Collecting thoughts on Research

My first attempt to calculate the phase from the simulated reflectivity data did not seem to work so I am going back and reading as much as I can. I am looking for ways to improve my calculation of phase.

I attempted to implement a method in matlab that integrated reflectivity data to get the phase but the results are not very good. (I took reflectivity data for every 0.1 eV (apparently this is typical for real data), created a function that interpolated those points into a smooth curve (I used a cubic spline), then I numerically integrated that using an adaptive simpson's rule routine. In order to handle the principal part integral, I removed a symmetric neighborhood of the singularity).  For extrapolation, I assumed that the reflectivity was constant.

One problem: I am getting high reflectivity in regions where we wouldn't normally measure the reflectivity. I am looking at an aperiodic mirror from "Aperiodic multilayer mirrors for efficient broadband reflection
in the extreme ultraviolet". So there is reflectivity of about 0.15 in a wide region (30-80 eV) where the mirror was optimized, but there is also another peak near 10 to 20 eV. I am not sure if this is due to bad optical constants, or if this is just how their calculation worked out. Anyways this may be causing problems in my integration (and my way of extrapolating the reflectivity to low frequencies is not going to work)

So I looked at a lot of papers today. There is a ton of work that has been done in the area of using the KK relations.  Here is a summary of things to do / things that I am thinking about:

1. learn more optics
a. how is the reflection coefficient derived from known constants for a multi-layer (maybe this would help me understand where the zero reflectivity points are)
b. does the reflectivity coefficient have zeros in the complex frequency plane  (how can I predict their position... eg. the dielectric function for a semi-infinite substrate has its only zeros on the imaginary axis)
c. qualitatively, what is the reflectivity of a multilayer for high frequency/low frequency (so I can extrapolate correctly)
d. what effect does polarization, angle of incidence have on the calculations that I am doing?
e. is there a relationship between the complex reflectivity coefficient and the complex dielectric function

2. Kramers-Kronig relationship
a. which representation should I be using
i. real and imaginary part of a boundary condition for an analytic function are related by the Hilbert transform
ii. the transformation can be done as two Fourier transforms
b. zeros of reflectivity (for complex frequency) give rise to singularities in the integral above and result in extra terms being needed in the transformation
c. how is this derived? how can one use causality to show that reflectivity as a function of a complex variable is analytic?

3. Using matlab to do the hilbert transform numerically (there are MANY papers on this subject)
a. use of gaussian quadrature method (write integral as INT w(x) p(x)  where w(x) is the weight function log x^(-1)... this may require a different program as you need high precision arithmetic for large numbers of sample points
b. write the transform of a sequence of fourier transforms and approximate as a discrete fourier transform (cooley-turkey algorithm?)
c. look into algorithms that allow one to isolate the zeros of the reflectivity as a function of a complex variable (eg. find the blaschke factors that are needed to recover a function that has singularities when it is fourier transformed...one paper suggests using transmission data?)
d. look into methods that acceleration of convergence of the integral by knowing the phase (eg. multiply subtractive KK method)

Anyways, I feel like there is SO much that I can do at the moment.