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

尽管计算速度取得了进步，但光波在复杂材料中传播的模拟仍然面临着巨大的挑战。从根本上说，困难的出现是因为材料结构的变化与光学波长的变化是相同的。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，这在计算上往往是不可行的。研究人员和合作者的目标是开发新的数学技术，允许在不需要完全模拟的情况下提取系统的重要属性和特征。在医学成像中，初始图像因此可以实时重建。这样的初始图像本身可能很有用，或者用作输入以加快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，以便定性地了解材料的潜在适宜性，从而加速下一代光学器件的构建。几名研究生和本科生将参与相关的研究项目。该项目的目标是解决与非均质材料的光学和光子学相关的几个公开问题。该项目将涉及四个主要组成部分。第一个是寻找周期性光学材料中边界效应的完整表征。虽然均化理论中的边界修正是出了名的难以理解，但它们对所产生的场有很大的影响。一大类周期光学材料可以用新的渐近技术来研究，这将使我们能够充分描述边界的作用。二是推导出周期散射体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关于介质的信息。三是发展了一种新的椭圆算子求逆的降阶模型方法。该方法采用了模型降阶理论的思想，并起源于对以前的谱匹配网格方法的突破，该方法允许应用于更高维和一般几何形状。最后一个目标是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.