A fourth order finite difference method for waveguides with curved perfectly conducting boundaries

A novel high order finite difference method is introduced for optical waveguides with smoothly curved perfectly electric conducting (PEC) boundaries. The proposed method shares some similarities with our previous matched interface and boundary (MIB) methods developed for treating dielectric interfaces of optical waveguides, such as the use of a simple Cartesian grid, the standard finite difference schemes, and fictitious values. However, the PEC boundary conditions have a physical nature quite different from that of the jump conditions at the dielectric interfaces, i.e., all six electric and magnetic field components are prescribed in the jump conditions, while only three of them are known at the PEC walls. Consequently, the previously developed MIB methods are not applicable to deal with the perfectly conducting boundaries. To overcome this difficulty, a novel ray-casting fictitious domain method is constructed to enforce the PEC conditions along the normal direction. Such a boundary implementation couples the transverse magnetic field components so that the resulting ray-casting MIB method is a full vectorial approach for the modal analysis of optical waveguides. The new MIB method is validated by considering both homogeneous and inhomogeneous waveguides. Numerical results confirm the designed fourth order of accuracy.