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Theory of SHG in a medium with combined nonlinear response

Second-harmonic generation (SHG) is used in many practical applications such as substance diagnostics and imaging of various processes as well as for frequency conversion. It is well known that frequency doubling may also be used for the generation of a tripled frequency wave due to mixing of optical radiation at basic frequency with optical radiation at doubled frequency. In this case, the relationship between phases of interacting waves with basic frequency and doubled frequency plays an important role. Therefore, the solution of the corresponding problem for wave phase evolutions is an important issue. In this paper, we provide such a solution for the SHG of high-intensity femtosecond pulses, taking into account the influence of a cubic nonlinear response on frequency doubling by using an original approach together with the long pulse duration approximation and plane wave approximation. The main feature of our approach is the absence of using the basic wave energy non-depletion approximation and using the problem invariants. It should be stressed that the frequency conversion process under consideration possesses a multi-stability property: there are two modes of SHG at certain values of intensity at basic frequency. We derive the wave intensity and phase evolution for both interacting waves for each of the modes. The derived formulas are verified by computer simulation based on using corresponding nonlinear Schrodinger equations. It is very important that we demonstrate the robustness of the derived formulas for the inhomogeneous shape of the pulse. They can be applied to computing of pulse shapes and beam profiles, even if reverse energy conversion occurs. As an example, we consider SHG for an incident Gaussian pulse with basic frequency.