Exact transparent boundary conditions for the parabolic wave equations with linear and quadratic potentials

TL;DR

This paper derives exact transparent boundary conditions for 2D parabolic wave equations with linear or quadratic potentials, simplifying numerical simulations of optical waveguides and quantum systems.

Contribution

It introduces elementary-function-based TBCs for 2D parabolic equations with linear/quadratic potentials, improving computational efficiency and accuracy.

Findings

Boundary conditions are explicitly expressed with elementary functions.

Applicable to optical waveguides with variable curvature.

Facilitate numerical solutions of 1D Schrödinger equations with specific potentials.

Abstract

In this paper exact 1D transparent boundary conditions (TBC) for the 2D parabolic wave equation with a linear or a quadratic dependence of the dielectric permittivity on the transversal coordinate are reported. Unlike the previously derived TBCs they contain only elementary functions. The obtained boundary conditions can be used to numerically solve the 2D parabolic equation describing the propagation of light in weakly bent optical waveguides and fibers including waveguides with variable curvature. They also are useful when solving the equivalent 1D Schr\"odinger equation with a potential depending linearly or quadratically on the coordinate. The prospects and problems of discretization of the derived transparent boundary conditions are discussed.