1. Introduction

In recent years, different groups have studied the interactions between dissipative solitons in mode-locked fiber lasers [1–4]. Such interactions can lead to the formation of bound states of solitons (doublet) through dispersive waves [5] or through a direct overlap of the solitons [6], depending on the round-trip dispersion of the cavity. Two groups independently reported that a bound state of solitons and a single soliton propagate according to different group velocities in a stretched-pulse fiber laser [7, 8]. As a conseuence, periodic collisions between a soliton doublet and a single soliton take place in the laser cavity.

To better understand this phenomenon, two issues have to be addressed. First, why would a soliton doublet and a single soliton move with different velocities? Second, what happens when the soliton doublet collides with the single soliton? A theoretical answer, based on the fact that the group velocity of a doublet is sensitive to the phase difference between the solitons in the bound state, was proposed by Grelu et al. [8]. In another paper [2], they presented some numerical scenarios of collisions having different outcomes. Later on, we confirmed experimentally the existence of such scenarios [9]. Although this explanation is certainly part of the answer, we believe that other mechanisms could also contribute to the relative group velocity between the bound state of solitons and the single soliton. In fact, it is important to note that the bound states in a fiber laser do not necessarily have a fixed phase difference [10]. Moreover, when they do, this phase difference can be varied by changing the parameters of the cavity. In our own experiments, we observed that even when the phase difference does not seem fixed, there could be a relative group velocity between a soliton doublet and a single soliton. We also show that the parameters of the bound state presented in this paper, both observed experimentally in our laser and simulated numerically, are such that the effect of the phase difference has a weak influence on the velocity of the doublet compared to other mechanisms.

In fact, due to the group-velocity dispersion (GVD) of the fibers in the cavity, a modification of the average frequency of the pulse structure is sufficient to explain a change of its group velocity. Such modification of the average frequency can originate from any type of nonlinear effect represented by “skew” dissipative terms in the propagation equation. The phenomenon of collisions was observed only in the stretched-pulse laser and not in the soliton laser. This fact suggests that this effect depends critically on the pulse peak power and width since this is the major difference between the two laser regimes. Amongst the known nonlinear effects taking place in fibers, the Raman effect [11] seems to be an ideal candidate to produce the necessary frequency shift. The goal of this paper is thus to study the influence of the Raman effect on the evolution of bound states of dissipative solitons. We show numerically that the Raman effect leads to a relative group velocity between a soliton doublet and a single soliton. We then interpret this result analytically with the help of two mechanisms, the Raman induced self-frequency shift (RSFS) and cross-frequency shift (RXFS). Finally, we discuss the implications of this result on the phenomenon of pulse collisions observed in fiber lasers.

2. Numerical model and results

2.1 Model

In this section, we discuss a numerical model that represents a fiber ring laser cavity with pulse-additive mode-locking (P-APM) as shown in Fig. 1. The parameters of the laser cavity are chosen so as to match the experimental parameters used in reference [7]. We thus have a dispersion-managed laser cavity with two segments of optical fiber having group-velocity dispersions of opposite signs, plus some discrete components.

 

Fig. 1. The laser cavity.

Download Full Size | PDF

Consider a vector field with an envelope A⃗ represented in a circular polarization basis by the components A ₊ and A _-. The propagation in each fiber segment is modeled by a pair of differential equations taking into account group velocity dispersion (GVD), electronic Kerr effect, gain profile of the active medium and Raman effect [12,13]. The latter mechanism is represented by the last three terms on the right-hand side of Eq. (1); these terms were obtained from the theory presented by Menyuk et al. [13], which was developed for the propagation of a vector field in a birefringent optical fiber with an isotropic non-instantaneous nonlinear response. This theory is valid for pulses duration of the order of one hundred femtoseconds and more. It is thus applicable to the solitons found in the stretched-pulse fiber laser for which the minimum pulsewidth is about 100 fs. Note that all the coherent terms in the original equation were preserved since the approximation of large birefringence is not valid here. The resulting propagation equations are thus:

where the parameters of each fiber are the GVD (β ₂), the Kerr nonlinearity (γ) and the Raman parameter (T_R ). The saturated gain coefficient g_sat is evaluated as g_sat =g_ns /(1+E/E_s ) where E is the energy of the signal, E_s is the saturation energy set at 30 pJ, and gns is the small signal gain coefficient modeled as a lorentzian with peak amplitude g ₀ and FWHM Δf=5.0 THz.

Table 1. Parameters of the fibers used for the simulations.

View Table

The output coupling loss is accounted for by multiplication of the field by the appropriate coefficient. The mode-locking mechanism is modeled by the multiplication of the vector field through a matrix representing a polarizer combined with a quarter-wave plate and a half-wave plate having given angles with respect to the polarizer axis. The values of these angles are not known experimentally so they represent “free” parameters in the simulations. They are adjusted so as to get self-starting mode locking and results that are consistent with our experiments. The parameters of the fibers used in the simulations are listed in Table 1.

When a single soliton is propagating in the cavity, its group velocity is sensitive to the value of the Raman parameter T_R . This is due to the soliton self-frequency shift [11], the Raman effect downshifting the frequency of the soliton. In the presence of gain, this downshift is balanced by the limited gain bandwidth, resulting in a lower soliton central frequency [14]. Then, because of the GVD, the group velocity of the soliton at this frequency is different from the group velocity when the soliton is not downshifted. Obviously, we expect this phenomenon to be present also for solitons in a doublet. However, what is shown in this paper is that the overall frequency downshift due to the Raman effect is different for a soliton doublet than for a single soliton, leading to a difference in their respective group velocities.

2.2 A stable bound state in the absence of Raman effect

We first simulated the formation of a doublet in the absence of Raman effect in our cavity. To do so, we used a gain g ₀=3.5 m^-1 and we injected pairs of solitons with different initial separations and phase differences. All the cases studied converge toward the same fixed solution which shows that only one bound state exists in the cavity. This bound state, or doublet, is stable and shows a separation of 1.82 ps and a phase difference of zero between the solitons. This result is different from the results obtained in the context of the Ginzburg-Landau equation [1], where a phase difference of π/2 radians was found. It differs also from the results presented in [2] in the case of the Ginzburg-Landau equation with parameter management where a phase difference of π/2 radians was also found. However, it is in agreement with the bound state found in [6] where polarized Ginzburg-Landau equations with parameter management were considered and phase differences of 0 or π radians were obtained depending on the values of the parameters being used. The bound state we obtained is shown in Fig. 2 at two different locations in the cavity, its point of maximum compression and its point of maximum stretch.

The fact that the bound state has a zero phase difference implies that the average frequency of the bound state is the same as that of a single soliton. Hence, if they are present simultaneously in the dispersive cavity, they shall move with the same velocity and never collide. To explain why collisions are observed experimentally, we incorporated the Raman effect in the simulations.

2.3 Influence of the Raman effect on the doublet

To see how the Raman effect affects the propagation of the doublet, we performed several simulations in which we varied the value of the Raman parameter. The stable bound state found in the previous section was injected as the initial signal and the same value of gain was used.

When the Raman parameter is small, its only effect is to modify the average frequency of the doublet and, consequently, its velocity. This was expected because each soliton is subjected to the soliton self-frequency shift introduced by Gordon [11]. When the Raman parameter is further increased, we find that, on top of the previous effect, it alters other properties of the bound state. The phase difference is not zero as before, it reaches a different value which remains constant afterward. This value is relatively small, for instance it is 0.2 rad when the Raman parameter is 0.5 fs. In addition, the separation of the bound state is modified slightly, it becomes 1.84 ps for the above-mentioned case, but it is stable as in the case without Raman effect.

 

Fig. 2. Doublet obtained numerically in the absence of Raman effect. The red curve is obtained at the point of maximum compression in the SMF fiber while the black curve is obtained at the point of maximum stretch in the erbium-doped fiber.

Download Full Size | PDF

When the Raman parameter reaches larger values (e. g. 1 fs and higher), the bound state does not show a fixed separation and phase difference anymore. The phase difference decreases linearly as a function of the roundtrip number, such that it evolves from 2π to 0 radians periodically with a period ranging from a thousand to a few hundred roundtrips as the Raman parameter is varied from 1 fs to 5 fs. The coherence of the doublet is lost. The evolution of the separation and the phase difference in the doublet is shown on Fig. 3 for the case where the Raman parameter is 5 fs. The separation oscillates about 1.85 ps with a period of 500 roundtrips and with an amplitude of about 15 fs. Note that experimentally, such variations of the separation would go unnoticed on the autocorrelation trace. However, the cyclic variations of the phase difference should make the modulation of the optical spectrum disappear. In fact, depending on the relative amount of time spent for each given separation and phase difference, it can lead to a partial modulation of the spectrum. This is a phenomenon that we observed often in the lab for certain regimes of operation. Sometimes, a bound state which seems to have a fixed separation on the autocorrelation trace does not show a complete modulation of its spectrum as would be expected for a coherent doublet.

A quick analysis of the velocity of the doublet as a function of the Raman parameter reveals that it seems to be affected in the same way as a single soliton would be under the influence of the Raman effect. To verify this fact with better accuracy, we decided to inject a doublet and a single soliton simultaneously in the cavity. The results are described in the next paragraph.

 

Fig. 3. Evolution of the separation and the phase difference for the doublet in the case of a Raman parameter of 5 fs.

Download Full Size | PDF

2.4 Relative group velocity between a doublet and a single soliton

We start with a signal made up of three solitons. Two of them form a doublet and are well separated from the third one (this is the 2+1 case [8]). The gain is increased enough to support three solitons, i. e. it is fixed to g ₀=4.3 m^-1. For the case of T_R =5.0 fs, the results are shown in Fig. 4; the graph is a contour plot showing the signal as a function of time (horizontal axis) and roundtrip number (vertical axis). The soliton doublet gradually approaches the single soliton since it moves faster. Eventually, this leads to a collision during which a new soliton doublet is formed and moves away with the same velocity as the incoming doublet. We will discuss the collision and its implications later in this paper. Note that the signal at each roundtrip is represented close to the point of minimum pulse width in the dispersion-managed cavity, which explains why the solitons do not appear to overlap significantly. However, the width of the solitons at their point of maximum stretch in the cavity is of the order of a picosecond which means that the solitons in the doublet clearly overlap.

Let us examine the relative group velocity between the soliton doublet and the single soliton. In the above example, the value of the Raman parameter was fixed to 5 fs, which corresponds to the commonly used value for silica [12]. In order to clearly establish the role of the Raman effect in this example, we performed a series of simulations where the Raman parameter was varied while keeping all the other parameters fixed (g ₀=5.1 m^-1). We then computed the relative group velocity Δv_g using the following formula:

where c is the speed of light, n_g is the group index in silica at 1550 nm, L=6.1 m is the total length of the cavity, and Δt is the change per roundtrip of the delay between the single soliton and the soliton doublet. A graph of the relative group velocity as a function of the Raman parameter is shown in Fig. 5.

 

Fig. 4. A collision between a single soliton and a soliton doublet.

Download Full Size | PDF

 

Fig. 5. Relative group velocity as a function of the Raman parameter.

Download Full Size | PDF

The relative velocity varies almost linearly with the Raman parameter. When there is no Raman effect (T_R =0) the single soliton and the soliton doublet are at rest with respect to each other, in agreement with the observation that their phase difference is zero. When the Raman parameter is 5 fs, the relative velocity is about 3.2 m/s, which compares well with the measurements reported in [7]. The Raman effect clearly plays a role in this phenomenon by fixing the relative group velocity between the single soliton and the soliton doublet. In view of this result, one expects that the overlap of the solitons in the doublet and their nonlinear binding interaction are responsible for the modification of their group velocity.

So far, the common explanation for this effect relied on the non-zero phase difference between the solitons in the doublet. In fact, this phase difference renders the spectrum of the doublet asymmetric and thus modifies its average frequency [1]. Because of the dispersion in the cavity, this affects the group velocity of the doublet at the same time. In our case, the phase difference in the doublet is zero, meaning that its average frequency is the same as that of a single soliton. Therefore, in the absence of the Raman effect, no collision occurs. In fact, it can be shown that the modification of the average frequency Δf_avg of a doublet of Gaussian pulses is Δf_avg ≈-π(Δf)² T sinϕexp[-(πTΔf)²] where Δf is the halfwidth of the power spectrum of the solitons, T their separation and ϕ their phase difference. In ref. [8], the phase difference is π/2 radians and TΔf is of the order of unity. Obviously, in that case, the phase difference has an important effect on the average frequency and, consequently, on the group velocity of the doublet. In the case considered here, the separation is about 1.8 ps and the width of the spectrum is about 4.0 THz. Due to the exponential factor, the effect of the phase difference on the average frequency of the doublet is negligible whatever the value of this phase difference is. Moreover, for larger values of the Raman parameter, the doublet is not coherent anymore as its phase difference decreases linearly in time. In this case, the group-velocity difference would equal zero on average. During one cycle of oscillation, the relative velocity would go from positive to negative values and we should have observed an oscillation in the relative group velocity with a period of a few hundred roundtrips. This effect was not observed in the simulations. The conclusion is that the mechanism based on the phase difference [1] does not lead to a significant modification of the group velocity of the doublet for the conditions examined in this paper. This leads naturally to consider the Raman effect in our simulations.

3. Different frequencies lead to different group velocities

Due to the presence of dispersion in the cavity, any modification of the average frequency of a signal leads to a modification of its velocity. Having that in mind, we come back to the case of the collision presented in Fig. 4. We computed the frequencies of the single soliton and the soliton doublet separately during one roundtrip (in the steady-state regime, the propagation repeats itself every roundtrip so these results will be the same for any roundtrip) by using the first moment of the instantaneous frequency chirp along the signal. The top part of Fig. 6 shows the frequency of the single soliton and the frequency of the bound state of solitons as a function of their position in the cavity. We clearly see that the frequencies of both structures change as they propagate through the different components of the cavity. In the initial portion of the erbium-doped fiber, the frequencies increase (part 1 on the graph) because of the finite gain bandwidth. Since the frequencies closer to the center of the gain spectrum (which is chosen to be at f=0 in the simulations) are more amplified, the overall effect on the structure is to pull its average frequency toward the gain peak. Then, as the solitons are more and more compressed by the effect of GVD, they eventually reach their minimum width and maximum peak power. At that point, they undergo a frequency downshift due to the Raman effect (part 2). As they start to stretch again, the Raman effect is reduced and the bandwidth effect pulls the average frequency to higher values again (part 3). Finally, they undergo another Raman downshift at their point of maximum compression in the SMF fiber (part 4).

 

Fig. 6. Evolution of the frequencies of the single soliton and soliton doublet during one cavity roundtrip (top part). The evolution of their frequency difference is also shown (bottom part).

Download Full Size | PDF

At the bottom of Fig. 6, the evolution of the frequency difference between the single soliton and the soliton doublet is shown, along with its average value Δf=0.022 THz. The fact that the frequency of the soliton doublet is different from the frequency of the single soliton explains their velocity difference. We can estimate their relative velocity using the following equation:

where β _{2 ,AVG}=(β _2,SMF l_SMF +β _{2 ,Er}l_Er )/(l_SMF +l_Er ) is the average GVD of the cavity. Using the known values for the different parameters, we get a relative velocity of 4.6 m/s. The value obtained using the numerical trajectories in this case is 1.3 m/s. The discrepancy between this value and the one estimated above can be explained by the local variations of the frequency difference combined with the local variations of the GVD. If, instead, the group-velocity difference is calculated using $\Delta v_g=\frac{2\pi c^2}{n_g^{2}L}\mp {}_0{}^L\beta _2^{}\left(z\right)\Delta f\left(z\right)\mathit{dz}$, we get 0.9 m/s. Ideally, an analytical model would need to include intra-cavity dynamics to get the precise value of the relative velocity. However, for the sake of simplicity, this fact is not taken into account in the following analysis since its purpose is to focus on a global understanding of the phenomenon.

4. Mechanisms leading to frequency shifts

Thus far, we have seen that the frequency of any pulse structure propagating in the cavity reaches an equilibrium value that is a balance between its Raman frequency downshift and the effect of the gain bandwidth that pulls it back toward the gain peak. The equilibrium frequency reached by a bound state of pulses is different from the equilibrium frequency of a single pulse. This is due to the fact that the Raman downshifts underwent by these pulse structures are different. Two distinct mechanisms that we call the modified self-frequency shift and the cross-frequency shift contribute to this difference.

4.1 Raman Effect for two Gaussian Pulses

The Raman effect acting on a linearly polarized field A(z,t) is represented by the following propagation equation [12]:

The average angular frequency of the signal A(z,t) can be calculated using:

where E is the total energy of the field. Using Eqs. (4) and (5), the rate of change of ω as the field is propagated along z is computed as:

Let us consider the sum of two identical linearly-chirped pulses A ₁ (z,t) and A ₂ (z,t) separated by a time interval much longer than their pulse width. The previous equation then becomes:

In this equation, the second and third integrals can be ignored since their integrands contains oscillating terms that will average out to be negligible compared to the other terms. The first integral leads to the Raman-induced self-frequency shift of the pulses (RSFS). In fact, since the pulse width is much smaller than the separation of the pulses, it can be shown that the terms of the form |A_i |²${\partial }_t^2$ (|A_j )² with i≠j have negligible contribution. The first integral thus reduces to the case discussed by Gordon [11]. The fourth integral is responsible for what we call the Raman-induced cross-frequency shift (RXFS); it will be discussed in more details below.

In a stretched-pulse fiber laser, the soliton width varies considerably during a roundtrip because of the linear chirp that arises from the large local GVD. Usually, the soliton has a chirp close to zero somewhere in the middle of each fiber segment in the dispersion-managed cavity; at such positions, the pulse reaches its minimum width (~100 fs). At the junction between the segments, the soliton is linearly chirped and attains its maximum width (~1 ps). The overlap between nearby solitons in this region of the cavity causes the formation of bound states in the stretched-pulse laser [6]. In what follows, we will analyze the perturbative effects of RSFS and RXFS acting on two Gaussian pulses propagating in a fiber under the influence of group-velocity dispersion (β ₂) only, an ansatz that represents the solitons formed in such lasers. We consider that the pulses are chirp-free at z=0 and have peak power P ₀ and pulse width T ₀. The delay T between both pulses remains constant during the propagation. Their evolution along z is then given by [12]:

4.2 Raman self-frequency shift (RSFS)

Using Eq. (8) into the first integral of Eq. (7), we get the rate of change of the angular frequency due to the RSFS:

where L_D =$T_0^2$/|β ₂| is the dispersion length. Integrating over z from -∞ to +∞, we get the cumulative self-frequency shift of pulses:

The self-frequency shift is proportional to the peak power and the inverse of the pulse width squared at z=0. Since the peak power decreases and the pulse width increases as the pulses move away from z=0, the frequency shift will occur mostly over one dispersion length centered at z=0. This explains the presence of L_D in the self-frequency shift. It also justifies the fact that we integrate over all values of z since, in our laser, the fiber segments are much longer than one dispersion length.

In the laser cavity we simulated, the properties of a soliton are changed slightly when the soliton is bound in a soliton doublet. In Fig 7, we represented a single soliton and a bound soliton at their respective chirp-free locations in the SMF fiber. First, we note that these locations are not the same, they differ by 2 cm in this case. Second, the pulse peak powers and widths are not the same either. Since the self-frequency shift depends critically on these parameters, the RSFS observed by a soliton in a doublet will be different from the RSFS undergone when it is alone. Thus, we refer to this mechanism as the modified self-frequency shift.

 

Fig. 7. Power profiles of the single soliton and one of the solitons in a doublet at their chirp-free locations in the SMF fiber.

Download Full Size | PDF

4.3 Raman cross-frequency shift (RXFS)

Inserting Eq. (8) into the fourth integral of Eq. (7) and assuming that T≫T ₀, we get the rate of change of the angular frequency due to the RXFS:

This effect takes place when the pulses start to overlap significantly. Since we assume T≫T ₀, this will occur for z≫L_D . Following the last approximation, we can integrate Eq. (11) in order to obtain the cumulative cross-frequency shift over a distance z :

This frequency shift depends on the peak power and the pulse width as well as on the ratio of distance z over the dispersion length L_D . But, most importantly, it depends critically on the ratio of the pulse separation to the pulse width. Two pulses too widely separated do not overlap significantly and then there is no cross-frequency shift. Of course, this effect does not occur for a single pulse.

The cross-frequency shift can be interpreted as follows. When the two pulses propagate from their chirp-free location, the frequency chirp increases gradually. As two linearly chirped pulses overlap, beats are formed in the overlap region. This can be seen in Fig. 2 which illustrates the doublet obtained numerically (g ₀=3.5 m^-1 and T_R =0) at the location where the pulses achieve their largest duration in the cavity. These beats arise from the coherent superposition of the pulses characterized by time-varying instantaneous frequencies. For linearly chirped pulses, the difference in their instantaneous frequencies is constant in time and is fixed by the magnitude of the linear chirp and the separation between the pulses.

The instantaneous Raman frequency shift depends on the curvature of the power profile. At the positions where the curvature is negative, it leads to a frequency downshift. At the positions where the curvature is positive, it gives a frequency upshift. In order to obtain the average frequency, we integrate the frequency shift multiplied by the power of the signal. Given that the points where the curvature is positive are associated with less power than the points where the curvature is negative, the overall effect is a downshift of the average frequency of the doublet.

4.4 Effect of gain bandwidth

As a pulse structure propagates in the gain fiber, its frequency is pulled toward the gain peak by the finite bandwidth of the gain medium. It can be shown that a Gaussian pulse of frequency f entering a Gaussian-like gain medium with maximum gain at f_gain will undergo a frequency shift

where κ is a constant that depends on the spectral width of the pulse and the bandwidth of the gain medium. In our case the pulses have a spectral width comparable to the gain bandwidth which fixes the value of this constant to approximately 1/2.

4.5 Cumulative effect

The three mechanisms summarized by Eqs. (10), (12) and (13) will affect the frequency of the pulse structure during one roundtrip. Thus, the total frequency shift Δf_RT during one roundtrip will be:

where Δf_RSFS,RT and Δf_RXFS,RT are the sum of the self-frequency shifts and cross-frequency shifts occurring in the erbium-doped fiber and the SMF fiber. At steady-state Δf_RT =0, thus the equilibrium frequencies f ₁ and f ₂ reached respectively by a single soliton and a soliton doublet will differ because their Raman shifts are different. Using Eqs. (13) and (14), this frequency difference is

where the subscripts “1” and “2” refer to the single soliton and the soliton doublet respectively (we also used the fact that Δf_{RXFS,RT, 1}=0). The expression between the parentheses is the modified self-frequency shift discussed before while the second term in the brackets is the cross-frequency shift that occurs in the bound state of solitons. We see that, in theory, both these shifts contribute to the frequency difference between the single soliton and the soliton doublet.

Using Eqs. (10) and (12) and the values for the soliton parameters P ₀, T ₀ and T inferred from the numerical simulations, we estimate the modified self-frequency shift to be -0.011 THz and the cross-frequency shift to be -0.028 THz. We see that both effects have approximately the same magnitude. Inserting these values in Eq. (16) yields a frequency difference of 0.034 THz between the single soliton and the doublet of solitons, in relative agreement with the numerical result of 0.022 THz. The discrepancy lies in the fact that the solitons parameters may not be estimated so easily since the gain of the erbium-doped fiber modifies considerably the temporal shape of the soliton.

As was discussed in section 3, the frequency difference between the single soliton and the soliton doublet leads to unequal group velocities because of cavity dispersion. Equations (10), (12) and (13) result in a frequency difference that depends linearly upon the Raman parameter. The consequence is that, in view of Eq. (3), the relative velocity should also depend linearly upon the Raman parameter T_R which is in agreement with the numerical results shown in Fig. 4. Also, since the soliton properties depend on the parameters of the cavity, we expect that the relative velocity will depend also on these parameters as it was observed experimentally. Note also that, since the modified self-frequency shift could be either positive or negative, we expect that the soliton doublet can move either faster or slower than the single soliton depending upon the cavity parameters

5. Collisions

In a laser cavity such as the one we simulated, collisions may be observed whenever there is a difference in group velocity between the single soliton and the soliton doublet. Such collisions may result in several outcomes as explained by Akhmediev et al. [2] and Roy et al. [9]. Taking into account the Raman effect, we were able to reproduce numerically two different scenarios.

5.1 Elastic collision

In the first scenario, depicted in Fig. 4, we start with a single soliton and a soliton doublet and we set g ₀=4.3 m^-1 and T_R =5.0 fs. Due to the Raman effect, the doublet is moving faster. Eventually this leads to a collision with the single soliton. During the collision, the bond between the solitons in the doublet is broken while a new bond is formed between the single soliton and one of the solitons in the doublet. This new doublet starts to move away from the third soliton with a relative speed similar to the speed of the incoming doublet. This type of collision has been named “elastic collision” by Akhmediev et al. [2] and a time-resolved sampling of an actual collision [15] agrees with this numerical scenario. In a real laser cavity, this process repeats every time the soliton doublet completes one more roundtrip than the single pulse; this leads to a series of periodic collisions. The relative group velocity in that case is about 1 m/s which means that for this cavity of 6 m we would have one collision every 6 s or so. This is of the same order of magnitude as the rate of collisions observed experimentally [7, 8].

5.2 Formation of a triplet

Changing the gain to g ₀=5.1 m^-1, we obtained a different scenario. As can be seen in Fig. 8, the doublet becomes bound to the single soliton after the collision and they form a bound state of three solitons, or “triplet”. Of course, this type of collision would happen only once and would not repeat periodically in a fiber laser. The initial state is thus not stable and can only be created when the laser is perturbed externally as was done in [9] to study the different scenarios.

Finally, let us mention that even though the two mechanisms leading to a relative group velocity between a single soliton and a soliton doublet are important in a stretched-pulse laser, they are negligible in the case of an anomalous dispersion cavity (soliton laser). The cross-frequency shift does not take place because solitons in a doublet are usually interacting through dispersive waves [5] and do not significantly overlap. The modified self-frequency shift is much smaller because of the lower peak power and larger durations of the solitons. Also, their properties are not modified significantly by the interaction through dispersive waves. This explains why periodic collisions are not observed in soliton lasers.

 

Fig. 8. Formation of a triplet from a soliton doublet and a single soliton.

Download Full Size | PDF

6. Conclusion

To sum up, through numerical modeling, we have shown that the Raman effect can induce a relative group velocity between a single soliton and a soliton doublet. The results from the simulations were explained in terms of two mechanisms. The first mechanism is the modified self-frequency shift. It is due to the fact that the solitons in a doublet have their properties slightly modified compared to a single soliton and thus undergo a different self-frequency shift. The second mechanism, called the cross-frequency shift, leads to an increased Raman downshift in a soliton doublet because of the beats resulting from the overlap of the chirped solitons. We discussed how this relative group velocity leads to collisions in a fiber laser cavity and how the outcome of such collisions depends on the cavity parameters.

Finally, more work needs to be done to better understand the influence of the Raman effect on the dynamics of solitons in a stretched-pulse laser. In particular, the stability of a single soliton [16] and the stability of a soliton doublet [17] as a function of the Raman parameter would require further investigation. As well, a detailed understanding of the dynamics during the collision itself would be useful in order to investigate the influence of the Raman parameter on this process.