Modes of a nonlinear planar waveguide with a dielectric layer immersed in a hyperbolic medium

The guided waves of a symmetric planar waveguide formed by an isotropic dielectric placed in a hyperbolic medium and having a cubic-nonlinear response are studied theoretically. The optical axis of the hyperbolic medium is directed along the normal to the interfaces between the media. If the permittivity of the waveguide core exceeds the main permittivity for an extraordinary wave in the hyperbolic medium, then each TM mode is characterised by two cut-off frequencies. Dispersion relations for these modes are found in the case of focusing and defocusing core media. The number of the modes possible at a given frequency depends on the radiation intensity. It is shown that zero values of the mode propagation constants are possible in the waveguide, which corresponds to the formation of a standing wave between the boundaries of the waveguide. In addition, in the case of a defocusing waveguide layer, such stopped modes can be obtained with increasing field intensity. The dependences of the propagation constant and the width of the transverse distribution of the mode field on the radiation intensity are found and analysed.