|Department of Mathematics and|
Ph.D. Student in Applied Mathematics
Department of Mathematics & Computer Science
233E Mathematical Sciences Bldg. - 1345 Summit Rd.
Kent State University - P.O. Box 5190
Kent, Ohio 44242-0001, U.S.A.
|Office: 272 Mathematical Sciences Building|
For detailed information regarding my daily activities, see the plan page. For information regarding content, grading, quizzes etc. for the courses I'm teaching, see the Beginning Algebra page.
Some information for prospective employers can be found in my resumé. A text-only version of this resumé can be obtained separately. The curriculum vitae is a similar document describing my qualifications for prfessional academic positions.
The mathematical problem of predicting the response of light to inhomogeneous media is a fairly old one. A medium is inhomogeneous whenever the speed at which it permits light to travel (i.e. its refractive index) varies with different locations in the medium. Common media of this sort, like the Earth's atmosphere and liquid-crystal films, often exhibit such variation in layers; these are stratified media. Fortunately, the mathematical formulation for the problem is a linear ordinary differential equation. Unfortunately, the o.d.e. is a complex-stiff oscillatory one. A sense for the history of the problem can be obtained by looking at Propagation of light through an inhomogeneous medium, a paper by R. Gans which appeared in Annalen der Physik back in 1915.
Yet another wrinkle to this problem is anisotropic media. Even homogeneous media can exhibit dielectric anisotropy where the index of refraction varies according to how the light is polarized. One example of an anisotropic medium is a calcite crystal. When looking at printed letters through a calcite crystal, two images of the letters appear because light of differing polarizations passing through the crystal is refracted differently. Most liquid-crystal media used in information display devices are not only anisotropic but also the direction of the anisotropy changes with depth inside the liquid-crystal film. Such stratified anisotropic media were what Dwight Berreman investigated in the early 1970's using a matrix method to solve the first-order linear differential equation which describes the optics of such a situation.
Matrix methods for the Berreman o.d.e. became extremely popular as desktop computers made their way into laboratories and offices. Since most optical modeling requires hundreds or thousands of solutions of the problem for various wavelengths and angles of incidence, speedy solutions of the o.d.e. are a priority. One of the best o.d.e. solvers to be used in these situations is the mid-point optical eigenmode method (informally, o.e.m.) Part of my research has been focussed on describing the advantages and disadvantages of several variants of o.e.m. In an attempt to gain quicker solutions, my advisor and I turned to the calculation of eigenvalues and eigenvectors for the central matrix of the o.d.e. as a way of factoring out the rapid oscillation in the solution functions and enabling the use of a more coarse grid in the numerical treatment of the problem. You can view a couple of pictures I've created which depict the variation of the ordinary and extraordinary eigenvectors for the Berreman 1-d liquid crystal optics o.d.e.
A different sort of graph I've created shows the problem eigenvalues as a function of the liquid-crystal director for the problem. Under the assumption of a particular incidence angle, many different individual l.c. director possibilities are summarized in this one graph.
Another analysis of the problem seeks to isolate the complex amplitudes for the four problem eigenmodes. Using the bend mode of a pi-cell, where the director field for the liquid-crystal exhibits varying degress of tilt alone, I've both worked the calculations by hand and allowed my software to do its job and graph for me the complex amplitudes. The ordinary eigenmodes for this situation do not vary at all, and are in fact constant to the precision of my machine. Their companions, the extraordinary eigenmodes appear also to be nearly constant. Their variation in this particular situation can be observed only in the fourth decimal place, which is a bit surprising since this director field gives an eigenvalue coalescence at z = 0.5 . The conclusion here seems to be that the coalescence does not contribute much to the variation of these eigenmode amplitudes, and in fact a "connection formula" for the o.d.e. propagator in the neighborhood of this particular coalescence ought to be close to an identity matrix.
The use of a helical director field shows how informative the "amplitude" vector approach can be. In a simple twist cell where an electric field has been imposed, the l.c. director partially aligns with the vertical in the interior of the cell. In this case light is incident at approximately 30° from the vertical, and the partial alignment of the director halfway through the medium again gives two eigenvalues which coalesce at z = 0.5 . The ordinary eigenmodes and the extraordinary eigenmodes for this problem vary quite a bit, though the variation occurs at scales related to the l.c. medium, rather than being associated with the much more rapid oscillation of the light waves themselves. In this sense our method stands to speed up computations of this sort because it requires a mesh which resolves only changes in the medium itself, not the more rapid variation of the electromagnetic fields within the light waves. Unfortunately, close attention to the continuity of the eigenvectors, along with a fair amount of computational overhead associated with the couplings between modes makes these "amplitude" vector methods somewhat slow by comparison with the midpoint o.e.m. method.
Because of the a problem with the numerical conditioning of eigenvector calculations at computational nodes where coalescence of eigenvalues occurs, we've experimented recently with a another variant of the midpoint optical eigenmode method. The "Cayley-Hamilton" version of midpoint o.e.m., described by Wöhler/Haas/Fritsch/Mlynski in 1988, does not need to handle the propagator in any special way when eigenvalues coalesce. Since it seems to require essentially the same amount of computational time as the "eigenvector" o.e.m., the improved handling of coalescence seems at this writing to be its only advantage. Its ease of use and comparisons with the "eigenvector" o.e.m. is described in my dissertation