Simplified numerical method for analyzing TE-like modes in a three-dimensional circularly bent dielectric rib waveguide by solving two one-dimensional eigenvalue equations

We present a method to solve the TE-like modes for a 3D circularly bent dielectric rib waveguide. The radial part of the Helmholtz equation in cylindrical coordinates was reduced to contain only the radial component of the electric field. The effective index method and the conformal transformation were applied to transfer the structure into two straight slab waveguides. The problem of a 3D circularly bent waveguide is reduced to the problem of solving two 2D straight slab waveguides and each one is characterized by a 1D eigenvalue equation that is solvable numerically by the beam propagation method (BPM). The bending loss, propagation constant, and radial component of the electric field distribution of the modes can be obtained directly. In contrast to many commercial 3D BPM methods our method can be applied to the bent waveguide without restriction to a large radius. Examples were given to demonstrate our method on the bent rib waveguide and compared with the results of other methods. (C) 2008 Optical Society of America.