guassianbeams

ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 22 R. Victor Jones, February 7, 2000III.THE PARAXIAL WAVE EQUATION -- PROPAGATION OF GAUSSIAN BEAMS IN UNIFORM MEDIA D ERIVATION OF P ARAXIAL W AVE E QUATION : In point-to-point communication, we may think of the electromagnetic field as propagating in a kind of "searchlight" mode -- i.e. a beam of finite width that propagates in some particular direction. In analyzing this mode of wave propagation, we make use of an important solution to the so call paraxial approximation of the electromagnetic wave equation (or, more precisely, the paraxial approximation of the Helmholz equation). To that end, we first derive the paraxial approximation and then examine the free-space Gaussian Beam solution(s). We start with the homogeneous Helmholz equation for the vector potential in the form -- see Equation [ I-13a ] ∇2r A r r ,ω()+ω2µ0εω()r A r r ,ω()=∇2r A r r ,ω()+k2r A r r ,ω()=0[ III-1 ] We are looking for a wave propagating in, say, the z-direction, so we write a particular component of the potential in the form Aαr r ,ω()=Ψr r ,ω()exp−ikz() [ III-2 ] The function Ψr r ,ω() represents a spatial modulation or "masking" of a plane wave propagating in the z-direction. The z-direction is obviously special and it is useful to appropriately parse the differential operators. For the grad operator we may write grad{ }=r ∇ { }=r ∇ t{ }+ˆ z ∂ ∂z{ }[ III-3 ] where, for example, ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 23 R. Victor Jones, February 7, 2000 r ∇ t{ }=ˆ x ∂ ∂x{ }+ˆ y ∂ ∂y{ } . [ III-4 ] so that r ∇ Aαr r ,ω()=r ∇ tΨr r ,ω()+ˆ z ∂ ∂zΨr r ,ω()−ikˆ z Ψr r ,ω()      exp−ikz() [ III-5 ] For the Laplacian operator we may write ∇2Aαr r ,ω()=∇t2Ψr r ,ω()exp−ikz()+∂ ∂z∂ ∂zΨr r ,ω()−ikΨr r ,ω()      exp−ikz()      [ III-6 ] where, for example, ∇t2{ }=∂2 ∂x2{ }+∂2 ∂y2{ } [ III-7 ] Therefore, ∇2Aαr r ,ω()=∇t2Ψr r ,ω()+∂2 ∂z2Ψr r ,ω()−2ik∂ ∂zΨr r ,ω()−k2Ψr r ,ω()      exp−ikz() [ III-8 ] and the parsed Helmholz equation ( without approximation ) becomes ∇t2Ψr r ,ω()−2ik∂ ∂zΨr r ,ω()+∂2 ∂z2Ψr r ,ω()=0 [ III-9 ] The paraxial approximation is precisely defined by the condition 2ik∂ ∂zΨr r ,ω()>>∂2 ∂z2Ψr r ,ω() [ III-10 ] ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 24 R. Victor Jones, February 7, 2000which means that the longitudinal variation in the modulation function, Ψr r ,ω(), changes very little in the wavelength associated with beam -- i.e. 2πk. In this approximation, we neglect the third term and obtain the equation ∇t2Ψr r ,ω()−2ik∂ ∂zΨr r ,ω()=0 [ III-11 ] which is called the paraxial approximation of the wave equation.15 S OLUTIONS OF THE P ARAXIAL W AVE E QUATION The Gaussian beam To inform or motivate our next step, we consider the paraxial approximation of a known solution of the Helmholz equation -- i.e. a spherical wave exp−ikr() r=exp−ikx2+y2+z2( ) x2+y2+z2=exp−ikz1+x2+y2 z2      z1+x2+y2 z2≈exp−ikz()exp−ikx2+y2() 2z      z[ III-12 ] Reflecting on the "quadratic" form of this approximate expression, it is reasonable to look for an axially symmetric solution of the paraxial wave equation in the following form -- i.e. a Gaussian beam : ΨGρ,z,ω()=AGexp−iPz()[]exp−ikρ2 2qz()      [ III-13 ] 15Obvious the paraxial equation has the same mathematical form as the Schrödinger equation and, thus, all that is know about solutions of that equation may be directly applied to understand issues in light propagation (or visa versa). ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 25 R. Victor Jones, February 7, 2000where ρ2=x2+y2 . We test our conjecture by substituting the Gaussian beam function -- i.e. Equation [ III-13 ] -- into the paraxial wave equation -- i.e. Equation [ III-11 ] -- to wit exp−iPz()[]∇t2exp−ikρ2 2qz()            −2ik∂ ∂zexp−iPz()[]exp−ikρ2 2qz()            =0 .[ III-14 ] Executing the indicated operations, we obtain exp−iPz()[]exp−ikρ2 2qz()      −2ik qz()−k2ρ2 qz()[]2−2ik−i∂ ∂zPz()+ikρ2 2qz()[]2∂ ∂zqz()            =0[ III-15a ] or simplifying 2k∂ ∂zPz()+2ik qz()+k2ρ2 qz()[]21−∂ ∂zqz()      =0 . [ III-15b ] Hence, for an arbitrary ρ this equation is separable into two parts -- viz. k2ρ2 qz()[]21−∂ ∂zqz()      =0 ∂ ∂zqz()=1[ III-15c ] and 2k∂ ∂zPz()+2ik qz()=0 ∂ ∂zPz()=−i qz()[ III-15d ] which are satisfied by the simple solutions qz()=z+q0 [ III-16a ] ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 26 R. Victor Jones, February 7, 2000and ∂ ∂zPz()=−i z+q0=−i∂ ∂zlnz+q0[] Pz()=−ilnz+q0[] .[ III-16b ] On comparison with the paraxial approximation of a spherical wave -- i.e Equation [ III-12 ] -- we may write qz() in terms of a radius of curvature Rz() and a width wz() -- viz. 1 qz()=1 z+q0=1 Rz()+−i2 kw2z() . [ III-17 ] To standardize the constants of integration we assume a plane wavefront at an arbitrary reference point z=0 -- i.e. we take R0()≡∞. Thus, 1 R0()=0 [ III-18a ] and−i2 kw20()≡1 q0 ⇒ q0=ikw20() 2=iπw20() λ=iLF [ III-18b ] whereLF=kw20()2=πw20()λis the critical Gaussian beam scaling parameter which is called variously the Fresnel length, the diffraction length, or the confocal parameter . In terms of this parameter, Equation [ III-17 ] may be written 1 qz()=1 Rz()+−i2 kw2z()=1 z+iLF=z−iLF z2+LF2 [ III-19 ] Equating real and imaginary parts, we obtain 1 Rz()=z z2+LF2 and −i2 kw2z()=−iLF z2+LF2 ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 27 R. Victor Jones, February 7, 2000or, finally, in standardized form Rz()=z1+LF2z2[] w2z()=w20()1+z2LF2[] LF=πw20()λ where[ III-20 ] Now since Equation [ III-16b ] may be written Pz()=−ilnz+q0[]=−ilnz+iLF[]=−ilnz2+LF2[]+itan−1LFz[] { }l[ III-16b' ] we may write exp−iPz()[]=exp−itan−1LFz[] [ ] z2+LF2=exp−itan−1LFz[] [ ] z1+LF2z2 to obtain the usual, officially approved form of the Gaussian Beam ΨGρ,z,ω()=AGexp−iPz()[]exp−ikρ2 2qz()      =AGw0() wz()exp−itan−1LFz[] [ ]exp−ikρ2 2Rz()      exp−ρ2 w2z()      [ III-21 ] ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 28 R. Victor Jones, February 7, 2000The following kind of picture is sometimes found to be helpful in understanding the propagation of a Gaussian Beam (the bold curve depicts the spatial variation of the beam width and the light curve the beam curvature at particular points in space): Higher order Hermite-Gaussian beams In order to study the propagation of higher order beams, we substitute the following trial solution: ΨH-Gρ,z,ω()=Fx,y,z()ΨGρ,z,ω() =fx w    gy w    exp−iΦz()[]ΨGρ,z,ω()[ III-22 ] into the paraxial wave equation -- viz. Equation [ III-11 ] -- and obtain ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 29 R. Victor Jones, February 7, 2000 Fx,y,z()∇t2ΨGρ,z,ω()+2r ∇ tFx,y,z()⋅r ∇ tΨGρ,z,ω() [ ]+ΨGρ,z,ω()∇t2Fx,y,z() −2ikΨGρ,z,ω()∂ ∂zFx,y,z()−2ikFx,y,z()∂ ∂zΨGρ,z,ω()=0[ III-23 ] Since the sum of the first and fifth terms already satisfies the paraxial wave equation, Equation [ III-23 ] reduces to ′ ′ f f+2ikdw dz−w q      x′ f f+′ ′ g g+2ikdw dz−w q      y′ g g−2kw2dΦ dz=0 [ III-24 ] From Equations [ III-19 ] and [ III-20 ] we see that dw dz−w q=w R−w R+−i2 kw      =i2 kw so that the reduced equation -- i.e. Equation [ III-24 ] -- becomes ′ ′ f f−4ξ′ f f+′ ′ g g−4ς′ g g−2kw2dΦ dz=0 [ III-25 ] where ξ=xw and ς=yw. A Hermite polynomial of order n16 has the following differential equation: d2 dτ2Hnτ()−2τd dτHnτ()+2nHnτ()=0. [ III-26 ] 16The Hermite polynomials have the generator Hnτ()=−1()nexpτ2()dn dτnexp−τ2() . ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 30 R. Victor Jones, February 7, 2000With the simple change in variables τ=2ξ=2xw and σ=2ς=2yw Equation [ III-25 ] may be written 1 fd2f dτ2−2τdf dτ      +1 gd2g dσ2−2σdg dσ      −2kw2dΦ dz=0[ III-27 ] Thus, it is apparent that we can write the functions fx w    and gy w    as Hermite polynomials -- viz. fx w    =Hnτ()=Hn2x w    and gx w    =Hmσ()=Hm2y w    if we require that 2kw2dΦ dz=−2n+m(). Hence dΦ dz=−n+m() kw2=−n+m()LF 2LF2+z2[] or Φz()=−n+m()atanzLF(). Finally we may write a general solution for the paraxial equation as ΨH-Gnmρ,z,ω()=AH-Gnmw0() wz()Hn2x w    Hm2y w    ×expin+m+1[]tan−1zLF() [ ]exp−ikρ2 2Rz()      exp−ρ2 w2z()      [ III-28 ] ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 31 R. Victor Jones, February 7, 2000A GALLERY OF HERMITE-GAUSSIAN FIELD DISTRIBUTIONS [0, 0] Hermite-Gaussian [0, 1] Hermite-Gaussian ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 32 R. Victor Jones, February 7, 2000 [1, 1] Hermite-Gaussian [2, 2] Hermite-Gaussian ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 33 R. Victor Jones, February 7, 2000 G AUSSIAN B EAM T RANSFORMATION M ATRICES What we have shown above is that a given Hermite-Gaussian beam is essentially completely specified or defined by the complex function qz(). In propagating through an optical system, the beams are transformed by various optical components. The amazing fact is that the transformation produced by a given component follows a simple ABCD transformation law -- viz. q2=Aq1+B Cq1+D[ III-29 ] where A, B, C, D are the matrix elements found in our analysis of geometric optics!! To "prove" this, we argue by example. For example, the transformation through a uniform dielectric region is given by q2=q1+L so that A=1;B=L;C=0;D=1 { } and the transformation through a thin lens is given by 1 q2=1 q1−1 f so that A=1;B=0;C=−1 f;D=1      . Further "justification" of this transformation law may be found in terms of the so called "ρ′ ρ argument"-- viz ON CLASSICAL ELECTROMAGNETIC FIELDS PAGE 34 R. Victor Jones, February 7, 2000 From geometric optics and, in particular, Equation [ II-11 ], we may write ρout ′ ρ out=Aρin ′ ρ in+B      Cρin ′ ρ in+D      −1 or Δzout=AΔzin+B( )CΔzin+D( )−1 which is identical to transformation equation [ III-29 ] if we interpret qz() as the wave optics generalization of Δz=ρ′ ρ .