**Ekaterina Romanova, Saratov State Technical University, RussiaL.A.Melnikov, Saratov State Technical University, Russia**

Recently long fiber Raman lasers having the length of resonator as long as few hundreds km become a subject of intensive researches [1-3]. Its features includes low beat frequency of longitudinal modes, small cavity bandwidth, huge number of longitudinal modes, feedback due to light scattering. For various applications of such lasers it is necessary to study the dynamics of the laser modes, dynamics of total field, influence on scattered radiation of the laser behavior the laser.

The main complexity in the description of dynamics of the laser is the high order of system of the equations for amplitudes of modes. In problems of the description of dynamics of the laser with the long resonator demanded number of modes nearby ΔΩ / (c/2nL) ≈107 at length of fiber L = 1 km, n ≈ 1.5, ΔΩ ≈ 1 ТГц that is far from possibilities of modern computers .

In present paper the dynamical model of these lasers is presented. Corresponding equations for pumping and generation pulse amplitudes are derived together with corresponding boundary conditions, demonstrating different regimes of operation.

Where κp, κ factors of the compelled combinational strengthening/absorption, γp, γ - factors of linear absorption on frequency of a rating and a signal, accordingly, and also distinction of group speeds and dispersions of group speeds (DGS) impulses on frequency of a rating and a signal is considered. The given equations describe rating and signal evolution on length of a fiber. It is necessary to add them with periodic boundary conditions, for example on a target element of communication. Possible frequency shifts are considered in complex bending around E (z, t).The numerical decision of these equations in case of a pulse regimes is shown in Fig. 1.Was considered that group speeds of impulses of a rating and generation are equal, DGS also are equal and correspond solitons to a mode. The rating impulse corresponds fundamental soliton, and generation starts from an impulse 0.04 sech (t). It is possible to note influence of combinational strengthening, compression of an impulse and a target element of communication (reduction of amplitude of an impulse at transition by the entrance end of a fiber).

Fig.1.

For simplification of model it is possible to use the generalized method of the moments which allows to receive the ordinary differential equations for parameters of pulses and the equation for systems with gain and losses. The corresponding equations turn out at substitution of the trial decision of type E (z, t) =A (z) exp [-P (z) t2/2] in the equation and linear transformations turned out are nonviscous to this trial decision and the following higher modes constructed on the trial decision [4]:

Ekaterina Romanova

Saratov State Technical University

Russia

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