FDTD Method with Explicit Non-Iterative and Second Order Treatment for Kerr Nonlinearities

The finite-difference time-domain (FDTD) method is a very popular numerical method used to solve Maxwell's equations in various types of materials, including those with nonlinear properties. When solving the nonlinear constitutive equation that models Kerr media, Newton's iterative method is accurate but computationally expensive, while the conventional explicit non-iterative method is less expensive but not very accurate. In this work, we propose a new explicit non-iterative algorithm to solve the Kerr nonlinear constitutive equation that achieves a quadratic convergence rate. This method attains a similar accuracy to Newton's method but does with a significant reduction in computational cost. To demonstrate the accuracy and efficiency of our method, we provide several numerical examples, including the simulations of four-wave mixing and soliton propagation in one and two dimensions.