Nonconservation of power is a perplexing problem in the propagating beam analysis of transverse magnetic (TM) waves in a z -variant step-index optical waveguide. To conserve the power in terms of a squared norm, a modified finite-difference (FD) formula is introduced that allows a general position of a core-cladding interface. The use of the modified formula contributes to a reduction in a field profile error caused by a staircase approximation with subsequent conservation of power, particularly for a symmetrical waveguide. To obtain the power conservation even in the analysis of an asymmetrical waveguide, a z-derivative of the refractive index is taken into account. An asymmetrical taper and tilted waveguides placed in parallel are investigated to validate the present technique.
Junji Yamauchi, Kenji Matsubara, Takeshi Tsuda, and Hisamatsu Nakano, "Norm-Conserving Finite-Difference Beam-Propagation Method for TM Wave Analysis in Step-Index Optical Waveguides," J. Lightwave Technol. 18, 721- (2000)