Abstract

We have earlier developed a method-the Collocation method for propagating beams or total fields through optical waveguiding structures^1,2 which is numerically more efficient and capable of giving better accuracy in comparison to commonly used propagating beam method (PBM). The collocation method, which converts the Helmholtz equation into a matrix total differential equation to facilitate direct numerical solution, suffers, however, from one drawback. For moderately large extrapolation intervals (typically 10 μm) the method becomes unstable and error increases exponentially. This kind of behavior is typical of simultaneous equations. Special methods have to be used to circumvent this problem which may not be very convenient and the advantage of computational efficiency may be lost. We have, therefore, developed an alternative method to solve the matrix differential equation. The new formulation involves only unitary operators, like in PBM, and is unconditionally stable with respect to the values of the extrapolation intervals. The method still has its computational advantage over PBM.

© 1991 Optical Society of America