZ-FZT computer software. The code already applies the desired second-order Mur ABC at the boundaries of the solution volume using the procedure described earlier (see 6.1.6). However, duringthe computation of the field samples, the existing code makes use of formulations basedon scattered-fields only and thus requires a specified incident E-field as its excitation.Because our application requires us to model a radiation structure that is the sole sourceof any field disturbances, we proceed to modify the code so that it uses the total-field 225formulations (described in the preceding sections) instead to update all field samples. Furthermore, the FDTD field solver must now have the ability to handle localized fieldsources so that the modeled radiation structure can be excited. This capability isnecessary for our particular application because we are not using a specified incidentplane wave as the excitation mechanism. Included in this basic feature is the ability touse a variety of spatial configurations and temporal waveforms in order to model almostany localized field source (electric and/or magnetic) that presents itself. Regarding the interface between the field solver and our NZ-FZT computer software, it is accomplished by spatially sampling the near-zone E-field along theprolate/oblate spheroidal transformation surface using the spheroidal sample gridspecified in equations (2-106a)-(2-109a) (see also Figure 2.2). However, the FDTDalgorithm delivers E-field samples corresponding to a Cartesian lattice that conforms tothe geometry specified in Figure 6.1. Fortunately, we are able to reconcile this differencein sample grids by using a multidimensional interpolation scheme to extract the near-zoneE-field samples along the spheroidal surface from their neighboring Cartesian samples.The extracted near-zone E-field samples are then converted to the frequency-domain for aspecified frequency by applying a complex running discrete Fourier transform (DFT). Atthe end of the FDTD simulation, the field solver renders samples of the near-zone E-fieldalong the prolate/oblate spheroidal surface at this specified frequency. (Note thatmultiple concurrent running DFTs can be used if additional frequencies are desired.)This near-zone information is then passed from the field solver to the NZ-FZT computersoftware by way of a single precision ASCII data file. Further details behind thisinvolved extraction process are given in the section that follows. Depicted in Figure 6.3 is a top-level flowchart indicating the overall process implemented by the FDTD electromagnetic field solver. Initially, the developed softwarereads the simulation setup (specified by the user) from multiple ASCII input data files.Contained in these files is the following setup information: (1) dimensions of the cell sizeand the solution volume; (2) definition of dielectric materials used; (3) configuration ofelectrical materials to form the radiation structure; (4) configuration of electric and 226StartRead in Simulation SetupCompute Constants Update Equations Are all Time-Steps Computed? Write Extracted Near-Zone to Output FileYes NoStop E-Field Samples Update E-field Samples Apply Electric Sources Apply Second-Order Mur ABC Apply Magnetic Sources Extract Near-Zone E-field Samples Spheroidal Surfacealongfor (including ABC)Compute Interpolation Arrays & Pointers Build Radiation StructureCheck Simulation SetupStructure & Update H-field Samples (Time-Domain)(Frequency-Domain) Figure 6.3 Flowchart of FDTD electromagnetic field solver. 227magnetic sources (spatially and temporally); (5) number of time-steps in the simulation; (6) choice of prolate or oblate spheroidal transformation surface; (7) location and shape ofthe prolate/oblate spheroidal surface; (8) fineness of corresponding spheroidal samplegrid; and (9) configuration of the DFT (viz., observation interval, window type, andfrequency index at which the near-zone E-field samples are to be extracted). After reading all of this information, the field solver begins by computing the parameters necessary to launch a successful FDTD simulation of the radiation structure.First, the interpolation arrays and pointers associated with the near-zone E-fieldextraction process must be computed and stored. This step is followed by computing andstoring the constants required to implement the FDTD update equations (described earlierin this chapter) as well as those required for the first- and second-order Mur ABCs. Forobvious reasons, storage of all of this information at the start of the process has adramatic effect on reducing total computation time. The field solver then proceeds to build the specified radiation structure by locating differently defined electric materials at various points of the lattice as per the user’sspecifications. This step is accomplished by creating a set of pointers for each Cartesiancomponent that indicates the material located at each lattice point. When updating the E-field samples, the field solver looks at these material pointers to determine the properFDTD formulation to use. In addition, the field solver also utilizes these pointers toefficiently retrieve from storage the constants necessary to implement each respective E-field formulation. The last step before starting the computation of the E- and H-field samples is to check that the modeled radiation structure and corresponding simulation setup are free oferrors due to invalid input data. By checking the data at the start of a simulation, the fieldsolver is able to flag the user of obvious errors before any lengthy computations are made.Toward this end, this step in the process can help avoid many needless hours ofcomputation time. With the preliminary computations and the simulation setup completed, the field solver is ready to implement the two-step finite-difference process described earlier in 228Figure 6.2. For our application, it is assumed that all E- and H-fields are at rest before the start of the simulation. The field solver begins the computations associated with eachtime-step by first updating the set of E-field samples. For each E-field sample, the properformulation must be used based upon the material present at its respective lattice point(specified by the material pointers). If any electric sources are present, they are at thistime applied to the simulation by forcing all E-fields that make up a given spatialconfiguration to values that depend on a user specified temporal waveform. Followingthis, the field solver then proceeds to apply the second-order Mur ABC to update the E-field samples situated at the boundaries of the solution volume. We recall that the E-fieldsamples located at the edges of the solution volume (i.e., the intersection of two planarboundaries) must be computed using the corresponding first-order Mur expressions. Thesecond part of this two-step finite-difference process requires computation of thecorresponding H-field samples. Because our application only addresses non-magneticmedia, the field solver always uses the non-magnetic H-field formulation to update all H-field samples. It is at this point in the procedural sequence that magnetic sources, if any,are applied to the corresponding H-field samples in much of the same way that theelectric sources are handled. With the computation of all E- and H-field samples for agiven time-step completed, the field solver now proceeds to interpolate the near-zone E-field samples along the spheroidal surface using neighboring Cartesian E-field samples.All interpolated values then contribute to their own individual running complexsummation that ultimately implements a DFT at the end of the simulation run. Thisentire two-step finite-difference process is then repeated until the desired number ofcomputed time-steps has been performed. At the completion of this loop, the field solverproceeds to place the extracted near-zone E-field phasor data (along the prolate/oblatespheroidal transformation surface) in a single precision ASCII data file that is suitable forprocessing by our NZ-FZT software. 2296.2.2 Extraction of Near-Zone E-Field Samples Along the Prolate/Oblate Spheroidal Surface As previously mentioned, a multidimensional interpolation scheme is used to extract near-zone E-field samples along the spheroidal surface from the Cartesian samplesprovided by the FDTD algorithm. Our field solver evokes this interpolation scheme eachand every time-step to form the required set of near-zone E-field samples that coincideswith the spheroidal sample grid. Each of these samples is then converted to thefrequency-domain for a particular frequency through the use of a complex running DFT.In short, all three vector components (Cartesian) of each spheroidal E-field samplecontribute on-the-fly (at each time-step) to their own individual complex runningsummations such that their DFTs (for a specified frequency) are effectively implementedat the end of the simulation. As required by our NZ-FZT software, the final result of thisextraction process is a phasor representation (corresponding to the specified frequency) ofthe near-zone E-field samples along the spheroidal transformation surface (coincidingwith the user specified spheroidal sample grid). The first step in the interpolation process is to assign Cartesian coordinates to the spheroidal sample grid as well as to the FDTD three-dimensional lattice of the E-field.Initially, the user defines the shape of the prolate/oblate spheroidal surface by providing the parameters ξo and a (both introduced in Chapter 2), and specifies its location by defining its center with respect to the reference corner of the solution volume, viz., the corner that corresponds to the origin. In addition to these parameters, the number ofsamples in the azimuthal direction and the number of samples in the angular direction arethen supplied in order to specify the desired fineness of the spheroidal sample grid.Using this information together with the respective prolate/oblate spheroidal-to-Cartesiantransformation provided in equations (2-1)-(2-5), the field solver then computes theCartesian coordinates of the spheroidal sample grid in relation to the reference corner ofthe solution volume (using the proper translation vector to account for the location of thespheroidal surface). Whereas only one set of Cartesian coordinates is required to describethe spheroidal sample grid, three sets of Cartesian coordinates are necessary to describe 230the FDTD E-field lattice. Referring to Figure 6.1, one can see that the need for three sets of coordinates arises from the fact that the spatial locations of the E-field samples areoffset with respect to one another. That is, the three-dimensional E-field lattice can, ineffect, be viewed as three separate grids (offset from one another) with eachcorresponding to a vector component of the computed E-field. Cartesian coordinates foreach of these grids are easily found by employing the definition of the Yee cell geometrydiscussed earlier in this chapter (see 6.1.1). In order to maintain a coordinate system thatis consistent with the spheroidal sample grid, the coordinates of these three separate gridsare also computed in relation to the reference corner of the solution volume. Mathematically speaking, the FDTD algorithm delivers sampled representations of three distinct scalar functions (for a given time-step) that happen to correspond to the vector components of the computed E-field: Ei j kxn++1 1 2(, , ) , Ei j kyn++1 1 2 ( , , ) , and Ei j kzn++1 1 2 ( , , ) . Meanwhile, the desired spheroidal samples (in the time-domain), i.e., Exn ij+′′1(, )ϑϕ , Eyn ij+′′1(, )ϑϕ , and Ezn ij+′′1(, )ϑϕ , represent samples of the same three scalar functions that are spatially situated in-between these computed data points. (Note that although both sets of E-field samples are referenced using the indices i and j, the symbols represent entirely different sample indices. Ambiguity is easily avoided byconsidering the context of usage.) Accordingly, we are able to utilize an interpolationscheme in three-dimensions to approximate these in-between values for each vector component at a given time-step: more precisely, we find E xn ij+′′1(, )ϑϕ through interpolation of Ei j kxn++1 1 2(, , ) , Eyn ij+′′1(, )ϑϕ through interpolation of Ei j kyn++1 1 2 (, , ) , and Ezn ij+′′1(, )ϑϕ through interpolation of Ei j kzn++1 1 2 (, , ). Because each vector component makes use of the same interpolation process, we need only discuss the method with respect to a given sampled scalar function g (where g represents any one of the field components Ex, Ey, and Ez computed by the field solver). The interpolation process continues by overlaying the spheroidal sample grid with the Cartesian grid corresponding to the field component under consideration. By 231comparing the Cartesian coordinates of these two overlaid grids, the field solver is able to determine which eight FDTD data points immediately surround each spheroidal sample.For the sake of clarity, the geometry of this three-dimensional interpolation scheme for a given spheroidal sample gg x y z sn ss sn ++=11( , , )| is illustrated in Figure 6.4. Employing a shorthand notation, we designate the values of the eight neighboring data points (for a given time-step) with the variables gn 11+, gn 21+, ... , gn 81+ and define them as follows with respect to the scalar function g: gg x y zn ij kn 111++=(, , ) | , gg x y zn ij kn 21 11 + ++=(, , ) | , gg x y zn ij kn 31 111 + +++=(, , ) | , gg x y zn ij kn 41 11 + ++=(, , ) | , gg x y zn ij kn 51 11 + ++=(, , ) | , gg x y zn ij kn 61 111 + +++=(, ,) | , gg x y zn ij kn 71 11 11 + ++ ++=(, , ) | , gg x y zn ij kn 81 111 + +++=(, , ) | . (Note how the configuration of these eight data points corresponds to the arrangement shown in Figure 6.4.) The next step in the interpolation process is to determine where each spheroidal sample is situated in relation to its set of eight neighboring data points. For eachspheroidal sample point, we do this by creating the following three auxiliary parameters: txx xxxx xsi iisi≡− −≡− +1 ∆(6-35) uyy yyyy ysj jjsj≡− −≡− +1 ∆(6-36) vzz zzzz zsk kksk≡− −≡− +1 ∆. (6-37) (Note that the variable t does not refer to time in this context.) In essence, these auxiliary parameters represent the fractional distance (with respect to all eight data points) between 232∆x∆yy zx z∆ zk+1zk xixi+1yjyj+1g3n+1 g1n+1g7n+1 g6n+1g2n+1g8n+1 g5n+1g4n+1 gsn+1(x ,y ,z )ss s = gn+1 Figure 6.4 Geometry of the three-dimensional interpolation scheme for a given spheroidal sample gsn+1. 233the spheroidal sample 1+n sg and the reference data point gn 11+. Furthermore, parameters t, u, and v respectively represent all three Cartesian components of this fractional distance. As a result, these auxiliary parameters must always assume values on the interval betweenzero and unity including at least one of these two endpoints. Use of an endpoint isessential to representing a spheroidal sample point that happens to lie in any of the planesthat coincide with the Cartesian sample grid under consideration. Because our FDTDfield solver uses a uniform E-field lattice, we can replace the denominators of equations(6-35)-(6-37) with their respective spatial sample periods, viz., ∆x, ∆y, and ∆z. Using these auxiliary parameters along with the set of eight neighboring data points, we are able to define the following function to interpolate a spheroidal sample at agiven time-step: = = + + 1 1|),,(n s s sn s zyxg g 1 21 1 )1)(1( )1)(1)(1(+ +− −+ − − −n ngv u t gv u t 1 51 41 3 )1)(1( )1()1( )1(+ + +− −+ − −+ − +n n ngvu t gv ut gv ut 1 81 71 6 )1( )1(+ + +−+ + −+n n ngvut gvut gvut . (6-38) This three-dimensional interpolation scheme is the natural extension of the two- dimensional version provided in Numerical Recipes in FORTRAN [37]. It can be rigorously derived by fitting a three-variable polynomial of the third degree to the eight neighboring data points. For the special case where the spheroidal sample gsn+1 is situated equidistant from all eight data points, parameters t, u, and v all assume a value of one-half, and the result is the sum of gn 11+, gn 21+, ... , gn 81+ divided by eight. Although there are higher-order interpolation techniques that can provide more accurate results (at the expense of having to conduct substantially more computations), our experience hasshown that this particular one performs quite well for our purposes. Once again, the field solver must individually carryout the described interpolation process for all three vector components E x, Ey, and Ez during each time-step. For each 234vector component, the field solver begins by finding which eight FDTD data points immediately surround each spheroidal sample (note that the spatial indices for gn 11+, gn 21+, ... , gn 81+ must be made to properly correspond to each respective FDTD sample grid). Consequently, the field solver must somehow store the information of which twenty-four data points (i.e., eight data points per vector component) are necessary tointerpolate each spheroidal E-field sample. Even though the same twenty-four Cartesiansample points are used to interpolate each spheroidal sample regardless of the time-step,storage of this information can become quite cumbersome depending upon the desiredfineness of the spheroidal sample grid. Fortunately, the arrangement of each set of eightdata points is always the same by definition (see Figure 6.4); thus, knowledge of which FDTD samples constitute g n 21+, gn 31+, ... , gn 81+ is implied based upon the reference data point gn 11+. As a result, only one pointer per Cartesian component is needed to convey the given set of surrounding eight data points. In total, the field solver only has to store three pointers per spheroidal sample point to indicate all three sets of surrounding eightdata points. As one can see, this approach is a significant improvement over storingtwenty-four pointers per spheroidal sample point. Finally, each of these pointers naturallymust contain three indices in order to properly point to a given reference data point. Having determined which twenty-four data points are necessary to interpolate each spheroidal E-field sample, the field solver must then determine where eachspheroidal sample point lies in relation to these known data points. It does this bycomputing and storing three sets of auxiliary parameters t, u, and v (using equations (6- 35)-(6-37)) with each set corresponding to the vector components E x, Ey, and Ez. In summary, the field solver requires a total of three pointers and nine auxiliary parameters per spheroidal sample point in order to conduct the interpolation for all time-steps duringa given simulation. As indicated by the flowchart shown in Figure 6.3, the field solver computes and stores this interpolation information only once during a particular FDTD simulation. Thisapproach is possible because the set of all pointers and auxiliary parameters is the same 235for all time-steps. At the end of every time-step, the field solver uses this stored information together with equation (6-38) to interpolate the near-zone E-field samplesalong the prolate/oblate spheroidal surface from the computed FDTD lattice. Inprocedural terms, the field solver performs this operation for each time-step by computing g sn+1 for all three vector components and then assigning the results to their respective spheroidal samples Exn ij+′′1(, )ϑϕ, Eyn ij+′′1(, )ϑϕ, and Ezn ij+′′1(, )ϑϕ. It is worth noting that equation (6-38) can be more efficiently implemented than that which is written by factoring and then locally storing some of its constituent subexpressions. With a grasp of the interpolation part of the extraction process, we now proceed to focus our attention on converting the interpolated time-domain data to the frequency-domain for a specified frequency. Specifically, this is done for every spheroidal gridpoint by individually applying a complex running DFT to the temporal sequences of the interpolated vector components: E xn ij+′′1(, )ϑϕ, Eyn ij+′′1(, )ϑϕ, and Ezn ij+′′1(, )ϑϕ. Thus, three separate running DFTs must be conducted for each spheroidal grid point ( , ) ′′ϑϕij . Once again, because this part of the extraction process is the same for each vector component, we need only examine it in terms of a given time-sampled sequence gsn+1 (where gsn+1 represents either Exn ij+′′1(, )ϑϕ, Eyn ij+′′1(, )ϑϕ, or Ezn ij+′′1(, )ϑϕ). The purpose of using a DFT during our extraction process is to compute the analog spectrum of the spheroidal sample gs from its corresponding time-sampled sequence gsn+1. However, it must be emphasized that the DFT will only yield an approximation of the analog spectrum belonging to gs. As we shall see, the accuracy of this approximation shall depend upon the spectral content of the sequence being processed. Consequently, the user cannot blindly apply the transform to a given samplesequence and confidently expect reasonable results. An understanding of the transformmethod as well as the sequence under consideration is required. In order to avoid the pitfalls typically associated with DFT processing, it is necessary for the user to have total control over the transform operation. Accordingly, the 236user must configure the DFT operation during the simulation setup by specifying the following transform parameters: the observation interval, the window type, and thefrequency index at which the near-zone E-field samples are to be extracted (all of whichare included in the simulation setup files). As for the sample period ∆t used during the DFT process, it naturally conforms to the time-step duration employed by the FDTD simulation (i.e., computed using the equality of (6-34)). Because g sn+1 is a time-sampled version of the scalar function gs, its spectrum actually corresponds to the spectrum of gs repeated about integer multiples of the sample rate fs (where fts=1∆) [52]. Furthermore, the spectrum of gsn+1 is also scaled in amplitude by a multiplying factor of fs as compared to the original spectrum of gs (due to the sampling process; see [52]). In other words, the periodic spectrum of gsn+1 is made up of a superposition of the scaled spectrum of gs (multiplied by fs) and an infinite number of its translated images which are each centered about integer multiples of fs. Unfortunately, this superposition can lead to unwanted overlapping of the spectrum of gs with its translated images. In cases where this unwanted effect crops up, the spectrum of gsn+1 becomes a corrupted version of the original spectrum of gs in the region where the overlapping occurs. Degradation of the spectrum due to this particular effect is often referred to as aliasing. Fundamental digital signal processing theory indicates that onemay avoid the problem of aliasing by satisfying the Nyquist sampling theorem andensuring that the sample rate f s is always greater than twice the highest frequency component contained within the original spectrum of gs (assuming that the original spectrum is bandlimited) [52]. Practical application of the DFT requires that the sequence under consideration be finite in length over some observation interval T with some sort of window function wn applied. Consequently, the resulting spectrum computed by the DFT is a version of the analog spectrum of gs that is further modified from the spectrum of gsn+1. More specifically, the spectrum delivered by the DFT is the convolution (in the frequency- 237domain) of the Fourier transform of the window function with the periodic spectrum of gsn+1 [52]. In essence, the DFT process delivers a sampled version of this convolved spectrum with samples at a discrete set of frequencies separated by the reciprocal of the observation interval, i.e., ∆fT=1 [52]. As one can see, DFT processing significantly alters the original spectrum of gs. Fortunately, with prudent and careful application of this transform, we are able to compute a fairly accurate approximation of the analog spectrum belonging to the spheroidal spatial sample gs. As always, the accuracy of the approximation can be improved at the expense of performing significantly more calculations. The user indirectly specifies the observation interval T by indicating the number of samples N to use during DFT processing. For all of our computational models (in this chapter and the next), the number of DFT samples is always made to equal thenumber of time-steps in the FDTD simulation. Thus, we shall not discriminate betweenthese two parameters for the remainder of our discussion. However, the values of bothparameters can be made to differ if the use of zero padding were to become necessary. Unfortunately, truncating (or windowing) the time-domain response of the spheroidal sample g s can cause an additional unwanted effect on the results of our DFT processing: spectral leakage. We begin by understanding that spectral components that are not part of the DFT basis set are not periodic in the observation window (that is, theirrespective periods do not fit exactly into the observation interval T ). These non-basis set spectral components cause the periodic extension of the spheroidal sample g s to exhibit discontinuities at the boundaries of the observation interval. In turn, these discontinuities are then responsible for unwanted spectral contributions (or leakage) over the entire basisset. Depending upon the sequence being processed, windowing of the spheroidal sample g s can cause spectral energy to leak from one frequency to another when using the DFT to approximate its spectrum. For a more detailed understanding of the theory behind this unwanted effect, we direct the interested reader to the comprehensive survey article onwindowing by Harris [53]. 238The basic window that is often used is the rectangular (or Dirichlet) and is defined as unity over the observation interval [53]: wn=1 where nN=−01 1,, ,/G15 . (6-39) Unfortunately, this particular window does not provide much in the way of mitigating spectral leakage. In order to minimize this unwanted processing effect, the sampledsequence can be weighted by a non-rectangular window that smoothly approaches zero atthe beginning and end of the observation interval [52]. The non-rectangular windowmust be designed to match as many orders of derivative (of the weighted data) as possibleat the boundaries of the observation interval [53]. Furthermore, the window should alsobe designed to be an even function centered about the N2 sequence position (i.e., including the implied sample at nN=; see [53]). For our particular application, we need to truncate the time-domain E-field samples along the spheroidal transformation surface (computed using the FDTDalgorithm) in a manner that is as smooth as possible. Although Harris [53] cataloguesover twenty windows able to successfully reduce spectral leakage, we will only make useof one of them to derive the benefits of non-rectangular windowing: the Hammingwindow. We choose this particular window because Ziemer et al. [52] recommend using it when wanting to resolve closely spaced frequency components while minimizingleakage from one component to another (note that it has a highest sidelobe level of -43dB and a 3 dB bandwidth of 1.30 bins [53]). Typically, the Hamming window is definedas follows over the observation interval [53]: wNnn=−  054 0462.. c o sπwhere nN=−01 1,, ,/G15 . (6-40) For each of these window functions, we must consider its normalized coherent gain ( NCG ). Harris [53] defines this parameter as the sum of all window terms normalized with respect to the total number of terms N: 239NCGNwn n=∑1where nN=−01 1,, ,/G15 . (6-41) Applying this definition to the windows given in (6-39) and (6-40), we are able to determine their respective NCG : rectangular window, NCG =10. ; and Hamming window, NCG =054. . Because all windows are defined in a manner where the weighting coefficients are between zero and unity, i.e., 0 1 ≤≤wn, the rectangular window intrinsically has the largest NCG . Thus, in general, the NCG for any other window is reduced (from unity) due to the window function smoothly approaching zeronear the boundaries. This reduction of the NCG is important because it represents a known bias on spectral amplitudes [53]. As we shall see, we will need to make use ofthis parameter to remove this bias when approximating the analog spectrum of thespheroidal sample g s. In the process of converting the interpolated time-domain data to the frequency- domain, we make use of the following definition of the DFT [52]: ()XNxekn jN n k nN =− =− ∑1 2 01 πwhere kN=−01 1,, ,/G15 . (6-42) We have elected to use this particular form of the DFT where there is a negative sign in the complex exponential because it remains consistent with our phasor (or time- harmonic) convention that assumes a tjeω+ time dependence. Secondly, the 1N factor is included outside the finite summation in order to account for the difference in amplitude between the analog spectrum and its corresponding windowed/time-sampledversion (i.e., obtained by convolving in the frequency-domain the periodic spectrum ofthe time-sampled sequence with the Fourier transform of the rectangular windowfunction). It is important to understand that the DFT operation defined in (6-42) computes the spectrum in the following format: zero frequency corresponds to k=0 ; positive 240frequencies 01 2 <<ffs correspond to values where 12≤<kN ; negative frequencies −< <1 2 0 ffs correspond to values where Nk N21<≤ − ; and the spectral position at kN= 2 corresponds to both ffs =1 2 and ffs =−1 2. Because our NZ-FZT algorithms have been designed to operate on E-field samples conforming to a tjeω+ time-harmonic convention (i.e., a positive-rotating phasor), our FDTD field solver only has to address non-negative frequencies. Recalling that the sampled frequencies are separated by thereciprocal of the observation interval, i.e., ∆∆fT N t==11 () , we use the following expression to determine which index k corresponds to the desired non-negative spectral component: fk fk Ntk==∆∆where 02≤<kN and 01 2 ≤<ffks . (6-43) (Note that all of the discussed expressions apply to both even and odd values of N. In the case where N is even, N2 is of integer value and as such, has a sequence position that coincides with an actual sample. To the contrary, when N is odd, N2 is not an integer and the sequence position actually exists in-between two samples.) We continue our discussion by understanding that the time-sampled sequence gsn+1 is a real sequence with no imaginary components. Consequently, the resulting spectrum computed by the DFT is double-sided and includes both positive and negative frequencies. However, our NZ-FZT algorithms have been designed to accept E-field samples in the frequency-domain that are single-sided and comply with the tjeω+ time- harmonic convention. Fortunately, we have a simple way to reconcile this difference in time-domain representations: double the amplitude of the DFT results and only use non- negative frequencies where 01 2 ≤<ffks . The reason for applying the factor of two can be easily understood by decomposing a real sinusoidal signal into its complex exponentials and comparing it to its time-harmonic counterpart (i.e., a positive-rotatingphasor). 241Equipped with an understanding of fundamental DFT processing, we are now able to approximate the analog spectrum of the spheroidal sample gs using the DFT operation defined in (6-42): ()GN NCGwg eskn sn jN n k nN =− =− ∑2 2 01 ()π(6-44) where 02≤<kN and 01 2 ≤<ffks . As we can see, the DFT process of (6-44) allows the user to apply additional window functions other than rectangular so that spectral leakage can be reduced.Correspondingly, the expression must be then divided by the NCG of the applied window function in order to remove its resulting bias on the spectral amplitude. All of ourcomputational models (in this chapter and the next) utilize the Hamming window asdefined in (6-40) for this part of the extraction process. As for the factor of two that isoutside the finite summation of (6-44), it is included to account for the difference in time-domain representations (i.e., phasor versus real signal format). The field solver uses the described DFT process to carry out the latter portion of the extraction process that converts the interpolated time-domain data to the frequency-domain for a single frequency. The software performs this conversion by individuallyapplying (6-44) for a given frequency index k (specified by the user) to the temporal sequences of the interpolated vector components: E xn ij+′′1(, )ϑϕ, Eyn ij+′′1(, )ϑϕ, and Ezn ij+′′1(, )ϑϕ (where in the process gsn+1 represents each vector component). This procedure is then performed for each grid point ( , ) ′′ϑϕij situated along the spheroidal transformation surface. Experience has shown that in order to get single precision results from the DFT process, the complex summation must be conducted in double precision.At the conclusion of the simulation, the results of the individual DFT processes yield theextracted near-zone E-field samples along the spheroidal surface: E xij(, )′′ϑϕ , 242Eyi j(, )′′ϑϕ , and Ezi j(, )′′ϑϕ (where in the process Gsk represents each vector component). The time-step superscript has been dropped from the notation because the extracted quantities have been converted to the frequency domain; more specifically, thenear-zone E-field samples along the spheroidal surface are now phasor quantities at thefrequency corresponding to the user specified index k (found using equation (6-43)). These extracted near-zone E-field samples are then passed from the FDTD field solver tothe NZ-FZT computer software by way of a single precision ASCII data file. Regarding the implementation of the time- to frequency-domain conversion process, one may wonder as to why we elected to use a DFT instead of a fast Fouriertransform (FFT). Conventional wisdom indicates that the FFT algorithm is always morecomputationally efficient than the DFT. This statement is true if our application requiredthe entire spectrum corresponding to all indices of k. However, this is not the case. As we have already seen, our NZ-FZT process only calls for the near-zone E-field samplesalong the spheroidal transformation surface for a single frequency. Conducting a full FFTwould involve many additional computations in order to obtain the same result as theDFT for a single frequency index. Ultimately, the field solver would end up wasting theFFT computed results for the other N−1 frequencies. Another important advantage that the DFT approach has is that it can be computed in an on-the-fly manner. For each time-step, the field solver applies therespective window coefficient to the interpolated spheroidal sample and then adds it to acomplex partial sum that is stored in a running accumulator. At the end of the simulation after all time-samples of g sn+1 have contributed to the finite summation of (6-44), the outcome of the running DFT is inherently found sitting in the accumulator. The reason for using this on-the-fly DFT method is that it greatly improves memory storageefficiency: the field solver is only required to store one complex quantity per each time- sampled sequence g sn+1 instead of N. (Note that with three individual DFTs for each spheroidal grid point ( , ) ′′ϑϕij , memory storage of N complex quantities per each time- sampled sequence can become quite cumbersome.) Finally, illustrated in Figure 6.5 is a 243Contribute to Own Running DFT Using (6-44) for Each Spheroidal Grid PointInterpolate Each Near-Zone E-field Sample along Spheroidal Surface Using (6-38)Interpolate Each Near-Zone E-field Sample along Spheroidal Surface Using (6-38)Update E- and H-field Samples FDTD Algorithmas per Interpolate Each Near-Zone E-field Sample along Spheroidal Surface Using (6-38) Contribute to Own Running DFT Using (6-44) for Each Spheroidal Grid PointContribute to Own Running DFT Using (6-44) for Each Spheroidal Grid Point YesAre all Time-Steps Computed?NoNEi j kxn++1 1 2(, , ) Ei j kyn++1 1 2 (, , ) Ei j kzn++1 1 2 (, , ) Exn ij+′′1(, )ϑϕ Eyn ij+′′1(, )ϑϕ Ezn ij+′′1(, )ϑϕ (time-domain) (time-domain) (time-domain) Partial Sum of DFTs Partial Sum of DFTs Partial Sum of DFTs Completed DFTs Completed DFTs Completed DFTs Exi j(, )′′ϑϕ Eyij(, )′′ϑϕ Ezi j(, )′′ϑϕ (frequency-domain for a single frequency)(frequency-domain for a single frequency)(frequency-domain for a single frequency) Figure 6.5 Block diagram depicting the overall extraction process of near- zone E-field samples along the prolate/oblate spheroidaltransformation surface. 244block diagram depicting the overall extraction process of near-zone E-field samples along the prolate/oblate spheroidal transformation surface. 6.3 NZ-FZT/FDTD END-TO-END TEST EMPLOYING THE ANALYTICAL BENCHMARK 6.3.1 NZ-FZT/FDTD Model of the Analytical Benchmark Once again, the objective of this end-to-end test is to validate and assess the overall performance of the combined NZ-FZT/FDTD software. As we shall see, thecombined software performs extremely well when computing the far-zone E-field of theanalytical benchmark dipole. It is because of this level of performance that we have agreat deal of confidence when modeling other radiation structures with our combinedsoftware package. In addition, by characterizing the accuracy of the NZ-FZT/FDTDmodel of the analytical benchmark, we also gain some insight into how well thecombined software package models radiation from our empirical benchmark (see Chapter7). In order to maintain a control in our numerical experiment, we choose to make use of the same test setups that are employed during our centered dipole numerical tests(see 5.4.1 for the specifics). However, this time the near-zone E-field along thespheroidal surface is computed by our FDTD model of the analytical benchmark dipole(using the developed field solver) as opposed to the analytical benchmark software driver.Conducting the end-to-end test in this fashion allows us to understand how much of theresulting error in the combined process truly stems from the FDTD portion of our NZ-FZT/FDTD model. We continue assembling our end-to-end test by developing the required NZ- FZT/FDTD model of the benchmark dipole. From Chapter 5, the z-directed dipole is ofinfinitesimal radius with a sinusoidal current distribution. To be more specific, thedistribution of the current along the filament dipole of length l is defined in terms of the free-space propagation constant k as follows [27]: 245IzIklzz l Iklzl zo o()sin , sin .=−    ≤≤ +    −≤ ≤   202 220(6-45) Note that it is this particular current distribution that leads to the closed form representation of the near-zone E-field presented in equations (5-1)-(5-6). Thus, in orderto simulate the analytical benchmark dipole, our model must emulate a filament ofcurrent that obeys the distribution of (6-45). We do so by representing samples of currentalong the dipole with a series of five magnetic loops that surrounds a perfect electricconductor (PEC) core. However, before we can proceed, the layout of the FDTD solutionvolume must first be considered. The first step in developing the setup of the solution volume is determining the size of the Yee cells. For the sake of simplicity, we shall choose to make the free-spacewavelength of our NZ-FZT/FDTD model equal to unity, i.e., λo m=1 . Since we want five magnetic loops to exactly represent the tenth of a wavelength dipole, it is necessary to employ 4 cells over its entire length in the z-direction. To achieve this resolution, theFDTD spatial sample period in the z-direction must be as follows: ∆zmo== ×− (. ).01 42 500000 102 λ. (6-46) Regarding the spatial sample periods in the x- and y-directions, they have been made to equal one another and are taken as follows: ∆∆xy mo=== ×− λ 156 666667 102. . (6-47) Note how all three spatial sample periods satisfy the modeling guideline of (6-33) that suggests always using a cell size that corresponds to ten or more samples (cells) perwavelength. 246Having addressed the cell size for our NZ-FZT/FDTD model of the benchmark dipole, we now must determine the overall size of the solution volume. In doing so, wemust ensure that the FDTD solution volume is broad enough to accommodate the largestspheroidal surface used during our end-to-end test, viz., the oblate spheroidal transformation surface of a o =10.λ and ξo=080. . In order to keep the test setup consistent with the centered dipole numerical tests of the previous chapter, the model of the dipole (five magnetic loops surrounding a PEC core) and the spheroidaltransformation surface both have their centers coincide with that of the rectangular FDTDsolution volume. Since we desire at least a 2 cell margin between the surface and theboundaries of the solution volume, our model makes use of the following number of cellsin each dimension: 44 cells in both the x- and y-directions, and 69 cells in the z-direction.As we can see, the modeled dipole has more than 20 cells between itself and theboundaries of the solution volume; this is well beyond the 10 cell minimumrecommended by Kunz and Luebbers [45]. At this point in our discussion, we now direct the reader to Figure 6.6 in order to get a better understanding of the FDTD model of the analytical benchmark dipole. Just asdescribed earlier, the current along the z-directed dipole is approximated by using a seriesof five magnetic loops that surrounds a PEC core. During the FDTD simulation, it isthese five magnetic loops that actually serve as the source of the radiating fields. Eachmagnetic loop is 1 cell in width in both the x- and y-directions and surrounds a z-directed E-field sample that is equal to zero, i.e., E Sn+=10 . We set this enclosed z-directed E- field sample to zero by telling the FDTD field solver that there is PEC material located at the corresponding grid point. The five magnetic loops are uniformly distributed along thefilament dipole (separated by ∆z) as follows: loops 1 and 5 are exactly at the ends of the dipole; loops 2 and 4 are in from the ends of the dipole by a quarter of its total length; and loop 3 is exactly at the midpoint of the dipole. Accordingly, the current through each ofthe magnetic loops is made to obey the sinusoidal current distribution defined in (6-45)where the central loop, i.e., loop 3, is taken as the center reference. As for the four H- 24744 cells44 cells69 cellsHSn 512+/ HSn 512+/HSn 512+/ HSn 512+/ HSn 412+/ HSn 412+/ HSn 412+/HSn 412+/ HSn 312+/HSn 312+/ HSn 312+/HSn 312+/ HSn 212+/ HSn 212+/ HSn 212+/HSn 212+/ HSn 112+/ HSn 112+/ HSn 112+/HSn 112+/ESn+1 ESn+1 ESn+1ESn+1ESn+1 z xy Figure 6.6 FDTD model of analytical benchmark dipole that is used during the NZ-FZT/FDTD end-to-end test (diagram not toscale). 248field samples that make up each magnetic loop, they are considered equal in magnitude because the dipole is rotationally symmetric and the loop forms a square contour. We arethen able to use the integral form of the Maxwell-Ampere equation (with thedisplacement current taken as zero due to the PEC core) to determine the four H-fieldtime-domain sources that make up each magnetic loop: H Sn 1120+=/(6-48) HI xlz f n tSn o oe x c 212 1 2422 2+=   −+/sin( ( )) cos( ( ) )∆∆∆ πλ πλ (6-49) HI xlf n tSn o exc 312 1 24222+=   +/sin( ( ) cos( ( ) )∆∆ ππλ (6-50) HI xlz f n tSn o oe x c 412 1 2422 2+=   −+/sin( ( )) cos( ( ) )∆∆∆ πλ πλ (6-51) HSn 5120+=/(6-52) where nN=−01 1,, ,/G15 . (Note that ∆t is computed using the equality of (6-34) and spatial sample periods defined in (6-46) and (6-47).) It is important to understand that these expressions are formulated assuming a source H-field that is uniform along the entire length of the square contour.The H-field sources associated with loops 1 and 5 are zero because the current mustvanish at both ends of the filament dipole in accordance with the distribution of (6-45).As for loops 2, 3, and 4, the sine term contained within each of their respectiveexpressions comes directly from the distribution of (6-45) and is based on the position ofthe magnetic loop with respect to the center of the dipole. Recalling that FDTD is a time- domain technique, we have included the cos( ( ) )21 2 πfn texc +∆ term in the expressions of (6-49)-(6-51) so that we may excite the NZ-FZT/FDTD model at a particular frequency of 249interest. Notice how the sampling of the cosine term is delayed by a half time-step; we do so in order to properly account for the half time-step difference between the E- and H-field samples. It is important to point out that disregarding this offset in time will causephase errors in the final results that could have easily been avoided. In addition, whenimplementing each magnetic loop, we must be careful to properly account for thedirection of the four H-field sources along the square contour in relation to the FDTDfield convention; that is, we must apply the proper sign to each H-field source inaccordance with the right-hand rule. Regarding the value of I o to use when implementing the H-field sources of (6-49)-(6-51), we have elected to use IAo=1 in order to remain consistent with the centered dipole numerical tests of Chapter 5. The next step in setting up our NZ-FZT/FDTD model of the benchmark dipole involves configuration of the DFT and the choice of excitation frequency. Using theequality of (6-34) together with (6-43), we obtain the following expression for finding thefree-space wavelength that corresponds to the center of the DFT frequency bin: λbink Nx y z=   ++  − − 1 22 21 2 111 ∆∆∆. (6-53) Our objective is to use equation (6-53) to find values of k (i.e., the frequency index of the DFT) and N that make the free-space wavelength corresponding to the center of the DFT frequency bin, i.e., λbin, as close as possible to the free-space wavelength of our NZ- FZT/FDTD model, i.e., λo m=1. Although it is actually the ratio of k to N that determines λbin, care must be taken to ensure that there are a sufficient number of time- steps to fully characterize the response of the FDTD simulation. Satisfying these requirements, our NZ-FZT/FDTD model uses a frequency index of k=14 and a total number of time-steps N=634 . Employing these values of k and N, the values of the spatial sample periods defined in (6-46) and (6-47), and the equation of (6-53), we findthat λbin m =1000194. (which is within 194 ppm of λo m=1). At last, we must 250Table 6.1 Summary of NZ-FZT/FDTD Model of Analytical Benchmark Dipole Dipole Model: Free-space wavelength: λo m=1. Length of benchmark dipole: lm=01.. Current distribution constant: IAo=1. Excitation frequency: fH zexc=×2 997382 108.. Description: Five magnetic loops surrounding a PEC core centered in the FDTD solution volume that obey the current distribution of (6-45). FDTD Solution Volume: Size of Yee cell: m x210 666667.6−× =∆ , m y210 666667.6−× =∆ , m z210 500000.2−× =∆ . Number of cells in each dimension ( x, y, z ): 44 44 69 ×× cells. Time-step duration: ∆t=×−7 367101 1011.s e c . Spheroidal Transformation Surface: Frequency index: k=14 . Total number of time-steps (and observation interval): N=634 . Free-space wavelength corresponding to center of bin: λbin m =1000194. . Window type: Hamming. Sample grid: ∆∆′= ′= ϑ ϕ 1o (IJ×= × 180 360 samples). Position: Centered in the solution volume and centered about the dipole model. Prolate tests: Oblate tests: am=01. am=10. ξo=1.5, 2.0, 3.0, 4.0, and 5.0. ξo=0.15, 0.2, 0.3, 0.4, 0.6, and 0.8. 251determine the frequency at which to excite the NZ-FZT/FDTD model, i.e., the parameter fexc in the expressions of (6-49)-(6-51). Initially, one might think that the best choice of frequency would be that which corresponds to the free-space wavelength of our NZ- FZT/FDTD model. However, because this frequency does not lie in the center of thefrequency bin, the results will incur a slight phase error due to the DFT processing.Instead, this potential phase error is eliminated by exciting the model using the frequencyat the center of the bin that corresponds to λbin: fk NtHzexc== ×∆2 997382 108. . (6-54) Although the modeled dipole no longer operates at exactly λo m=1, the slight shift in operating frequency is worth the numerical tradeoff. Finally, all related DFT processing during the end-to-end test exclusively makes use of the Hamming window. Provided inTable 6.1 is a summary of the NZ-FZT/FDTD model of our analytical benchmark dipole. 6.3.2 Comparison of NZ-FZT/FDTD Results with Theoretical Results We now use the just developed model of the benchmark dipole to carry out the end-to-end test of the combined NZ-FZT/FDTD software. The overall performance ofthe combined software package is characterized by comparing the computed far-zone E-field of the model to that which is theoretically known (described by the closed formexpressions of (5-7) and (5-8)). Still, it must be made clear that the performance of thisend-to-end test intrinsically depends on the accuracy of our NZ-FZT/FDTD model of thebenchmark dipole. To be more specific, the level of detail used to construct the dipolemodel and the distance it is from the ABC (situated at the boundaries of the solutionvolume) both have a direct bearing on the accuracy of the computed results. Bearing thisin mind, using this specific model of our analytical benchmark still allows us to drawsome valid conclusions about the overall performance of the combined NZ-FZT/FDTDsoftware. 252Performance of the combined NZ-FZT/FDTD software is evaluated much in the same way as that which is done in Chapter 5 for both of the developed NZ-FZTalgorithms. Essentially, the performance evaluation follows the same format utilized inthe centered dipole numerical tests (see 5.4.1) and is accomplished by conducting thefollowing numerical studies: (1) an error analysis of the NZ-FZT/FDTD computedresults, (2) an analysis of how well the model approximates theoretical values of zero,and (3) a convergence study of the computed results. So that we may avoid beingredundant, the reader is referred to Section 5.4.1 to understand the many details behindthis performance assessment process. However, this time around some of theperformance parameters have been slightly altered. First, we choose to quantify the relative error in magnitude of Efzθθϕ( , ) in terms of percent (%) instead of parts per million (ppm). This is done by using a slightly modified version of equation (5-9) which replaces the written 106 factor with a 102 factor. As we shall see, the percent error quantity more appropriately describes the performance of the combined software duringthe end-to-end test. Second, we have also chosen to include the relative error in magnitude of Efzθθϕ( , ) in terms of decibels (dB) so as to get a more intuitive feel of the model’s performance: Relative Error of |( , ) |Efzθθϕ (dB)= 20 logcomputed theoretical  . (6-55) Notice how (6-55) expresses the relative error in dB using both positive and negative quantities. As for the corresponding phase error of Efzθθϕ( , ) , its evaluation remains unchanged and is still computed in terms of degrees using equation (5-10). Presented in Tables 6.2a, 6.3a, 6.2b and 6.3b is a compilation of the error analysis for both the prolate and oblate test cases of the end-to-end test. Once again, it isimportant to emphasize that this error analysis makes use of computed and theoretical 253Table 6.2a Comparison of Prolate Spheroidal End-to-End Test Results from the NZ-FZT/FDTD Model of the Analytical Benchmark Dipole with Theoretical Results: ϕ=0o lm=01.; am=01.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (% and dB) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=0o ProlateξoConvrgce Lθ=10o θ=170oθ=20o θ=160oθ=30o θ=150oθ=40o θ=140oθ=50o θ=130oθ=60o θ=120oθ=70o θ=110oθ=80o θ=100oθ=90o 1.5 6 8.4% (-0.8 dB); 0.8 deg7.8% (-0.7 dB); 0.8 deg6.9% (-0.6 dB); 0.8 deg5.8% (-0.5 dB); 0.8 deg4.8% (-0.4 dB); 0.7 deg3.8% (-0.3 dB); 0.7 deg3.1% (-0.3 dB); 0.7 deg2.6% (-0.2 dB); 0.7 deg2.5% (-0.2 dB); 0.7 deg 2.0 6 2.0% (+0.2 dB); 0.3 deg1.8% (+0.2 dB); 0.3 deg1.3% (+0.1 dB); 0.3 deg0.8% (+0.1 dB); 0.3 deg0.2% (~ 0 dB); 0.3 deg0.3% (~ 0 dB); 0.4 deg0.8% (-0.1 dB); 0.4 deg1.1% (-0.1 dB); 0.4 deg1.2% (-0.1 dB); 0.4 deg 3.0 6 3.7% (+0.3 dB); 0.6 deg3.2% (+0.3 dB); 0.6 deg2.4% (+0.2 dB); 0.6 deg1.4% (+0.1 dB); 0.7 deg0.5% (~ 0 dB); 0.7 deg0.4% (~ 0 dB); 0.8 deg1.1% (-0.1 dB); 0.8 deg1.6% (-0.1 dB); 0.8 deg1.7% (-0.2 dB); 0.8 deg 4.0 6 4.5% (+0.4 dB); 0.2 deg3.9% (+0.3 dB); 0.2 deg2.9% (+0.3 dB); 0.2 deg1.9% (+0.2 dB); 0.2 deg0.7% (+0.1 dB); 0.2 deg0.3% (~ 0 dB); 0.2 deg1.1% (-0.1 dB); 0.2 deg1.6% (-0.1 dB); 0.3 deg1.8% (-0.2 dB); 0.3 deg 5.0 6 5.9% (+0.5 dB); 0.9 deg5.3% (+0.4 dB); 0.7 deg4.3% (+0.4 dB); 0.5 deg3.2% (+0.3 dB); 0.2 deg1.9% (+0.2 dB); < 0.1 deg0.8% (+0.1 dB); 0.1 deg0.1% (~ 0 dB); 0.2 deg0.7% (-0.1 dB); 0.2 deg0.9% (-0.1 dB); 0.2 deg 254Table 6.3a Comparison of Prolate Spheroidal End-to-End Test Results from the NZ-FZT/FDTD Model of the Analytical Benchmark Dipole with Theoretical Results: ϕ=45o lm=01.; am=01.; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (% and dB) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=45o ProlateξoConvrgce Lθ=10o θ=170oθ=20o θ=160oθ=30o θ=150oθ=40o θ=140oθ=50o θ=130oθ=60o θ=120oθ=70o θ=110oθ=80o θ=100oθ=90o 1.5 6 8.5% (-0.8 dB); 0.8 deg7.9% (-0.7 dB); 0.8 deg7.0% (-0.6 dB); 0.8 deg5.9 % (-0.5 dB); 0.8 deg4.9% (-0.4 dB); 0.7 deg3.9% (-0.3 dB); 0.7 deg3.1% (-0.3 dB); 0.7 deg2.6% (-0.2 dB); 0.7 deg2.4% (-0.2 dB); 0.7 deg 2.0 6 2.0% (+0.2 dB); 0.3 deg1.8% (+0.2 dB); 0.3 deg1.3% (+0.1 dB); 0.3 deg0.8% (+0.1 dB); 0.3 deg0.2% (~ 0 dB); 0.3 deg0.3% (~ 0 dB); 0.4 deg0.7% (-0.1 dB); 0.4 deg1.0% (-0.1 dB); 0.4 deg1.1% (-0.1 dB); 0.4 deg 3.0 6 3.7% (+0.3 dB); 0.6 deg3.2% (+0.3 dB); 0.6 deg2.4% (+0.2 dB); 0.6 deg1.4% (+0.1 dB); 0.7 deg0.4% (~ 0 dB); 0.7 deg0.5% (~ 0 dB); 0.8 deg1.2% (-0.1 dB); 0.8 deg1.7% (-0.2 dB); 0.8 deg1.9% (-0.2 dB); 0.8 deg 4.0 6 4.5% (+0.4 dB); 0.2 deg3.8% (+0.3 dB); 0.2 deg2.9% (+0.2 dB); 0.2 deg1.7% (+0.1 dB); 0.2 deg0.5% (~ 0 dB); 0.3 deg0.6% (-0.1 dB); 0.3 deg1.6% (-0.1 dB); 0.3 deg2.2% (-0.2 dB); 0.4 deg2.4% (-0.2 dB); 0.4 deg 5.0 6 5.9% (+0.5 dB); 0.9 deg5.2% (+0.4 dB); 0.8 deg4.1% (+0.3 dB); 0.6 deg2.7% (+0.2 dB); 0.4 deg1.1% (+0.1 dB); 0.3 deg0.3% (~ 0 dB); 0.3 deg1.5% (-0.1 dB); 0.3 deg2.3% (-0.2 dB); 0.3 deg2.6% (-0.2 dB); 0.3 deg 255Table 6.2b Comparison of Oblate Spheroidal End-to-End Test Results from the NZ-FZT/FDTD Model of the Analytical Benchmark Dipole with Theoretical Results: ϕ=0o lm=01.; am=1; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (% and dB) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=0o OblateξoConvrgce Lθ=10o θ=170oθ=20o θ=160oθ=30o θ=150oθ=40o θ=140oθ=50o θ=130oθ=60o θ=120oθ=70o θ=110oθ=80o θ=100oθ=90o 0.15 14 2.0% (+0.2 dB); 1.3 deg3.3% (+0.3 dB); 1.5 deg4.6% (+0.4 dB); 1.6 deg5.5% (+0.5 dB); 1.5 deg5.6% (+0.5 dB); 1.2 deg5.2% (+0.4 dB); 1.0 deg4.5% (+0.4 dB); 0.8 deg3.9% (+0.3 dB); 0.7 deg3.7% (+0.3 dB); 0.6 deg 0.20 14 1.0% (+0.1dB); 2.0 deg2.8% (+0.2 dB); 1.9 deg4.7% (+0.4 dB); 1.8 deg5.7% (+0.5 dB); 1.6 deg5.8% (+0.5 dB); 1.3 deg5.3% (+0.4 dB); 1.1 deg4.6% (+0.4 dB); 1.0 deg4.1% (+0.3 dB); 0.9 deg3.9% (+0.3 dB); 0.9 deg 0.30 14 2.8% (-0.2 dB); 3.2 deg1.1% (+0.1 dB); 2.7 deg4.5% (+0.4 dB); 2.2 deg6.2% (+0.5 dB); 1.9 deg6.1% (+0.5 dB); 1.6 deg5.3% (+0.5 dB); 1.5 deg4.5% (+0.4 dB); 1.4 deg4.0% (+0.3 dB); 1.4 deg3.8% (+0.3 dB); 1.4 deg 0.40 14 8.5% (-0.8 dB); 3.6 deg1.8% (-0.2 dB); 3.1 deg4.0% (+0.3 dB); 2.6 deg6.5% (+0.5 dB); 2.2 deg6.3% (+0.5 dB); 1.9 deg5.0% (+0.4 dB); 1.8 deg4.0% (+0.3 dB); 1.8 deg3.5% (+0.3 dB); 1.9 deg3.3% (+0.3 dB); 1.9 deg 0.60 14 19.7% (-1.9 dB); 2.6 deg 7.2% (-0.7 dB); 0.8 deg 3.1% (+0.3 dB); 2.6 deg 6.5% (+0.5 dB); 2.8 deg 4.9% (+0.4 dB); 2.3 deg 2.4% (+0.2 dB); 2.0 deg 1.0% (+0.1 dB); 2.0 deg 0.7% (+0.1 dB); 2.3 deg 0.7% (+0.1 dB); 2.4 deg 0.80 14 16.1% (-1.5 dB); 10.4 deg5.4% (-0.5 dB); 1.6 deg3.9% (+0.3 dB); 2.7 deg5.9% (+0.5 dB); 2.6 deg2.7% (+0.2 dB); 0.6 deg1.0% (-0.1 dB); 0.7 deg2.5% (-0.2 dB); 0.5 deg2.0% (-0.2 dB); 0.7 deg1.5% (-0.1 dB); 1.3 deg 256Table 6.3b Comparison of Oblate Spheroidal End-to-End Test Results from the NZ-FZT/FDTD Model of the Analytical Benchmark Dipole with Theoretical Results: ϕ=45o lm=01.; am=1; IJ×= × 180 360 ; ∆∆′= ′= ϑϕ 1oRelative Error of |( , ) |Efzθθϕ (% and dB) and Phase Error of ∠Efzθθϕ( , ) (deg) where ϕ=45o OblateξoConvrgce Lθ=10o θ=170oθ=20o θ=160oθ=30o θ=150oθ=40o θ=140oθ=50o θ=130oθ=60o θ=120oθ=70o θ=110oθ=80o θ=100oθ=90o 0.15 14 1.6% (+0.1 dB); 1.4 deg1.2% (+0.1 dB); 1.6 deg0.5% (~ 0 dB); 1.6 deg3.2% (-0.3 dB); 1.4 deg5.8% (-0.5 dB); 0.9 deg7.6% (-0.7 dB); 0.4 deg8.6% (-0.8 dB); <0.1 deg8.9% (-0.8 dB); 0.3 deg9.0% (-0.8 dB); 0.4 deg 0.20 14 0.4% (~ 0 dB); 2.0 deg0.2% (~ 0 dB); 1.9 deg1.1% (-0.1 dB); 1.7 deg3.6% (-0.3 dB); 1.2 deg6.2% (-0.6 dB); 0.6 deg8.0% (-0.7 dB); <0.1 deg8.9% (-0.8 dB); 0.5 deg9.3% (-0.8 dB); 0.7 deg9.3% (-0.8 dB); 0.8 deg 0.30 14 3.7% (-0.3 dB); 3.1 deg2.5% (-0.2 dB); 2.3 deg2.6% (-0.2 dB); 1.4 deg4.3% (-0.4 dB); 0.5 deg6.6% (-0.6 dB); 0.4 deg8.4% (-0.8 dB); 1.1 deg9.3% (-0.8 dB); 1.6 deg9.7% (-0.9 dB); 1.9 deg9.7% (-0.9 dB); 2.0 deg 0.40 14 9.8% (-0.9 dB); 3.2 deg6.2% (-0.6 dB); 1.9 deg3.9% (-0.3 dB); 0.6 deg4.2% (-0.4 dB); 0.5 deg6.0% (-0.5 dB); 1.5 deg7.8% (-0.7 dB); 2.3 deg8.9% (-0.8 dB); 3.0 deg9.3% (-0.9 dB); 3.4 deg9.4% (-0.9 dB); 3.5 deg 0.60 14 20.8% (-2.0 dB); 4.3 deg 10.8% (-1.0 dB); 3.3 deg2.7% (-0.2 dB); 2.2 deg<0.1% (~ 0 dB); 1.6 deg 1.1% (-0.1 dB); 1.9 deg 3.3% (-0.3 dB); 3.1 deg 4.9% (-0.4 dB); 4.6 deg 5.6% (-0.5 dB); 5.7 deg 5.8% (-0.5 dB); 6.1 deg 0.80 14 15.3% (-1.4 dB); 12.9 deg5.3% (-0.5 dB); 7.2 deg2.0% (+0.2 dB); 1.8 deg3.5% (+0.3 dB); 1.7 deg1.7% (+0.1 dB); 2.2 deg0.3% (~ 0 dB); 0.1 deg0.7% (-0.1 dB); 2.9 deg0.3% (~ 0 dB); 5.2 deg<0.1% (~ 0 dB); 6.0 deg 257data that are unnormalized with respect to both magnitude and phase. Notice how the NZ-FZT/FDTD model of the benchmark dipole performs the same when computing theE-field in the upper and lower hemisphere of the far-zone. This occurrence is naturallywhat should be expected due to the fact that the spheroidal transformation surface, themodel of the dipole, and the rectangular FDTD solution volume all have centers thatcoincide with each other. Yet, this is where the symmetry of the model’s performanceends. Because the benchmark dipole is modeled in FDTD using a Cartesian lattice and arectangular solution volume, the computed far-zone E-field for all of the prolate andoblate test cases does not possess the azimuthal (rotational) symmetry that is theoreticallyexpected for the analytical benchmark. As a result, we have chosen to include in all of our test cases both the ϕ=0o and ϕ=45o planar cuts so as to demonstrate these asymmetrical differences. The other six cardinal and intercardinal planes have not been included in our numerical studies since the model will essentially yield the same results as the ϕ=0o and ϕ=45o planar cuts (a single test case for each type of spheroidal surface has been conducted to corroborate this assertion). Furthermore, by invoking symmetry arguments, one can intuitively understand as to why the model performs in thismanner: (1) the FDTD sample lattice is square in the x-y plane, and (2) the dipole modeland the spheroidal transformation surface are both situated exactly in the middle of therectangular FDTD solution volume. Inspection of Tables 6.2a and 6.3a reveals that the NZ-FZT/FDTD model of the benchmark dipole computes the far-zone E-field within ±85. % (or ±08. dB) and ±09. deg for the prolate test cases. In addition, all of the prolate tests converge with L=6 (i.e., the degree at which the spheroidal wave-harmonic expansion is truncated). The reason for excluding the ξo=12. test case from the prolate numerical study is that some sample points along the spheroidal surface lie within one spatial sample period from the modeled dipole. As a result, the simulated radiating field is not able to properly formalong the transformation surface in so little a number of Yee cells. If we wanted toaccommodate this specific test case, resolution of the entire NZ-FZT/FDTD model would 258have to be increased to a point that would ultimately preclude the model from addressing the larger of the oblate surfaces under consideration. Regarding the oblate test results, Tables 6.2b and 6.3b indicate that the NZ- FZT/FDTD model of the benchmark dipole performs much better when the oblatespheroidal surface is situated closer to the modeled source. For the oblate test cases where ξo= 0.15, 0.20, 0.30, and 0.40, the model of the dipole computes the far-zone E- field within ±98. % (or ±09. dB) and ±36. deg. As for the outermost test cases where ξo= 0.60 and 0.80, the NZ-FZT/FDTD model computes the far-zone E-field within ±208. % (or ±20. dB) and ±12 9. deg. All of the oblate test cases converge with L=14 . Upon review of these results, we can gather that the combined NZ-FZT/FDTD model does not perform as well as either spheroidal NZ-FZT algorithm operating in astand-alone manner. Logically speaking, this degradation of performance must beexpected because the FDTD portion of the model only approximates the radiation fromthe benchmark dipole; to the contrary, the analytical benchmark software driver providesthe exact near-zone E-field for the benchmark dipole using closed form expressions (5-1)-(5-6). In Chapter 5, both NZ-FZT algorithms (while operating on theoretically generatednear-zone E-field samples for the benchmark dipole) deliver the magnitude of the far-zone E-field within ±01. % for the same exact test cases that are used in this end-to-end test. As a result, these findings imply that most of the resulting error of the NZ- FZT/FDTD model stems from the FDTD portion of the model rather than the developedNZ-FZT algorithms. We are only able to draw this conclusion because the centereddipole numerical tests of Chapter 5 establish isolated performances of the NZ-FZTalgorithms for comparison (i.e., stand-alone from the FDTD field solver). Having concluded that most of the error arises from the FDTD portion of the NZ- FZT/FDTD model, we now turn our attention to the differences in performance betweenthe prolate and oblate test cases. As we can see, the NZ-FZT/FDTD model overallperforms better for the prolate cases than it does for the oblate cases. Yet, both NZ-FZT 259algorithms have been shown to compute the magnitude of the far-zone E-field within ±01. % when isolated from the FDTD portion of the model. Although a certain amount of error can be attributed to the discretizing of the benchmark dipole, the bulk of the resulting error does not appear to originate from this approximation. We draw thisconclusion because all of our end-to-end test cases use the same exact FDTD simulationand still achieve a wide range of performances (see Tables 6.2a, 6.3a, 6.2b and 6.3b).Indeed, the only difference between all of the end-to-end test cases is the actual locationof the spheroidal sample points. Also note that the prolate spheroidal transformationsurfaces are closer on average to the modeled dipole than their oblate counterparts. All ofthese facts together suggest that the performance of the NZ-FZT/FDTD model mustsomehow be linked to the position of the spheroidal sample points with respect to boththe modeled dipole and the solution volume boundaries. We have already concluded thatour NZ-FZT/FDTD model does not function well when using a transformation surfacesituated within a limited number of Yee cells from the modeled radiation source.Likewise, performance of the model appears to degrade as the transformation surface getsfarther away from the modeled dipole and closer to the ABC situated at the boundary of the solution volume, viz., the oblate test cases where ξo= 0.60 and 0.80. All things considered, there are several possible reasons as to why the NZ- FZT/FDTD model behaves in this manner. First, we must consider the effects ofnumerical dispersion on a FDTD simulation. In short, the phase velocity of a simulatedwave can differ from that of free-space. Moreover, the speed at which the simulatedwave may propagate varies with the modal wavelength, the direction of propagation inthe grid, and the grid discretization [46]. As a result, fields emanating from our modeleddipole can experience different free-space phase velocities depending upon the directionof propagation: the result being unwanted phase errors along our spheroidaltransformation surface. According to Taflove [46], these phase errors are cumulative andincrease linearly with wave propagation distance. Second, we consider the effects ofunwanted reflections from the ABC. In essence, these unwanted reflections from theboundary of the solution volume can produce erroneous traveling waves that pass through 260a given spheroidal transformation surface and ultimately corrupt the NZ-FZT process. As the transformation surface gets closer to the ABC, the true outgoing waves have tocompete with unwanted reflections that are closer in magnitude (i.e., the outgoing wavesdecrease in intensity at the spheroidal transformation surface as the surface gets fartherfrom the modeled dipole due to spherical spreading loss). Note that the lack of rotational symmetry observed in the relative error for all of the test cases is also consistent with the presence of both of these unwanted effects. TheABC does not uniformly absorb outgoing waves at the boundaries of the solution volume.Simulated waves that are perpendicularly incident upon the planar boundaries are betterabsorbed by this particular ABC than those that are incident upon the corners and edgesof the solution volume. Likewise, the effects of numerical dispersion also vary withregard to the direction of outgoing waves. Once again, we examine how well the NZ-FZT/FDTD model of the benchmark dipole approximates the null at the far-zone observation angles θ=0o and θ=180o. For all of the prolate and oblate test cases, the model computes the null as values that are below -110 dB from the main beam peak (i.e., which exists at θ=90o). Regarding the cross-polarized far-zone E-field, the model computes the theoretical zero of Efzϕθϕ(, ) as values that are well below -120 dB from the main beam peak. For all practical purposes, the NZ-FZT/FDTD model of the benchmark dipole approximates theoreticalvalues of zero comparable to either NZ-FZT algorithm operating in a stand-alone manner.Like before, the corresponding phase is ignored because it essentially represents someform of numerical phase noise. Overall, the results of the end-to-end test indicate that the combined NZ- FZT/FDTD software performs extremely well. With the exception of the outermost oblate test cases where ξo= 0.60 and 0.80, the fact that this specific NZ-FZT/FDTD model of the analytical benchmark dipole computes the magnitude of the far-zone E-field within ±10% error demonstrates the validity and robustness of the overall NZ- FZT/FDTD combined process. However, like all numerical processes, a full 261understanding of its inner-workings as well as its limitations is essential to its successful application. As we have discovered, the user must be judicious when placing thespheroidal transformation surface with respect to both the modeled radiation structure andthe boundaries of the solution volume. Note that in addition to this consideration, theuser must not forget to observe guidelines for the FDTD algorithm (see 6.1.7) as well asthose for the DFT portion of the extraction process (see 6.2.2). 262CHAPTER 7. FURTHER VALIDATION VIA AN EMPIRICAL BENCHMARK We are now ready to proceed with the final step of this research project: applying the developed NZ-FZT process to a computational model of a physically constructedradiating structure. In doing so, we are able to provide further validation of our newlydeveloped process as well as demonstrate a practical example of its application. First,this task requires us to select and fabricate a radiation structure that is to serve as ourempirical benchmark. Once constructed, this benchmark structure proceeds to undergopattern testing at the NASA Langley research facility and the Virginia Tech antennalaboratory in order to empirically characterize its far-zone performance. With both sets ofpattern measurements established as a basis for comparison, we then move forward toconstruct our computational model of the benchmark structure using the developed NZ-FZT/FDTD software. Computed results are then compared to those obtained from theNASA Langley and Virginia Tech measurements, and conclusions are drawn as to howwell the combined software package is able to model radiation from the developedempirical benchmark. An important consideration when selecting the radiating structure to use as our empirical benchmark is its ability to be physically realized within tight electrical andmechanical tolerances. Practically speaking, a radiating structure that cannot beaccurately constructed is not worth much as a comparison standard; the last thing wewant to do is introduce additional significant sources of error into our comparative study.Moreover, we desire a radiating structure that is not too complicated to build, excite, ortest. Satisfying all of these prerequisites, an edge-fed microstrip patch antenna makes anexcellent choice for our empirical benchmark. Its construction is relatively simple andemploys standard fabrication techniques for printed circuit boards that are presentlyconsidered quite mature in the field of antenna design. 2637.1 MICROSTRIP PATCH ANTENNA 7.1.1 Design Layout and Fabrication Having selected an edge-fed microstrip patch antenna as our empirical benchmark, we now turn our attention toward its design. When considering its frequencyof operation, we prefer that our empirical benchmark not operate below microwavefrequencies. The reason for imposing such a requirement is essentially twofold. First,from a fabrication and material perspective, we naturally want to limit the physical size ofthe antenna. Second, pattern testing at these lower frequencies usually requires a rangefacility that is quite large in size and, more often than not, is limited in terms ofavailability. Similarly, we also cannot permit our benchmark structure to operate at toohigh a frequency. For obvious reasons, fabrication errors have a much greater effect onthe overall accuracy of measured results as mechanical tolerances get electrically largerwith respect to the operating modal wavelength (which not only depends on thefabrication material, but also the radiating structure itself). Accordingly, designing ourmicrostrip patch antenna to operate at 3.0 GHz allows us to create an empiricalbenchmark that achieves a reasonable balance between all of these considerations. We choose to fabricate our microstrip patch antenna with a substrate of RT/duroid 5870 which is manufactured and made available from the Rogers Corporation. Theadvantage of this particular material is that it maintains close tolerances on all of itsmechanical and electrical parameters. As a result, RT/duroid 5870 makes an excellentchoice of substrate for our comparison standard. From a mechanical standpoint, its glassmicrofiber reinforced polytetrafluoroethylene (PTFE) composite is specifically designedfor exacting microstrip circuit applications [54]. Furthermore, the material has excellentdimensional stability and is resistant to all solvents and reagents hot or cold, normallyused in etching printed circuits [54]. With regard to electrical properties, its dielectricconstant remains quite uniform throughout the material, and its low dissipation factorextends its usefulness to X-band applications and above [54]. The material comes in avariety of thicknesses for both the dielectric substrate and the double-sided copper 264Table 7.1 Summary of Pertinent Mechanical and Electrical Parameters for the RT/duroid 5870 Substrate Used in theConstruction of the Empirical Benchmark [54] Parameter Nominal Value Relative Dielectric Constant, εr 1 MHz: 2.35 10 GHz: 2.33 Dissipation Factor (Loss Tangent), tanδ 1 MHz: 0.00510 GHz: 0.0012 Thickness of Dielectric Substrate, h 45 mils (0.045 in.) Weight/Thickness of Copper Cladding 1 oz./ft.2 or 1.4 mils (0.0014 in.) cladding. For our particular application, the thickness of the dielectric substrate is 45 mils, and the weight of the copper cladding is 1 oz./ft.2 (i.e., a thickness of 1.4 mils [55]). Displayed in Table 7.1 is a summary of all pertinent electrical and mechanical parametersfor the RT/duroid 5870 substrate specific to the construction of our microstrip patchantenna. With the substrate and operating frequency selected, we now shift our focus to the actual design of our 3.0 GHz edge-fed microstrip patch antenna. We begin this effort byconsidering the simple, yet useful, transmission line model of a rectangular patch antennadescribed by Bhartia et al. [56]. Although there are other more accurate models that describe this type of antenna, this specific one, as we shall see, is quite sufficient for ourpurpose of determining the dimensions of our empirical benchmark (note that theinterested reader is directed to [56] for the details regarding the other models available forthis particular radiating structure). In accordance with the transmission line model, theresonant frequency of a rectangular patch antenna is determined using the followingexpressions [56]: 265fc Llr oc re= +22() ∆ ε(7-1) where εεε rerr Wh=++−+  −1 21 21101 2 ()(7-2) )813.0 )(258.0 ()264.0 )(3.0 (412.0+ −+ += ∆hWhWh l rere ocεε. (7-3) Within these expressions the parameters are defined as follows: L and W, the length and width of the patch antenna; h, the thickness of the substrate (and subsequently the distance between the etched metalization and the ground plane); εr, the relative dielectric constant of the substrate; εre, the effective relative dielectric constant of the microstrip patch; and ∆loc, the line extension (due to fringing fields at the radiating edges of the patch antenna). For the sake of simplicity, we have chosen to make our patch antenna square in shape. Given the fact that the length and width of the patch antenna areconsidered equal, expressions (7-1)-(7-3) are then used to calculate the dimensions of thesquare patch antenna necessary to achieve resonance at 3.0 GHz for the given RT/duroid5870 substrate. Results of these calculations yield the following dimensions for the patchantenna: LW== 1252 mils. Note that these calculations utilize the εr=23 3. value (see Table 7.1) since the antenna is designed to operate at microwave frequencies. Next, we follow with the design of the antenna feed. An edge feed is selected to excite our patch antenna because of its ability to be constructed with a great deal ofprecision. It is this feature that allows the dimensional differences between our NZ-FZT/FDTD model and the fabricated benchmark structure to be minimized, thus reducingthe amount of additional error introduced. The feed design makes use of a microstriptransmission line that perpendicularly feeds the square patch in the middle of one of itsradiating edges. Furthermore, the characteristic impedance of this transmission line isdesigned to be 50 Ω in order to maintain commonality with the reference impedance of 266the range measurement system (note that 50 Ω is considered the industry standard for microwave measurement systems). As a result, we must determine the width Wfeed necessary to construct the 50 Ω microstrip feed for the given RT/duroid 5870 substrate. To do so, we apply the following formula for impedance of a microstrip transmission line[55]: []ZWh Whore feed feed=++ +−120 1393 0 667 14441πε .. l n ( . ) for Whfeed ≥1. (7-4) It should be understood that when applying this equation the effective dielectric constant εre of the 50 Ω microstrip feed is also found by using equation (7-2); however, W(the width of the patch) must now be replaced with Wfeed (the width of the microstrip feed). Employing both (7-2) and (7-4), we determine that a line width of Wfeed=133 mils yields approximately the desired 50 Ω microstrip transmission line for our particular substrate (i.e., precisely 50.2 Ω as calculated by (7-2) and (7-4)). At this point, it must be mentioned that a typical microstrip patch antenna design would operate with a matching network between itself and the 50 Ω transmission line feed. However, our objective is to accurately characterize the performance of theempirical benchmark, not to make it achieve the most efficient radiation. Accordingly,we directly excite the radiation structure from the standard reference impedance in orderto isolate its performance from the effects of additional microwave circuitry.Unfortunately, we still cannot get away from the fact that the 50 Ω microstrip feed does contribute some radiation to the overall far-zone performance of the empirical benchmarkstructure. The empirical benchmark is fabricated on a finite piece of substrate that maintains at least a half-wavelength (in the substrate) between the printed patch antenna and theedge of the ground-plane backed substrate. Consequently, the 50 Ω transmission line is required to travel 1417 mils to reach the edge of the substrate. It is here where the printed 2674780 mils 5417 milsW = 1252 mils1417 milsSMA Connector 2813 mils133 mils y xL=1252 milsRT/duroid 5870 559.5 mils -1 oz./ft.2of Cu Figure 7.1 Overall layout of the fabricated empirical benchmark (top-view; not to scale). 268circuit assembly interfaces with the measurement system by way of a standard subminiature (SMA) coaxial connector. This interface between the SMA connector andthe microstrip transmission line is physically realized as follows: first, the centerconductor of the connector is soldered to the upper metalization of the microstrip feed;and second, the outer conductor of the connector (specifically the flange) is soldered tothe ground-plane. Depicted in Figure 7.1 is the overall layout of the fabricated empiricalbenchmark. Note that the fabrication tolerances for the constructed benchmark structureare on the order of ±1 mil for the printed dimensions, i.e., the etched metalization, and ±20 mils for dimensions describing the overall size (or edges) of the ground-plane backed substrate. 7.1.2 NASA Langley and Virginia Tech Measurements of Far-Zone Radiation Patterns Pattern testing of the empirical benchmark structure is conducted in an anechoic chamber at the NASA Langley research facility and on an outdoor range at the VirginiaTech antenna laboratory. Measurements of the far-zone radiation patterns are taken at the3.0 GHz resonant frequency of the microstrip patch and consist of both E- and H-planelinear co-polarization cuts. (Note that the E-plane for a microstrip patch antenna isparallel to its length and perpendicular to the substrate; meanwhile, the corresponding H-plane is parallel to its width and also perpendicular to the substrate. See Figure 7.1.) Theresulting pattern data is comprised of relative power measurements (i.e., no phase) thatare expressed in terms of decibels (dB) and are normalized with respect to the peak of its main lobe. Pattern measurements are made over the full 360 o for both planar cuts and are taken with the following angular resolution: in 100 .o increments at NASA Langley and in 0 352 .o increments at Virginia Tech. Both sets of measured radiation patterns are subsequently shown in the section that compares them to the computed results of our NZ-FZT/FDTD model (see 7.2.2). 2697.2 EMPIRICAL VALIDATION OF THE NZ-FZT PROCESS 7.2.1 NZ-FZT/FDTD Model of the Empirical Benchmark In creating a NZ-FZT/FDTD model of our benchmark patch antenna, we must first consider the layout of the FDTD solution volume. Once again, the first step indeveloping the setup of the solution volume is determining the size of the Yee cells. Aswe can see from the design layout of our fabricated empirical benchmark (see Figure 7.1),there are many geometrical details necessary to physically describe its construction.Consequently, we need to select a cell size that is best able to represent the geometry ofthe entire radiating structure as defined by the coordinate system of Figure 7.1. We beginthis process by first establishing critical dimensions of the structure in each Cartesiandirection that would best be represented exactly by the model. The size of the Yee cell isthen designed to fit into each of these three critical dimensions using an integral numberof cell widths; in other words, the critical dimension in each Cartesian direction is exactlydivisible by its respective FDTD spatial sample period. Unfortunately, choosing the cellsize in this manner generally precludes the remainder of the geometrical features frombeing modeled exactly. Nevertheless, this difficulty can be easily overcome by choosinga cell size that is able to closely approximate all of the other less critical dimensions. The first critical dimension that we consider is the thickness h of the dielectric substrate. Referring to expressions (7-1)-(7-3), one can plainly see that the distance atwhich the upper metalization is situated above the ground plane has a profound impact onthe operation of a microstrip patch antenna. In addition, our experience shows that usingthree cells (in the z-direction) to represent the thickness of the substrate is quite sufficientto accurately model a microstrip structure. Accordingly, we have selected the FDTDspatial sample period in the z-direction to be as follows: ∆zmilsmils m == = ×− 45 315 3810000 104. . (7-5) 270Focusing our attention on the remaining two critical dimensions, we naturally conclude that they should be the length L and width W of our microstrip patch antenna. Again referring to expressions (7-1)-(7-3), we are able to recognize that both of thesedimensions are also crucial to the operation of a microstrip patch antenna. Because bothof these critical dimensions are equal, the spatial sample periods in the x- and y-directionscan be made to equal one another. Having chosen to represent both the length and widthof the patch using 29 cells in each direction, we determine the spatial sample periods inthe x- and y-directions as follows: ∆∆xymilsmils m == = = ×− 1252 29432 1096579 103. . . (7-6) It is important to note that partitioning the length and width of the patch into exactly 29 cells is not completely arbitrary. First, it is necessary to have a significant number ofspatial samples along the patch antenna in order to accurately simulate its operation.Second, using 3 cells (in the y-direction) of this particular spatial sample period enablesus to closely approximate the line width of the microstrip feed (i.e., W feed=133 mils). Our experience has shown that using 3 cells along the width of the microstrip transmission line is quite sufficient. Due to the intricate geometrical features of ouroverall model, all three spatial sample periods do turn out to be much less than the tenthof a wavelength maximum cell width recommended by the modeling guideline of (6-33). With the size of the Yee cell designed to exactly meet the critical dimensions of the microstrip patch antenna, we now proceed to model the remainder of the fabricatedempirical benchmark. Each of the other less critical dimensions is represented by usingthe integral number of cells that most closely approximates the given spatialmeasurement. Thus, the spatial extent of the modeled radiation structure without thefree-space buffer is determined to be as follows: 111 cells in the x-direction, 125 cells inthe y-direction, and 3 cells in the z-direction. Regarding the surrounding free-spacebuffer, we have chosen to equidistantly space the modeled radiation structure from theboundaries of the solution volume as follows: a 37 cell border in the x-direction, a 30 cell 271111 cells 125 cells29 cells33 cells 65 cells3 cells29 cells 13 cells30 cell free-space border free-space37 cell border 30 cell free-space border xy 185 cells 185 cells - PEC thin sheet - Air-dielectric interface thin sheetsubstrate materialfree-space37 cell border Figure 7.2 Overall layout of NZ-FZT/FDTD model of the fabricated empirical benchmark (top-view; not to scale). 27233 cells 29 cells21 cell free-space border 21 cell free-space border xz 45 cells 185 cells3 cells49 cells 2 cells1 cell Esn37 cell borderfree- space37 cell borderfree- spacesubstrate material(patch) (feed) - PEC thin sheet - Air-dielectric interface thin sheet - Lumped load resistive material(ground-plane) Figure 7.3 Overall layout of NZ-FZT/FDTD model of the fabricated empirical benchmark (side-view along mid-section ofsolution volume; not to scale). 273border in the y-direction, and a 21 cell border in the z-direction. As we can see, this choice of free-space buffer allows the modeled radiation structure to be situated exactly inthe center of a FDTD solution volume with 185 cells in both the x- and y-directions, and45 cells in the z-direction. Furthermore, notice how the shortest distance between theradiation structure and all boundaries of the solution volume well exceed the 10 cellminimum suggested by the FDTD modeling guideline (see 6.1.7). Illustrated in Figures7.2 and 7.3 is the overall layout (depicting different views) of our NZ-FZT/FDTD modelof the fabricated empirical benchmark. As shown in Figures 7.2 and 7.3, the upper metalization and the ground-plane of the patch antenna are both modeled using thin sheets of perfect electric conductor (PEC)that are respectively situated at the top and bottom of the modeled substrate. Note thatboth PEC thin sheets are of infinitesimal thickness and are modeled by setting thetangential components of the E-field samples in their respective planes to zero (see 6.1.3). Regarding the RT/duroid 5870 substrate, it is modeled as a lossy dielectric. As mentioned earlier, we use εr=23 3. as the relative dielectric constant of the substrate because our radiating structure operates at microwave frequencies (see Table 7.1). Furthermore, the loss of the substrate is specified by the manufacturer in terms of its losstangent and has the following value at microwave frequencies: tan . δ=0 0012 (see Table 7.1). Yet, the FDTD lossy dielectric E-field formulation of our electromagnetic field solver requires us to represent loss in a material using a finite conductivity σ (see 6.1.4). Fortunately, we are able to reconcile this difference in the representation of loss. First, we must understand that loss (or power dissipation) in a dielectric material can stem fromeither of the two following sources or a combination of both: (1) a time lag in the D-fieldrelative to the E-field that is attributable to damping effects of the medium (which thusleads to loss resulting from a complex permittivity); and (2) the finite conductivity σ of the medium (which thus leads to ohmic loss) [57]. Basic electromagnetic theory reveals that the two loss mechanisms are virtually indistinguishable as far as external effectsrelated to power dissipation are concerned [57]. As a result, Collin [57] states that thecombination of both loss effects can be lumped under a total effective conductivity or 274under an effective imaginary part of the permittivity. In order to maintain consistency with our electromagnetic field solver, we opt to represent the total loss in our substrateusing an effective conductivity σeff. At the 3.0 GHz resonant frequency, the total effective conductivity of the substrate is determined from its loss tangent tan δ and relative dielectric constant εr as follows [57]: σπ ε ε δeff r o fS m == ×−2 4 666355 104tan . . (7-7) Modeling of the air-dielectric interface of the substrate is achieved by placing a thin sheet (i.e., of infinitesimal thickness) of a special dielectric along all substratesurfaces absent of metalization. Specifically, the relative dielectric constant of this air-dielectric interface thin sheet is taken as the average of the two dielectric constants: εεint().=+ =1 2 1665r . (7-8) This modeling technique allows the tangential FDTD E-field samples that are on the air- dielectric interface to simultaneously account for the tangential E-field on both sides ofthe interface. The air-dielectric interface thin sheet is approximated as lossless. Treatingthe air-dielectric interface in this manner is consistent with that which is done in [58] andis further explained in [59]. As stated earlier, the fabricated empirical benchmark is actually excited by way of a SMA coaxial connector. Unfortunately, incorporating the fine geometrical details of theconnector into our overall model would ultimately cause our entire NZ-FZT/FDTDsimulation to become quite cumbersome. Consequently, we have chosen to simulateexcitation of the radiating structure using an alternative approach that is much morecomputationally efficient. As revealed in Figures 7.2 and 7.3, our approach makes use oftwo adjacent, z-directed, E-field sources that are situated internal to the modeled radiatingstructure as follows: both are 1 cell (in the x-direction) in from the edge of the substrate; both are 1 2 cell (in the z-direction) up above the ground-plane; and each is situated 1 cell 275(in the y-direction) in from either of the two respective edges of the 50 Ω microstrip feed (the width of which is modeled using 3 cells). The overall NZ-FZT/FDTD model of ourempirical benchmark is excited by driving both of these E-field sources using thefollowing sinusoidal waveform: Ef n t sn exc =cos( )2π ∆ where nN=−01 1,, ,/G15 . (7-9) (Note that the time-step duration ∆t is computed using the equality of (6-34) and spatial sample periods defined in (7-5) and (7-6).) Even though we have conveniently chosen to excite our model using a sinusoidal waveform of unity amplitude, for all practicalpurposes, the choice of amplitude has no real bearing on the computed results. This istrue for this particular application because the computed far-zone radiation patterns areultimately normalized with respect to the peak of its main lobe. In this instance, the NZ-FZT/FDTD model performs pattern normalization so that its computed results can becompared directly to both sets of measured far-zone radiation patterns (which are alsonormalized; see 7.1.2). Included in our source model is a 50 Ω lumped load sandwiched between both z- directed E-field sources and the upper metalization of the microstrip feed. The load ismodeled as a lossy dielectric and is part of the source model because it simulates the50Ω output impedance of the range measurement system. In accordance with Figure 7.3, our lumped load is form-factored as follows: 1 cell in the x-direction, 3 cells in the y-direction, and 2 cells in the z-direction. Naturally, we want this size block of resistivematerial to simulate 50 Ω in the z-direction. To do so, the conductivity of the lumped load resistive material must be as follows: σloadz xyRS m ==− () () ( ).2 34 2245811 ∆ ∆∆. (7-10) Since the 50 Ω lumped load is imbedded within the substrate of our modeled structure, we have chosen to make its relative dielectric constant match that of the substrate. 276The next step in constructing the NZ-FZT/FDTD model of our fabricated empirical benchmark involves configuration of the DFT (see 6.2.2) and the choice ofexcitation frequency. Employing the time-step duration ∆t, our objective is to find values of k (the frequency index of the DFT) and N (the number of time-steps in the simulation to be processed by the DFT) that allow the center of the resulting DFTfrequency bin to be situated as close as possible to the 3.0 GHz resonant frequency of themicrostrip patch. In doing so, we are able to excite the model at the frequencycorresponding to the center of the bin so that we can minimize the unwanted effects ofDFT processing. Although the modeled patch antenna no longer operates at exactly 3.0GHz, the trivial shift in operating frequency is worth the numerical tradeoff. Once again,care must be taken to ensure that there are a sufficient number of time-steps to fullycharacterize the response of the FDTD simulation. Satisfying all of these requirements,our NZ-FZT/FDTD model uses a frequency index k=14 and a total number of time- steps N=4091. Employing these values of k and N, the time-step duration ∆t, and equation (6-43), we find that the center of the resulting DFT frequency bin and, hence, the excitation frequency of (7-9) is as follows: fk NtGHzexc==∆3000282. . (7-11) Note that the excitation frequency of the model is within 94 ppm of the 3.0 GHz resonant frequency. As for the window function applied during DFT processing, the NZ-FZT/FDTD model of our fabricated empirical benchmark exclusively makes use of theHamming window (see 6.2.2). When selecting the spheroidal transformation surface to use during the NZ- FZT/FDTD simulation of our fabricated empirical benchmark, we must consider severalkey elements. First, the spheroidal transformation surface must be able to completelysurround the entire benchmark structure without intersecting any part of it. Due to theshape of our empirical benchmark, one can easily see that the oblate spheroid is muchmore suitable for this particular model than its prolate counterpart. Furthermore, 277Table 7.2 Summary of NZ-FZT/FDTD Model of the Fabricated Empirical Benchmark Microstrip Patch Model: Size of entire radiation structure ( x, y, z ): 111 125 3 ×× cells. Thickness of modeled substrate: 3 cells.Length and width of patch antenna: 29 cells.Width of microstrip feed: 3 cells.Length of microstrip feed: 33 cells.Substrate: Modeled with a lossy dielectric using εr=23 3. and σeff Sm =×−4 666355 104.. Metalization: Modeled using a PEC thin sheet (infinitesimal thickness). Air-dielectric interface: Modeled using a lossless dielectric thin sheet (infinitesimal thickness) of εεint().=+ =1 2 1665r . Excitation: Modeled using two E-field sources that drive the microstrip feed against the ground-plane through a 50 Ω lumped load. Lumped load: Modeled with 1 3 2 ×× cells ( x, y, z ) of lossy dielectric with εr=23 3. and σload Sm =4 224581. to yield 50 Ω in z-direction. Excitation frequency: f GHzexc=3000282.. Position: Entire radiation structure is centered in the FDTD solution volume. FDTD Solution Volume: Size of Yee cell: ∆∆xy m== ×−1096579 103. and ∆zm=×−3810000 104.. Number of cells in each dimension ( x, y, z ): 185 185 45 ×× cells. Time-step duration: ∆t=×−1140608 1012.s e c . Oblate Spheroidal Transformation Surface: Size of oblate spheroidal surface: 24 cells in total height (in the z-direction) and 170 cells in diameter (in the x-y plane). Corresponding oblate spheroidal parameters: am=×−9 309704 102. a n d ξo=×−4 911005 102.. Frequency index: k=14. Total number of time-steps (and observation interval): N=4091. Window type: Hamming. Sample grid: ∆∆′= ′= ϑϕ 1o (IJ×= × 180 360 samples). Position: Centered in the solution volume and about the entire radiation structure. 278experience gained from our end-to-end test (see 6.3.2) indicates that it is best not to situate the transformation surface too close to the boundaries of the solution volume nortoo close to the radiating structure. Taking into account all of these factors, we choose touse an oblate spheroidal transformation surface that is 24 cells in total height (in the z-direction) and 170 cells in diameter (in the x-y plane). This particular oblate spheroid canbe described in terms of the following oblate spheroidal parameters: am=× −9 309704 102. and ξo=×−4 911005 102. (found by employing the oblate coordinate transformation of (2-1)-(2-5) and the FDTD spatial sample periods defined in (7-5) and (7-6)). Regarding the placement of the oblate spheroidal transformationsurface, it is centered both in the solution volume and about the entire radiation structure.Lastly, the NZ-FZT/FDTD model of our empirical benchmark makes use of a uniform sample grid along the spheroidal surface with 1 o intervals in both the ϑ and ϕ dimensions (i.e., IJ×= × 180 360 ). Provided in Table 7.2 is a summary of the NZ- FZT/FDTD model of our fabricated empirical benchmark. 7.2.2 Comparison of NZ-FZT/FDTD Model Results with Measured Results At last, we are now able to compare the far-zone radiation patterns computed by the just developed NZ-FZT/FDTD model with the patterns measured at NASA Langley and Virginia Tech. The computed results are in 1o increments over the full 360o for both E- and H-plane linear co-polarization cuts. Like the measured data, the computedradiation patterns also consist of relative power measurements that are normalized withrespect to the peak of the main lobe. Specifically, the computed results are found byexpressing the relative magnitude of the far-zone E-field in decibels (dB). Regarding theconvergence of our NZ-FZT/FDTD model, numerical tests indicate that the computed far-zone radiation patterns converge at the point where L=16 (i.e., the degree at which the spheroidal wave-harmonic expansion is truncated; see 2.3.1). Both Figures 7.4 and 7.5 graphically depict a direct comparison of all three sets of pattern data for both the E- and H-plane linear co-polarization cuts. Notice that there are 279Figure 7.4 Comparison of NZ-FZT/FDTD model results (of the empirical benchmark) with NASA Langley and Virginia Tech measuredresults for the E-plane linear co-polarization pattern at 3.0 GHz. 280Figure 7.5 Comparison of NZ-FZT/FDTD model results (of the empirical benchmark) with NASA Langley and Virginia Tech measuredresults for the H-plane linear co-polarization pattern at 3.0 GHz. 281some inconsistencies between the patterns measured at NASA Langley and those measured at Virginia Tech. In essence, these differences demonstrate that a certainamount of measurement error is introduced by each range measurement system togetherwith its respective range design. Part of this measurement error can be attributed tounwanted reflections from both the antenna positioner and the range surroundings.Consequently, we have included both sets of measured data in order to give us a morecomplete picture of the actual far-zone behavior of the fabricated empirical benchmark. For the most part with a few exceptions, the far-zone radiation patterns computed by our NZ-FZT/FDTD model are in fairly good agreement with both sets of measured data for the forward-looking region where −≤ ≤120 120 ooθ . Naturally, we exclude the region behind the benchmark microstrip patch from consideration because of the measurement error introduced by the presence of an antenna positioner and its associatedmeasurement equipment. To put it briefly, our NZ-FZT/FDTD model simulates thefabricated benchmark antenna situated in free-space, not mounted on a positioner.Furthermore, inspection of Figure 7.4 reveals that both sets of measured E-plane datahave an asymmetry with respect to the far-zone observation angle θ that is more pronounced than the slight asymmetry exhibited by the computed results. Although the computed results are in good agreement with the measured results for the positive half of the forward-looking region and in the broadside direction where −≤ ≤22 5 22 5..ooθ , some differences do exist over some of the observation angles in the corresponding negative half of the forward-looking region. The cause of these differences can mostlikely be attributed to the fact that the SMA connector and its attached coaxial cable arepresent during the actual measurements whereas both microwave structures are not partof our NZ-FZT/FDTD simulation (see 7.2.1). As a result, unwanted coupling, scattering,and radiation from both the connector (including its flange) and the coaxial cable allultimately alter the true far-zone performance of the fabricated empirical benchmark. Thefact that the actual test setup has a connector and a cable on only one side of thefabricated structure is consistent with the asymmetry found in the E-plane measured 282results. In addition, the slight asymmetry present in the computed results is consistent with the microstrip patch being fed from only one side. Finally, Figure 7.5 indicates thatthe computed results for the H-plane pattern cut are in good agreement with the measured results for the entire forward looking region where −≤ ≤120 120 ooθ . The fact that no significant asymmetries are found in either of the measured H-plane pattern results or in the computed results is also consistent with the configuration of the actual test setup andthe layout of the fabricated empirical benchmark. On the whole, our NZ-FZT/FDTDmodel does a good job of numerically simulating radiation from the empirical benchmark. 283CHAPTER 8. SUMMARY AND CONCLUSIONS There are several important conclusions that can be drawn from this research. First, our findings show that it is possible to construct a practical NZ-FZT process thatmakes use of a spheroidal transformation surface and requires knowledge of only one ofthe two near-zone electromagnetic fields, not both. Furthermore, it should be emphasizedthat the process is quite successful despite the fact that both spheroidal transformationsurfaces do not support orthogonal vector wave functions. As shown in Chapter 2, we areable to sidestep this difficulty by employing a differential equation approach that solvesthe three-dimensional vector Helmholtz equation using a hybrid of coordinate systems.Although our algorithms are defined throughout the dissertation in terms of the radiatingE-field (both near- and far-zone), it is important to realize that the overall NZ-FZTprocess also applies to the radiating H-field (this assertion is easily understood byevoking the duality theorem of electromagnetics). Throughout our investigation the newly developed NZ-FZT process is shown to have several noteworthy features that are considered quite beneficial. Since the entireprocess is very efficient from a computational standpoint, the developed algorithms arecapable of being successfully mapped to an IBM compatible personal computer (seeChapters 3 and 4). As a result, we need not have the computational power of asupercomputer or of some other type of specialized hardware to carry out our numericalprocess. Also, because our NZ-FZT process is formulated in a generalized manner, it isnot just limited to addressing radiation problems with rotational symmetry. The capacityto handle radiation problems that are rotationally asymmetric is numerically demonstratedwith the offset dipole numerical tests (in Chapter 5) and with the NZ-FZT/FDTD modelof the empirical benchmark (in Chapter 7). Lastly, an additional feature that is affordedto us by our NZ-FZT process is the ability to compute the far-zone in terms of its correctabsolute quantity, and not some relative quantity; that is, the far-zone is accuratelyrendered without any magnitude and phase normalization. Note that being able to 284compute far-zone field quantities in an unnormalized format is often necessary depending upon the given application (e.g., computation of antenna gain). Regarding the performance of our NZ-FZT process, the numerical studies of Chapter 5 indicate that the algorithms (operating on the analytical benchmark radiationstructure for a given set of spheroidal transformation surfaces) provide unnormalized far-zone results that match theory within ±01. % in magnitude and ±012. deg in phase. In addition, both algorithms demonstrate that they provide at least 78 dB of numerical dynamic range with regard to pattern computations. And finally, the rate of convergenceis shown to be more than acceptable for algorithmic expansions of this nature. Citingthese results, we conclude that our newly developed NZ-FZT process is extremely soundboth in theory and implementation. However, although both algorithms perform extremely well during the numerical tests of Chapter 5 for a specific radiating structure and a specific set of spheroidaltransformation surfaces, the possibility exists that the numerical process may breakdownin some extreme cases. It is important to emphasize that this shortcoming is not causedby the underlying theory behind the developed NZ-FZT process, but rather the inability ofthe algorithms to accurately compute all orders and degrees of the required specialfunctions necessary to achieve convergence. Naturally, this particular situation can arisewhen applying the algorithms to very large radiating structures. In Chapter 6, we are able to successfully integrate our NZ-FZT process with a FDTD electromagnetic field solver. We do so in order to create a tool that allows us tofurther validate our newly developed process. The entire combined process is initiallyverified by using a computational model of our analytical benchmark to conduct an end-to-end test. From this numerical study we also gain an understanding of both thelimitations and expected performance of the NZ-FZT/FDTD combined process. Finally,the linking of both processes allows us to successfully demonstrate a novel application ofour newly developed NZ-FZT process. Our research concludes in Chapter 7 by applying the developed NZ-FZT process to a computational model of our fabricated empirical benchmark. Far-zone radiation 285patterns computed by the NZ-FZT/FDTD model of the empirical benchmark are then compared to those measured at NASA Langley and Virginia Tech. Overall with a fewexceptions, both the computed and measured sets of far-zone radiation patterns are infairly good agreement for the forward-looking region. Ultimately, it is this additionalnumerical/empirical experiment that provides further validation of our newly developedNZ-FZT process. In the end, it is our hope that the work presented in this dissertation, in part or as a whole, is able to further serve the research community as both a tool and a springboardfor a variety of other uses. Regardless of their application, the numerical routinesdeveloped in Chapter 3 provide an efficient way to compute all of the special functionspertaining to prolate and oblate spheroidal wave-harmonics. In addition, both developedbenchmarks are able to provide other researchers with a means through which to test theirown computational algorithms. Finally, computational strategies and approaches used todevelop our NZ-FZT process may also be of aid in addressing other electromagneticproblems with spheroidal based geometries. 286APPENDIX A. PROLATE AND OBLATE SPHEROIDAL COORDINATE SYSTEMS Prolate Spheroidal Coordinates (with z-axis preference) Independent coordinates and their domain: (, , )ξηϕ or ( , , )ξϑϕ, where ηϑ=cos and 1≤< ∞ξ , −≤ ≤11η or 0≤≤ϑπ , 0 2≤≤ϕπ. Prolate spheroidal-to-Cartesian transformation [23]: xa a=− − =−ξ η ϕ ξ ϑϕ22 211 1 cos sin cos (A-1) ya a=− − =−ξ η ϕ ξ ϑϕ22 211 1 sin sin sin (A-2) za a==ξηξϑcos (A-3) where a is the distance between the focal point of the coordinate system and the origin. Prolate spheroidal scale factors (metric coefficients) [22]: haξξη ξ=− −22 21=− −aξ ϑ ξ22 21cos,( A - 4 ) haηξη η=− −22 21 or (A-5) haϑ ξ ϑ =−22cos , (A-6) haϕ ξ η =− −2211 =−aξ ϑ21sin . (A-7) 287 Prolate spheroidal unit vector representations: /G04ξ=− −+− −+− −ξη ξηϕ ξη ξηϕ ηξ ξη11 12 222 222 22cos /G04 sin /G04/G04 xy z = −+ −+− −ξϑ ξ ϑϕξϑ ξ ϑϕξ ξ ϑϑsin coscos /G04sin cossin /G04 coscos /G04 22 222 221xyz ,( A - 8 ) /G04η=−− −−− −+− −ηξ ξηϕ ηξ ξηϕ ξη ξη2 222 222 2211 1cos /G04 sin /G04/G04 xy z or (A-9) /G04ϑ=− −ξ ξ ϑϑϕ2 221 coscos cos /G04x +− −− −ξ ξ ϑϑϕξϑ ξ ϑ2 22221 coscos sin /G04sin cos/G04 yz (note that /G04 /G04ηϑ=− ), (A-10) /G04ϕ=− +sin /G04cos /G04 ϕ ϕ xy , (A-11) and /G04x=− −−− −− ξη ξηϕξηξ ξηϕη ϕϕ112 222 22cos/G04cos /G04sin /G04 = −+− −−ξϑ ξ ϑϕξξ ξ ϑϑϕϑ ϕϕsin coscos/G04 coscos cos/G04sin /G04 222 221, (A-12) /G04y=− −−− −+ ξη ξηϕξηξ ξηϕη ϕϕ1 12 222 22sin/G04sin /G04cos /G04 = −+− −+ξϑ ξ ϑϕξξ ξ ϑϑϕϑ ϕϕsin cossin/G04 coscos sin/G04cos /G04 222 221, (A-13) /G04z=− −+− −ηξ ξηξξη ξηη2 222 2211/G04 /G04 =− −− −ξ ξ ϑϑξξϑ ξ ϑϑ2 22221 coscos/G04 sin cos/G04. (A-14) 288Definition of cross product in prolate spheroidal coordinates: The unit vectors /G04ξ, /G04ϑ, and /G04ϕ form a right-handed coordinate system that is defined by the cross products /G04 /G04 /G04 ξϑϕ ×= (A-15) /G04 /G04 /G04ϕξϑ ×= (A-16) /G04 /G04 /G04ϑϕξ ×= (A-17) where /G04 /G04ηϑ=− . Far-zone behavior ( ξ→∞ ) of prolate spheroidal coordinates [23]: In the far-zone, viz., large arguments of ξ, the prolate spheroidal coordinate system approaches the spherical coordinate system in the following manner: ξ→ra, (A-18) ηθ→cos or (A-19) ϑθ→ , (A-20) ϕϕ≡, (A-21) and/G04/G04ξ→r, (A-22) /G04 /G04ηθ→− or (A-23) /G04/G04ϑθ→ , (A-24) /G04/G04ϕϕ≡. (A-25) 289Oblate Spheroidal Coordinates (with z-axis preference) Independent coordinates and their domain: (, , )ξηϕ or ( , , )ξϑϕ, where ηϑ=cos and 0≤< ∞ξ , −≤ ≤11η or 0≤≤ϑπ , 0 2≤≤ϕπ. Oblate spheroidal-to-Cartesian transformation [23]: xa a=+ − =+ξ η ϕ ξ ϑϕ22 211 1 cos sin cos (A-26) ya a=+ − =+ξ η ϕ ξ ϑϕ22 211 1 sin sin sin (A-27) za a==ξηξϑcos (A-28) where a is the distance between the focal point of the coordinate system and the origin. Oblate spheroidal scale factors (metric coefficients) [22]: haξξη ξ=+ +22 21=+ +aξ ϑ ξ22 21cos, (A-29) haηξη η=+ −22 21 or (A-30) haϑ ξ ϑ =+22cos , (A-31) haϕ ξ η =+ −2211 =+aξ ϑ21sin . (A-32) 290Oblate spheroidal unit vector representations: /G04ξ=− ++− +++ +ξη ξηϕ ξη ξηϕ ηξ ξη11 12 222 222 22cos /G04 sin /G04/G04 xy z = ++ +++ +ξϑ ξ ϑϕξϑ ξ ϑϕξ ξ ϑϑsin coscos /G04sin cossin /G04 coscos /G04 22 222 221xyz , (A-33) /G04η=−+ +−+ ++− +ηξ ξηϕ ηξ ξηϕ ξη ξη2 222 222 2211 1cos /G04 sin /G04/G04 xy z or (A-34) /G04ϑ=+ +ξ ξ ϑϑϕ2 221 coscos cos /G04x ++ +− +ξ ξ ϑϑϕξϑ ξ ϑ2 22221 coscos sin /G04sin cos/G04 yz (note that /G04 /G04ηϑ=− ), (A-35) /G04ϕ=− +sin /G04cos /G04 ϕ ϕ xy , (A-36) and /G04x=− +−+ +− ξη ξηϕξηξ ξηϕη ϕϕ112 222 22cos/G04cos /G04sin /G04 = +++ +−ξϑ ξ ϑϕξξ ξ ϑϑϕϑ ϕϕsin coscos/G04 coscos cos/G04sin /G04 222 221, (A-37) /G04y=− +−+ ++ ξη ξηϕξηξ ξηϕη ϕϕ1 12 222 22sin/G04sin /G04cos /G04 = +++ ++ξϑ ξ ϑϕξξ ξ ϑϑϕϑ ϕϕsin cossin/G04 coscos sin/G04cos /G04 222 221, (A-38) /G04z=+ ++− +ηξ ξηξξη ξηη2 222 2211/G04 /G04 =+ +− +ξ ξ ϑϑξξϑ ξ ϑϑ2 22221 coscos/G04 sin cos/G04. (A-39) 291Definition of cross product in oblate spheroidal coordinates: The unit vectors /G04ξ, /G04ϑ, and /G04ϕ form a right-handed coordinate system that is also defined by the cross-products specified in expressions (A-15)-(A-17). Far-zone behavior ( ξ→∞ ) of oblate spheroidal coordinates [23]: In the far-zone, viz., large arguments of ξ, the oblate spheroidal coordinate system approaches the spherical coordinate system in a manner that is similar to its prolate counterpart.Accordingly, the far-zone expressions provided in (A-18)-(A-25)also apply to the oblate spheroidal coordinate system. 292APPENDIX B. SCALAR LAPLACIAN IN SPHEROIDAL COORDINATES The generalized Laplacian for a scalar function defined in terms of an orthogonal curvilinear coordinate system is as follows [60]: ∇=2 123fuu u(, , ) 1 123 123 11 231 22 312 33 hhh uhh hf uuhh hf uuhh hf u∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂∂ ∂  +  +     (B-1) where h1, h2, and h3 are the respective scale factors. In the prolate spheroidal case, we employ the scale factors of (A-4)-(A-7) with (B-1) to yield the following representation of the Laplacian in prolate spheroidal coordinates [22]: ∇2ψ ξηϕP(,, ) =  +  +      1 hhhhh hhh hhh hPPP ξηϕηϕ ξϕξ ηξη ϕ∂ ∂ξ∂ψ ∂ξ∂ ∂η∂ψ ∂η∂ ∂ϕ∂ψ ∂ϕ () ()() () ()=−−  +−  +− −−    111 1122 22222 222 2aPP P ()ξ η∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξ η ξ η∂ψ ∂ϕ. (B-2) Likewise, we employ the scale factors of (A-29)-(A-32) with (B-1) to yield the following representation of the Laplacian in oblate spheroidal coordinates [22]: 293 ∇2ψξ η ϕO(,, ) =  +  +      1 hhhhh hhh hhh hOOO ξηϕηϕ ξϕξ ηξη ϕ∂ ∂ξ∂ψ ∂ξ∂ ∂η∂ψ ∂η∂ ∂ϕ∂ψ ∂ϕ () ()() () ()=++  +−  ++ +−    111 1122 22222 222 2aOO O ()ξη∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξη ξη∂ψ ∂ϕ. (B-3) 294APPENDIX C. SEPARATION OF VARIABLES OF THE SCALAR HELMHOLTZ EQUATION INSPHEROIDAL COORDINATES For the prolate spheroidal case, we apply the separation of variables method to the scalar Helmholtz partial differential equation expressed in terms of coordinate specificdifferential operators [23]: () ()() () ()∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξ η ξ η∂ψ ∂ϕξ ηψ2222 222 222 211 110 −  +−  +− −−+− =PP P Ph() (C-1) where hk a a ao ≡≡ ≡ ()22πλ πλ. We begin this process by assuming the solution of the partial differential equation to be the product of three distinct functions with each being only dependent on its respectivevariable: ψξηϕ ξ η ϕP PP PRS (, , ) () () () =Φ (C-2) where RP()ξ is the radial solution, SP()η is the angular solution, and ΦP()ϕ is the azimuthal (rotational) solution. Substituting (C-2) into the partial differential equation of (C-1), we obtain the following: () ()111122 RR SS pP pp ∂ ∂ξξ∂ ∂ξ∂ ∂ηη∂ ∂η−  +−   () () ()+− −−  +− =ξ η ξ η∂ ∂ϕξ η22 222 222 2 1110ΦΦ pph() . ( C - 3 ) 295Upon review of (C-3), one can see that ΦP, which is a function of ϕ only, can be made to equal a function of the other two variables: 12 2ΦΦ pp ∂ ∂ϕ= () () ()() () −−− −−  +−  +−    ξη ξη∂ ∂ξξ∂ ∂ξ∂ ∂ηη∂ ∂ηξη22 2222 2 2 211 1111RR SSh pP pp() . (C-4) However, all three variables are suppose to be independent of one another. This paradox can be resolved by setting each side of (C-4) equal to the same separation constant.Although theory allows this constant to be represented by any mathematical symbol, it is convenient to select the separation constant as −m 2. Consequently, we are able to reduce the partial differential equation of (C-1) into an ordinary differential equation thatdepends on ϕ only, viz., d dmPP2 220ϕϕϕΦΦ() ()+= where 0 2 ≤≤ϕπ ,( C - 5 ) and a partial differential equation that depends on both ξ and η only, viz., () ()111122 RR SS pP pp ∂ ∂ξξ∂ ∂ξ∂ ∂ηη∂ ∂η−  +−   () () ()−− −−+− =ξη ξηξη22 2222 22 110 mh () . ( C - 6 ) With some mathematical manipulation, the partial differential equation of (C-6) can be written as follows: 296()11122 222 RR mh pP ∂ ∂ξξ∂ ∂ξ ξξ −  −−+= () −−  +−+11122 222 SS mh pp ∂ ∂ηη∂ ∂η ηη.( C - 7 ) At this point, we have the two remaining independent variables represented as a function of one another. Once again, we resolve the paradox by setting both sides equal to asecond separation constant λ (not to be confused with free-space wavelength). As a result, we are able to reduce the intermediate partial differential equation of (C-6) into two separate ordinary differential equations: ()()d dd dRhmRPPξξξξ λ ξ ξξ22 22 21 10 −  −− + −    = () () w h e r e 1 ≤< ∞ξ , (C-8) ()()d dd dShmSPPηηηηλ η ηη 1 1022 22 2−  +− − −    = () () where −≤ ≤11η . (C-9) In short, the partial differential equation of (C-1) is reduced to three separate ordinary differential equations, namely (C-4), (C-8), and (C-9), by way of introducing two separation constants, namely −m2 and λ. Similarly, we can apply the same method to reduce the scalar Helmholtz partial differential equation in oblate spheroidal coordinates [23]: () ()() () ()∂ ∂ξξ∂ψ ∂ξ∂ ∂ηη∂ψ ∂ηξ η ξ η∂ψ ∂ϕξ ηψ2222 222 222 211 110 +  +−  ++ +−++ =OO O Oh() (C-10) where hk a a ao ≡≡ ≡ ()22πλ πλ. 297Adopting the same separation constants as the prolate spheroidal case, we are able to reduce (C-10) to the following three ordinary differential equations: d dmOO2 220ϕϕ ϕ ΦΦ() ()+= where 0 2 ≤≤ϕ π, (C-11) ()()d dd dRhmROOξξξξ λ ξ ξξ22 22 21 10 +  −− − +    = () () w h e r e 0 ≤< ∞ξ , (C-12) ()()d dd dShmSOOηηηηλ η ηη 1 1022 22 2−  ++ − −    = () () w h e r e −≤ ≤11η . (C-13) 298APPENDIX D. LEGENDRE FUNCTIONS Legendre Polynomials and Functions Legendre’s differential equation (where order m=0 ) [29], [40]: d dzzdy dzll y′−′′  ++ = () ( )11 02 where l is an integer and ′z is (D-1) complex; regular singularities exist at ′=±z 1 and ′=∞z . Definition of domain: In the following, z represents any arbitrary point in the finite complex plane with the exception of those points on the realaxis that lie on the −< <11x branch cut; meanwhile, x represents all points on the branch cut. Independent solutions of Legendre’s differential equation: Legendre polynomials (or Legendre functions of the first kind): Pz l( ) (off the branch cut) and Pxl( ) (on the branch cut); Legendre functions of the second kind: Qzl() (off the branch cut) and Qxl() (on the branch cut). 299Definition of Legendre polynomials [29], [40]: [] Pzlq ql q l qzlq l qInt l lq() ( )() ! !( )!( )!/ =−− −−=−∑ 122 2202 2(D-2) Pzld dzzl ll ll()!() =−1 212 (Rodrigues ’ formula) (D-3) where both equations are valid on and off the branch cut, i.e., z can be replaced with x; and Int x[ ] is the greatest integer function, e.g., Int[]72 3 = and Int[]22=. Table D.1 Legendre Polynomials Pz0 1 ()= Pz z1()= Pz z21 2231 () ( )=− Pz z z31 2353 () ( )=− Pz z z41 84235 30 3 () ( )=− + Pz z z z51 85363 70 15 () ( )=− + Pz z z z61 16642231 315 105 5 () ( )=− + − (Note: Polynomials are valid on and off the branch cut, i.e., z can be replaced with x.) Definition of Legendre functions of the second kind [29], [40]: Off the branch cut: Qzl()= 1 21 1Pzz zl() l n+ −−−− +−=−   −− ∑() () ( )()() 24 1 2101 21 21lq ql qPz qInt l lq for l≥1, (D-4) 300and Qz0()= 1 21 1lnz z+ − for l=0 ; (D-5) On the branch cut: Qxl()= 1 21 1Pxx xl() l n+ −−−− +−=−   −− ∑() () ( )()() 24 1 2101 21 21lq ql qPx qInt l lq for l≥1, (D-6) and Qx0()= 1 21 1ln+ −x x for l=0 . (D-7) From these expressions, one can see that the Legendre functions of the second kind exhibit singularities at z=±1. Special values and relationships between Legendre polynomials [28], [29], [40]: Pz P zll l ()() ( )−= − 1( D - 8 ) PzP zll−− =1( ) ( ) (D-9) where both equations are valid on and off the branch cut, i.e., z can be replaced with x; and Pl l ll llll l()()()()() ! ! !!00 1135 1 2461122 =−⋅ ⋅ ⋅⋅⋅ − ⋅⋅⋅ ⋅ ⋅=−−   odd, even;(D-10) Pl()11=; (D-11) Pl ll()−=−  11 odd, 1 even.(D-12) 301Special values and relationships between Legendre functions of the second kind [28], [40]: Qz Q zll l ()() ( )−= −+11(D-13) where the equation remains valid on and off the branch cut, i.e., z can be replaced with x; and Ql()1=+∞ ; (D-14) Ql ll()−=+∞ ∞  1 odd, - even.(D-15) Orthogonality relation [29]: Px P xd xll() ()′ −∫ 11 =+′2 21lllδ,. (D-16) Recurrence relations [29], [40]: ( ) () ( ) () ()21 111 lz P zlP z l P zll l += + ++− (D-17) ()( )zd dzPzl21−= + − ++ () ( ) () ( )lP zlz P zll111 =−− lz P z l P zll() ()1 (D-18) where both recurrence relations are also valid for Qzl(); in addition, both equations are valid on and off the branch cut, i.e., z can be replaced with x. 302Associated Legendre Functions Associated Legendre ’s differential equation (where order m≠0 ) [29], [40]: d dzzdy dzllm zy′−′′  ++ −−′  = () ( )111022 2 where l and m are (D-19) integers and ′z is complex; regular singularities exist at ′=±z 1 and ′=∞z . Definition of domain: As before, z represents any arbitrary point in the finite complex plane with the exception of those points on the real axisthat lie on the −< <11x branch cut; meanwhile, x represents all points on the branch cut. Independent solutions of the associated Legendre ’s differential equation: Associated Legendre functions of the first kind: Pz lm( ) (off the branch cut) and Pxlm( ) (on the branch cut) where Pz P zll0() ()= and Px P xll0() ()= (by definition); Associated Legendre functions of the second kind: Qzlm( ) (off the branch cut) and Qxlm( ) (on the branch cut) where Qz Q zll0() ()= and Qx Q xll0() ()= (by definition). 303Definition of associated Legendre functions of the first kind [29], [40], [41]: Off the branch cut: Pzlm() =−() ( )zd dzPzmm m l221 (Hobson ’s definition) =−−+ +() !()z ld dzzm lml mll22 2 1 21 ; (D-20) On the branch cut: Pxlm() =−() ( )122xd dxPxmm m l (Ferrer ’s definition) =−−+ +() !()1 2122 2 x ld dxxm lml mll. (D-21) (Note that Pxlm( ) is sometimes alternatively defined with an additional factor of ( ) −1m. As a consequence, some of the expressions that follow (namely, those that pertain to arguments on the branch cut) may or may not directly apply when using this otherdefinition. However, with some slight modifications, thoseexpressions that are affected by this alternative definition can be made to account for the additional ( ) −1 m factor.) 304 Table D.2 Associated Legendre Functions of the First Kind Off the branch cut On the branch cut Pz z112 1 21 () ( )=− Px x112 1 21 () ( )=− Pz z z212 1 231 () ( )=− Px x x212 1 231 () ( )=− Pz z22231 () ( )=− Px x22231 () ( )=− Pz z z31 3 222 1 251 1 () ( ) ( )=− − Px x x31 3 222 1 251 1 () ( ) ( )=− − Pz z z32215 1 () ( )=− Px x x32215 1 () ( )=− Pz z332 3 215 1 () ( )=− Px x332 3 215 1 () ( )=− Pz z zz41 5 232 1 273 1 () ( ) ( )=−− Px x x x41 5 232 1 273 1 () ( ) ( )=− − Pz z z42 15 22271 1 () ( ) ( )=− − Px x x42 15 22271 1 () ( ) ( )=− − Pz z z432 3 2105 1 () ( )=− Px x x432 3 2105 1 () ( )=− Pz z442 2105 1 () ( )=− Px x442 2105 1 () ( )=− Definition of associated Legendre functions of the second kind [40], [61]: Off the branch cut: Qz zd dzQzlmmm m l () ( ) ()=−221 ; (D-22) On the branch cut: Qx xd dxQxlmmm m l () ( ) ()=−122; (D-23) 305note that the associated Legendre functions of the second kind exhibit singularities at z=±1. (Just like Pxlm() , Qxlm() i s sometimes alternatively defined with an additional factor of ( ) −1m. Likewise, some of the following expressions (namely, those that pertain to arguments on the branch cut) also may or may notdirectly apply when using this other definition. Nevertheless, thoseexpressions that are affected by this alternative definition can also be made to account for the additional ( ) −1 m factor). Special values and relationships between associated Legendre functions [23], [29], [40], [61]: Pz P zlml m lm()() ( )−= −−1 (D-24) Qz Q zlml m lm()() ( )−= −++11(D-25) PzP zlm lm −− =1( ) ( ) (D-26) where all three equations are valid on and off the branch cut, i.e., z can be replaced with x. Off the branch cut: Pzlm lmPzlm lm −=− +()() ! () !( ) (D-27) Pzm mzm zmm mmm()() ! !()( ) ! ! () =− = − −2 212 1 122 22; (D-28) 306On the branch cut: Pxlm lmPxlmm lm −=−− +() ( )() ! () !() 1 (D-29) Pxm mxm xmm mmm()() ! !() ( ) ! ! () =− = − −2 212 1 122 22. (D-30) Additionally, Plm lm lm lm lm lmlmlml m l lm()() ()() ! !! ()() ! ! () ! !)() ()00 1 222 112 2=− −+ −  +   =−+− −−       − − odd, ( even;(D-31) Plm()±=1 0 for m≠0 . (D-32) Orthogonality relation [29]: Px Px d xlm lm() ()′ −∫ 11 =++ −′2 21llm lmll() ! () !,δ. (D-33) Recurrence relations [29], [35], [40]: ( ) () ( ) () ( ) ()21 111 l z P z l m Pz l m Pzlm lm lm+= − + + ++− (D-34) d dzPzlm()=−−+ −−lz zPzlm zPzlm lm 22 111()()( ) (D-35) 307where both recurrence relations are valid on and off the branch cut, i.e., z can be replaced with x. Off the branch cut: d dzPzlm()=+− + −−−− () ( )() ()lm lm zPzmz zPzlm lm 1 1 121 2 = −+−+Pz zmz zPzlm lm1 221 1()( ) ; (D-36) On the branch cut: d dxPxlm()=−+− + −+−− () ( )() ()lm lm xPxmx xPxlm lm 1 1 121 2 = −−−+Px xmx xPxlm lm1 221 1()( ) . (D-37) Note that all recurrence relations, i.e., (D-34)-(D-37), are also valid for Qzlm( ) and Qxlm() . Useful Formulas and Definitions Double factorial notation [29]: () ! ! () !22 4 6 2 2nn nn= ⋅ ⋅ ⋅⋅⋅ = (D-38) () ! ! ()() ! !21 1 3 5212 2nnn nn−= ⋅ ⋅ ⋅ ⋅ ⋅ − = . (D-39) 308Leibnitz ’s formula for the nth derivative of a product [29]: []d dxAxBxn sd dxAxd dxBxn nns ns sn s s() () () () =  − − =∑ 0(D-40) where n sn sn s  =−! !( )! is a binomial coefficient. 309APPENDIX E. DOUBLE- TO SINGLE-SIDED SPHEROIDAL WAVE-HARMONIC IDENTITY Employing the prolate spheroidal wave-harmonic defined in Chapter 2 (see 2.2.7), we can write the following: Yh Yh Rhml ml ml mll ,, ,()(, , ) (, , ) * (,)ϑϕ ϑϕ ξ11 2 2 4 =−+ ∑ [] =        +−− =−+ ∑Sh NheSh NheR hml mljm ml mljm ml mll , ,, ,,() (, c o s ) ()(, c o s ) ()(,)ϑ πϑ πξϕ ϕ 12 41 2212 =+− =−+ ∑1 212 412 πϑϑ ξϕϕSh Sh Nh Rheml ml ml mljm mll ,, ,,()()(, c o s ) (, c o s ) () (,).( E - 1 ) We now express the right-hand side of (E-1) in terms of non-negative indices of m: Yh Yh Rhml ml ml mll ,, ,()(, , ) (, , ) * (,)ϑϕ ϑϕ ξ11 2 2 4 =−+ ∑ = + +      +− −− −−−− =+ ∑1 2 1 201 02 004 12 4 12 4112 12πϑϑ ξ πϑϑ ξ ϑϑ ξϕϕ ϕϕSh Sh Nh Rh Sh Sh Nh Rhe Sh Sh Nh Rhell ll ml ml ml mljm ml ml ml mljm ml,, ,,() ,, ,,()() ,, ,,()()(, c o s ) (, c o s ) () (,) (, c o s ) (, c o s ) () (,) (, c o s ) (, c o s ) () (,).( E - 2 ) By definition (see Chapter 2), Sh S hml ml− ≡,,( ,cos ) ( ,cos ) ϑϑ , Rh R hml ml− ≡,() ,()(,) (,)44ξ ξ, and Nh N hml ml− ≡,,( ) ( ) . Accordingly, we are able to simplify (E-2) and thus put forth the 310following double- to single-sided spheroidal wave-harmonic identity for the prolate spheroidal case: Yh Yh Rhml ml ml mll ,, ,()(, , ) (, , ) * (,)ϑϕ ϑϕ ξ11 2 2 4 =−+ ∑ = +−       =∑1 2201 02 004 12 4 112πϑϑ ξ ϑϑ ξϕ ϕSh Sh Nh Rh Sh Sh Nh Rhmll ll ml ml ml ml ml,, ,,() ,, ,,()(, c o s ) (, c o s ) () (,) (, c o s ) (, c o s ) () (,)cos ( ).( E - 3 ) In a manner that is quite similar, one can show that the form of (E-3) is also valid for oblate spheroidal wave-harmonics. 311REFERENCES [1] R. C. Johnson, H. A. Ecker, and J. S. Hollis, “Determination of far-field antenna patterns from near-field measurements,” Proc. IEEE , vol. 61, pp. 1668-1694, Dec. 1973. [2] A. D. Yaghjian, “An overview of near-field antenna measurements,” IEEE Trans. Antennas Propagat. , vol. AP-34, pp. 30-45, Jan. 1986. [3] J. H. Richmond and T. E. Tice, “Probes for microwave near-field measurements,” IRE Trans. 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York, “FDTD analysis of CPW-fed folded-slot and multiple- slot antennas on thin substrates, ” IEEE Trans. Antennas Propagat., vol. AP-44, pp. 217-226, Feb. 1996. [59] G. C. Liang, Y. W. Liu, and K. K. Mei, “Full-wave analysis of coplanar waveguide and slotline using the time-domain finite-difference method, ” IEEE Trans. Microwave Theory Tech., vol. MTT-37, pp. 1949-1957, Dec. 1989. 316[60] M. R. Spiegel, Vector Analysis and an Introduction to Tensor Analysis, Schaum ’s Outline Series. New York: McGraw-Hill, 1959. [61] E. Jahnke and F. Emde, Tables of Functions with Formulae and Curves , Fourth Edition. New York: Dover, 1945. 317VITA Gerald F. Ricciardi was born in Bronx, NY, in 1965. He received the B.S. degree in electrical engineering from Columbia University, New York, NY, in 1987, and theM.S. degree in electrical engineering from the Johns Hopkins University, Baltimore, MD,in 1991. From 1987 to 1992, he worked for Westinghouse Electric Corporation,Baltimore, MD, where he developed algorithms used in the testing of prototype defensemissile systems. In August 1992, he came to Virginia Tech to pursue the Ph.D. degree inelectrical engineering. Gerald Ricciardi is a member of IEEE. His current research interests are in the areas of computational electromagnetics, numerical analysis, and antenna design.