Perturbative model for nonstationary second-order cascaded effects

We report a semianalytical solution describing the type I second-order nonlinear interaction of the fundamental and the second-harmonic fields in a nonlinear crystal, which accounts for the phase- and group-velocity mismatch of the interacting pulses. The method uses a series-development solution of the propagation equations in respect to the second-harmonic conversion efficiency. The method describes the self-phase and self-amplitude modulation experienced by the fundamental pulse in single- and double-pass (i.e., reinjecting into the nonlinear crystal the outgoing pulses) interaction geometries, following better with respect to a numerical analysis, the dependence from the propagation parameters such as the crystal length, the pulse duration, and the phase- and group-velocity mismatch. It appears that it is possible to obtain an efficient self-phase modulation on the fundamental field even in nonstationary conditions. This paper describes the advantages of a double-pass configuration, which, for a given crystal length, allows a stronger nonlinear phase modulation of the fundamental field and minimizes its losses toward the second harmonic.