Computer Simulation of Transition Regimes of Solitons in Four-Photon Resonant Parametric Processes in Case of Two-Photon Resonance

Abstract

The transition regimes of solitons in four-photon resonant processes in the case of two-photon absorption of the fundamental radiation are numerically investigated. The standard system of equations for the amplitudes of probability of finding the system in state with certain energy is used to derive the expression for the induced polarization in the nonlinear medium. As for the equations for the amplitudes of the optical pulses, the general case is considered in which both the amplitudes and phases are space-time dependent. We focus on the finite difference methods and the case of simultaneously propagating solitons at all frequencies of the interacting waves (simultons). The obtained results indicate that upon certain threshold conditions all interacting pulses become the solitons of Lorentzian shape. The numerical analysis has also shown that the soliton amplitudes significantly depend on the ratio between the nonlinear polarizability at the fundamental frequency <i>&omega;</i>₀ and that of combination of <i>&omega;</i>₀ and the trigger-field frequency <i>&omega;</i>₁(2<i>&omega;</i>₀ + <i>&omega;</i>₁). In the second part of the paper, we apply the method of phase planes to show that at typical values of parameters, the solitons are stable.