Full-vectorial finite-difference beam propagation method based on the modified alternating direction implicit method

A modified alternating direction implicit algorithm is proposed to solve thefull-vectorial finite-difference beam propagation method formulation basedon $H$ fields. The cross-coupling terms are neglected in the first sub-step, butevaluated and doubly used in the second sub-step. The order of two sub-stepsis reversed for each transverse magnetic field component so that thecross-coupling terms are always expressed in implicit form, thus thecalculation is very efficient and stable. Moreover, an improved six-pointfinite-difference scheme with high accuracy independent of specificstructures of waveguide is also constructed to approximate thecross-coupling terms along the transverse directions. The imaginary-distanceprocedure is used to assess the validity and utility of the present method.The field patterns and the normalized propagation constants of thefundamental mode for a buried rectangular waveguide and a rib waveguide arepresented. Solutions are in excellent agreement with the benchmark results fromthe modal transverse resonance method.