Resonance Phenomena in Wave Scattering

ShipmanDMS-1411393 光学和电子设备，因为它们的多样性和复杂性，是基于一个共同的基本物理原理，称为谐振。 这一原理存在于激光器、滤波器、天线和发光二极管以及其组件在纳米尺度下工作的设备背后。 研究人员研究有效和稳健的方法来利用设备设计中的共振现象，以及如何控制由于不可避免的制造缺陷而导致的设备性能变化。 共振理论有着深厚的数学基础，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，物理和工程的接口开发技能，特别注意弥合学科之间的沟通差距。 本科生参与创建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)双周期波导的电磁谐振与麦克斯韦方程的复本征值相关联，该复本征值对谐振的频率和谱宽进行编码。 研究人员考虑了新的波导结构，该结构允许通过改变源波的入射角来调谐谐振的两个特性。 在谐振波导模式的现有构造中，由于它们利用的对称性，排除了这种调谐能力。 通过利用非互易材料、特殊几何形状和对称导模到反对称模的分叉来打破这种对称性，以构建非对称可调谐波导谐振。 (2)嵌入特征值在局部扰动的周期量子图负责共振现象的二维分子结构，如双层石墨烯。 基于他最近关于嵌入本征值的结果，研究人员采用不对称性来控制量子图中的共振散射特性。 (3)即使在非线性系数很小的材料中，由于共振引起的场放大也会引起强非线性效应，如双稳态和磁滞。 调查量化这些影响在一个易于处理的模型中，光子传输线耦合到本地激子场，产生一个非线性场的光子激子准粒子称为极化激元。 (4)波导的材料特性和几何形状的随机波动将可变性引入到谐振的特征中，例如它们的中心频率和频谱宽度。 研究人员计算这种变异性使用微扰分析的散射矩阵的微扰参数是随机变量。

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.