A 100x Faster Beam Propagation Method for Nonlinear Optical Wave Propagation Based on Discrete Exterior Calculus

Abstract

Efficient simulation of nonlinear light propagation in complex photonic structures remains a major challenge because these systems combine intricate transverse geometries with propagation over distances spanning many diffraction lengths. Existing numerical methods often require computationally intensive uniform discretizations or struggle to accurately represent complex material boundaries, limiting the practical simulation of multiscale nonlinear photonic devices. Here we introduce a computational framework for solving the nonlinear Schr\"odinger equation that accelerates simulations by more than two orders of magnitude (over 100x) compared with conventional approaches while maintaining spectral-level accuracy, thereby reducing the computational time required to solve large-scale and multiscale problems from days to hours. The method is based on discrete exterior calculus, enabling geometry-conforming discretization directly on unstructured meshes without the weak formulations required by conventional finite-element methods. In contrast to Fourier-based spectral solvers, it avoids global oversampling, eliminates Gibbs-type oscillations at material interfaces, naturally incorporates absorbing boundary conditions, and preserves the topological structure of the underlying differential operators. A key feature of the framework is that higher-order propagation operators are generated algorithmically from lower-order discrete operators, providing a systematic route to extending simulations beyond the standard nonlinear Schr\"odinger equation. Benchmarks on fundamental solitons, asymmetric beams, and optical vortices confirm spectral-level accuracy while demonstrating computational speedups exceeding 100x.