Propagation characteristics of optical waveguiding structures by direct solution of the Helmholtz equation for total fields

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Abstract
A method for obtaining total field solutions from the Helmholtz equation for optical waveguiding structures is presented. The method is based on a conversion of the Helmholtz equation into a matrix total differential equation and direct solution of the resulting equation by means of standard numerical methods. The principle of the method is discussed in detail, and numerical results are presented for planar guiding structures. The method has been extended to three-dimensional guiding structures. Comparisons with the propagating-beam method show that the present method yields better accuracy and is more efficient numerically for planar as well as for three-dimensional waveguiding structures. Further, unlike the propagating-beam method, the present method needs no inherent approximation and hence can be used to obtain any desired accuracy, if the increase in computation time is not a consideration.