OP: Monge-Ampere type equations and geometric optics

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Optical devices play very important roles in multiple practical applications, and many of these devices are built with lenses and mirrors. Traditionally, most optical designs have been rotationally symmetric 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 advances in additive or three-dimensional manufacturing, free-form lenses can also be made. The design of lenses and mirrors is currently made computationally with ray tracing (i.e., by software calculating the trajectory of the light rays), in which choices are based upon educated guesses. This project is concerned with the development of mathematical methods for the design of lenses, mirrors, and antennas. It provides a better understanding than past approaches of the design problems and enables one to adapt the solutions to the needs required for any specific application. It develops models that are physically more accurate than earlier ones and takes into account, for example, internal reflection. The project addresses theoretical aspects of nonlinear partial differential equations with concrete problems in optics and photonics. In addition, the principal investigator is concerned with the mathematical descriptions of the loss of energy due to bending in waveguides, chromatic aberration for lenses, and variations of refractive indices in materials, in particular, negative refraction. A part of the research concerns the development of an algorithm for the numerical calculation of lenses.The problems are described by partial differential equations involving the output and input intensities and other parameters depending on the models. The goals are concerned with existence, uniqueness, and regularity properties of the solutions. The methods used derive from the mathematical fields of optimal mass transportation, optimization, and nonlinear partial differential equations of a specific type, called Monge-Ampere-type equations. The techniques include lower and upper estimates of various gradient maps and strict convexity analysis. Partial differential equations appear naturally because the interface surface one looks 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 energy densities. This yields an equation involving second derivatives of the unknown surface. The development of an algorithm for the numerical calculation of lenses requires establishing Lipschitz estimates for the associated functionals. Furthermore, as waveguides of size comparable to the wavelength of radiation are electromagnetic in nature, the proposal blends with the study of Maxwell's equations in discontinuous and anisotropic media.