Abstract
Electromagnetic TE wave propagation in an inhomogeneous nonlinear cylindrical waveguide is considered. The permittivity inside the waveguide is described by the Kerr law. Inhomogeneity of the waveguide is modeled by a nonconstant term in the Kerr law. Physical problem is reduced to a nonlinear eigenvalue problem for ordinary differential equations. Existence of propagating waves is proved with the help of fixed point theorem and contracting mapping method. For numerical solution, an iteration method is suggested and its convergence is proved. Existence of eigenvalues of the problem (propagation constants) is proved and their localization is found. Conditions of k waves existence are found.
Highlights
Electromagnetic wave propagation in linear waveguide plane layers and cylindrical waveguides with circular cross section is of particular interest in linear optics
We suggest and develop a method to investigate the problem of existence of electromagnetic waves that propagate along axis of an inhomogeneous nonlinear cylindrical waveguide
The nonlinearity inside the waveguide is described by the Kerr law; the inhomogeneity is described by a function that depends on radius of the waveguide
Summary
Electromagnetic wave propagation in linear (homogeneous and inhomogeneous) waveguide plane layers and cylindrical waveguides with circular cross section is of particular interest in linear optics (see, e.g., [1, 2]). Problems of electromagnetic wave propagation in nonlinear waveguides (plane and cylindrical) lead to nonlinear boundary and transmission eigenvalue problems for ordinary differential equations. The dispersion equation of the nonlinear inhomogeneous case can be written as DElin + Tnonlin = 0 In this case the term DElin is written in an implicit form as opposed to the case of a homogeneous waveguide, and its roots are unknown. In study [26] authors apply integral equation approach in the way as they would solve the problem for a homogeneous waveguide. We emphasize that for inhomogeneous waveguides important and general results can be obtained with the method we use in this paper in which the Green function has implicit form. In spite of the fact that the method here looks similar to the method in [21,22,23,24], we solve radically different problem, as we consider inhomogeneous nonlinear waveguide
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