Investigation of adiabatic waveguide modes model for smoothly irregular integrated optical waveguides

The model of adiabatic waveguide modes (AWMs) in a smoothly irregular integrated optical waveguide is studied. The model explicitly takes into account the dependence on the rapidly varying transverse coordinate and on the slowly varying horizontal coordinates. Equations are formulated for the strengths of the AWM fields in the approximations of zero and first order of smallness. The contributions of the first order of smallness introduce depolarization and complex values characteristic of leaky modes into the expressions of the AWM electromagnetic fields. A stable method is proposed for calculating the vertical distribution of the electromagnetic field of guided modes in regular multilayer waveguides, including those with a variable number of layers. A stable method for solving a nonlinear equation in partial derivatives of the first order (dispersion equation) for the thickness profile of a smoothly irregular integrated optical waveguide in models of adiabatic waveguide modes of zero and first orders of smallness is described. Stable regularized methods for calculating the AWM field strengths depending on vertical and horizontal coordinates are described. Within the framework of the listed matrix models, the same methods and algorithms for the approximate solution of problems arising in these models are used. Verification of approximate solutions of models of adiabatic waveguide modes of the first and zero orders is proposed; we compare them with the results obtained by other authors in the study of more crude models.