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

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 0 ω and that of combination of 0 ω and the trigger-field frequency ( ) 2 + 1 0 1 ω ω ω . 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.

ω and that of combination of 0 ω and the trigger-field frequency ( )

Introduction
Solitons or self-reinforcing solitary waves can emerge spontaneously in a physical system in which some energy is fed in, for instance as thermal energy or by an excitation with an electromagnetic wave, even if the excitation does not match exactly the soliton solution. Therefore, if a system possesses the necessary properties to allow the existence of solitons, it is highly likely that any large excitation will indeed lead to their formation [1]- [3]. The field of solitons and related nonlinear phenomena has been substantially advanced and enriched by research and discoveries in nonlinear optics [4]- [7]. In our previous research [8], we established the possibility of the existence of simultons (simultaneously propagating solitons at different frequencies) in the case of nonstationary Raman scattering with excitation of polar optical phonons under the conditions of the interaction of ultrashort pulses of exciting and Stokes radiation in nonlinear crystals. The relevance of this study is connected both with the fact that one can extract additional information on the optical characteristics of matter, and with the possibility of obtaining of ultrashort pulses.
The second topical problem in modern nonlinear optics is the production of coherent and frequency-tunable radiation in the far ultraviolet (UV) and infrared (IR). In these spectral areas, solid materials have broad absorption bands and this narrows down the application of nonlinear crystals for the generation of electromagnetic radiation. Possible ways of overcoming those difficulties are related with the utilization of nonlinear phenomena in gases and metal vapors. The resonant four-photon interaction (RFPI) in the case of two-photon resonance is one of them. Among the advantages of gases are the presence of narrow resonances and possibility of continuous variation of density, width of spectral line, length of the medium, etc. [9]- [11]. Ultrashort pulse propagation in the case of two-photon resonance was first examined in [12] where the two-photon self-induced transparency effect was predicted. This prediction was subsequently confirmed experimentally [13] and by numerical studies [14]. Third-harmonic generation (THG) in media exhibiting resonance behaviour has also attracted considerable attention [15]- [21]. However, RFPIs that are not frequency degenerate are of no less interest; they can be used to transfer the tuning of radiation from one range to another [22] [23].
The present paper is devoted to the computer simulation of transition regimes of RFPI solitons in the case of two-photon resonance. The basic equations describing this process are given in Section 2 [12] [24]. The results of computer simulation are shown in Section 3. The stability of solitons is considered in Section 4.

Fundamental Principle
Let us assume that two optical pulses with frequencies 0,1 ω propagate in the nonlinear medium at the angles ω are considered to be in the transparent range of frequencies. We also assume that all electromagnetic waves have the same polarizations.
To find the system of equations that governs the processes of propagation of optical pulses with frequencies 0,1,2,3 ω in the medium we take the standard system of equations for the amplitudes of probability k a of finding the system in state with energy k E [25] 1 e , ϕ are the frequencies, wave vectors, real "slowly-varying amplitudes", and phases of the interacting waves, respectively. We next use (1) and the theory of perturbations [26] to find l a ( ) To obtain the system of equations for 1,2 a we introduce the expression (3) into the Equation (1), which becomes 1   3  1  2  3  2  2  1  11  1  12  0  12  1 2  12  1 3  2  0   1 e e e , 4 4 The expression for the polarization induced by the superposition of nonlinear waves is defined by We introduce (3) into (7) and find that the expression for the induced polarization becomes where ( ) The system of Equations (4) and (5) can now be rewritten in terms of 1,2 P and n as follows  To make the system (8) -(10) complete we add Maxwell's equations for the all real "slowly-varying amplitudes" 0,1,2,3 E and their phases 0,1,2,3 ϕ . We obtain

Transition Regime
We carried out the computer simulation of the system (8) - (10) and (11)  [27]- [29]. The optical pulses on the pump and trigger frequencies 0,1 ω were chosen to be of Gaussian shape. The accuracy of numerical results was based upon monitoring the conservation of energy of the system at every cross-sectional area in the medium. We considered the following trends: the phase locking and confinement; the length of medium needed for soliton formation; the distribution of energy during the transition regime; the energy limitations on solitons formation; the effect of slowing down of solitons in resonant systems; the relationship between the soliton speed and characteristics of incoming electromagnetic waves and nonlinear medium; the conditions leading to formation solitary wave at one frequency (instead of generation of sequence of them); the connection between the amplitude of soliton and parameters upon consideration. The space-time evolution of the normalized intensities ( ( )

Stability
To investigate the stability of solitons we perform the summation of the Equations (12) (14) (16) and (18) for phases 0,1,2,3 ϕ and transform them to the equations for 1,2 ∆ . We assume that both processes 2,3 0 1 2 ω ω ω = ± occur at the conditions of synchronism, so that 7 This condition is usually satisfied in gases [29]. Moreover, we also suggest that the phase differences 1,2 ∆ are locked to 2πn or ( ) 2 1 π n + due to the nonlinear effects [30]. In this case 1 0 P = ( 0 1 τ ∆Ω The behaviour of the latter system is analyzed in terms of phase planes. As an example, Figure 5 shows the phase plane of the following system ( )  ( ) at 11 different initial conditions for 1,2 ∆ .

Conclusion
The space-time evolution of the optical pulses by using the computer simulation of transition regimes of fourphoton resonant parametric processes in case of two-photon resonance is investigated. The computer simulation was based on application of the finite difference methods to the system of nonlinear equations modeling the foregoing interactions. It is shown that at certain boundary conditions (those result from the "area theorem" (see, e.g. [30])) the incoming laser pulses at frequencies 0,1 ω first generate new waves at 2,3 ω , and then all become simultons of Lorentzian shape. It has also been shown that upon the conditions of phase locking ( 1,2 1,2 2πn ∆ = + ∆