Improved finite-difference beam-propagation method based on the generalized Douglas scheme and its application to semivectorial analysis

Abstract

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.

Efficient time-domain finite-difference beam propagation methods for the analysis of slab and circularly symmetric waveguides

“…As stated in [12], (13) can be equivalently expressed as the following equations regarding the and directions:…”

“…For TE-mode propagation, (5) can be written as follows [4]- [9]: (6) Here, we discretize (6) using the alternating-direction implicit method (ADIM) as was shown in [12]. The first half-step in ADIM factorization for (6) can be separated into (7) (8) The GD scheme can directly be applied to the second derivatives with respect to both the and directions.…”

“…For the difference equation corresponding to (11) becomes (12) where and are central difference operators for the first and second derivatives, respectively. We divide the propagation step into two steps of size (13) and (14) Since the treatment regarding the two half-steps is the same, we only deal with the first half-step.…”

“…This results in a truncation error of where is the sampling width in space. We should recall that, in the BPM the generalized Douglas (GD) scheme [12] has been successfully applied to the second derivatives with respect to space. The GD scheme attains a truncation error of provided that uniform or graded-index materials are treated [13].…”

Accuracy of three-point finite difference approximations for optical waveguides with step-wise refractive index discontinuities

“…The standard 3−point FD approxima− tions were used to improve the computational efficiency through the application of the Douglas scheme [7][8][9]. Fur− ther, the method of expanding field values at both sides of the discontinuity and matching them at the interface was used in for the derivation of the 5−point [5,10] and more recently even 7− and 9−point FD schemes [11].…”