Mark Meilstrup and Dr. Scott Glasgow, Mathematics
Many physical processes involve the loss or dissipation of energy. For example, when pushing a box across the floor, some of the energy that is spent will be lost to frictional forces. Our research is concerned with the study of the energy which is lost, and also the energy which is recoverable, in the more complex setting of a linearly dissipative optical system. We consider some electromagnetic pulse, such as a laser beam, entering and interacting with an absorbing medium, such as glass or some gas. When the pulse exits the medium, some of the energy has been absorbed, and this amount varies dependant upon the evolution of the pulse. While our research centers on such electromagnetic systems, the results of our research also apply directly to other linear systems, such as lumped LRC (inductor, resistor, capacitor) circuits and also springmass systems.
In order to determine how much energy was actually lost at a given time during our process, we defined the irrecoverable energy (or losses) as the amount of energy that would never be recovered, no matter what the future evolution of the electromagnetic pulse might be. Using this definition and previous theorems and equations describing such systems, we were able to find the future pulse for optimal energy recovery, and consequently the amount of energy that was recoverable and that which was irrecoverably lost.
The solution to this problem dictated a recovery pulse that never ceased in time. That is, for any pulse ending at some time t, there is another pulse that recovers more energy from the system, but which does not cease until after time t. This infinitely long recovery leads us to the question which is the center of this continued research: How much energy is irrecoverably lost, if we require that our recovery happen within some specified period of time?
Adding this deadline to the energy recovery made the problem significantly harder. For simple enough systems, where we are given an explicit representation of how the medium responds to excitation, we were able to take advantage of this extra knowledge to adequately solve the problem by means of factorization. This factorization method works (for simple media) both with and without the deadline, being more complicated in the case with a deadline.
When not considering a deadline for recovery, we were able to solve the problem for arbitrary media, with the solution usable in cases where no explicit formula describing the response is given, but rather there are only experimental data. This method of solving our problem is related to solving a scalar Riemann-Hilbert problem, in which we have two unknown variables, but only one equation. We are able to solve this because of some other information that we know about the variables from the physical problem. However, with the deadline attached, this method gives us four unknown variables, with only two equations; equivalently, we now have a vector Riemann-Hilbert problem.
Unfortunately, the vector Riemann-Hilbert problem has not been solved in generality. One way that we have attempted to solve our instance of the vector Riemann-Hilbert problem is to compare it to those which have been solved by others describing certain physical phenomena. But this has not proved very useful, as our problem seems to be significantly different than those discussed and solved by these other authors.
We have also looked in a textbook on complex analysis, the branch of mathematics that covers the Riemann-Hilbert problem. This has provided more success. One section of the book discusses the general solution of the problem for a class of vector problems, of which ours is a specific case. Using this method, we have a solution to our vector problem, given the solution to five simpler scalar Riemann-Hilbert problems. The difficulty comes in finding correct solutions to these scalar problems so that the combined answer to the vector problem satisfies the correct physical properties that our system requires.
In addition to these methods of directly attacking the vector problem, we have also considered the factorization method, for a simple medium. While this factorization method does not directly apply to the general case which we are trying to solve, we hope to find some insight from this approach that might apply to solving the vector problem. Specifically, we hope that by obtaining a solution by this method for a certain simple medium, we will learn how to solve either the scalar problems that arise in our partial solution to the vector problem, or will otherwise help us in the matrix factorization that is inherent in the vector problem.