Excellence in Research: Optimal Transport and Freeform Refractive Surfaces in Geometric Optics

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The project aims to study mathematical models to design freeform (asymmetric) optical surfaces (lenses) that direct light intensity distribution in a controlled and efficient way. This project has vast applications in the design of optical components in antennas, solar panels, medical devices, military equipment, and many other related areas. The research will help reduce the amount and size of optical components required in optical devices, thereby improving the manufacturing of lighter, more compact, and more efficient optical systems. The project will enhance the research infrastructure at Howard University and provide training opportunities for students from underrepresented groups in interdisciplinary research activities that intertwine mathematics, physics, and engineering. The project aims to study geometric optics problems that formulate relationships between the intensities of light before and after transport via reflection or refraction. This research will introduce new numerical approaches and further discuss analytical aspects of solutions to problems related to refractive optical surfaces in both isotropic and anisotropic media. The numerical techniques will be developed by exploiting entropic optimal transport, which involves adding a penalty term to make the problem convex and also by using Minkowski-type arguments. The analytical investigation will be conducted by using the intrinsic geometric properties of the redirecting surfaces. Additionally, techniques from the closely related nonlinear partial differential equations known as generated Jacobian equations, one of which is the Monge-Ampere equation, will be used for further analysis.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.