Propagation of electromagnetic waves in planar waveguide with non-linear properties: a repulsive Casimir-Lifshitz force and optical bistability

In this paper, we investigate the propagation of electromagnetic radiation in a model planar three-layer waveguide with a nonlinear dielectric layer (film) with accounting for the absorption of radiation in each layer and the second-order dependence of the dielectric permittivity on the electric field amplitude. In this model, absorption was taken into account as an imaginary component of the dielectric permittivity and electric field strength. In order to compute the electric field strength, Maxwell's equations, the finite-difference method and matrix sweep method with simple iterations were used. The energy flux of guided waveguide modes was determined using the Poynting vector, which depends on the amplitude of the electric field strength. A nonlinear dependence of the electromagnetic radiation flux on the effective refractive index, which characterizes the wave velocity, was pointed out and optical bistability, where the same value of the flux corresponds to two values of the effective refractive index, was revealed. The developed model, which explicitly account for the impact of the absorption coefficient on the transmission of electromagnetic radiation, can be used to design optoelectronics and integrated optics, such as nonlinear waveguides, optical switches, and various optical devices based on the use of optical bistability. It was shown how the regions of optical bistability change depending on the thickness of the nonlinear layer and the dielectric constants of the waveguide components. Since waveguide layers considered here are of micro- and nano-size, the Casimir force can play an important role in the propagation of electromagnetic waves, which, at a certain ratio of the dielectric constants of the waveguide layers, manifests itself as a repulsive force arising between the dielectric layers of the waveguide. The nonlinear nature of the transfer of electromagnetic waves was found to lead to a bistable dependence of the Casimir-Lifshitz repulsive force on the effective refractive index. The change in the film layer thickness due to the Casimir-Lifshitz repulsive force was also investigated. It was found that the presence of multimode waveguide modes for wavelengths of ~ 10 nm can significantly reduce energy losses during the propagation of electromagnetic waves in nanostructures.