Rapid and Robust High Order Spectral Solvers for Learning Photonic Structures

在过去的二十年里，工程师们已经制造出了具有惊人的探测、传感和成像能力的光学设备。 虽然直觉一直是推动可能性的宝贵指南，但设计的日益复杂性和各种可用组件使这种方法变得更加困难。因此，数值模拟已经成为一种非常有用的工具。随着这种方法变得越来越普遍，对快速、鲁棒和高度准确的算法的搜索变得非常尖锐。 PI将通过探索数值近似的新途径，执行控制方程的新途径，以及吸引强大的机器学习方法（特别是深度学习）来发现器件设计的最佳参数值，来增强他的仿真工具类。PI还将对这些新的数值方案进行严格的分析，以评估和验证其真实世界的性能。在过去的二十年里，工程师们已经建造了具有惊人的检测，传感和成像能力的光学设备。 虽然直觉和线性化模型在推动可能性方面起到了宝贵的指导作用，但设计的日益复杂性和各种可用组件使这种方法变得更加困难。 因此，数值模拟已经成为一种非常有用的工具。 随着这种方法变得越来越普遍，对快速、鲁棒和高度准确的算法的搜索变得非常尖锐。 主要研究者将提高他的高阶谱解算器的水平，其目标如下：（1）将二维材料简化为他的快速和高精度的三维“场展开”矢量麦克斯韦方程解算器;（2）将他的“高阶表面扰动/渐近波形评估”算法扩展为三维矢量麦克斯韦方程的联合几何/频率扰动;（3）将他的“高阶包络摄动法”推广到三维情形，（4）对后两种方法进行了严格的分析论证;以及（5）利用强大的深度学习新工具来发现设备设计的最佳参数值。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的知识价值和更广泛的影响审查标准进行评估。

Over the past two decades engineers have built optical devices with stunning capabilities of detection, sensing, and imaging. While intuition has been an invaluable guide in pushing the envelope of what is possible, the increasing complexity of design and the wide array of available components is rendering this approach ever more difficult. For this reason, numerical simulation has become an invaluable tool. As this approach becomes ever more pervasive, the search for rapid, robust, and highly accurate algorithms has become quite acute. The PI will enhance his class of simulation tools by exploring new avenues of numerical approximation, novel paths to enforcing the governing equations, and appealing to the powerful methods of Machine Learning (in particular, Deep Learning) to discover optimal parameter values for device design. The PI will also develop a rigorous analysis of these new numerical schemes in order to evaluate and validate their real-world performance. This project will provide support for one graduate student each year of the three year award.Over the past two decades engineers have built optical devices with stunning capabilities of detection, sensing, and imaging. While intuition and linearized models have been invaluable guides in pushing the envelope of what is possible, the increasing complexity of design and the wide array of available components is rendering this approach ever more difficult. For this reason, numerical simulation has become an invaluable tool. As this approach becomes ever more pervasive, the search for rapid, robust, and highly accurate algorithms has become quite acute. The PI will enhance his class of High-Order Spectral solvers with the following objectives: (1) Incorporating two-dimensional materials into his rapid and highly accurate three-dimensional "Field Expansions" vector Maxwell equation solver; (2) Expand his "High-Order Perturbation of Surfaces/Asymptotic Waveform Evaluation" algorithm for joint geometry/frequency perturbation to the three-dimensional vector Maxwell equations; (3) Extend his "High-Order Perturbation of Envelopes" algorithm for structural perturbations to this three--dimensional case; (4) Provide rigorous analytical justification for each of the latter two; and (5) Bring to bear the powerful new tools of Deep Learning to discover optimal parameter values for device design.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.