Another Day of Research

So I am really getting started on my research. I am pleased that I will be doing a challenging project.  To review, there is a mathematical relationship between the reflectivity and the delay (phase delay) of a mirror. Strictly speaking, this relationship requires that we know the reflectivity for all frequencies in order to determine the phase for a particular frequency. As it is not practical to measure the reflectivity for all phases, I am looking to approximate that relation so that the reflectivity only needs to be measured over a band. Also our lab builds mirrors by making layers of different metals. The lab uses a matlab program to optimize the depths of the layers. I am interested in learning how this program works.

Things to do:
1. continue to try to understand papers given to me and look at other papers that I have found on this subject
2. understand the matlab code used to simulate the reflectivity (this is more of a personal goal)
2a. learn about the simulated annealing algorithm
3. use simulated reflectivity data to estimate phase
3a. understand better what typical reflectivity data looks like for multilayer mirrors and see where the major contributions come from in terms of the

Today, I continued to read over the papers that one of the professors gave me. Now I am going to be more specific. So given a mirror, we can define a complex number called reflectivity, r(omega). This is a function of frequency. The magnitude of this complex number is the amplitude of the reflected wave divided by the input amplitude. The phase of this complex number is equal to the phase delay of the pulse after it is reflected. So it turns out that defining a complex reflectivity in this way has a number of nice properties. This is somewhat obvious, but if we write the input pulse in complex number form, then to transform that pulse (in the frequency domain), we just multiply by the reflectivity. So if we have a pulse in the time domain, we just Fourier transform it, multiply by the reflectivity, and then inverse fourier transform.

The other property that is really nice is that r(omega) is an analytic function (we extend omega to allow for complex frequencies) except at isolated points for Im(omega)>0. I am not quite sure why this is so. It may be arguable due to "causality arguments" but I am not sure. Anyways, experimentally, it is relatively easy to calculate R(omega) := |r|^2. So we know the magnitude but not the phase. We do a bit of a trick, we note that   f(w) = ln r(w) = (1/2) ln R(w)+i phi(w)   where phi(w) denotes the phase as a function of frequency. Since r is an analytic function, so is its logarithm. Now we know that the real an imaginary parts of a complex analytic function are related. This relation involves an integral from w=0 to infinity.

The formula that lets us calculate the imaginary from the real part requires that the real part be well behaved. In general, this is not a valid assumption so we need to do some normalization procedures in order to make the integral converge. As far as I can see, doing this requires some general knowledge of the function. Ex. for a semi infinite medium, the reflectivity for large frequencies is a real constant. Here, the integral diverges as ln(constant not equal to 1) is nonzero. So we divide out this constant and use that normalized reflectance to calculate the phase. I will write more about this when I get to it.