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In this paper exact 1D transparent boundary conditions (TBC) for the 2Dparabolic wave equation with a linear or a quadratic dependence of thedielectric permittivity on the transversal coordinate are reported. Unlike thepreviously derived TBCs they contain only elementary functions. The obtainedboundary conditions can be used to numerically solve the 2D parabolic equationdescribing the propagation of light in weakly bent optical waveguides andfibers including waveguides with variable curvature. They also are useful whensolving the equivalent 1D Schr\"odinger equation with a potential dependinglinearly or quadratically on the coordinate. The prospects and problems ofdiscretization of the derived transparent boundary conditions are discussed.

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