9610444 Kuchment The problem of creating photonic crystals has two major components: a mathematical one and a technological one. The mathematical part consists of non-standard spectral problems for the periodic Maxwell operator and some of its scalar counterparts. The aim of the proposed research is to develop novel analytic and numerical methods of analysis of frequency spectra depending on geometric and physical characteristics of the band-gap material. In particular, it is planned to develop new asymptotic models of the problem in 2D and 3D, to study numerically and analytically the asymptotic models for different geometries of PBG materials in 2D and 3D, to study relations between spectral properties of photonic crystals and of other mesoscopic physical systems, to study novel possibilities of localization of light, to study geometries that promise to open large spectral gaps, and to obtain new analytic results about effects caused by localized impurities. Photonic band-gap materials, often referred to as photonic crystals, have attracted a lot of attention since the idea was suggested in 1987. A photonic crystal is an artificial composite optical material that is an optical analog of a semiconductor. The main property of interest is existence of a range of frequencies (``colors") of light that cannot propagate in this medium. It is expected that creation of such a material would bring about a technological revolution in optics, laser technology, computing, information transmission, and other areas. Significant increase in efficiency of laser devices, highly efficient planar antennas, optics protecting against enemy's laser beams, perfect mirrors, laser diodes, memories, switches, and lossless optical information transmission lines are among numerous applications. This award is supported by the Division of Mathematical Science and the Experimental Program to Stimulate Competitive Research (EPSCoR).
