Numerical-Analytical Investigation of the Model of Gradient Optical Waveguide for Obtaining Equidistant Spectrum of Waveguide Modes

A model of the planar optical waveguide with linear, exponential and modified exponential profiles of the refractive index has been studied on the basis of numerical and analytical approaches to solving a parametric inverse Sturm-Liouville problem. The goal of the investigation is to determine the profile parameters which provide proximity of the waveguide modes spectrum to an equidistant one. A software complex which we have developed in the MAPLE system is used for a numerical analysis. For solving a direct spectral problem with predetermined parameters of the model, a scheme has been proposed which uses an analytical representation of the general solution to the wave differential equation. The scheme is used for a supplementary accuracy control of the results, if a correct analytical general solution can be obtained by MAPLE tools. For the linear profile model, a parameter domain has been defined in which the Sturm-Liouville problem for description of the waveguide mode spectrum has three solutions. This domain borders on the domain where the Sturm-Liouville problem has two solutions only, and a bifurcation point is calculated over parameters. In a vicinity of this point we have calculated parameters that provide approximate equidistance of the waveguide mode spectrum. The results for exponential and modified exponential profiles have been recalculated in view of the calculated value of the parameter obtained for a linear profile model. This parameter corresponds to the height of the waveguide layer. The characteristics of spectrum equidistance have been improved.

wave equation, model potential, model parameters, waveguide modes spectrum, spectrum equidistance, quadratic functional, minimization