Monge-Ampere-type equations and geometric optics

在这个项目中考虑的主要问题是关于表面的设计，将从一个源点（或一组点）发出的辐射重定向到一组方向或目标屏幕上，以这样一种方式，辐射的输入和输出强度是规定的。光源和目标位于不同的材料中，具有不同的折射率，所寻求的表面是两种介质之间的界面。为了尽可能准确地描述这些现象，提出了几种模型，考虑了各种重要因素，如内反射的能量损失和目标到源的距离。所提出的问题涉及这些不同模型解的存在性、唯一性和规律性。这些问题的表面解满足完全非线性的蒙日-安培型偏微分方程，该方程涉及输出和输入强度以及取决于模型的其他参数。这样的方程很自然地出现了，因为我们所寻找的界面表面具有这样的性质，即在小面积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。本项目所进行的研究自然出现在光学器件（如非球面透镜、反射镜、天线）的设计中，这些器件在许多光学和传输仪器的构造中有多种应用。所研究的大部分问题都具有实际意义，例如，设计将光聚焦到期望的目标目的地的透镜。这个项目的影响在于数学理论的发展，它将使这个设计更有效、更精确，并且更容易和快速地适应不断变化的情况。它包含了对这些设备的构造具有潜在变革意义的想法和模型。该研究对工程设计和制造具有潜在的应用价值；特别是，在设计非球面透镜的照明应用，其中能量优化是重要的。此外，所开发的数学理论的数值实现将在光学器件的实际构造中有用。这项研究还结合了其他数学和物理领域的最新发展，如变换光学和隐形。该项目涉及与美国和国外科学家的合作，并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.