Numerical scheme with external complex scaling for 2D Schroedinger equation in paraxial optics

We propose a stable numerical scheme for solving Schrodinger equations that arise in the scalar paraxial theory of optical waveguides as well as in the quantum mechanics of a particle in a potential field. The scheme makes use of a multi-coordinate version of the split-step approach combined with the complex scaling that assures the implementation of boundary conditions. Beam propagation in a Gaussian waveguide is considered to illustrate the ability of the scheme to account for both waveguide and radiation modes.