Abstract

An exact analytical expression for the mode propagation constants does not exist for most of the index profile functions. Thus, numerical or graphical methods are employed even for the solution of the simplest step-index slab. The derivation of the approximate analytical formula presented is based on two facts: (1) Two waveguides can be made equivalent by changing the profile’s parameters in such a way that moments of the profile’s functions are made identical.¹ (2) The cosh^–2(x) profile has a particularly simple analytical solution for its eigenmode equation. The results of the approximate formula are compared with numerical results for several profiles, namely, the exponential, the Gaussian, the rectangular, and the cladded parabolic. The comparison shows agreement better than 97% for the normalized propagation constant in the single-mode range. The formula obtained can also be easily inverted to yield a simple expression of the propagation constant as a function of the profile’s width. This would be of use in cases where the width of the guide has to be designed to furnish a required value of the propagation constant.

© 1985 Optical Society of America