# 1D Finite Planar Photonic Structure - Transfer Matrix Method

1. Nov 16, 2012

### aphirst

1. The problem statement, all variables and given/known data
I'm implementing the transfer matrix method (manually) for an EM wave through a 1D layered structure. Basically I'm just considering a plane wave in the positive-x direction, conserving E and H across each material interface, and constructing interface matrices, the appropriate matrix product of which allows you to work out forward/reverse coefficients in each region, and the reflectivity/transmittivity.

I have it set up such that I just make a big list of (complex) refractive indices, and the widths of each sublayer (or the positions of each interface); also specifying the angle of incidence (in the xz plane), and the polarisation (TE or TM, i.e. either E or H respectively entirely in the y-direction).

Whenever the refractive index on the far LHS (i.e. the material that extends to $-\infty$) is equal to that on the far RHS (i.e. extending to $+\infty$), reflectivity and transmittivity both work perfectly.

However when they differ, for instance in the case with only one interface; reflectivity behaves perfectly, while transmittivity is no longer bounded under 1.

2. Relevant equations
E-mail to my supervisor, in which I walk through the cases that do and don't work. Includes plots.
https://dl.dropbox.com/u/3219541/Project/email.pdf [Broken], or the attached email.pdf

A transcript of my working for this method, including some explicit calculations to demonstrate the fact that R behaves, while T does not.
https://dl.dropbox.com/u/3219541/Project/calculations.pdf [Broken], or the attached calculations.pdf

3. The attempt at a solution
I went to see him to discuss the problem yesterday: he told me it was to do with that RHS/LHS difference (rather than an artefact of only performing the method for a single interface, which is what I had thought). He then said something about phase velocities, and about having to scale something according to the ratio of permittivities (or, complex refractive index squared), but wasn't massively clear where that was supposed to come up in the maths. I've read over the relevant sections of some EM textbooks, but I can't see how that's supposed to change what I've worked out.

If any of you have the time to read over what I've done, and (if I'm fortunate) point me in the right direction, I'd really appreciate it.

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Last edited by a moderator: May 6, 2017
2. Nov 25, 2012

### aphirst

After a lengthy dialogue with my supervisor, I solved the problem!

I had been implicitly assuming that since I was conserving the values of E and H at each interface, I was therefore automatically conserving energy flow through the structure. This of course was not the case.

So I constructed a function to calculate the time-averaged Poynting vector, and took the ratio of the energy flow just after the final interface and the energy flow of only the incident light. And what do you know, this gave me exactly what I expected for T.

For instance, here's the output for TM light at the interface between air and glass.