1. Introduction

Nowadays the problem of generating high-energy pulses in fiber lasers is still one of great importance, although this problem is widely addressed by researchers. As it is known all-normal fiber lasers may generate a dissipative soliton of energy up to orders of magnitude more compared to that of the classic soliton [1–6]. One of key properties of the dissipative soliton is that it experiences small deviations when propagating along the cavity, i.e. when passing through various intracavity devices [7–9]. In this case it is also observed that the increase of cavity length (up to several kilometers) corresponds to the increase of the output pulse energy.

To generate the signal that meets certain criteria, the complex optimization problem is to be solved [5, 8, 10]. Namely, to optimize the output signal properties, it is necessary to vary many cavity parameters over a wide range. In general it is impossible to solve this problem analytically, and the methods of numerical simulations are widely used to address the issue. However, the mathematical modeling has its disadvantages, one of which is that the modeling of long fiber lasers that generate dissipative solitons is a time-consuming and computationally complex problem [11]. This disadvantage is related to approach to the modeling when the signal passes through various devices consequently over many round trips. This normally takes the significant amount of time to simulate, especially in the case of long cavity with the significant pulse width and pulse energy. In addition to this, much more time is required to simulate the signal generation from the initial noise: the problem is of practical interest, but its numerical study requires to propagate the signal over many round trips to obtain the output stable pulses, i.e. those that preserve their shape at the laser cavity output [12].

As a result, there is a need to develop the analytical methods that describe physical processes that affect the signal propagation along the cavity. This may be applied to many problems that come from practice, such as the determining of the stable generation area and the optimization of pulse properties. Such analytical methods are based on distributed mathematical models, where physical effects affecting the signal are counted in a single equation. To improve the process of optimization of fiber lasers, the complex distributed nonlinear Ginzburg-Landau equation was considered [7, 13, 14]. This equation takes into account the signal gain and losses, the dispersion, the nonlinearity, and the mode-locking. Over the years, there have been found several stable highly chirped analytical solutions to this equation [7, 13, 14].

In this paper we investigate the properties of analytical solutions of Ginzburg-Landau equation [14]. We consider here the model that includes the gain saturation and show that this significantly affects the properties of the solutions. It will also be shown that taking into account the gain saturation improves the prediction of output pulses characteristics without carrying out full numerical modeling. The results are verified on parameters of fiber laser systems [12]; based on the analysis, we also propose the method to estimate the stable generation area.

2. Analytical models

The equation that describes the evolution of a slowly varying electromagnetic field envelope $E\left(z,t\right)=\left(A_{z,t}{e}^{i\left(\omega t-kz\right)}+A^{*}{ }_{z,t}{e}^{-i\left(\omega t-kz\right)}\right)/\sqrt{2}$ can be written in the following general form [14, 15]:

This equation is well known as complex generalized cubic-quintic Ginzburg-Landau equation (CQGLE − Cubic-Quintic Ginzburg-Landau Equation), it describes the dynamics of optical pulses in the fiber laser cavity. Equation (1) is written in the dimensionless form; here $\tilde{\beta }$ is the dispersion coefficient, $\tilde{\gamma }$ is the self-phase modulation parameter (nonlinearity), $\tilde{\sigma }$ is the difference between the losses and gain, $\tilde{\alpha }$ is the spectral filtering parameter, $\tilde{\kappa }$ is the amplitude self-modulation parameter, and $\tilde{\zeta }$ is the amplitude self-modulation saturation parameter. In this notation, all parameters are positive. Passive mode-locking in laser cavities that are covered by Eq. (1) can be implemented experimentally either by using the saturable absorber [16] or by nonlinear polarization rotation [6, 10].

To estimate the possibility of the stable pulsed generation in ultra-long dissipative fiber lasers let us study the properties of analytical solutions of Eq. (1). They have recently been found in [14], where CQGLE with constant$\tilde{\sigma }$ has been considered. We consider here the more general case that takes into account the gain saturation, i.e. in our consideration $\tilde{\sigma }$ is not constant. As we show below, the gain model significantly affects the properties of analytical solutions of CQGLE [14].

To study the stable generation area of dissipative soliton and to determine its characteristics let us find analytically the coefficients of Eq. (1) as functions of the parameters of intracavity devices. Equation (1) is dimensionless, the distance of z = 1 corresponds to the distance of one full cavity round-trip. This approach allows to simplify the task from the mathematical point of view, and, if necessary, one can easily revert the formulae back to the dimensional form.

Mathematical model of the saturable absorber as the passive mode-locking device in the ultra-long fiber laser can be described by the following equation:

Equation (1) in its dimensionalized form can be expressed as follows:

The pulse shape averaged along the laser cavity, $P\left(t\right)$, can be found by solving Eq. (3) with the condition $\partial P/\partial z=0$. To describe the pulse, the exact solution from [17] can be presented as $A\left(t\right)=P{(t)}^{\left(1+if\right)/2}$ [18], where f. is the dimensionless chirp parameter. The limit case of $f\to \infty$, when the phase changes faster than the amplitude does, can be achieved if the amplitude self-modulation is much less than the self-phase modulation $\left(\frac{\kappa }{P_{sam}}\ll \gamma L\right)$; this condition defines the pulse chirp. In this limit case the chirp parameter can be easily expressed as a function of cavity parameters: $\mathrm{f}\simeq 3\frac{\gamma L{P}_{sam}}{\kappa }\gg 1$ (see, for example, [18]). Physically, the chirp parameter is the product of spectrum half-width Δ and the pulse half-width T ($f\simeq \mathrm{\Delta }\cdot T$). The exact solution presented in [17] becomes singular for large f and thus does not have physical sense in this case.

To estimate the stable generation area of dissipative all-normal soliton lasers, the stability of analytical solution against the amplitude and phase perturbations is required. The first attempts to analytically describe highly chirped dissipative solitons in the scope of CQGLE [14] have shown the instability of solutions against the small variations of amplitude. The reasons of such a behavior in case of positive self-phase modulation is that the increase of intensity corresponds to the further increase of the gain. This fact leads to the unlimited grows of the peak power. To overcome such an instability we propose here the more realistic mathematical model that takes into account the gain saturation. In this case one can express σ as follows:

 

Fig. 1 a) The output energy and pulse width evolution as a function of number of round-trips (the total length of one round-trip, i.e. the total cavity length is 1 km); b) and c) show the pulse and spectral shape of stable pulsed regime, respectively. Black line corresponds to the numerical results, green line corresponds to the theoretical results.

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Let us analyze the stability of analytical solution from [19]. For this purpose we solve Eq. (3) numerically. As an example, we use in modeling the ring-cavity ultra-long fiber laser, its parameters are described in [12]. By means of the mathematical modeling we have compared the results obtained from Eq. (3) with analytical results from [14]. We use pulsed initial distribution in order to start the simulation from the point with $\sigma >0$. From the analytical solution one can see that the most difficult for numerical simulation is the rectangular pulse shape that can be obtained in the case of $\frac{G\left(\epsilon \right)}{2{\mathrm{\Omega }}_G^2}+\frac{1}{2{\mathrm{\Omega }}^2}=0$ (without spectral filtering). Figure 1 shows the results of numerical modeling in this case. Figure 1(a) depicts the energy and pulse width dynamics as a function of the round-trip number. The final pulse energy of stable regime $\approx 0.85$ nJ. Figures 1(b) and (c) show the pulse and spectral shape of stable pulsed regime, respectively. Here black line corresponds to the numerical results, and green line corresponds to the negative branch of analytical solution from [14]. From the numerical results we can see that the negative branch of analytical solution is the stable solution in the case of saturated gain. The obtained results indicate that even without the spectral filtering one can achieve the stable pulse generation.

First of all, note that in case of a highly chirped pulse the absolute values of the term $i\cdot \left(...\right)A$ on the right side of Eq. (3) are much more compared to the absolute values of the second term on the right in Eq. (3). Then ${\mathrm{\Omega }}_G\gg \mathrm{\Omega }>0.1$ nm. And then we can rewrite Eq. (3) as follows:

To investigate the domain of definition for Eq. (5), let us estimate the parameters of Eq. (5). To study the analytical solution let us introduce the variable $\mathrm{\Theta }=2-\frac{\gamma }{2{\Omega }^2\beta \kappa }$. Using the cavity parameter set from [12], one can see that all values of Θ are located in the narrow range

Obviously, inequality 6 holds for any reasonable value of a spectral half-width Ω. From inequality 6 one can see that there exists only the negative branch of the analytical solution. If condition 6 is met, the domain of definition of σ can be described as follows:

For further evaluation of definition area of analytical solution, let us rewrite the analytical solution from [14] in dimensional form. The peak power P_m can be expressed as follows:

In this case the implicit solution for the spectral pulse shape is as follows [14]:

Here $\mathrm{\Delta }=\sqrt{\left(\gamma /\beta \right)P_m}$ is the spectral half-width of a highly chirped dissipative soliton, $T=\left(3{\gamma }^{2}L{P}_{\text{sam}}^2\right)/\left(\beta \kappa {\mathrm{\Delta }}^3\left(1+R^2\right)\right)$ is the pulse half-width, and R is the only parameter that defines the pulse shape and is described by the following equation:

The most important characteristic to study the stable generation area of dissipative soliton in the ultra-long fiber lasers is the energy of the analytical solution presented below:

Let us examine the area of existence of the stable soliton solution of distributed Eq. (3). Since the pulse power is the positive real value, analysis of Eq. (8) indicates that there is no solution for $\sigma <0$. Another condition results from the restriction on the pulse spectrum: $R^2>0$, that is equivalent to inequality 7.

Numerical simulations in [19] show that the numerical and analytical results coincide when chirp parameter $f=T\cdot \mathrm{\Delta }>10$. This means that the analytical solution is able to describe the pulse evolution in dissipative fiber lasers with the cavity length $L>10\kappa /\gamma P_{\text{sam}}$.

Besides the suppression of the instability, the gain saturation allows to derive the self-starting condition of a dissipative soliton fiber laser of any configuration. From Eq. (4), the condition $\sigma \left(\epsilon =0\right)<0$ provides the stable generation from the initial noise. Let us explain this statement. The balance equation that describes the intracavity dynamics of the signal energy that is derived from CQGLE can be written in the following form:$\frac{d\epsilon }{dz}=-2\sigma \epsilon +\int \frac{2\kappa }{P_{\text{sam}}}|A{|}^2\left(1-\frac{1}{P_{\text{sam}}}{\left|A\right|}^2\right)A\left(t\right)dt.$

When the signal power ${\left|A\right|}^2\ll 1$ the integral in the right side of the equation becomes negligible. The balance equation that describes the start of generation in this case is

It is obvious that to make possible the stable pulsed generation the nonsaturated gain should be greater than the total round-trip nonsaturated losses; the latter is achieved if the following equation is met:

Note that the self-starting condition 12 is necessary, but not sufficient for the stable pulsed generation of the laser.

Let us transform Eq. (4) by the explicit expression of the energy ε:

From the right sides of 11 and 13 one can obtain the following equation that can be solved in the interval $0<\sigma <\mathrm{\Psi }$:

 

Fig. 2 The intracavity energy as a function of σ. The black line corresponds here to Eq. (11), the blue line corresponds to Eq. (13). The intersection of two lines corresponds to the solution of Eq. (14) in the interval $0<\sigma <\mathrm{\Psi }$.

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3. Results and Discussion

Figure 2 shows the dependence between the soliton energy and σ. Here the black line corresponds to Eq. (11), the blue line corresponds to Eq. (13). The intersection of two lines corresponds to the solution of Eq. (14) in the interval $0<\sigma <\mathrm{\Psi }$. From the figure one can see the existence of the single solution when ε (see Eq. (13)) is positive, provided that σ = 0. This condition coincides with the condition of the generation start (inequality 12). Thus, the self-starting condition is the sufficient condition to the pulsed generation in an ultra-long dissipative fiber laser, because the condition $\mathrm{\Psi }<\kappa +\mathrm{\Sigma }/2$ is always met with condition 6.

Let us consider the ring-cavity dissipative fiber laser described in [12] that consists of an active Er-doped fiber with the pump power P_pump, passive normal-dispersion fiber of a variable length L (from several tens of meters), passive mode-locking device, and the output coupler that provides the output energy $\left(1-R_{\text{in}}\right)E_{\text{in}}$ where E_in is the input energy.

To optimize the output energy in terms of pump power, cavity length, and output coupler reflectivity, we use the following parameters of losses: nonsaturated losses in active fiber ${\alpha }_A=0.75$ dB/m [20], nonsaturated losses in passive fiber $\alpha =0.2$ dB/km, the modulation depth of the saturable absorber $\kappa =0.1$, other intracavity losses are l = 7 dB (e.g., connection losses between intracavity devices), the active fiber length L_A = 2 m. The dependences of small signal gain g and saturation power P_sat on the active fiber length (L_A) and pump power (P_pump) can be described as follows [20]:

CQGLE is not applicable to this case when, during the soliton evolution, the energy is conveyed into larger wavelengths, and Raman dissipative soliton is formed. To simulate the evolution of Raman solitons one has to use the model based on the coupled equations with the integral nonlinear term. To avoid such a situation it is necessary to bound the stable soliton energy to 20 nJ [21].

In Fig. 3 the stable generation areas of the dissipative soliton fiber laser is shown for different passive fiber lengths in (R_in, P_pump) plane limited by the interval $0<R_{\text{in}}<1$ and conditions 12 (lowerleft border) and the energy limitation (upper right border).

 

Fig. 3 Stable generation areas of the dissipative soliton fiber laser with different passive fiber lengths as a function of (R_in, P_pump). The optimal value of output energy is observed for the cavity length of 2 km and is about 9.2 nJ.

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In Fig. 3 it is shown that, when the cavity length grows, the stable generation area narrows. The optimal value of output energy is observed for the cavity length of 2 km and is about 9.2 nJ. In Fig. 3 the points A, B and C indicate the energy maxima for cavity length of 1, 2 and 3 km, respectively. Obviously, the energy maximum corresponds to the maximum pump power. If the cavity length is more than 1 km, the optimal energy corresponds to the maximum of coupler reflectivity and is situated on the boundary of the generation area (see points B and C). On the contrary, if the cavity length is less than 1 km, the stable generation area becomes wider and the optimal generation corresponds to the reflectivity of $R_{\text{in}}\approx 0.5$ and minimizes the functional of total linear losses $\exp \{-2\kappa -\alpha L-{\alpha }_A{L}_A-l\}\cdot R_{\text{in}}\cdot \left(1-R_{\text{in}}\right)$ over the range $0<R_{\text{in}}<1$ (see the point A).

Only the parameters of gain and losses define the area of stable pulse generation that is covered by the analytical solution of CQGLE. By means of the optimization of the dissipative soliton output energy in the ultra-long laser cavity the optimal values of the cavity length and the coupler reflectivity may be found. The drawback of such an approach is that the dissipative soliton properties are estimated only in average, because the mathematical model considered does not take into account the signal dynamics in intracavity devices; also, it does not takes into account the arrangement of intracavity devices. Because of this, we intend to study further the differential equation system obtained in the approximation of a negative branch of highly chirped family of solutions of CQGLE.

4. Conclusions

We have proposed the analytical method of estimating the stable generation area in a long cavity fiber laser that generates the dissipative highly chirped soliton. The method is based on the analytical approximation of the solution of complex nonlinear Ginburg-Landau equation with taking into account the gain saturation. It has been shown that if the self-start condition is met, this corresponds to the pulsed generation in a long dissipative fiber laser.

We have found the areas of stable pulsed generation that exclude the formation of Raman dissipative soliton. These theoretical results enable to reduce the time required to investigate numerically the long-cavity fiber laser systems. We have shown that out of two branches of GLE solution the “negative” branch is stable and only it may be observed in practice due to the presence of saturated gain in real laser systems.