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.
