Exact analytical solutions for the steady-state transmission and reflection characteristics of a nonlinear Fabry-Perot resonator applicable to bistable optical devices are derived. The resonator consists of a Kerr-like nonlinear film sandwiched by reflection mirrors made of a quarter-wave dielectric stack. An equivalent mirrorless model has been introduced to facilitate the analysis. For both positive and negative nonlinear coefficients, the rigorous solutions have been simply expressed in terms of Jacobian elliptic functions.

This paper presents a simple numerical method for calculating the stationary transmission and reflection characteristics of a variety of nonlinear Fably-Perot resonators. In nonlinear media, Maxwell's equations are directly solved by using a numerical integration of complex variables. The input-output characteristics of the Kerr-like nonlinear film without reflection mirrors and with multilayer mirrors have been calculated to demonstrate the usefulness and versatility of the proposed method and to find out resonator configurations exhibiting optical bistability at low incident-power levels. The effects of saturation in the nonlinear permittivity on the input-output characteristics have also been investigated. It has been found that a single nonlinear film with oblique incidence exhibits optical bistability without using reflection mirrors even if the refractive index of the film is low. This offers a simple method for measuring third-order nonlinearities of optical materials.