Uniform asymptotic solution of guided modes in inhomogeneous slab waveguides with an even polynomial refractive index profile

Uniform asymptotic solutions are obtained for the propagation constants and field distributions for guided modes inside slab waveguides with inhomogeneous profiles of refractive index. Uniform asymptotic solutions are formed with an arbitrary order by a recursive algorithm for a related equation method. Through a quantum condition the propagation constant is determined. The parabolic cylinder functions describe the field distributions. By the expansion coefficients of polynomials which approximately represent the refractive index profiles, the asymptotic solutions can be represented analytically. This paper examines the analytic solution for a squared hyperbolic tangent profile, and for a Gaussian profile, it was analyzed numerically. The effectiveness of the uniform asymptotic solutions was verified.