Things are Coming Together...

Things that I did today:

(Got Principal Value Integral Package to work)
So today, I got a matlab package to work that integrates principal value integrals. The code was written poorly but I adjusted it so that I could pass a function handle to the method that did the integration. From what I can see, the integral does not contribute much to the phase. The larger contributions seem to come from the extrapolation terms.

(Thoughts about extra terms due to complex zeros of R)
In the actual phase graph, there seems to be a jump discontinuity. Looking at the equation for the phase, it seems like this is due to a complex zero in the reflectivity. (The arctan term gives a jump discontinuity if the real part is small).

(Learning how to Calculate Reflectivity)
Anyways, now I've been looking into the code that lets you calculate the reflectivity. The relevant topic is called "the transfer matrix method"   (see http://en.wikipedia.org/wiki/Transfer-matrix_method_(optics) ). The idea is that if you have a layered structure, you look at the electric field and the derivative of E with respect to the normal to the surface and the propagation of the waves can be described by multiplying the E, dE/dz vector by a matrix.

(Derivation of KK Relations)
Also one question that came up yesterday is why are the complex zeros of reflectance relevant in the equation. So the derivation of the KK relationship using complex variables is very nice and can be found here:  http://en.wikipedia.org/wiki/Kramers%E2%80%93Kronig_relation (see derivation). The basic idea is that you take a contour integral of f(omega')/(omega'-omega) that goes along the real axis and then a semicircle in the upper half of the complex plane (with a little indent around omega). If there are singularities, you have to surround those with little loops to take them into account when you deform the integration contour.

(Algorithm to find complex zeros)
So I adjusted the program that is used to calculate reflectivity given a frequency so that it can pass it into a root finding algorithm. I wonder if I will be able to find the zeros of this function.

Anyways, I feel pretty good about my project because I am understanding things and I have a lot of things that I can do.