Efficient Solution of the Nonlinear Helmholtz Equation in 2D Photonic Crystals

Abstract

The nonlinear Helmholtz equation models nonparaxial electromagnetic wave propagation in Kerr-type media and presents numerical challenges in two-dimensional photonic crystals due to strong nonlinearities, high wave numbers, and material discontinuities. This work presents an efficient and robust numerical framework combining a fourth-order finite difference scheme with multiple iterative methods, including fixed-point, frozen nonlinearity, Newton’s, and modified Newton’s methods. A key novelty contribution is the high-order discretization of nonlinear interfaces, enabling accurate treatment of discontinuous coefficients. Enhanced boundary conditions further ensure stability at high frequencies. Numerical experiments validate the accuracy of the scheme using analytical solutions, and demonstrate that the modified Newton’s method provides superior convergence performance. The results offer an efficient and robust approach for simulating nonlinear wave propagation in complex periodic media and designing nonlinear photonic device.