The main question considered in this project is concerned with the design of surfaces redirecting radiation emanating from a source point (or a set of points) onto a set of directions or a target screen in such a way that the input and output intensities of the radiation are prescribed. The sources and the target lie in different materials having different refractive indices, and the surface sought is the interface between the two media. Several models are proposed to describe these phenomena as accurately as possible, taking into account various important factors such us energy loss in internal reflection and distance of the target to the sources. The questions proposed concern existence, uniqueness, and regularity of solutions for these various models. The surface solutions to these problems satisfy fully nonlinear partial differential equations of Monge-Ampere type involving the output and input intensities and other parameters depending on the model. Such equations appear naturally because the interface surface we seek has the property such that the ratio between the changes of energy sent and received in a small area can be expressed in terms of the input and output intensities.
The research pursued in this project arises naturally in the design of optical devices (e.g., aspherical lenses, mirrors, antennas) that have multiple applications in the construction of many optical and transmission instruments. A large portion of the problems under study have practical interest, for example, in the design of lenses focusing light into a desired targeted destination. The impact of the project lies in the development of a mathematical theory that would render this design more efficient, precise, and easily and quickly adaptable to changing situations. It contains ideas and models that are potentially transformative for the construction of those devices. This research is potentially useful for engineering design and manufacturing; in particular, in the design of aspherical lenses for illumination applications where energy optimization is important. In addition, numerical implementation of the mathematical theory developed will be useful in the actual construction of optical devices. The research also blends with recent developments in other areas of mathematics and physics, such as transformation optics and cloaking. The project involves collaborations with scientists in the US and abroad, and it will contribute to the training of graduate students.
