Analytical consideration and computer simulation of DFWM

Degenerate four-wave mixing (DFWM) for co-propagating femtosecond laser pulses is considered in the axial-symmetric case for bulk medium with cubic nonlinear response, if pump-wave amplitudes are being equal. Computer simulation is based on the set of nonlinear Schrödinger equations describing this process. For the analytical consideration we use the frame-work of both plane wave approximation and long pulse duration approximation taking into account the phase matching. In opposite to widely used approach, based on the pump-waves non-depletion, the problem invariants are used for analytical solution developing. The solution demonstrates various DFWM modes existence and allows us to provide full analysis of the problem in dependence of its parameters. Analytical solution and derived pulse interaction modes can explain complicated regime of DFWM, which may appear at different intensities of interacting waves.

Obviously there are many physical factors which influence on FWM modes. To understand the FWM features one needs to make computer simulation of this problem or various analytical approaches developing. Obviously, the last is more preferable because it allows to see a visual dependence of a frequency conversion efficiency. Usually the analytical consideration is based on the pump-waves non-depletion approximation which is valid at the initial stage of the pulses interaction. This approach possesses the essential restrictions. For example, the interacting wave energy does not preserve. Its more significant limitation arises out of neglecting of self-modulation and crossmodulation of the pumping waves. As a result of this, the phase difference between pumping waves and other two waves becomes wrong and therefore an evolution of the interacting waves is wrong also. To revise this situation we develop new approach for analytical solution of the problem: we use conservation laws of the FWM and both plane wave approximation and long pulse duration approximation. In this case we accurate take into account the interacting waves phase changing. Hence, the analytical solution of the problem becomes more adequate to a physical situation if pulse dispersion and beam diffraction is weak at a medium length under consideration. In this paper we compare the problem analytical solution made in [20,21] with computer simulation results obtained using the solution of four nonlinear Schrödinger equations describing FWM of axially-symmetry beams with taking into consideration a second order dispersion of interacting pulses, their self-and cross-modulation and beam diffraction. Let us notice that early we have used with success the similar approach for analysis of frequency conversion efficiency when the laser pulses interact in a medium with quadratic nonlinear response [22][23][24][25][26]).

Problem statement
We consider the FWM of co-propagating laser pulses with current frequencies denoted as in a medium with cubic nonlinear response. In the framework of slowly varying envelope the considered process is governed by the set of dimensionless nonlinear Schrödinger equations with respect to complex amplitudes 1 2 3 4 , , , A A A A of interacting waves: with the following initial and boundary conditions for complex amplitudes We assume that there is a group velocity matching. In Eqs. (1) the dimensionless parameters and functions can be expressed through the physical ones in the following manner:    ) of a medium on the optical frequency is neglected. The similar assumption is valid for the refractive index of interacting waves. Parameters j  characterize self-action of the laser beams due to a cubic nonlinear response.
Below we will take into account the following relation for nonlinear coefficients of the problem: Taking into account the equality of nonlinear coefficients for the pump waves, the equality (3) reduces to the next equality:

Conservation laws (invariants) for the case of interacting waves phase matching (
In this paper we compare computer simulation results with an analytical solution of the problem (1) in the case of a phase matching for the interacting waves. Therefore, we write invariants (conservation laws) for this case.
First invariant is energy one and it is written in the form: However, as it is well-known, there are other invariants which are similar to (5) and they are written as:  (1) and it is seen as 22 4 ** 3 1 2 3 4 1 00 To obtain the analytical solution we essentially use these invariants. Moreover, they are used for a control of computer simulation results: we develop conservative finite-difference scheme for a numerical solution of the problem (1).

Analytical solution f the problem in the frame-work of long pulse duration and plane wave approximation
Many special features of the FWM process can be clarified by using the long pulse and plane wave approximation. Taking into account the well-known representation of complex amplitude: where an amplitude j a and phase j  of the waves are real functions. In this case, the waves interacting can be described by the following equations: The intensity 4 () pz evolution is defined by roots of the equation ( ) 0 fx . We see from expression (9) and (10) that the root properties depend strongly on the problem parameters. One of the roots is always equal to 0 and others can be found from a cubic equation, which may possess either three real roots or one real root and two complex roots. Moreover, the fourth wave changing also depends on the sign of the parameter  because of its influence on the radicand sign. Obviously, for the solution existence it is necessary a validity of the inequality: In general case, the fourth wave intensity changing is described by an elliptical function. Thus, the FWM mode is governed by the relationship between the parameters 10 20 2 ( , , ) aa . Below we write only two different solutions because in our previous papers we describe various solutions of the problem in detail. xx  ] at the fourth wave propagation (this generation mode is named as low efficiency mode). In this case, the fourth wave intensity evolution is described by: where sn denotes the elliptical sinus. If four real roots satisfy the following relation To clarify conditions of the used approach for the analytical solution developing we construct the conservative finite-difference scheme for the problem (1).

Conservative finite-difference scheme
The conservative finite-difference scheme for problem (1-2) is constructed as follows. In the domain Using the notations introduced above, the finite-difference scheme for the problem (1) in the case of the phase matching can be written in the form: (2( ) ), are positive parameters which defines the iteration process accuracy.

Comparison of computer simulation results with analytical solution
In figure 1 the analytical solution (12)    a beam diffraction in breaking of the long pulse duration approximation and plane wave approximation we make computer simulation with unchangeable value of the diffraction coefficient with the dispersion coefficient changing and vice versa. The corresponding comparison is shown in figures 3, 4. We see that a second order dispersion influence results in the pulse compression and consequently, the maximal intensity of the pulse increases and this leads to the intensity oscillation period decreasing (see formula (12)). The beam diffraction also leads to the maximal intensity enhancing. However, this enhancing is not pronounced along a pulse propagation distance under consideration.

Conclusions
Thus, in this paper we made a comparison of the analytical solution, obtained in the frame-work of long pulse and plane wave approximation for FWM problem, with computer simulation results, obtained by using the conservative finite-difference scheme proposed for this problem solution.
We consider a case of equality of the pump wave intensities at co-propagating FWM under phase matching condition. We showed that along big distance of a laser pulse propagation both solutions coincide. Main reason of a deviation of the solutions is an influence of second order dispersion. Beam diffraction influences less than pulse dispersion till propagation distance under analysis.