Dispersion Relation of Leaky Modes in Nonhomogeneous Waveguides and Its Applications

For a nonhomogeneous waveguide, whose refractive index is not a constant, the problem is very complicated since the nonlinear eigenvalue problems are unable to reduce to algebraic equations yet. When the refractive index is varied, the dispersion relation cannot be derived by using the analytic expressions of the solutions in each layer. In this paper, this problem is solved by using the differential transfer matrix method, which is introduced to deduce the dispersion relations of leaky modes for TE and TM cases, respectively. Moreover, for the waveguide whose refractive index is gradually varied, the dispersion relations can be approximated by some simpler algebraic equations, which are close to the exact relations and very easy to analyze. Asymptotic solutions are used as initial guesses, and followed by Newton's method, to give very accurate solutions. This paper is a generalization of the asymptotic method of slab waveguides; all the results therein are consistent with the analysis here.