Discrete spectrum of waveguide modes of a linearly graded-index film introduced into a medium with a stepwise nonlinearity

Even and odd modes propagating along the symmetrical planar waveguide, which is a film embedded in a nonlinear medium, are described analytically. The dielectric function of the film is characterized by a linear spatial profile symmetrical about its center in transverse direction. The dielectric function of the nonlinear medium changes stepwise its value when the electric field amplitude exceeds the threshold value. Exact solutions of even and odd symmetry to the wave equation with such compound dielectric function are found. The solutions describe the waveguide modes existing with the discrete spectrum of the effective refractive index. Waveguide modes can be excited at certain discrete values of the angle of incidence of the exciting beam. Low-order modes are excited at large angles of incidence, and high-order modes are excited at small ones. The number of excited modes is limited and is determined by the parameters of the waveguide system, in particular, by the film thickness and the wavelength of the exciting radiation. Excitation of an even mode of certain order requires a wavelength longer than that of an odd mode of the same order. The intensity of the mode increases with an increasing film thickness, and it decreases with an increasing wavelength. The power flux and the radiation confinement factor are calculated analytically. The radiation confinement factor behaves in the same way as the intensity of the waveguide mode. It is shown that the minimum value of the radiation power required to excite the mode is reached at the critical value of the film thickness (or the critical wavelength), at which the radiation confinement factor is equal to a half.