Exact analytic solutions to the nonlinear wave equation for a saturable Kerr-like medium: modes of nonlinear optical waveguides and couplers

Exact analytic solutions to the nonlinear wave equation for a lossless saturable Kerr-like medium are found. The nonlinearity is an intensity-dependent refractive index which saturates according to the model n²=n₀²±a|E|²/(1+b|E| ²). The solutions to the wave equation are used to determine the TE power-dependent dispersion relations, and therefore the guided modes, of a single waveguide structure and of a directional coupler whose coupling medium is the nonlinear saturable material. Although the interaction between the sinh- and cosh-like modes of the coupler, and therefore the coupling length, is not treated in this work, it is shown that the critical power, defined in the literature to be the level of input power above which 100% of the power cannot be switched between the guides of a coupler, is a mathematical misinterpretation in both saturable media as well as in nonsaturable Kerr media. The absence of a critical power in single waveguide structures is also demonstrated. Parameters of GaAs-based waveguides and of a GaAs-GaAlAs MQW coupling medium are used in the numerical analysis