  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.
