Uniform asymptotic analysis of guided modes in diffused slab waveguides with polynomial refractive-index profiles

Abstract

The propagation constant and the field distribution of the guided modes in a diffused slab waveguide with an inhomogeneous refractive index profile are analyzed by means of the uniform asymptotic series. We used the Airy differential equation as the reference equation to which the Langer transform and the Liouville transform are used repeatedly to construct a uniform asymptotic series solution. If this recursive algorithm is used, an asymptotic solution of an arbitrary approximate degree can be obtained. This uniform asymptotic series solution can be represented analytically by the expansion coefficients of the polynomial approximating the refractive index profile. The propagation characteristics of the guided modes in a diffused waveguide with an index of refraction expressed in terms of an exponential function or an error function are studied and convergence and accuracy of the uniform asymptotic series solution are investigated. The numerical results confirm that the uniform asymptotic series solution agrees well with the rigorous solution or the numerical solution. The propagation characteristics of the guided modes in a diffused waveguide with an exponential refractive index profile have been found with high accuracy. The determination of the propagating modal index is discussed.