This project investigates nonlinear partial differential equations of Monge-Ampere type (i.e., equations involving the Jacobian determinant of a map). A large portion of the research is concerned with lens and reflector antenna design. A lens is basically an optical surface that separates two materials with different indices of refraction. Two types of situations are considered: the far field problem, in which light or radiation needs to be received in a prescribed set of directions; and the near field problem, in which a target or screen needs to be illuminated (or radiated) in a prescribed way. In both cases, radiation emanates from a source point. Since the phenomena of refraction and reflection always occur simultaneously, when energy is refracted (or transmitted) there is always a fraction of this energy that is lost in internal reflection. It is important in the applications to optimize the energy refracted and so we are interested in the development and treatment of models that take into account this loss of energy. For the mathematical treatment of these problems, a fundamental difference appears: far field problems can be cast in the frame of optimal transportation (an area of mathematics dealing with the optimal allocation of resources). By contrast, since near field problems are not variational, they cannot be cast in those terms. This makes near field problems more difficult. The problems in the project range from questions of existence and uniqueness of solutions to various equations that model these problems to the study of their geometric and regularity properties. They offer various degrees of difficulty. Recent major breakthroughs for Monge-Ampere-type equations make these problems mathematically sound and challenging. A large portion of them have practical interest and, in addition, are aesthetically beautiful. The ideas proposed for their solution will improve the theoretica lunderstanding of fully nonlinear partial differential equations and will have an impact on applications in geometric optics.
The research in this project arises in the mathematical description of numerous optical, acoustic, and electromagnetic applications, as well as in global positioning systems (GPS). If successful, it could be of great benefit for engineering design and manufacturing. The project has connections, interactions, and applications within several areas in mathematics and outside. In addition to what was mentioned earlier, questions in mass transportation have applications to differential and convex geometry, optimization, economics, and quality control. The understanding of the properties of optimal maps also has possible implications for numerical computations. The work will involve collaborations with mathematicians in the US and abroad and will contribute to the training of graduate students.
Optical devices play a very important role in practical multiple applications and many are built with lenses and mirrors. Traditionally, most lens and mirror designs have been rotationally symmetric with conic sections profile because of manufacturing limitations and production costs. With the development of computer controlled precision machines, optical devices that are not necessarily symmetric, called free-form or aspherical, can be manufactured. In particular, with the recent and very quick development of 3d-printing or additive manufacturing, free-form lenses can also be made. A main part of this research is concerned with building a bridge between partial differential equations (PDEs), and it's applications to geometric optics including the desing of lenses, mirrors and antennas. In particular, we develop theories providing a better understanding of the design problems that enable us to adapt the solution to the needs required for the specific application. The methods used to develop our theories derive from the mathematical fields of optimal mass transportation, optimization, and non linear partial differential equations of a specific type, called Monge-Ampere type equations. Mathematically, the design of several optical devices can be succinctly formulated as follows: We have two homogenous materials I and II with refractive indices nI and nII. From a point source O surrounded by medium I, radiation is being emitted with directions in a set D, and with a given energy distribution f. We are given a surface screen destination T with a distribution of energy g, where T is surrounded by medium II. Assuming conservation of energy in the process, the question is how to design an interface surface between media I and II redirecting the radiation emanating from O into the destination T in such a way that distributions f and g are preserved. In other words, and for example if I is glass and II is air, how to design a lens such that the energy f over each little patch of directions in D is refracted into a subset of the target T with energy g. Similar problems can be formulated in the design of reflector antennas, for example in mirror design for cars. PDEs appear naturally because the interface surface we are looking for has the property that the ratio between the energy sent and received over a small area can be expressed in terms of the input and output functions f and g. This yields an equation involving second derivatives of the unknown surface. We have also developed models that are physically more accurate to enable the design of lenses that take into account the loss of energy due to internal reflection. In all cases, we prove existence of the interface surface desired with a pretty explicit construction that in some cases can be effectively calculated numerically.
