Inverse design methods for optical applications

该项目涉及光学中出现的两个问题的分析研究和计算方法的发展。第一个问题出现在渐进镜片的设计中，渐进镜片必须具有期望的矫正能力和最小的像差。在这种情况下的光学现象是很好地描述了几何光学。第二个问题出现在利用光子带隙现象的光学器件的设计中。在这个问题中，光的波长与结构的大小大致相同，并且需要使用麦克斯韦方程对光的传播进行建模。在这两种情况下，设计问题可以作为一个逆问题，其中所需的性能可以被视为数据和设计参数作为未知数。 本研究的目的是使用分析工具来指导有效的计算方法的发展，以解决这些反问题。许多45岁以上的人患有一种叫做老花眼的疾病，这是眼睛聚焦近物的能力丧失。渐进式镜片，它可以同时矫正远近物体，经常被开给老花眼患者。考虑到患者的需求和舒适度，这种镜片的设计是一个涉及光学、微分几何和优化的数学问题。创造具有高处理速度能力的光学计算机的梦想需要开发与电子设备相对应的光学设备。光学器件的设计需要对光在复杂纳米结构中的传播进行建模，这可以通过偏微分方程来描述。 为了创建执行某些功能的设备，有必要解决优化设计的数学问题。在这两个光学器件的设计统一的主题是数学的逆问题。即将进行的研究将有助于为未来的光学计算机创造前所未有的性能和光学计算子系统。

This project concerns analytical studies and the development of computational approaches to two problems arising in optics. The first problem arises in the design of progressive lenses which must have desired corrective powers and minimal aberrations. The optical phenomenon in this case is well described by geometrical optics. The second problem arises in the design of optical devices that exploit photonic bandgap phenomena. In this problem, the wavelength of light is about the same size as the structure, and calls for modeling of the light's propagation using Maxwell's equation. In both cases, the design problem may be posed as an inverse problem where the desired performance can be viewed as data and the design parameters as unknowns. The aim of this research is to use analytical tools to guide in the development of effective computational methods to solve these inverse problems.Many people over 45 years old suffer a condition called presbyopia, which is the loss of the ability of the eye to focus on near objects. Progressive lenses, which correct for seeing both far and near objects, are often prescribed to presbyopia patients. The design of such lenses, taking into account the patient's need and the comfort, is a mathematical problem that involves optics, differential geometry, and optimization. The dream of creating optical computers that have the capability of high processing speeds requires the development of optical devices that are counterparts of electronic devices. The design of optical devices requires modeling of light propagation in complex nanostructures, which can be described by partial differential equations. In order to create devices that perform certain functions, it is necessary to solve a mathematical problem of optimal design. The unifying theme in the design of these two optical devices is the mathematics of inverse problems. The research to be carried out will help create lenses of unprecedented performance and optical computing subsystems for future optical computers.