Research Statement
George F. Barrick, Ph.D.

January, 2007

My dissertation research investigated analytic and numeric techniques for solving systems of ordinary differential equations for liquid crystal display optics in one spatial dimension. Summaries of the dependence of optical contrast for a display upon the angle at which light impinges upon the device are an important application of display optics. Mathematical modeling that describes the strength and polarization of light waves passing through a liquid crystal display cell requires many solutions of a differential system boundary value problem for various wavelengths and angles of incidence. Because the model is implemented repeatedly for single analyses of a device, efficient solutions of the system are an important goal.

The primary difficulty in solving this system (often referred to as the Berreman problem) is the rapid and non-uniform oscillation exhibited by the functions that are solutions to the problem. Common quadrature methods suffer from a need to take small computational steps when solving for the oscillating functions that occur in this context. My approach to this difficulty focused upon a specialized kind of midpoint quadrature rule for differential equations having harmonically oscillating solutions. The specialized quadrature rule permits computations that proceed accurately using only one computational step per wavelength of light in the liquid crystal portion of the device.

My recent research is directed at numeric techniques for analyzing optical phased arrays for signal switching systems. This research applies to beam steering devices used in laser optical communication systems, but it also gave me the opportunity to use parallel computing techniques when solving energy minimization and optical transmission parts of the problem. A newer project, in which I have so far read only three articles, is leading toward computational simulations of photonic crystals and the (still speculative) applications of these constructs to negative-index materials.

During the past three years I have been involved with the small company Rocketcalc that supplies parallel computing hardware and software solutions to academic and industrial concerns. My efforts there have been directed towards creating flexible software systems for use in high-performance computing environments for both Windows and GNU/Linux systems, though I’ve also spent time assembling and testing HPC systems.

George Barrick