OP: Monge-Ampere type equations and geometric optics

OP：Monge-Ampere 型方程和几何光学

• 批准号: 1600578
• 负责人: Cristian Gutierrez
• 金额: $ 24万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2016
• 资助国家: 美国
• 起止时间: 2016-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1600578&HistoricalAwards=false
• 关键词: OP; Monge; Ampere; type; equations

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.

光学器件在许多实际应用中起着非常重要的作用，并且这些器件中的许多都是用透镜和镜子构建的。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，可以制造出不一定对称的光学器件，称为自由曲面或非球面。特别地，随着增材或三维制造的最新进展，还可以制造自由形式的透镜。透镜和反射镜的设计目前是通过光线跟踪（即，通过软件计算光线的轨迹），其中选择基于有根据的猜测。这个项目关注的是透镜、反射镜和天线设计的数学方法的发展。它提供了一个更好的理解比过去的方法的设计问题，并使一个适应的解决方案所需的任何特定的应用程序。它开发的模型在物理上比以前的模型更准确，并考虑到了内部反射等因素。 该项目涉及非线性偏微分方程的理论方面，以及光学和光子学中的具体问题。 此外，主要研究者关注的是由于波导弯曲导致的能量损失的数学描述，透镜的色差，以及材料折射率的变化，特别是负折射。研究的一部分涉及透镜数值计算的算法的发展。问题由包含输出和输入强度以及依赖于模型的其他参数的偏微分方程描述。目标是关注的存在性，唯一性和正则性的解决方案。所使用的方法来自数学领域的最佳质量运输，优化和非线性偏微分方程的一个特定类型，称为蒙格安培型方程。该技术包括各种梯度映射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。这就产生了一个涉及未知曲面的二阶导数的方程。透镜数值计算的算法的发展需要建立相关泛函的Lipschitz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，因此该提案与不连续和各向异性介质中的麦克斯韦方程组的研究融为一体。