Cross‐impact of beam local wavenumbers on the efficiency of self‐adaptive artificial boundary conditions for two‐dimensional nonlinear Schrödinger equation

Abstract

In this paper, we propose self‐adaptive artificial boundary conditions (ABCs) for the two‐dimensional (2D) time‐dependent nonlinear Schrödinger equation and studied their efficiency. This equation describes the laser pulse interaction with a semiconductor layer under laser‐induced plasma generation. Because of its nonlinearity, the interaction process is very sensitive to the appearance of the optical beam spurious reflection; therefore, highly transparent ABCs are urgently required to avoid disturbing the interaction process by an unphysical reflected wave. This can be achieved by using time‐ and spatial‐dependent local wavenumbers computed near the artificial boundaries. We claim that it is necessary to account for both wavenumbers in each of the adaptive ABCs stated on both spatial coordinates. Based on computer simulation, we demonstrate an increase in the efficiency of the adaptive ABC stated on the coordinate along which the optical pulse propagates by involving both local wavenumbers. The accuracy of the developed approach is verified using a well‐known analytical solution of the linear Schrödinger equation as well as the reference problem solution, obtained by solving the problem with zero‐value boundary conditions on the extended spatial domain. For computer modeling, the finite‐difference scheme, possessing asymptotic stability, is applied together with a two‐stage iterative process for its realization. The algorithm and peculiarities of the local wavenumber computation are discussed in detail. The applicability of the developed approach is also demonstrated via the solution of both linear and nonlinear problems. The proposed method has widespread application for solving various laser physics problems governed by the Schrödinger equation.