^{1}

^{1}

^{2}

^{1}

Dissipative soliton resonance (DSR) is a phenomenon where the energy of a soliton in a dissipative system increases without limit at certain values of the system parameters. Using the method of collective variable approach, we have found an approximate relation between the parameters of the normalized complex cubic-quintic Ginzburg-Landau equation where the resonance manifests itself. Comparisons between the results obtained by collective variable approach, and those obtained by the method of moments show good qualitative agreement. This choice also helps to see the influence of the active terms on the resonance curve, so can be very useful in constructing passively mode-locked laser that generate solitons with the highest possible energies.

One of the most perceptible features of solitons in dissipative systems is that they exist only when there is a continuous energy supply from an external source. Dissipative solitons have an internal energy exchange mechanism [

The dynamic and stability of dissipative soliton are governed by parameters fluctuations and noise, which can lead to the formation of complex structures like stationary, pulsating solutions [

More specifically, the shape, amplitude, width and all other parameters of dissipative soliton and its energy are defined in advance by the initial conditions and the equation parameters. Nevertheless, the profile of these structures is indeed fixed for a given set of parameters, but one can obtain a wide range of dissipative solitons profiles by tuning each equation parameter.

It has recently been demonstrated that for certain values of the system parameters, the dissipative soliton energy can increase indefinitely and so the process resembles the resonance phenomenon in the theory of oscillators. In such specific situation, the solitons increase their width indefinitely while keeping their amplitude constant. This feature of dissipative soliton was labeled “dissipative soliton resonance”.

These main features of that localized structures offer highly desirable properties for applications, such as the generation of stable trains of laser pulses by mode-locked cavities, or the in-line regeneration of optical data streams [

One of the master equations for wave dynamic in dissipative nonlinear media can be based on a cubic-quintic complex Ginzburg-Landau equation (CGLE) [

In recent past, Chang et al. have also predicted a novel soliton formation, dissipative-soliton-resonance, in the frame of CGLE [

Indeed, finding the solutions of the CGLE or the region of parameters where DSR exists for a given set of parameters and a given initial conditions is an extremely lengthy and costly task. It requires an enormous number of numerical simulations. In this context, it is helpful to develop theoretical tools that can perceive CGLE soliton solutions more efficiently. In the case of DSR, this task can be simplified with approximations of the pulse shape, finding the set of parameters which predicts the areas where resonances can be found.

In this present study, we devoted, in Section 2 to find the resonance curve by applying the collective variable approach and then we compare our result to the reference [

We consider, as in the work [

The left-hand-side contains the conservative terms which describe gain and loss of the pulse in the cavity and the right-hand-side includes all dissipative terms. The optical envelope

As in these works [

The function

As in the example in [

where

The collective variables

Applying the bare approximation by setting the residual field to zero

(

we get easily the four collective variables evolve according to the following set of four coupled ordinary differential equations. It gives the dynamical system for the evolution of the soliton amplitude

The main advantage of the trial function lies in its simplicity, which makes the procedure of derivation of the variational equations relatively easy. Thus it allows seeing clearly the influence of each parameters of the CGLE on the various physical parameters of the soliton. To a certain extent the natural control parameter of the soliton is the total energy

Likewise the energy density

And the corresponding flux,

Then the evolution equations (Equation (4)) for the soliton parameters can also be expressed as a function of the total energy

Here the master equation (Equation (1)) is reduced to an ordinary differential equation given by the soliton energy

The results clearly confirm that the collective variable approach is a useful tool for studying the dynamic of the soliton evolution. It also helps to research the resonance curve analytically. We know that near the resonance curve, that soliton energy

So the terms in Equation (8) of the order of

It is recognised that near the resonance curve the fixed points are defined by the zeros of the right-hand side of the equation presented above. As well as the values

where,

Substituting these equations into the third equation in Equation (9), we obtain the expression for the resonance curve in terms of the system parameters

The expression Equation (10) depends on all coefficients of the propagation equation Equation (1). It has the same terms as that found by the method of moments (see Equation (10) in [

The design of a particular laser with specific features requires numerous modeling

and numerical simulations [

curve retains almost the same value. The different curves have the same distinctive features. First, they kept an almost constant value but when the dispersion value is greater than −6 the resonance curves differentiate and increase steadily. In this case the solitons have different amplitudes, widths and therefore energies which basically provide the highest distinguishing features.

The other active term,

The spectral filtering or gain dispersion (if positive) in the cavity is described by the coefficient

when the dispersion is negative. However, for positive dispersion values, the coefficient

Finally, in this present study, we have presented collective variable approach for the phenomenon of dissipative soliton resonance found earlier using numerical simulations and confirmed by the method of moments. In particular, using the method of collective variable approach, we have found an approximate relation between the parameters of the normalized complex cubic-quintic Ginzburg- Landau equation and simple analytic expression for the resonance curve which is in good qualitative agreement with the results of the method of moments. But also as the master equation that describes laser systems has many parameters, it’s so difficult to study the influence of all of them on the soliton properties. Here we describe the influence of dissipative terms of CGLE on resonance curve. Namely how the linear loss, the gain and its saturation in the system and the spectral response of the cavity can be suitably chosen for generation of high-energy pulses. Clearly this work can help to optimize the active parameters of cavities that generate solitons with the highest possible energies. We think that these results can be helpful for the design of the mode-locked laser systems that generate record-high energy short pulses, without the need for additional amplifiers.

Kamagaté, A., Konaté, A., Soro, P.A. and Asseu, O. (2017) Effects of Dissipative Terms on Dissipative Soliton Resonance Curve. Optics and Photonics Journal, 7, 57-66. https://doi.org/10.4236/opj.2017.73005