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Origin of the bound states of pulses in the stretched-pulse fiber laser

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Abstract

A simple analytical model of the mechanism responsible for the formation of bound states of pulses in the stretched-pulse fiber laser is given. The proposed model is based on a noncoherent interaction occurring between the pulses near their position of maximum stretch within the dispersion-managed cavity, where the pulses possess a large linear chirp. This nonlinear interaction is due to the combined effects of the cross-phase modulation and the cross-amplitude modulation caused by the nonlinear gain associated with the mode-locking mechanism used in the laser. This model predicts the existence of a single bound state with a separation of the order of the pulsewidth at maximum stretch, a result consistent with simulations and experiments.

©2009 Optical Society of America

1. Introduction

It is well known that pulses in a fiber laser cavity can interact due to nonlinear effects in the optical fibers. This interaction can lead to the formation of stable bound states of pulses as discussed in several experimental and theoretical investigations reported in recent years (see [1,2] and references therein). This phenomenon is also referred to as temporal soliton molecules or molecules of light pulses [3]. In a stretched-pulse fiber laser operating in the normal-dispersion regime, the interaction leading to the bound states is due to the direct overlap of the pulses that are stretched considerably during a cavity roundtrip [4]. Simulations using the complex cubic-quintic Ginzburg-Landau equation predict the existence of bound states of pulses [5]. In other words, the interaction in such lasers seems to occur via nonlinear phase modulation, nonlinear gain and saturation of nonlinear gain. The same conclusion was reached recently by authors that considered the interaction of parabolic pulses in fiber lasers using a vectorial laser model solved numerically [6]. Of course, the introduction of dispersion-management in fiber lasers makes the situation more complex but it has been shown that stable bound states can still exist [1].

In this paper, we study the interaction of pulses in a stretched-pulse fiber laser in the normal-dispersion regime. Numerical simulations based on vector propagation equations taking into account saturated gain with finite bandwidth, dispersion (GVD), Kerr nonlinearity and the mode-locking mechanism based on nonlinear rotation of polarization (sometimes referred to as polarization additive-pulse modelocking, or PAPM) show that a single bound state can form in such a laser. The bound state occurs via a direct interaction involving the overlap of the pulses. We interpret this result in terms of a simplified analytical model in which we consider the noncoherent interaction through nonlinear phase modulation, nonlinear gain and saturation of nonlinear gain that occurs at the position where the overlap of the pulses is maximized within the cavity and where the pulses are largely chirped. This model predicts the existence of a single bound state having a separation comparable to the pulse duration at this position, in good agreement with simulations and experiments. We believe this model could also be used to explain the formation of bound states in parabolic-pulse fiber lasers [6,7].

2. Numerical model and results

2.1 Laser cavity

To study the formation of bound states of pulses in the stretched-pulse fiber laser, we use a numerical model that incorporates the most important physical effects in such a laser cavity. The laser considered is shown in Fig. 1 and is representative of an experimental laser that we used previously to study the collisions of bound states of pulses [8]. We also used the same cavity layout in a theoretical paper showing that the Raman effect can explain the collisions of bound states of pulses observed in stretched-pulse fiber lasers [9].

The all-fiber cavity is made up of two fiber segments having group-velocity dispersions (GVD) of opposite sign. Using the values of the parameters listed in table 1, the roundtrip dispersion of the cavity is estimated to be +0.005 ps2. Because this value is slightly positive, the laser will produce ultrashort pulses with a broad spectrum. Dispersion management within the cavity leads to large variations of the pulsewidth during a roundtrip [10]. The gain is provided by an erbium-doped fiber with a spectrum centered at 1550 nm. The PAPM system is a polarizer followed by a quarter wave plate and a half wave plate making angles α and θ with the axis of the polarizer respectively. This system, combined with nonlinear polarization rotation that occurs within the fibers leads to a nonlinear gain that, in turn, leads to mode locking in the laser [11]. Once the cavity is built, the parameters that can be varied experimentally are the angles of the waveplates and the gain coefficient in the erbium-doped fiber which is controlled by pump power.

 figure: Fig. 1.

Fig. 1. The laser cavity considered in our model. The PAPM system is made up of two waveplates and a polarizer.

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Tables Icon

Table 1. Parameters of the fibers used in the laser cavity considered above.

2.2 Numerical Model

To simulate numerically the behavior of this laser, we start with an arbitrary signal and let it evolve over several roundtrips in the cavity as explained by Siegman [12]. Since this laser is a dissipative system, the signal eventually converges toward a steady-state solution. Because the PAPM mechanism acts on the polarization state, the propagation equations must involve a vector electric field:

Ert=12[x̂FxyAxzteiβ0ziω0t+ŷFxyAyzteiβ0ziω0t],

where x̂ and ŷ are orthonormal vectors in the transverse direction, F(x, y) is the transverse profile of the fundamental mode of the fiber, ω 0 is the reference frequency chosen to correspond to the peak of the laser gain and β 0 is the propagation constant at this frequency. We concentrate on the evolution of the slowly-varying envelopes Ax(z,t) and Ay(z,t), or, equivalently, on the circular polarization basis components given by A ± = Ax ± iAy. The propagation equations for the components A + and A - in the fibers are expressed as [12, 13]:

A±z=gsat2A±i2β22A±t2+iγ3(A±2+2A±2)A±.

The coefficients associated with the fibers are the GVD(β 2) and the Kerr nonlinearity (γ). The saturated gain coefficient g sat is evaluated as g sat = g ns/(1 + E/Es) where E is the energy of the signal, E s is the saturation energy set at 30 pJ, and g ns is the small signal gain curve modeled as a lorentzian with peak amplitude g 0 and a FWHM of 5.0 THz. This modeling does not take into account the change in saturation energy with pump power; the consideration of this effect should not affect the conclusions of our work, since gain saturation is a cumulative effect that does not take place within a single pulse.Note that this numerical model is similar to the models presented in [4] and [6]. It is used here to put some emphasis on the properties of the bound states formed in such a laser and to have a comparative basis for the results of our analytical model presented later.

2.3 Single-pulse regime

When the gain is set to g0 = 1.4 m-1, a single pulse circulates in the laser cavity. The pulse power profile and power spectrum at the point of maximum compression of the pulse within the cavity are shown in Fig. 2. The spectrum, seen on the right, is almost gaussian. Although the oscillations in the wings of the spectrum look like Kelly sidebands, they are not. They are a typical signature of dispersion-managed solitons [14]. This is a major difference with respect to laser cavities having a global anomalous average dispersion as discussed in [15].

 figure: Fig. 2.

Fig. 2. Pulse power profile and pulse spectrum in the single-pulse regime.

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As expected for a stretched-pulse fiber laser, the pulse undergoes large variations within a roundtrip due to the large local value of the GVD. The pulse stretches by a factor of almost 10 as can be seen on the left portion of Fig. 3. When the pulse is at its maximum compression, halfway through each segment of fiber, it is almost chirp-free. However, at the junctions between the fibers, especially at the end of the erbium-doped fiber, the pulse reaches its maximum stretch and it is linearly chirped as seen on the right portion of Fig. 3. For future reference, at maximum stretch, the pulse FWHM is 1.30 ps, its peak power is 140 W and the pulse is linearly chirped with a chirp factor C = 10 (C will be defined in Eq. (7)).

 figure: Fig. 3.

Fig. 3. On the left, the pulse profile at the locations where its duration is maximum and minimum within the laser cavity. On the right, the pulse profile and instantaneous frequency at its point of maximum duration, i.e. at the output of the erbium-doped fiber.

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2.4 Double-pulse regime

It was found that two pulses can form from noise by setting the gain at g0 = 3.5 m-1. Moreover, the two pulses are always separated by 1.8 ps. In order to understand the dynamics of this regime, simulations starting from two pulses are better indicated. Consequently, we ran simulations using as the initial signal the sum of two pulses similar to the one obtained numerically in the previous section. These simulations were run using different values of initial separation and phase difference between the pulses. The results are shown in Fig. 4 where the evolution of the pulse separation and the phase difference is plotted as a function of roundtrip number. First of all, we observe that two pulses initially separated by 3.5 ps seem to keep a constant separation whatever their phase difference is. This shows that the pulses do not interact at all when their separation is too large. Two pulses initially separated by 1.5 ps repel each other until their separation becomes fixed at 1.8 ps, after a quick overshoot. Two pulses initially separated by 2.5 ps initially attract each other until their separation becomes also fixed at 1.8 ps. This points out to the existence of a stable bound state of pulses at 1.8 ps. Furthermore, the evolution of the separation seems to be almost independent of the phase difference between the pulses, except toward the final stages of the bound state formation.

The phase difference oscillates randomly with small amplitude during the initial cycle of attraction/repulsion. Then, once the final separation is attained, the phase difference starts to evolve towards its final value which is zero in all cases.

These curves are in qualitative agreement with the phase diagram trajectories presented earlier by Grelu et al. [4]. The interaction leads to the existence of a single bound state with a fixed separation of 1.8 ps and a fixed phase difference of zero. However, an important point we want to underline is that the interaction fixing the separation appears to be to a large extent independent of the phase difference as seen by comparing the curves for different initial phase differences but equal initial separations. In fact, the coherent nature of the interaction begins to show up only when the final separation is reached and then fixes the final phase difference.

The upper portion of Fig. 5 shows the evolution of the bound state temporal profile during one roundtrip. The two pulses separated by 1.8 ps overlap strongly at the positions where they are stretched within the laser cavity. However, their overlapping is negligible when they are compressed. This is seen clearly on the lower portion of Fig. 5 which shows the bound state profile at the point of maximum compression (middle of the SMF-28 fiber) and at the point of maximum stretch (end of the erbium-doped fiber) within the cavity. Again, this behavior is in agreement with the results of Grelu et al. [4]. This tends to show that the interaction between the pulses occurs mainly near the point of maximum stretch, where the pulses overlap significantly and where they possess a large linear chirp.

 figure: Fig. 4.

Fig. 4. Evolution of the separation and phase difference between two pulses for different initial conditions. The evolution from given initial conditions (separation and phase difference) is identified by the same color on top and bottom.

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3. Analytical model for the interaction of pulses

Following this numerical analysis, we are in a position to propose an analytical model that will reproduce the most important features of the dynamics discussed above. This model is based on a few simplifying assumptions that will render the analysis tractable algebraically but still representative of experimental conditions.

3.1 The interaction is a small perturbation affecting the frequency of each pulse

The first assumption is that the interaction between the pulses is just a small perturbation to be included in the dynamics of each pulse. In fact, the evolution of each pulse is controlled mostly by the local group velocity dispersion (GVD), the self-phase modulation (SPM) and the PAPM mechanism that leads to self-amplitude modulation (SAM). In the stretched-pulse laser, these effects lead collectively to the formation of gaussian-shaped-dispersion-managed pulses that are periodically compressed and stretched within the laser cavity [16]. We assume that, when two pulses are interacting, they keep their gaussian shape to the first order. Their power and phase profile are affected slightly and, consequently, the average frequency of each pulse is modified by the presence of the other pulse. Because the average dispersion of the laser cavity is not zero, this modification of the frequency results in a change in the group velocity of each pulse and, consequently, it could lead to a pulse attraction or repulsion.

 figure: Fig. 5.

Fig. 5. Intracavity dynamics of the bound state and pulse profile at the positions of maximum compression and maximum stretch within the laser cavity.

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3.2 An interaction through cross-phase modulation and nonlinear gain

As two pulses overlap each other, their superposition changes the overall power profile and, consequently, it influences the nonlinear effects to which they are subject. The two most important nonlinear effects in such a laser cavity are the phase modulation (Kerr effect) and the nonlinear gain (PAPM mechanism). In this paper, we will ignore the influence of the Raman effect that has been studied in another paper [9]. To simplify the analysis, we will consider the evolution of a scalar electric field subject to distributed nonlinear effects in the cavity. In a fiber laser, the evolution of the slowly varying envelope A(z,t) of the field can be modeled using the complex Ginzburg-Landau equation [17]. Treating the interaction as a perturbation and keeping only the relevant nonlinear terms that will be of importance during the interaction, this equation becomes:

Az=iγA2A+δA2AvA4A,

where γ, δ and v are positive constants representing respectively the nonlinear phase modulation produced by the Kerr effect, the nonlinear gain and the saturation of the nonlinear gain due to the PAPM mechanism. The last term will be important later on and its presence can be explained as follows. The PAPM mechanism in the laser cavity leads to a transmission curve that is sinusoidal as a function of power. This originates from the interferometric nature of the PAPM mechanism. It could also be understood in terms of the nonlinear rotation of the polarization ellipse during a roundtrip in the cavity. From either point of view, the peak of the transmission curve will fix the maximum peak power attained by the pulses formed in the laser and so, the signal will always remain within the first half period of the sinusoidal transmission curve. This first half period of the sinusoidal curve can then be represented by a parabola as a function of power which leads to the last two terms in Eq. (3). At low power, the nonlinear gain increases linearly with power leading to mode locking. As the power increases, the quadratic term in power becomes important and the nonlinear gain increases more slowly and eventually starts to decrease. This mechanism is at the origin of the saturation of the nonlinear gain.

As explained in the discussion of the numerical results, the interaction that fixes the separation appears to be independent of the phase difference between the pulses. The second assumption made in our analytical model is that the interaction between the pulses is noncoherent. It can be justified as follows. In the stretched-pulse fiber laser, the pulses are stretched and compressed periodically within the cavity. As explained previously, this can lead to significant overlap of the pulses in the region where their duration is important. In this region the pulses possess an important linear chirp. When they overlap, the resultant power profile is modified. Because of the linear chirp, the instantaneous frequencies of the pulses differ by a constant value in the overlapping interval and this leads to the presence of beats [9]. This can clearly be seen in Fig. 5 by examining the power profile between the two pulses. It can be shown that the period of these beats depends on the chirp parameter and the separation between the pulses. The absolute position of the beats with respect to the pulse peaks depends on the phase difference between the pulses. Hence, a priori, we would expect that the nonlinear effects generated by the modification of the power profile in the overlapping interval would depend on the phase difference between the pulses. However, as the pulses travel within the laser, their chirp parameter evolves really quickly due to the local GVD. Consequently, the period of the beats changes rapidly as a function of the position in the laser cavity. We can thus assume that, on average, the beat structure is washed out in the interacting zone of the cavity near the junction between the erbium-doped fiber and the SMF-28 fiber. By doing so, we consider that the interaction is noncoherent since the only effect of the phase difference on the power profile is to modify the absolute position of the beats.

Considering two interacting pulses, i.e. writing the resultant signal as the sum of two individual pulses A(z,t) = A 1(z,t) + A 1(z,t), and keeping only the interaction terms between the two pulses that do not depend on the phase difference between the pulses, Eq. (3) reduces to the following two propagation equations (one for each pulse, where kj):

δAjz=iγAk2AjXPMinteraction+δAk2Ajv(Ak4+2Aj2Ak2)AjNonlineargaininterction.

The first term on the right-hand side of Eq. (4) represents the nonlinear cross phase modulation between the pulses (XPM) and the last two terms represent an interaction due to the nonlinear gain within the cavity. More precisely, since the second term is proportional to the power of the neighboring pulse we refer to it as cross amplitude modulation (XAM). The last term is due to the saturation of the nonlinear gain and so is referred to as saturation of the cross amplitude modulation (SXAM).

The effect of XPM is well understood as it has been described in the context of the interaction of classical solitons found in the nonlinear Schrödinger equation [18]. The effect of XPM on the evolution of the first pulse can be summarized as follows. The presence of the second pulse modifies the power profile in the tail of the first pulse. In fact, considering a noncoherent superposition, it adds a component for which the derivative of the power versus time is positive. Since the XPM is proportional to the negative of the time derivative of the power, it will tend to reduce the instantaneous frequency content of the first pulse in that overlapping interval thus leading to a reduction of the its average frequency. This effect is represented schematically in Fig. 6. In the stretched-pulse laser cavity, the average GVD is normal, and this reduction of the average frequency leads to an increase of the group velocity. For the second pulse, the effect is exactly the opposite. The time derivative of the first pulse’s tail in the overlapping interval is negative. The XPM on the second pulse will be negative, its average frequency will become larger and its group velocity will decrease. We can thus conclude that the XPM leads to repulsion between the pulses.

 figure: Fig. 6.

Fig. 6. Mechanism of the non-coherent XPM interaction.

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The effect of the nonlinear gain interaction (XAM and SXAM) is explained as follows. As the pulses are brought closer to each other, XAM leads to a nonlinear gain that is more important in their overlapping wings as pictured in Fig. 7. The consequence is that each pulse undergoes an asymmetric temporal gain. Moreover, at the position of overlap, the pulses are linearly chirped. As a consequence of this chirp, the asymmetric temporal gain becomes associated with an asymmetric spectral amplification of the pulse that will affect the average frequency of each pulse. For instance, consider two positively chirped pulses that overlap. XAM will lead to a greater amplification of the later portion of the first pulse compared to its earlier portion. This effect, combined with the positive chirp of the pulse, yields a positive spectral content that is more amplified than the negative spectral content of the pulse resulting in an increase of its average frequency. This first pulse will thus slow down due to the normal average GVD of the laser cavity. The effect for the second pulse will be the opposite. Thus, the XAM interaction will result in an attraction between the pulses. The effect of SXAM is similar to the effect of XAM but is repulsive due to the opposite sign of the SXAM. Also, since it depends upon the square of the power, its effect is negligible when the pulses are largely separated but it becomes more important as they are brought closer and closer.

 figure: Fig. 7.

Fig. 7. Mechanism of the interaction due to the nonlinear gain. The asymmetric nonlinear gain resulting from the presence of a second pulse combined with the linear chirp of the pulse during the interaction leads to a modification of its average frequency and thus its speed.

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The XPM and XAM terms vary essentially in the same way with respect to the separation of the pulses. One of them will thus overcome the other and their combined effect will be attractive or repulsive for any separation. Obviously, this could not lead to the formation of a bound state. The SXAM term is thus extremely important. Let us assume the particular situation where the attractive XAM term is more important than the repulsive XPM term at large separation. As the pulses attract each other, the SXAM term becomes more and more important. Eventually, its repulsive effect becomes larger that the combined attractive effect of the other two contributions. Thus, there should be a separation for which its repulsive effect exactly compensates for the attractive effect resulting from the combination of XAM and XPM, hence leading to the formation of a bound state with that particular separation.

At this point, it is important to note that, as the pulses with different frequencies propagate in the laser, the interaction will alternate between repulsion and attraction. This is due to the fact that the GVD has opposite signs in the SMF and in the erbium-doped fiber. In this simplified analytical model, we consider only the effect of the average round-trip dispersion. In other words, we assume that the round-trip average interaction will follow the sign of the path-average cavity dispersion. This works well if the frequency difference between the pulses does not change too much during a roundtrip after it has been fixed at the end of the erbium-doped fiber since then the attractive and repulsive effect will counteract each other leaving an overall effect that is proportional to the average dispersion. It would be difficult to prove that it is actually what happens in the cavity considered here. However, in view of the results discussed later, this assumption is a good approximation of the actual behavior of the laser.

To support the statement that a bound state should exist in this model, we will analyze the effect of the interaction between two pulses quantitatively. First of all, the average angular frequency of each pulse can be computed using:

ω̄j=12Ej(Aj*AjtAjAj*t)dt,

where the index j = 1 or 2 refers to the first and second pulse and Ej is the energy of each pulse. Using Eq. (5) combined with Eq. (4) representing the propagation of each pulse, the difference between the average frequencies of the two pulses Δω̄ ≡ ω̄1 - ω̄2 will evolve as:

d(Δω̄)dz=i2Ejdt{4iγA12t(A22)
+(2δA224vA12A22)(A1*A1tA1A1*t)
(2δA124vA12A22)(A2*A2tA2A2*t)}.

As discussed in the analysis of the numerical results, the interaction between the pulses occurs mainly in the regions where they overlap. In principle, there are two regions in the laser cavity where the pulse overlap and thus their interaction could be important: at the SMF-28 - erbium-doped fiber junction and also at the erbium-doped - SMF-28 junction because this is where the pulses are maximally stretched. However, since the pulses are amplified within the erbium-doped fiber and the output coupling occurs just after propagation in this fiber, their power and the nonlinear effects they undergo will be more important at that position. This is the third important assumption made in our analytical model: the two pulses interact only near the output of the erbium-doped fiber. Again, considering the interaction to be a perturbation as in Haus et al. [16], we can assume that the pulses are described by chirped gaussians near the output of the erbium-doped fiber:

A1,2(t)=P0exp[(1+iC)2(t±T/2)2T02],

where P 0 is the peak power of the pulses, T 0 is their halfwidth, C is their normalized chirp parameter and T is their separation. Using this ansatz in Eq. (6), the evolution of their frequency difference becomes:

d(Δω̄)dz=P0TT02exp[T22T02]{2(δCγ)893vCP0exp[T26T02]}.

3.3 A single bound state

For the simulations presented in section 2, the angles of the waveplates were α = π/8 rad and θ = -π/16 rad. It is possible to estimate the values of the different parameters in the preceding equation. At the output of the erbium-doped fiber, the pulse parameters are P 0 = 140 W, T 0 = 0.76 ps and C = 10. The parameters of the laser system are γ = 5.5×10-3 m-1W-1, δ = 5.8×10-4 m-1W-1 and v = 5.8×10-7 m-1W-2. For these values, the graph of the rate of change of the frequency difference Δf̄ = Δω̄/2π as a function of their separation is shown in Fig. 8.

 figure: Fig. 8.

Fig. 8. Rate of change of the frequency difference between the two pulses as a function of their separation according to the analytical model. Due to the normal average GVD of the laser cavity, it leads to a repulsion for small separation and an attraction for large separation with a stable equilibrium point near 1.9 ps which corresponds to a single bound state.

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For a large separation (> 3 ps), the interaction is negligible. For a separation between 1.9 ps and 3 ps, the rate of change of the frequency difference is positive and this eventually leads to an attraction due to the average normal dispersion in the laser cavity. For a separation smaller than 1.9 ps, the rate of change of the frequency difference is negative, leading eventually to repulsion. Note that the approximations we used in our analytical model do not hold for separations that are too small. Hence, the portion of the curve for values smaller than 1 ps should be taken with caution. It should be noted that as soon as the pulses do not have the same frequency, the ansatz we have used does not seem to apply anymore. The precise dynamics of the interaction as the frequency difference evolves could be affected. However, because of the limited bandwidth of the gain, the frequency difference will probably be compensated when the pulses propagates in the erbium-doped fiber. In other words, at the erbium-doped fiber - SMF-28 fiber junction the pulses undergo a frequency shift that affects their velocities during the roundtrip until they reach the erbium-doped fiber again where their frequency shift is reduced close to zero. Note also that the conclusions concerning the separation at equilibrium should be correct since in that case the frequency difference will remain at zero if it starts at zero. Consequently, we conclude that there should be a single bound state at a separation of 1.9 ps in such a system. More generally, fixing the rate of change of the frequency difference to zero in Eq. (8) leads to the following pulse separation at equilibrium:

Tboundstate=6ln[9283(δCγ)vCP0]T0.

These conclusions concerning the bound state drawn from the simple analytical model presented here are in relative agreement with the results obtained numerically.

Of course, since this analytical model neglects the coherent nature of the interaction, it cannot explain why the bound state should possess a phase difference of zero as found numerically. We believe that if all the coherent terms were kept in the analysis, then it could be shown that the equilibrium phase difference for the bound state would be zero. However such an analysis would be vastly more complicated than the one presented here.

It is important to note that the conclusions obtained using the analytical model shown here are not only in agreement with results from simulations but also in agreement with the experimental behavior observed in stretched-pulse fiber lasers. For example, in our experiments presented in a previous paper [8], we found a single bound state with a separation of a few picoseconds, a behavior very close to the predictions of this analysis. Also, we noticed that the separation in this experimental bound state could be varied by changing either the pump power or the parameters of the PAPM mechanism in the laser. Again, this is qualitatively in agreement with Eq. (9). In fact, when the gain is varied, the pulse parameters such as peak power, width and chirp are modified, leading to a different pulse separation at equilibrium. Similarly, changing the configuration of the PAPM mechanism modifies the values of the nonlinear gain parameters δ and v which, through Eq. (9), should affect the separation of the bound state. Also, it was noted experimentally that for some parameters the laser cavity does not form the close bound state discussed here but falls into the harmonic mode-locking regime. Again, this could be explained by Eq. (9) since for some parameters the argument of the square root becomes negative, indicating that the interaction does not lead to any physical solution at equilibrium. At the experimental level, other long range mechanisms such as gain depletion can then become important and lead to harmonic mode locking [19].

Finally, this model can probably be used to explain the existence of bound states in parabolic-pulse fiber lasers [6,7]. In such lasers, the pulses undergo linear propagation in the anomalous dispersion portion of the cavity, and then they undergo nonlinear propagation in the normal dispersion fiber. Thus, their nonlinear interaction will occur in this segment where they are largely stretched and also positively chirped. These conditions correspond exactly to the ones specified in the analytical model presented in this paper.

4. Conclusion

In this paper, we have presented a simple analytical model to explain the formation of bound states of pulses in the stretched-pulse fiber laser operated in the normal dispersion regime. Following numerical simulations showing that a single bound state exists in such a laser for a given set of parameters, the following observations were made. Firstly, the interaction is due to a direct overlap of the pulses occurring mainly at the output of the gain fiber where the pulses possess large values of linear chirp, pulsewidth and energy. Secondly, the interaction that fixes the pulse separation within the bound state appears to be independent of the phase difference between the pulses. Thirdly, the interaction seems to be weak compared to other effects in the laser cavity. Based on these observations, we have developed a model in which a nonlinear and noncoherent interaction occurs between the chirped pulses at the output of the normal dispersion gain fiber. More specifically, the model considers the average evolution of chirped gaussian pulses as they are affected by the perturbative influence of cross-phase modulation, cross-amplitude modulation and saturation of the cross-amplitude modulation modeled through the appropriate terms in the scalar Ginzburg-Landau equation. It was shown that this model leads to the existence of a single bound state with a separation in relative agreement with the results obtained through direct numerical integration of the full vector propagation equations. It was also shown that the pulse separation in the bound state depends on the characteristics of the pulses such as their chirp, peak power and duration and also on the parameters of the laser system defining the contributions of the Kerr effect, nonlinear gain and saturation of the nonlinear gain. These conclusions agree, at least qualitatively, with the observation that the pulse separation is influenced by the pump power and the adjustment of the PAPM mechanism in experimental systems.

The interacting mechanism can be interpreted as follows. Firstly, there is cross-phase modulation (XPM) between the pulses similar to the interaction between classical solitons which leads to a repulsion. Secondly, the nonlinear gain in the laser (PAPM mechanism) leads to a non-symmetric amplification of the pulses as they overlap each other. The non-symmetric gain, combined with the fact that the pulses are linearly chirped during their interaction, leads to an asymmetric spectral gain which affects the average frequency of each pulse. At large separation, only XAM is important and the interaction is attractive. However, at short separation, the SXAM, which leads to a repulsion, becomes more and more important. If the combined effect of XPM and XAM is attractive, the repulsive effect of SXAM can lead to the formation of a single bound state with stable separation. It should be noted that the fact that the pulses are linearly chirped during the interaction is of prime importance since it allows nonlinear gain to affect the spectral content of the pulses as they overlap. We believe this heuristic interpretation of the interaction between the pulses brings some new light on this phenomenon although the model should be extended further to incorporate the coherent part of the interaction that would explain why the phase difference becomes fixed in the bound state.

Acknowledgments

The authors are grateful to Mr. Vincent Roy for many fruitful discussions concerning pulse interactions in stretched-pulse fiber lasers. This work was supported by Exfo Electro-Optical Engineering Inc., the Natural Sciences and Engineering Research Council of Canada, the Fonds Québécois de la Recherche sur la Nature et les Technologies, the Canadian Institute for Photonic Innovations and the Femtotech Consortium.

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8. M. Olivier, V. Roy, M. Piché, and F. Babin, “Pulse collisions in the stretched-pulse fiber laser,” Opt. Lett. 29, 1461–1463 (2004). [CrossRef]   [PubMed]  

9. M. Olivier, V. Roy, and M. Piché, “Influence of the Raman effect on bound states of dissipative solitons,” Opt. Express 14, 9728–9742 (2006). [CrossRef]   [PubMed]  

10. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-femtosecond pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993). [CrossRef]   [PubMed]  

11. M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992). [CrossRef]  

12. A. E. Siegman, Lasers (University Science Books, 1986), Chap. 27.

13. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995), Chap. 7.

14. P. M. Lushnikov, “Oscillating tails of a dispersion-managed soliton,” J. Opt. Soc. Am. B 21, 1913–1918 (2004). [CrossRef]  

15. M. Olivier, V. Roy, and M. Piché, “Third-order dispersion and bound states of pulses in a fiber laser,” Opt. Lett. 31, 580–582 (2006). [CrossRef]   [PubMed]  

16. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995). [CrossRef]  

17. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 1–4 (2005). [CrossRef]  

18. F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 355–357 (1987). [CrossRef]   [PubMed]  

19. J. N. Kutz, B. C. Collings, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998). [CrossRef]  

References

  • View by:

  1. N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, “Dissipative soliton interactions inside a fiber laser cavity,” Opt. Fiber Technol. 11, 209–228 (2005).
    [Crossref]
  2. D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 1–10 (2005).
    [Crossref]
  3. M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett.,  95, 143902 (2005).
    [Crossref] [PubMed]
  4. Ph. Grelu, J. Béal, and J. M. Soto-Crespo, “Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime,” Opt. Express 11, 2238–2243 (2003).
    [Crossref] [PubMed]
  5. Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case 2+1,” Opt. Express 12, 3184–3189 (2004).
    [Crossref] [PubMed]
  6. G. Martel, C. Chédot, V. Réglier, A. Hideur, B. Ortaç, and Ph. Grelu, , “On the possibility of observing bound soliton pairs in a wave-breaking-free mode-locked fiber laser,” Opt. Lett. 32, 343–345 (2007).
    [Crossref] [PubMed]
  7. B. Ortaç, A. Hideur, M. Brunel, C. Chédot, J. Limpert, A. Tünnermann, and F. Ilday, “Generation of parabolic bound pulses from a Yb-fiber laser,” Opt. Express 14, 6075–6083 (2006).
    [Crossref] [PubMed]
  8. M. Olivier, V. Roy, M. Piché, and F. Babin, “Pulse collisions in the stretched-pulse fiber laser,” Opt. Lett. 29, 1461–1463 (2004).
    [Crossref] [PubMed]
  9. M. Olivier, V. Roy, and M. Piché, “Influence of the Raman effect on bound states of dissipative solitons,” Opt. Express 14, 9728–9742 (2006).
    [Crossref] [PubMed]
  10. K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-femtosecond pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
    [Crossref] [PubMed]
  11. M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
    [Crossref]
  12. A. E. Siegman, Lasers (University Science Books, 1986), Chap. 27.
  13. G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995), Chap. 7.
  14. P. M. Lushnikov, “Oscillating tails of a dispersion-managed soliton,” J. Opt. Soc. Am. B 21, 1913–1918 (2004).
    [Crossref]
  15. M. Olivier, V. Roy, and M. Piché, “Third-order dispersion and bound states of pulses in a fiber laser,” Opt. Lett. 31, 580–582 (2006).
    [Crossref] [PubMed]
  16. H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995).
    [Crossref]
  17. A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 1–4 (2005).
    [Crossref]
  18. F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 355–357 (1987).
    [Crossref] [PubMed]
  19. J. N. Kutz, B. C. Collings, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
    [Crossref]

2007 (1)

2006 (3)

2005 (4)

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 1–4 (2005).
[Crossref]

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, “Dissipative soliton interactions inside a fiber laser cavity,” Opt. Fiber Technol. 11, 209–228 (2005).
[Crossref]

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 1–10 (2005).
[Crossref]

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett.,  95, 143902 (2005).
[Crossref] [PubMed]

2004 (3)

2003 (1)

1998 (1)

J. N. Kutz, B. C. Collings, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[Crossref]

1995 (1)

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995).
[Crossref]

1993 (1)

1992 (1)

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

1987 (1)

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995), Chap. 7.

Akhmediev, N.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, “Dissipative soliton interactions inside a fiber laser cavity,” Opt. Fiber Technol. 11, 209–228 (2005).
[Crossref]

Ph. Grelu and N. Akhmediev, “Group interactions of dissipative solitons in a laser cavity: the case 2+1,” Opt. Express 12, 3184–3189 (2004).
[Crossref] [PubMed]

Babin, F.

Béal, J.

Brunel, M.

Chédot, C.

Collings, B. C.

J. N. Kutz, B. C. Collings, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[Crossref]

Fermann, M. E.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Grapinet, M.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, “Dissipative soliton interactions inside a fiber laser cavity,” Opt. Fiber Technol. 11, 209–228 (2005).
[Crossref]

Grelu, Ph.

Haberl, F.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Haus, H. A.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995).
[Crossref]

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-femtosecond pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
[Crossref] [PubMed]

Hideur, A.

Hofer, M.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Ilday, F.

Ippen, E. P.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995).
[Crossref]

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-femtosecond pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
[Crossref] [PubMed]

Knox, W. H.

J. N. Kutz, B. C. Collings, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[Crossref]

Komarov, A.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 1–4 (2005).
[Crossref]

Kutz, J. N.

J. N. Kutz, B. C. Collings, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[Crossref]

Leblond, H.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 1–4 (2005).
[Crossref]

Limpert, J.

Lushnikov, P. M.

Martel, G.

Mitschke, F.

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett.,  95, 143902 (2005).
[Crossref] [PubMed]

Mitschke, F. M.

Mollenauer, L. F.

Nelson, L. E.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995).
[Crossref]

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-femtosecond pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
[Crossref] [PubMed]

Ober, M. H.

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

Olivier, M.

Ortaç, B.

Pagel, T.

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett.,  95, 143902 (2005).
[Crossref] [PubMed]

Piché, M.

Réglier, V.

Roy, V.

Sanchez, F.

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 1–4 (2005).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 27.

Soto-Crespo, J. M.

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, “Dissipative soliton interactions inside a fiber laser cavity,” Opt. Fiber Technol. 11, 209–228 (2005).
[Crossref]

Ph. Grelu, J. Béal, and J. M. Soto-Crespo, “Soliton pairs in a fiber laser: from anomalous to normal average dispersion regime,” Opt. Express 11, 2238–2243 (2003).
[Crossref] [PubMed]

Stratmann, M.

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett.,  95, 143902 (2005).
[Crossref] [PubMed]

Tam, H. Y.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 1–10 (2005).
[Crossref]

Tamura, K.

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995).
[Crossref]

K. Tamura, E. P. Ippen, H. A. Haus, and L. E. Nelson, “77-femtosecond pulse generation from a stretched-pulse mode-locked all-fiber ring laser,” Opt. Lett. 18, 1080–1082 (1993).
[Crossref] [PubMed]

Tang, D. Y.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 1–10 (2005).
[Crossref]

Tünnermann, A.

Zhao, B.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 1–10 (2005).
[Crossref]

Zhao, L. M.

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 1–10 (2005).
[Crossref]

IEEE J. Quantum Electron. (2)

M. Hofer, M. E. Fermann, F. Haberl, and M. H. Ober, “Characterization of ultrashort pulse formation in passively mode-locked fiber lasers,” IEEE J. Quantum Electron. 28, 720–728 (1992).
[Crossref]

J. N. Kutz, B. C. Collings, and W. H. Knox, “Stabilized pulse spacing in soliton lasers due to gain depletion and recovery,” IEEE J. Quantum Electron. 34, 1749–1757 (1998).
[Crossref]

IEEE J. Quantum. Electron. (1)

H. A. Haus, K. Tamura, L. E. Nelson, and E. P. Ippen, “Stretched-pulse additive pulse mode-locking in fiber lasers: theory and experiment,” IEEE J. Quantum. Electron. 31, 591–598 (1995).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Express (4)

Opt. Fiber Technol. (1)

N. Akhmediev, J. M. Soto-Crespo, M. Grapinet, and Ph. Grelu, “Dissipative soliton interactions inside a fiber laser cavity,” Opt. Fiber Technol. 11, 209–228 (2005).
[Crossref]

Opt. Lett. (5)

Phys. Rev. E (2)

A. Komarov, H. Leblond, and F. Sanchez, “Quintic complex Ginzburg-Landau model for ring fiber lasers,” Phys. Rev. E 72, 1–4 (2005).
[Crossref]

D. Y. Tang, B. Zhao, L. M. Zhao, and H. Y. Tam, “Soliton interaction in a fiber ring laser,” Phys. Rev. E 72, 1–10 (2005).
[Crossref]

Phys. Rev. Lett. (1)

M. Stratmann, T. Pagel, and F. Mitschke, “Experimental Observation of Temporal Soliton Molecules,” Phys. Rev. Lett.,  95, 143902 (2005).
[Crossref] [PubMed]

Other (2)

A. E. Siegman, Lasers (University Science Books, 1986), Chap. 27.

G. P. Agrawal, Nonlinear Fiber Optics (Academic Press, 1995), Chap. 7.

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Figures (8)

Fig. 1.
Fig. 1. The laser cavity considered in our model. The PAPM system is made up of two waveplates and a polarizer.
Fig. 2.
Fig. 2. Pulse power profile and pulse spectrum in the single-pulse regime.
Fig. 3.
Fig. 3. On the left, the pulse profile at the locations where its duration is maximum and minimum within the laser cavity. On the right, the pulse profile and instantaneous frequency at its point of maximum duration, i.e. at the output of the erbium-doped fiber.
Fig. 4.
Fig. 4. Evolution of the separation and phase difference between two pulses for different initial conditions. The evolution from given initial conditions (separation and phase difference) is identified by the same color on top and bottom.
Fig. 5.
Fig. 5. Intracavity dynamics of the bound state and pulse profile at the positions of maximum compression and maximum stretch within the laser cavity.
Fig. 6.
Fig. 6. Mechanism of the non-coherent XPM interaction.
Fig. 7.
Fig. 7. Mechanism of the interaction due to the nonlinear gain. The asymmetric nonlinear gain resulting from the presence of a second pulse combined with the linear chirp of the pulse during the interaction leads to a modification of its average frequency and thus its speed.
Fig. 8.
Fig. 8. Rate of change of the frequency difference between the two pulses as a function of their separation according to the analytical model. Due to the normal average GVD of the laser cavity, it leads to a repulsion for small separation and an attraction for large separation with a stable equilibrium point near 1.9 ps which corresponds to a single bound state.

Tables (1)

Tables Icon

Table 1. Parameters of the fibers used in the laser cavity considered above.

Equations (11)

Equations on this page are rendered with MathJax. Learn more.

Ert=12 [x̂FxyAxzteiβ0ziω0t+ŷFxyAyzteiβ0ziω0t] ,
A±z=gsat2A±i2β22A±t2+iγ3(A±2+2A±2)A±.
Az=iγA2A+δA2AvA4A ,
δAjz=iγAk2AjXPMinteraction+δAk2Ajv(Ak4+2Aj2Ak2)AjNonlineargaininterction .
ω̄j=12Ej (Aj*AjtAjAj*t) d t ,
d(Δω̄)dz=i2Ej dt { 4 i γ A12 t (A22)
+ (2δA224vA12A22)(A1*A1tA1A1*t)
(2δA124vA12A22) (A2*A2tA2A2*t) } .
A1,2(t)=P0 exp [(1+iC)2(t±T/2)2T02] ,
d(Δω̄)dz=P0TT02 exp [T22T02] {2(δCγ)893vCP0exp[T26T02]} .
Tboundstate=6ln[9283(δCγ)vCP0] T0 .

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