OP: Monge-Ampere type equations and geometric optics

光学器件在多种实际应用中起着非常重要的作用，许多光学器件都是由透镜和反射镜组成的。传统上，由于制造限制和生产成本，大多数光学设计都是旋转对称的。随着计算机控制的精密机器的发展，不一定是对称的光学器件，称为自由形式或非球面，可以制造。特别是，随着增材制造或三维制造的最新进展，自由形状的透镜也可以制造出来。透镜和反射镜的设计目前是通过光线追踪（即通过软件计算光线轨迹）进行计算的，其中的选择是基于有根据的猜测。这个项目关注的是设计透镜、反射镜和天线的数学方法的发展。与过去的方法相比，它提供了对设计问题的更好理解，并使解决方案能够适应任何特定应用程序所需的需求。它开发的模型在物理上比以前的模型更准确，并考虑了内部反射等因素。该项目涉及光学和光子学中具体问题的非线性偏微分方程的理论方面。此外，首席研究员还关注由于波导弯曲，透镜色差和材料折射率变化，特别是负折射引起的能量损失的数学描述。研究的一部分涉及透镜数值计算算法的开发。这些问题是用偏微分方程来描述的，它涉及到输出和输入强度以及依赖于模型的其他参数。目标涉及解的存在性、唯一性和正则性。所使用的方法源自数学领域的最优质量传输，优化和非线性偏微分方程的一种特殊类型，称为蒙日-安培型方程。这些技术包括各种梯度图的上下估计和严格的凸性分析。偏微分方程很自然地出现，因为人们寻找的界面表面具有这样的性质，即在小面积上发送和接收的能量之间的比率可以用输入和输出能量密度来表示。这就得到了一个包含未知曲面二阶导数的方程。透镜数值计算算法的发展需要建立相关泛函的利普希茨估计。此外，由于尺寸与辐射波长相当的波导本质上是电磁的，因此该建议与不连续和各向异性介质中麦克斯韦方程组的研究相结合。

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.