Mie second attempt debug log

Mie second attempt debug log PhL 2.16.03 I am now working with the xxx1.doc paper which has many things stripped out. 1. Setup of the E fields. The setup on the three E equations agrees with the 4 page paper exactly. So we are defining the a,b,c,d coefficients as they are defining them. 2. Obtaining the H field. This is the area in which there is a large chance of an error. There are two aspects of this that need to be scrutinized. (a) my rule for getting B from E (b) my rule for generalization to and media of Jackson's work. None of the references I have state the B or H fields, so I have no way to directly check. I will come back and work in this area in a moment. 3. Statement of the Boundary Conditions. Continuity of both E and H tangential is what is claimed. This is the way Krugel states it. I could look at the other two BC's and try to debug that way, seeing if I get different equations from them. This is a possible debug approach. At least for non-conductors where there are no surface currents, I am pretty confident that these are the correct r x boundary conditions. So let's move on. 4. Calculation of the r x M and the r x N expressions. My 4-page paper does not really define these things in the even and odd sense, but shows the general form that M = x (f r) where f is a Bessel, and is an angle-only function. As noted elsewhere, we have x (f r) = (f ) x r + f x r = [ f + f ] x r + 0 = [ -hat and -hat things + f ' r-hat ] x r = f x r = -f r x So we can think of M either way: M = x (f r) M = -f r x The first form appears in the 4-page paper. The second form appears in Krugel, except he leaves off the factor of -f and talks about just M = r x by itself as a VSH function. I guess then the superscripted M functions are supposed to include a Bessel function. Now let's look in a general way at r x M. It is f * - r x r x where f is the generic Bessel. I don't really care about the details of the second factor, except to observe whether it has the even or odd in it. What is important is that r x M = constant * f . Thus, wherever we see r x M, we are going to replace with f. There is no derivative of f here, there is no possible index n involved here. is an angle function, it knows nothing about index. It is true that f = f(k,r) = j(kr) for example DOES know about the index because k is for either inside or outside. So we have to keep track of fo versus fi . Now what about r x N? We DO have to think in more detail about this one because the function f is so embedded. First of all, every one agrees that N = (1/k) x M and I suddenly see a light bulb go on: I may be done something wrong with this 1/k factor! Remember that it is an index-dependent factor! But let's continue for a while on our little systematic debug and at least finish this section. We then have r x N = (1/k) r x x M = (1/k) r x x [ -f r x ( r) ] This doe seem a bit formidable, as the French say. Parens are always implied in the usual grouping. In my appendix I show that x [ -f r x ( r) ] = - f 2 r + (fr)' Then we apply r x to this thing, and the first term drops out since r x r = 0, and we get r x N = (1/k) r x [(fr)' ] = (1/ki) (fr)' r x where k is a specific region k value. From later on we know that ki = ni k, so rewrite r x N = (fr)'/n * (1/k) r x The thing on the right is the inert common factor, what we care about is on the left! So this is my second huge mistake of today! Let's go back now to the xxxx1 paper and see if this fixes things. Probably not, but at least we will have another bug crushed. Result: As expected, fixing this error still has not given the right answers. We are now getting n2 factors in the solution. But at least we did fix something that was surely wrong. Summary of the corrected rules from this section: r x M = f * - r x r x r x N = (fr)'/n * (1/k) r x The factors on the right will be "common factors" for all terms of that type in an equation. STOP. Are these two VSH basis functions shown on the right orthogonal? From my appendix, the first one is r x r x = - r2 , and we recall that yes, r is in another direction from r x . That is also obvious from the r x (r x ) relation. So yes, the rx M and r x N terms must be balanced separately. STOP AGAIN. Is there some hidden k-dependence in that derivative?? Write it out in more detail: (fr)' = r [ f(kr) r ] = /(kr) [f(kr) kr ] = /x[f(x) x ] | x = kr = [f(x) x ]' This last thing must be evaluated at the correct x, either xi = ki a or the other. It and (fr)' are both dimensionless. Try again on this, be specific (fr)' = r [ f(ki r) r ] = r r f(ki r) + f(ki r) = r ki /( ki r) f(ki r) + f(ki r) = r ki f' (ki r) + f(ki r) = xi f' (xi) + f(xi) = /xi [xi f(xi) ] = /x [x f(x) ] | x = xi . I don't see any room for an error here, we end up with (fr)' = [f(x) x ]' | x = xi for internal, for example. This must be what is meant by the symbols in my 4 page paper. So, I guess we might now say: (ji r)' = [j(x) x ]' | x = xi BUT, now look at our expected answer (1.10) for coefficient 'a' in the 4 page paper. If we keep the factors together as above, then there is in fact NO index appearing in this result!!! The factor [ mx j(mx)]' is really a shorthand for [ x j(x) ]' | x = mx. In this case, we ARE supposed to get factors of n2 in the b coefficient! In fact, I see that now my answers for a and b agree with the 4-page paper, except we have a b, so that somehow these two coefficients are swapped. But c and d are exactly correct without swapping! So this is the closest we have been yet! What would happen to the results presented in the 4-page paper if we go on to define those new functions: (x) x j(x) (x) x h(x) In this case, you would simply replace things like so: [ x j(x) ]' | x = mx = '(mx) which gives a less ambiguous result! Therefore, (1.10) for 'a 'would have no index, and (1.11) would have the square of the index appearing! Then the results for "the scattering coefficients" a and b shown in (3.9) and (3.10) of my 3 page library book clipping DISAGREE with those of my 4-page paper! Why is this? Because in the 3-page library thing, these results have linear factors of m only, m being the "complex refractive index". But in my 4-page paper, a has NO factor of m, and b has only quadratic factors! What about Krugel? His results agree exactly with the 3-page clip, so they must both be quoting the same source! But both Krugel and the 3-page-clip give the same THREE references, so we don't know which one they are really quoting! I wish I had the Hulst book for $10 from Amazon, or from elsewhere. So now my sources are disagreeing among themselves. Let's now go back to the original paper on "second attempt" and clean up all the known errors and just see what it has to say. I have to fix these things: (1) I goofed up by adding an extra k in dealing with the type functions. (2) the replacement rule for the N terms has a 1/n index added to it. These are the two bugs I found today. 5. Extraction of the four equations. Once we have our two big r x X equations, we should be able to just "read off" the balances. The M and the N terms must separately balance, as noted above. This is just a question of writing things down correctly without making typos. I will recheck this one more time now on a printed copy. // DONE. 6. Entry into Maple and use of Maple. This SEEMS straightforward, but at this point I need to suspect every single step along the way. I just checked the transcription of the equations, did that right. 7. Reversing the bug? Look at the middle two equations for a and d in the Version 2 debug print. They have this form: A d = n B + n a C D n d = E + a F Mult top one by n D/A and get D n d = n2 G + n2 a H D n d = E + a F When we subtract, we find that a will be a function of n2 , as Maple confirms. Therefore, at least one of these two middle equations must be wrong.