Nanophotonic Simulation and Inverse Design using Fast High-order Chebyshev-based Nyström Methods

The majority of electromagnetic devices for practical nanophotonic applications are tedious and challenging to design and analyze due to the lack of analytical solutions, necessitating fully vectorial three-dimensional solutions of Maxwell’s equations. Unfortunately, these devices are often also difficult to solve numerically due to being electrically large (spanning many wavelengths in size) and often having subwavelength feature sizes. Time domain approaches, such as the Finite-Difference Time-Domain (FDTD) method, are often the only viable algorithms for simulating most nanophotonic devices, such as grating couplers, power and mode conversion splitters, tapers, and long waveguides. This is because volumetric frequency-domain methods such as the Finite Element Method (FEM) or the Finite-Difference Frequency-Domain (FDFD) method result in prohibitively large and poorly conditioned linear systems which become intractable even for modestly sized problems. In this work, we present a fast, high-order 3D integral equation solver based on the Nyström method [1], which can be used to simulate large electromagnetic devices efficiently and rapidly in the frequency domain, including nanophotonic devices with semi-infinite waveguides [2]. We will introduce the Chebyshev-based discretization scheme, which is leveraged by the solver to discretize a stable, well-conditioned N-Müller integral formulation, and discuss recent acceleration techniques that we have developed, including utilizing GPU computing, incorporating a sparse block-diagonal preconditioning strategy, and implementing a matrix-free approach for computing the transpose operator required by the discrete adjoint method. We will also present our most recent advancements, including automated high-order surface meshing of electromagnetic structures, adaptive integration of singular and near interactions for automated error control, and a new approach for efficiently launching unidirectional modes in waveguides.