Research Initiation Awards provide support for faculty at Historically Black Colleges and Universities who are building a research program. It is expected that the award helps to further the faculty member's research capability and effectiveness, improves research and teaching at his home institution, and involves undergraduate students in research experiences. The award to Howard University has potential broader and societal impact in a number of areas. The project seeks to investigate mathematical models that originate from geometrical optics. The research crosses the fields of mathematics, engineering and physics and has potential applications in the synthesis, design and manufacturing of optical systems in an efficient and cost effective manner. Undergraduate students will gain research experiences and the research will be integrated in a number of mathematics courses.
The refractor problem is an inverse problem in geometric optics that is related to modeling of interface surfaces between two media that have different optical properties. The goal of this work is to obtain and analyze surfaces which are capable of redirecting light rays emanating from a point source located in one of the media into a set of directions or onto a specific target located in the other media, in such a way that the input and output intensities of the rays are prespecified. The application of the laws of geometric optics and conservation of energy principles to formulate the mathematical model will naturally lead to second order nonlinear partial differential equations of Monge - Ampère type. The determination of the surfaces will incite problems of existence, uniqueness and regularity of solutions to nonlinear partial differential equations and mass transport problems. The specific objectives of this project include: developing a computational algorithm for the near field refractor problem in isotropic media, formulating mathematical models of geometrical optics problems in anisotropic media such as crystals, finding numerical and analytical solutions for near field and far field refractor problems in anisotropic media and investigating the regularity of analytical solutions for near field and far field refractor problems in anisotropic media. The consideration of anisotropic media will advance the mathematical theory of geometric optics problems to media of propagation that are more general and will make it more amenable to several optics applications including optical devices, design of freeform refractive beam shapers and laser optics.
