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.
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