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

高频腔本征模：快速计算方法、应用和渐进

• 批准号: 0507614
• 负责人: Alexander Barnett
• 金额: --
• 依托单位: New York University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2005-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0507614&HistoricalAwards=false
• 关键词: Frequency; Cavity; Eigenmodes; Rapid; Computational

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.

研究者试图开发有效且鲁棒的数值工具，以在两个或多个维度上计算腔的共振模式。 这些模式描述了多种技术重要性的线性波系统的行为，例如电磁，光学和声腔，弹性膜，金属波导的横向模式以及量子颗粒的界能状态。 很难高估对波浪在上个世纪（尤其是通信技术）对波浪的数学理解对社会产生的影响。 在高频率下，波长比空腔大小要小得多，并且常规的数值离散方法在不切实际的速度下变得不切实际，从而使问题在计算上具有挑战性。 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具有同时计算长序列的能力。 这是基于“缩放方法”，这是特定解决方案方法的鲜为人知的变体。 与其他边界方法相比，显着的效率增益量表，例如边界上的波长数量（其“电气区域”），并且已显示出比任何其他已知方法在非常高的频率下的速度快一千倍。 但是，控制该方法固有的误差大小的机制尚不清楚。 研究者建议应用数值分析的工具来了解这些错误，以扩大可以应用其应用的腔范围和边界条件的范围（例如，具有重新进入角或三个维度的腔体），并在其他有效的方法中受益于这些新的方法。微型腔电介质激光器的建模，可以为光纤通信提供更高的功能，并能够制造集成的光学元件“芯片上”，ii）``量子混乱''，``量子混沌''，奇异的（高频）腔共鸣的行为，其形状会导致Chaotic Ray运动。 了解这种模式中波强度的分布和统计数据对原子理和化学以及在自动形式和数字理论中产生的数学问题产生了影响。 量子混乱还可以对量子点，纳米级实验室中的电子传输和耗散进行建模，以进行连贯的波电子效应，这是量子计算机的有希望的候选者。 最后，对于谐音模式的快速求解器的开发，尤其是对于麦克斯韦方程，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。