Resonance Phenomena in Wave Scattering

尽管光学和电子设备种类繁多且复杂，但它们都是基于一个共同的基本物理原理，即共振。这一原理存在于激光器、滤光片、天线、发光二极管以及其组件在纳米级运行的设备背后。研究者研究有效和稳健的方法来利用共振现象的设备设计，以及如何控制设备性能的变化，导致不可避免的制造缺陷。共振理论有着深刻的数学根源，它的发展是为了更好地理解现代设备设计中的新物理现象。该项目培养研究生和本科生在数学、物理和工程方面的技能，特别注意弥合学科之间的沟通差距。本科学生参与创建一个用于虚拟实验和光学结构设计的在线计算工具。研究了共振的四个问题。(1)双周期波导的电磁共振与麦克斯韦方程组的复特征值相关联，复特征值编码了共振的频率和谱宽。研究者考虑了一种新的波导结构，这种结构允许通过改变源波的入射角来调节共振的两个特性。由于共振波导模式的对称性，这种调谐能力在现有结构中被排除在外。通过利用非互易材料、特殊几何形状和对称导模分岔到反对称模来打破这种对称性，以构建非对称可调谐波导共振。(2)局部摄动周期量子图中的嵌入特征值导致了双层石墨烯等二维分子结构中的共振现象。在他最近关于嵌入特征值的研究结果的基础上，研究者利用不对称性来控制量子图中的共振散射特性。(3)即使在非线性系数较小的材料中，共振引起的场放大也会引起强烈的非线性效应，如双稳性和迟滞性。研究者在一个易于处理的模型中量化了这些效应，在这个模型中，光子传输线与一个局部激子场耦合，产生一个被称为极化子的光子激子准粒子的非线性场。(4)波导的材料特性和几何结构的随机波动引入了共振特征的可变性，例如它们的中心频率和谱宽。研究者计算这种可变性使用散射矩阵的扰动分析，其中扰动参数是随机变量。

ShipmanDMS-1411393 Optical and electronic devices, as varied and complex as they come, are based on a common underlying physical principle called resonance. This principle lies behind lasers, filters, antennas, and light-emitting diodes, as well as devices whose components operate at the nanoscale. The investigator studies efficient and robust ways to exploit the phenomena of resonance in device design and how to control variations in device performance resulting from inevitable manufacturing imperfections. The theory of resonance has deep mathematical roots, which are developed to gain a better working understanding of new physical phenomena for the design of modern-day devices. The project trains graduate and undergraduate students to develop skills at the interfaces of mathematics, physics, and engineering, paying special attention to bridging communication gaps between the disciplines. Undergraduate students are involved in creating an online computational tool for virtual experimentation and design of optical structures. Four problems of resonance are studied. (1) An electromagnetic resonance of a doubly periodic waveguide is associated with a complex eigenvalue of the Maxwell equations, which encodes the frequency and spectral width of the resonance. The investigator considers new waveguide structures that allow both characteristics of a resonance to be tuned by varying the angle of incidence of a source wave. This ability to tune is precluded in existing constructions of resonant waveguide modes due to the symmetry that they utilize. This symmetry is broken by exploiting non-reciprocal materials, special geometries, and bifurcations of symmetric guided modes into antisymmetric modes, to construct asymmetric tunable waveguide resonances. (2) Embedded eigenvalues in locally perturbed periodic quantum graphs are responsible for resonance phenomena in two-dimensional molecular structures such as bi-layer graphene. Building on a recent result of his on embedded eigenvalues, the investigator employs asymmetry to control resonant scattering characteristics in quantum graphs. (3) Even in materials with small coefficients of nonlinearity, field amplification due to resonance causes strongly nonlinear effects, such as bistability and hysteresis. The investigator quantifies these effects in a tractable model in which a photon transmission line is coupled to a local exciton field, producing a nonlinear field of photon-exciton quasiparticles known as polaritons. (4) Random fluctuations of a waveguide's material properties and geometry introduce variability into features of resonance, such as their central frequency and spectral width. The investigator calculates this variability using perturbation analysis of the scattering matrix where the perturbative parameters are random variables.