The research supported with this award will develop the mathematical foundations of the physical phenomena that drive the functionality of photonic devices. Specific objectives include the development of (1) models, for example in slab structures, that allow a prediction of detailed outcomes of resonant electromagnetic scattering behavior based on certain measurable data for the device, (2) algorithms for optimization of certain output of devices, as the energy transmission or density of photonic states, and (3) exploration of new directions, in particular, waves in composite materials with negative-index components. These problems will be approached with rigorous mathematical methods, including perturbation theory, the theory of partial differential equations, homogenization, and complex function theory.
The research funded by this award is inspired by topical problems and challenges arising in the engineering and fabrication of existing and future opto-electronic devices, including imaging devices, lasers, and antennas. Many of the problems deal with physical phenomena that have been known for over a hundred years but which, because of recent recognition of the potential of photonic crystals and related structures to control the path of light, manifest themselves in new and largely unexplored ways. Other problems are concerned with the development of novel materials, or meta-materials, which have the capacity to maneuver electricity and magnetism in ways that cannot be achieved using naturally-occuring materials. The involvement of graduate students in the research agenda is an integral part of the project. Three students will be supported as research assistants as they are trained for careers in mathematical research. An undergraduate student mentored by the PI is involved as well in the investigations pertaining to meta-materials. The investigations will be strengthened and enriched through the PI's collaboration with scientists in the Physics and Electrical Engineering Departments at LSU, as well as with colleagues affiliated with the Center for Computation and Technology at LSU.
