Collaborative Research: Efficient High-Order Parallel Algorithms for Large-Scale Photonics Simulation

这一研究项目的重点是开发高效、准确和可扩展的计算技术，并为光电子行业提供急需的仿真设计工具。越来越多的现代生活是建立在快速和廉价的通信基础上的。由于电力问题，通过电路传输信息在延迟上受到限制，在带宽上受到成本的限制。光子电路实际上消除了这些限制，并提供了一种方法来制造高带宽、低延迟的互连，在许多应用中，这些互连远远优于它们的电子同行。但光子电路的设计极具挑战性。事实上，由于缺乏高效可靠的集成光子电路设计工具，目前最先进的集成光子电路芯片只包含数百个光子组件。这里的困难在于，集成光子器件的设计需要对传播的电磁波进行准确的模拟，而这反过来又需要极大量的未知数，即使在体积离散化中具有中等精度。该项目开发的工具将解决这些重要的挑战。大多数光子学应用的基本数学模型由麦克斯韦方程组成，在特征长度尺度的广泛变化下，具有复杂的结构材料系数。在设计工程师可行的计算规模下，现有技术对于复杂光子器件的设计来说太不准确了。这种不准确/低效率的权衡严重限制了模拟在设计反馈循环中的有用性。这进一步阻碍了光电子行业的快速发展，因为通过制造进行的设计迭代通常非常昂贵，而且单个设计可能需要数月的周转时间。解除这种低效率的限制是具有挑战性的，它有可能在雄心勃勃的光子电路和器件的发展中发挥关键作用。研究人员建议开发以下技术，以克服在实际的大规模光子学模拟中遇到的障碍。(1)基于边界积分方程法的光子器件模拟的模块化。具体地说，将所谓的模式计算转化为边界积分方程组的非线性本征值问题，将所谓的传播问题转化为标准的散射问题，然后用边界积分方程法求解。(2)扩展的QBX方法(“展开求积”)，这是一种通用的高阶求积格式，用于处理具有角、边和多尺度结构的区域的三维问题，以实现精确的光子学模拟。(3)QBX方法与一种新的快速多极子方法(FMM)的无缝结合，以近似最优的运算和存储要求，以无矩阵的形式求解积分方程组。

The focus of this research project is to develop efficient, accurate, and scalable computational techniques and provide much-needed simulation design tools for the photonics industry. More and more of modern life is based on fast and cheap communication. The transmission of information by electrical circuitry is limited in latency by power concerns and in bandwidth by cost. Photonic circuits virtually eliminate these constraints and provide a way to make high-bandwidth, low-latency interconnects that are, in many applications, far superior to their electric counterparts. But photonic circuits are extremely challenging to design. Indeed, the current state-of-art integrated photonic circuits chip contains only hundreds of photonic components due to the lack of efficient and reliable tools for the design of integrated photonic circuits. The difficulty here is that the design of integrated photonic devices requires accurate simulation of propagating electromagnetic waves which, in turn, requires extremely large numbers of unknowns even for modest accuracy in a volume discretization. The tools developed by this project will address these important challenges.The fundamental mathematical model for most photonics applications consists of Maxwell's equations with complex, structured material coefficients under wide variation of feature length scales. At computational scales feasible for a design engineer, existing techniques are too inaccurate for the design of complex photonic devices. This inaccuracy/inefficiency trade-off severely limits the usefulness of simulation in a design feedback loop. It further represents an impediment to the rapid development of the photonics industry since design iteration through manufacturing is typically very expensive and can take months of turn-around time on a single design. Lifting this inefficiency constraint is challenging, and it has the potential to play a pivotal role in the development of ambitious photonic circuits and devices. The investigators propose to develop the following techniques to overcome the obstacles encountered in practical, large-scale photonics simulation. (1) Modularization of photonic device simulation via boundary integral equation methods. Specifically, the so-called mode calculation will be converted to a nonlinear eigenvalue problem of boundary integral equations, and the so-called propagation problem will be converted to a standard scattering problem and then solved via boundary integral equation methods. (2) Extension of the QBX method ("Quadrature by Expansion"), a general-purpose, high order quadrature scheme to treat three-dimensional problems with domains having corners, edges and multi-scale structures for accurate photonics simulation. (3) Seamless combination of the QBX method with a novel variant of the Fast Multipole Method (FMM) to solve integral equations in a matrix-free form with near optimal operation and storage requirements.