OP: Heterogeneous Optical Media: Boundary Effects, Spectral Properties, and Inversion

尽管在计算速度方面取得了进步，但光波通过复杂材料传播的模拟仍然面临着重大挑战。从根本上说，困难的出现是因为材料结构的变化与光波长在同一尺度上。因此，蛮力方法需要非常精细的材料离散化和非常大量的参数，这通常在计算上是不可行的。研究人员和合作者的目标是开发新的数学技术，允许提取系统的重要属性和特征，而无需完全模拟。在医学成像中，因此可以在真实的时间中重建初始图像。这样的初始图像本身就很有用，或者用作输入来加速高分辨率方法的运行时间。在材料设计中，可以估计材料的能量损失，以提供对材料潜在适用性的定性洞察，从而加速下一代光学器件的构建。本计画的目标是解决数个与异质材料光学与光子学相关的开放性问题。该项目将包括四个主要组成部分。第一个目标是找到周期性光学材料中边界效应的完整表征。虽然均匀化理论中的边界修正是出了名的难以理解，但它们对结果场有很大的影响。一个大类的周期性光学材料可以研究新的渐近技术，使我们能够充分表征边界的作用。 第二是推导出周期散射体的各种光谱特性的显式公式。这包括传输本征值，它可以在远场中读取并提供有关介质的信息。三是发展了一种新的降阶模型方法来求解椭圆算子的反演问题。该方法采用模型简化理论的思想，并起源于一个突破，在以前的光谱匹配网格的方法，允许应用到更高的维度和一般的几何形状。 最后一个目标是推导出多个线性和非线性散射体相互作用的共振值的显式计算。研究人员和合作者使用光谱渐近微扰方法来理解线性材料和克尔效应的多散射体共振相互作用的行为。

Despite advances in computing speed, the simulation of the propagation of optical waves through complex materials continues to pose significant challenges. Fundamentally, the difficulty arises because variations in the material structure are on the same scale as the optical wavelength. Brute-force approaches thus require very fine discretization of the materials and a very large number of parameters, which is frequently computationally infeasible. The investigator and collaborators aim to develop new mathematical techniques that allow extraction of important properties and features of a system without the need for full simulation. In medical imaging, an initial image could thus be reconstructed in real time. Such an initial image can be useful in itself, or used as input to speed up the run time of high-resolution methods. In design of materials, the energy losses of a material can be estimated to provide qualitative insight into the potential suitability of the material and thus accelerate the construction of the next generation of optical devices. Several graduate and undergraduate students will participate in related research projects.The objective of this project is to solve several open problems related to optics and photonics of heterogeneous materials. The project will involve four main components. The first is to find a complete characterization of boundary effects in periodic optical materials. While boundary corrections in homogenization theory are notoriously difficult to understand, they have a large effect on the resulting fields. A large class of periodic optical materials can be studied by new asymptotic techniques that will allow us to fully characterize the role of the boundary. The second is the derivation of explicit formulae for various spectral properties of periodic scatterers. This includes the transmission eigenvalues, which can be read in the far field and provide information about the medium. The third is to develop a new reduced order model approach to inversion of elliptic operators. The method employs ideas from model reduction theory, and originates from a breakthrough in a previous spectrally matched grid approach that allows application to higher dimensions and general geometries. The last goal is to derive explicit calculations of the resonance values for multiple linear and nonlinear scatterer interactions. The investigator and collaborators use spectral asymptotic perturbation methods to understand the behavior of multiple-scatterer resonant interactions, both for linear materials and for Kerr effects.