High Frequency Cavity Eigenmodes: Rapid Computational Methods, Applications and Asymptotics

研究者寻求开发有效和强大的数值工具来计算二维或多维腔的谐振模式。这些模式描述了各种具有重要技术意义的线性波系统的行为，如电磁、光学和声学腔、弹性膜、金属波导的横向模式和量子粒子的束缚能态。在上个世纪，对波的数学理解对社会的影响，尤其是对通信技术的影响，怎么估计都不为过。在高频率下，波长变得比空腔尺寸小得多，传统的数值离散方法变得不切实际，使得计算问题具有挑战性。边界积分方程方法通过使用已知的在均匀介质中的传播，用更少的自由度（像边界面积而不是腔体体积那样缩放）来解决相同的问题，但需要单独定位每个谐振频率（特征值）。在量子物理界令人兴奋的发展的基础上，研究人员开创了将这种少量自由度与同时计算长序列模式的能力相结合的方法。这是基于“缩放法”，一种鲜为人知的特解方法的变体。与其他边界方法相比，显着的效率增益如边界上的波长数量（其“电区域”），并且已被证明在非常高的频率下比任何其他已知方法快一千倍。然而，控制该方法固有误差大小的机制尚不清楚。研究者建议应用数值分析工具来理解这些错误，寻求扩大可应用的空腔几何形状和边界条件的范围（例如，具有可重新进入角的空腔，或在三维空间中），并编写一个强大的软件包，使其他研究人员能够从这些新的有效方法中受益。研究人员将继续将该方法应用于两个令人兴奋的领域：1)微腔介质激光器的设计和建模，它可以为光纤通信提供更高的功率，并可以实现“芯片上”集成光学器件的制造；2)“量子混沌”，谐振腔的半经典（高频）行为，其形状导致混沌射线运动。理解这种模式下波强度的分布和统计对原子物理和化学，以及自同构形式和数论中出现的数学问题都有影响。量子混沌还可以模拟量子点中的电子输运和耗散，纳米级实验室用于相干波电子效应，这是量子计算机的有希望的候选者。最后，谐振模式的快速求解器，特别是麦克斯韦方程的快速求解器的发展，将对整个工程世界中使用的光学和微波谐振器的设计和形状优化具有深远的好处。

The investigator seeks to develop efficient and robust numerical tools for calculating the resonant modes of cavities in two or more dimensions. These modes describe behavior of a wide variety of linear wave systems of technological importance, such as electromagnetic, optical, and acoustic cavities, elastic membranes, transverse modes of metallic waveguides, and bound energy states of quantum particles. It is hard to overestimate the impact that mathematical understanding of waves has had on society over the last century, particularly in communications technology. At high frequencies the wavelength becomes much smaller than the cavity size, and conventional numerical discretization methods become impractically slow, making the problem computationally challenging. Boundary integral equation methods, by using the known propagation across a uniform medium, solve the same problem with a much smaller number of degrees of freedom (scaling like the boundary area rather than the cavity volume), but suffer from the need to locate each resonant frequency (eigenvalue) individually.Building on exciting developments which emerged from the quantum physics community, the investigator has pioneered methods which combine this small number of degrees of freedom with the ability to compute a long sequence of modes simultaneously. This is based upon the `scaling method', a little-known variant of the Method of Particular Solutions. The remarkable efficiency gain compared to other boundary methods scales like the number of wavelengths on the boundary (its `electrical area'), and has been shown to be a thousand times faster than any other known method at very high frequencies. However, the mechanism controlling the size of errors intrinsic to the method is unknown. The investigator proposes to apply the tools of numerical analysis to understand these errors, to seek to widen the range of cavity geometries, and boundary conditions, to which it can be applied (for instance, cavities with re-entrant corners, or in three dimensions), and to write a robust software package that can allow other researchers to benefit from these new efficient methods.The investigator will continue to apply the method in two exciting areas to which it is well adapted: i) the design and modeling of micro-cavity dielectric lasers, which can give higher powers for optical fiber communications and could enable fabrication of integrated optics `on a chip', and ii) `quantum chaos', the semiclassical (high frequency) behavior of resonances of cavities whose shape leads to chaotic ray motion. Understanding the distribution and statistics of wave intensities in such modes has impact in atomic physics and chemistry, as well as for mathematical questions arising in automorphic forms and number theory. Quantum chaos can also model electronic transport and dissipation in quantum dots, nanoscale laboratories for coherent wave electron effects, which are promising candidates for quantum computers. Finally, the development of rapid solvers for resonant modes, especially for the Maxwell equations, would have far-reaching benefits for the design and shape optimization of optical and microwave resonators used throughout the engineering world.