My recent research investigates analytic and numeric techniques for solving systems of ordinary differential equations for liquid crystal display optics in one spatial dimension. An important application of measurements derived from display optics is a graphical summary of the dependence of optical contrast for a display upon the angle at which light impinges upon the device. The optical response of a display device does not deal with focusing of light waves; rather it describes the strength and polarization of waves which are either prevented from passing through the device (in the on-state) or are permitted to go entirely through the device (when it's in the off-state). Mathematical modeling of this optical response for a single display device requires many solutions of a differential system boundary value problem for various wavelengths and angles of incidence, thus making efficient solutions of the system an important goal.
The primary difficulty in solving this system (often referred to as the Berreman problem) is the very rapid and non-uniform oscillation exhibited by the functions which are solutions to the problem. This difficulty is termed complex oscillatory stiffness of the system and is parametrized by the thickness of the display cell in units of the light wavelength in the external medium.
Numerous researchers during the latter half of the twentieth century addressed this kind of problem, most offering techniques which seek to alleviate the oscillation difficulty for a large class of related problems. My own approach has focused on optimizing the use of a unique sort of midpoint quadrature rule for o.d.e.'s, whose utility is restricted to a smaller class of problems. I also compared several other methods for solving the problem, indicating computational regimes in which they perform favorably, and those situations where each experiences difficulties. I was also able to document some of the properties of and relationships between two asymptotic-numeric methods sometimes used for the Berreman system.
My immediate aim is to publish some papers directly related to my dissertation research. In the area of longer-term goals, I'm beginning to familiarize myself with the mathematics of optical data transmission networks, all-optical switching and data multiplexing schemes. I hope to be able to solve some problems involved with the application of liquid crystals to optical switching.