A Unified Mode Solver for Optical Waveguides Based on Mapped Barycentric Rational Chebyshev Differentiation Matrix

A unified high-accuracy mode solver is proposed to compute the modes of planar optical waveguides. The advantage of this solver is composed of three aspects. The first is the high accuracy of mapped barycentric rational Chebyshev differentiation matrix (MBRCDM) for approximating differential operator. The second is the high efficiency of multidirectional Newton iteration for root finding on complex plane. The third is the outstanding performance of perfectly matched layer (PML) as an absorbing boundary condition (BC). Specifically, MBRCDM method is accurate enough to compute as many modes as necessary for waveguides bounded with regular Robin BCs. MBRCDM-PML method, where PML is used to truncate an unbounded waveguide followed by the MBRCDM method, performs very well. MBRCDM-Newton method, where multidirectional Newton iteration for a dispersion equation is implemented with some initial values obtained by MBRCDM method, is more efficient. By this combined solver, a large number of accurate modes can be easily calculated. This solver is particularly essential for the computing of high index modes.