Novel phase-integral equation for the computation of the exact number of guided modes in a graded-index waveguide

Abstract

We have recently reported a closed-form formalism that furnishes an expression for the exact number of guided electromagnetic TE and TM modes in a multilayered dielectric waveguide structure.1 This formalism is developed further here to give the exact number of guided modes for any case of graded-index dielectric waveguide. The computation of the number of modes is made by solving a first-order nonlinear phase integral equation (PIE). The PIE is actually an extension of the well-known WKB formula, and besides being exact, it has the advantage of taking into account not only the surface of the index profile but also its slope. In this way, the slope has no smoothness limitation relative to the wavelength. Another problem which is overcome here is that of the turning point boundary matching, since the integral is carried on the edge of the profile [i.e., β = K0n(∞)]. Moreover, we have also shown that for every case the result of the computation comes out to be ϕ∞ = (m + 1/2)π, where m stands for the number of guided modes. Numerical calculations and curve plottings were made for various profile shapes, such as exponential, Gaussian, inverse square cosh, etc.