Monge-Ampere-type equations and geometric optics

在这个项目中考虑的主要问题是与表面的设计有关，该表面将从源点（或一组点）发出的辐射重定向到一组方向或目标屏幕上，以这种方式规定辐射的输入和输出强度。源和目标位于具有不同折射率的不同材料中，并且所寻求的表面是两种介质之间的界面。提出了几种模型来尽可能准确地描述这些现象，考虑到各种重要因素，如我们的能量损失内反射和目标的距离源。所提出的问题涉及这些不同模型的解的存在性、唯一性和规律性。这些问题的表面解决方案满足完全非线性偏微分方程的Monge-Ampere型涉及的输出和输入强度和其他参数取决于模型。这样的方程自然出现，因为我们寻求的界面具有这样的性质，即在小区域内发送和接收的能量变化之间的比率可以用输入和输出强度来表示。本项目中所追求的研究自然出现在光学器件的设计中（例如，非球面透镜、反射镜、天线），其在许多光学和传输仪器的构造中具有多种应用。研究中的大部分问题具有实际意义，例如，在将光聚焦到期望的目标目的地的透镜的设计中。该项目的影响在于数学理论的发展，这将使这种设计更有效，更精确，更容易，更快地适应不断变化的情况。它包含的想法和模型可能会对这些设备的构建产生变革性影响。 这项研究是潜在的有用的工程设计和制造，特别是在非球面透镜的照明应用中的能量优化是很重要的设计。此外，数值实现的数学理论开发将是有用的，在实际建设的光学设备。这项研究还融合了数学和物理学其他领域的最新发展，如变换光学和隐身。该项目涉及与美国和国外科学家的合作，它将有助于培养研究生。

The main question considered in this project is concerned with the design of surfaces redirecting radiation emanating from a source point (or a set of points) onto a set of directions or a target screen in such a way that the input and output intensities of the radiation are prescribed. The sources and the target lie in different materials having different refractive indices, and the surface sought is the interface between the two media. Several models are proposed to describe these phenomena as accurately as possible, taking into account various important factors such us energy loss in internal reflection and distance of the target to the sources. The questions proposed concern existence, uniqueness, and regularity of solutions for these various models. The surface solutions to these problems satisfy fully nonlinear partial differential equations of Monge-Ampere type involving the output and input intensities and other parameters depending on the model. Such equations appear naturally because the interface surface we seek has the property such that the ratio between the changes of energy sent and received in a small area can be expressed in terms of the input and output intensities.The research pursued in this project arises naturally in the design of optical devices (e.g., aspherical lenses, mirrors, antennas) that have multiple applications in the construction of many optical and transmission instruments. A large portion of the problems under study have practical interest, for example, in the design of lenses focusing light into a desired targeted destination. The impact of the project lies in the development of a mathematical theory that would render this design more efficient, precise, and easily and quickly adaptable to changing situations. It contains ideas and models that are potentially transformative for the construction of those devices. This research is potentially useful for engineering design and manufacturing; in particular, in the design of aspherical lenses for illumination applications where energy optimization is important. In addition, numerical implementation of the mathematical theory developed will be useful in the actual construction of optical devices. The research also blends with recent developments in other areas of mathematics and physics, such as transformation optics and cloaking. The project involves collaborations with scientists in the US and abroad, and it will contribute to the training of graduate students.