Abstract-The modified finite-difference formula is presented for the second derivative of a semivectorial field in a step-index optical waveguide. The present formula achieves a truncation error of O(1x 2 ) provided the discontinuity coincides with a mesh point or lies midway between two mesh points. Furthermore, the formula allows a general position of the interface, when used with the beam-propagation method (BPM). To demonstrate the effectiveness of the formula, asymmetric step-index waveguides are analyzed using the imaginary-distance BPM.

Abstract-The generalized Douglas scheme for variable coefficients is applied to the propagating beam analysis. Once the alternating direction implicit method is used, the truncation error is reduced in the transverse directions compared with the conventional Crank-Nicholson scheme, maintaining a tridiagonal system of linear equations. Substantial improvement in the accuracy is achieved even in the TM mode propagation. As an example of the semivectorial analysis, the propagating field and the attenuation constant of a bent embedded waveguide with a trench section are calculated and discussed.