Continued Research Ideas

One paper that I have found seems to be helpful in that it discusses using the KK relations for a thin film. The generalized relation that takes into account the singularities that occur for complex frequencies is

Theta_R(omega)= - omega/Pi INT [0,Infinity]   (ln R(omega') )/(omega'^2-omega^2)d omega'

+ 2 SUM arctan (omega-a_j)/b_j

where a_j+i* b_j are the zeros of R(omega).

(1) These zeros are difficult to find experimentally.   I wonder if I can modify the code for the simulation in order to allow for complex frequencies.   (I do not know what the equations are like but it seems at least possible to do this).

Some other things that are important to do are:
(2) Think of an appropriate way to extrapolate in the LOW frequency region. The issue is that even outside the region where the mirror that I am working with was optimized, there is some significant reflection in the 10 to 30 eV range. My professor said that this isn't too surprising because in addition to the peaks that come from constructive interference, the metals that they use just reflect those frequencies. Experimentally, this could be problematic because you can only go so low with the energies in the synchrotron. Regardless, I hope to get some sort of bound on the error that this produces.

For now, it seems like it is appropriate to assume that reflectivity is a constant in the high frequency region since that is a small contribution. One thing that I am not sure about, however, is how the phase acts for large x. Regardless, I should be able to get a bound on the error that comes from the HIGH frequencies.


Another nice thing that I figured out today is that instead of writing a code to integrate the cauchy principal value integrals, I was just able to find a method that should be able to do it for me. The code is written somewhat poorly so I am working on getting that to work correctly.

All in all, I do feel good about this project. It is quite a challenge, but it is important for this lab to be able to measure the phase delay for their mirrors so I am contributing something. Hopefully I will be able to get my program to do a better job of predicting the phase.  If the few things that I am doing don't work, I will go to more papers and see how they approached this problem.