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

研究人员试图开发高效和稳健的数值工具来计算二维或更多维的腔体的共振模式。这些模式描述了各种具有重要技术意义的线性波动系统的行为，例如电磁、光学和声学腔、弹性膜、金属波导的横模以及量子粒子的束缚能态。在过去的一个世纪里，对波的数学理解对社会产生了很大的影响，特别是在通信技术方面，这一点怎么估计都不为过。在高频下，波长变得比腔体尺寸小得多，而传统的数值离散方法变得不切实际地慢，这使得这个问题在计算上具有挑战性。边界积分方程法利用已知的在均匀介质中的传播，以更少的自由度(像边界面积而不是腔体体积)解决同样的问题，但需要单独定位每个共振频率(本征值)。基于量子物理界出现的令人兴奋的发展，研究人员开创了将这种少量自由度与同时计算长模式序列的能力相结合的方法。这是以“比例法”为基础的，这是一种鲜为人知的特解方法的变体。与其他边界方法相比，效率的显著提高与边界上的波长数量(其“电子面积”)一样大小，并且已被证明在很高的频率下比任何其他已知的方法快一千倍。然而，该方法固有的误差大小的控制机制尚不清楚。研究人员建议应用数值分析工具来了解这些误差，寻求扩大可以应用该方法的腔几何和边界条件的范围(例如，具有折返角的腔，或三维的腔)，并编写一个健壮的软件包，使其他研究人员能够从这些新的有效方法中受益。研究人员将继续在它很适合的两个激动人心的领域应用该方法：i)微腔介质激光器的设计和建模，其可以为光纤通信提供更高的功率，并且能够制造集成光学器件，谐振腔的半经典(高频)行为，其形状导致光线的混沌运动。了解这些模中波强的分布和统计对原子物理和化学，以及在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，这是研究相干波电子效应的纳米实验室，是很有希望的量子计算机候选者。最后，谐振模，特别是麦克斯韦方程的快速解算器的发展，将对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.