Universal dispersion curves of a planar waveguide with an exponential graded-index guiding layer and a nonlinear cladding

A planar waveguide consisting of three layers is considered. The guiding layer is assumed of exponentially graded index of refraction. The cover layer is a nonlinear material of Kerr-type. The refractive index distribution of the film layer changes as an exponential function from the guiding layer to the substrate. The solutions of Helmholtz equation are found. They are written in terms of three parameters a, b and V. The solutions in the guiding layer and substrate are found as Bessel functions of order V ̅̅̅b√ . The characteristic equation is derived and the dispersion curves are plotted and analyzed. A set of attracting features are found such as there is no cut-off thickness corresponding to a symmetric waveguide structure. The b-values do not exceed unity. This means the dispersion curves refer to guided modes.A planar waveguide consisting of three layers is considered. The guiding layer is assumed of exponentially graded index of refraction. The cover layer is a nonlinear material of Kerr-type. The refractive index distribution of the film layer changes as an exponential function from the guiding layer to the substrate. The solutions of Helmholtz equation are found. They are written in terms of three parameters a, b and V. The solutions in the guiding layer and substrate are found as Bessel functions of order V ̅̅̅b√ . The characteristic equation is derived and the dispersion curves are plotted and analyzed. A set of attracting features are found such as there is no cut-off thickness corresponding to a symmetric waveguide structure. The b-values do not exceed unity. This means the dispersion curves refer to guided modes