1. Introduction

Soliton molecules in fibers and fiber lasers have attracted research interests for a long time because of their potential applications in lasers and communications. Two solitons bounded through Kerr-mediate interaction [1–4] or the dispersive waves (DWs) [5–8] are regarded as a soliton molecule or the so-called bound state (BS) pulse. The interaction between two adjacent solitons mediated by the Kerr nonlinearity in the soliton molecules described by the nonlinear Schrodinger equations (NLSEs) depends on their phase difference and temporal separation [1, 2]. In-phase results in attraction between two solitons while out-of-phase results in repulsion. The perturbed NLSE can have a stable two-bounded-solitons solution with in- or out-of-phase difference [3, 4], which depends on the minimal interaction potential between the two solitons with oscillating tails overlapping. Different from a conservative system such as a fiber with neglected gain and loss, a dissipative system such as a fiber laser, a fiber amplifier or an optical transmission link is described by the complex Ginzburg-Landau equations (CGLEs) with large gain and loss [9]. The stability of the soliton molecules in fiber lasers depends on the balance between dispersion and nonlinearity as well as the balance between gain and loss [10–12]. It has been predicted theoretically [12] and experimentally [13–15] that a stable soliton molecule with phase difference of $\pm$π /2 can be realized in mode-locked fiber lasers depending on the energy and momentum balances. Due to the complex physical effects in fiber lasers, soliton molecules with vibrating and shaking evolutions [16, 17], flipping phase evolution [18, 19], chaotic evolution [20], independent phase evolution [18, 21], and sliding phase evolution [18, 20, 22] have been realized in simulations and experiments. Solutions of the soliton molecules are nonlinear attractors of the dynamical dissipative systems [16].

Usually, researchers regard the soliton molecules as scalar pulses even when the coupled CGLEs are used to simulate the characteristics because polarizers are used to realize the nonlinear polarization rotation (NPR) mode-locking and thus fix the polarization accordingly [17, 23, 24]. It is interesting to know the polarization properties of the soliton molecules in fiber lasers mode-locked by polarization-insensitive saturable absorbers (SAs) such as carbon-nanotubes (CNTs) [25–27] and graphene [28–30] because the polarization is an additional freedom of the soliton molecule, which has potential in pulse shaping of fiber systems as well as the capacity increasing of optical communication systems based on the polarization division multiplexing (PDM) [31, 32]. Different from the scalar soliton molecule in a fiber laser [9–24], the mechanism of a vector soliton molecule is more complex because the coupling between the orthogonal components have strong effects on its molecular properties, which might result in various physical pictures of the vector soliton molecule. Recently, several experiments were conducted to investigate the polarization dynamics of the soliton molecules in the CNT mode-locked fiber lasers through polarimeters [25, 26]. Locked and precessing polarization evolutions of the vector soliton molecule are revealed in [25, 26]. Polarization splitting method together with an optical spectrum analyzer (OSA) was used to investigate the vector soliton molecule evolution in [28–30]. Vector soliton molecules with group velocity-locked [28], polarization-locked [29] and polarization-rotation-locked [29] states have been realized. Despite various and interesting phenomena in those experiments [25–30], some important information of the vector soliton molecule is still lacked due to the limited resolution of the measurement instruments. For example, the polarimeter used in [25] has a time resolution of 1us, which is about 40 round trips of the pulse circulation. The polarimeter with 1ns time resolution in [26] can only measure the averaged polarization across the time slot of the vector soliton molecule. A vector soliton molecule is formed by the leading and trailing vector solitons. It is interesting for us to know whether the polarizations of the leading and trailing vector solitons are the same or not as well as their evolutions. In [28–30], the real-time round-to-round dynamics of two orthogonal components were recorded by an oscilloscope while the spectra and autocorrelation traces of two orthogonal components were averagely measured by an OSA and an autocorrelator. The information of the phase difference and peak separation between the leading and trailing pulses measured by the OSA and autocorrelator is not direct when the soliton molecule is not stationary [20]. It is curious for us to know whether the internal interaction traces of the slow and fast modes of the vector soliton molecule are the same or not and how they evolve on the interaction plane. In a word, two important questions remain to be answered for the vector soliton molecule in a fiber laserone is the polarization dynamics of the leading and trailing vector solitons, the other is the internal interaction dynamics of the slow and fast modes of the vector soliton molecule. For convenience, we call the slow and fast components of a vector soliton molecule “slow soliton molecule” and “fast soliton molecule”, respectively. We call the leading and trailing vector solitons in the vector soliton molecule “vector L”and “vector T”, respectively.

In this paper, we investigate the polarization dynamics and internal interaction dynamics of the vector soliton molecules in an anomalous-dispersion mode-locked fiber laser. The slow and fast soliton molecules of the vector soliton molecule usually have different internal interaction dynamics. Soliton molecules with vibrating, sliding, chaotically vibrating and quasi-locked states can be obtained at orthogonal polarized components of the vector soliton molecule under different initial conditions and beat lengths. Vector L and vector T can have different polarization attractors. Rotating, oscillating, chaotically oscillating, quasi-locked and shaking polarization trajectories of vector L and vector T can be obtained under certain beat lengths and initial conditions. Intra-cavity dynamics of the vector soliton molecule reveal that the complex balance between gain and loss, the strong coupling between the slow and fast modes and the interaction between the leading and trailing vector solitons make the vector soliton molecule a robust solution for the coupled CGLEs. In Section 2, we describe our simulation model and parameters. The shot-to-shot evolutions of the slow and fast soliton molecules are revealed in Section 3.1. The shot-to-shot polarization dynamics of vector L and vector T are revealed in Section 3.2. The intra-cavity dynamics of the vector soliton molecules are presented in Section 3.3. Finally, we summarize the formation mechanism of the vector soliton molecule in the mode-locked fiber laser in Section 4.

2. Simulation model

The simulation model we use is schematically shown in Fig. 1. The main parts of the cavity include 1m Er-doped fiber (EDF), 4m single-mode fiber (SMF), a lumped polarization-insensitive saturable absorber (SA) and a 10:90 output coupler (OC). Our simulation runs from the EDF to the SMF in each round. We use the coupled CGLEs to describe the vector pulse propagating in the laser cavity:

 

Fig. 1 Schematic of the fiber laser in our simulation.

Download Full Size | PDF

Where, u and v are the envelopes of the pulses polarized along the slow and fast axes, respectively. $\beta =\pi \Delta n/\lambda$is half of the wave-number difference between the slow and fast modes and $\Delta n$is the difference of the effective refractive index between the slow and fast modes. $2\delta =2\lambda \beta /2\pi c$is the inverse group-velocity difference between the two modes. ${\beta }_2$is the second order dispersion which is −29ps^2/km and −21ps^2/km in the EDF and SMF, respectively. $\gamma$is nonlinear coefficient of the fiber which is 2W⁻¹km⁻¹ and 1.3W⁻¹km⁻¹ in the EDF and SMF, respectively. $\Omega$represents the gain bandwidth of the EDF and the full width at half maximum (FWHM) of the gain bandwidth is 20nm in our simulation. g is the gain of the EDF which is represented by:

Where Es is the saturable energy of the EDF, which also represents the pump strength.The transmission function of the amplitude of the lumped SA is represented by:

Where P is the instantaneous power of the pulse, which has a unit of Watt. The modulation depth and saturation power of the SA are 0.4 and 500W, respectively. In the simulation, we choose different linear beat length while other parameters are fixed. We use the symmetric split-step Fourier method to implement our simulations. We run the programs for 100000 rounds for each case. The results presented in this paper are the stable results of the final continuous hundreds or thousands rounds. The initial conditions are weak vector soliton molecules with equal slow and fast components. Different peak separations and phase differences are chosen to obtain different stable vector soliton molecules. In the simulations, the linear birefringence and the initial conditions are varied while other parameters are kept fixed.

The round-to-round evolution of the vector soliton molecule at the end of the laser cavity is described by the internal interaction trace on the interaction plane [12]. Each point of the interaction trace represents the temporal separation and phase difference between the leading and trailing pulses at their peaks. The round-to-round polarization dynamics of vector L and vector T are described by the normalized polarization trajectories on the polarization Poincare sphere. The normalized Sotkes parameters of each point of the trajectories on the Poincare sphere are:

Where u and v are the peak amplitude of the slow and fast components of vector L or vector T, respectively. $\phi$is the phase difference between the slow and fast components of vector L or vector T. $s_i$is the normalized Stokes parameter.

3. Results

3.1 internal interaction traces of the slow and fast soliton molecules

We set the pump strength and linear beat length to be 250pJ and 2m, respectively. Stable vector soliton molecules can be obtained by choosing proper initial conditions. We do not aim to find the parameter space for different stable solutions, but focus on revealing the typical characteristics of the vector soliton molecules. The initial conditions in our simulations are weak vector soliton molecules considering the weak seeding signal in real laser systems. The final states depend not only on the initial conditions but also on the initial complex evolution in the first hundreds of round trips, especially when the initial separation is less than 1ps, in which case the vector soliton molecule experiences dramatic evolution such as merging and splitting. Figure 2 shows three kinds of stable solutions. Figures 2(a)-2(c) show the stable soliton molecule evolution when the pulse separation is ~800fs, in which the initial separation and leading-to-trailing phase difference between the leading and trailing pulses are 2ps and π, respectively. We can obtain from Fig. 2(a) that the slow soliton molecule is a typical vibrating soliton molecule [17, 18] according to its weakly oscillating separation and strongly oscillating phase difference. The oscillation of the separation and phase difference are synchronous, which is due to the slight amplitude oscillation of the leading and trailing pulses of the slow soliton molecule [18]. The fast soliton molecule in Fig. 2(b) has sliding phase difference and weakly oscillating separation during the evolution. Such a sliding phase difference has been observed in [18, 20, 22]. One should note that the decreasing of the phase difference of the fast soliton molecule is due to the strong interaction between the slow and fast modes through XPM as well as the intensity difference between the leading and trailing pulses in the fast soliton molecule, which will be discussed in section 3.3. One can see from Fig. 2(c) that the slow soliton molecule oscillates while the fast soliton molecule rotates on the interaction plane. Figures 2(d)-2(f) show the soliton molecule evolution with the initial temporal separation of 8ps and the initial leading-to-trailing phase difference of π. We can obtain from Figs. 2(d) and 2(e) that the slow and fast soliton molecules are vibrating soliton molecules. The separation and phase difference oscillations in Figs. 2(d) and 2(e) are both synchronous. For the slow or fast soliton molecule, the leading and trailing pulses interact with each other through direct overlapping mediated by the Kerr nonlinearity and indirect interaction mediated by the DW while the slow and fast soliton molecules interact with each other directly through XPM and FWM. Such complex nonlinear effects together with the balance between the gain and loss synchronize the oscillation of the slow and fast modes as shown in Figs. 2(d) and 2(e). The interaction traces of the slow and fast soliton molecules are shown in Fig. 2(f). The internal interaction traces of both modes oscillate but with different phases on the plane as one can see from Fig. 2(f). Figures 2(g)-2(i) show the soliton molecule evolution with the initial temporal separation of 17ps and the initial leading-to-trailing phase difference of π. For clarity, we show only 200 round trips in Figs. 2(g) and 2(h). The separation in Figs. 2(g) and 2(h) is almost invariable while the phase difference oscillates with small amplitudes. The weak oscillations of the separation and phase difference are due to the weak interaction mediated by the DW under large separation [6–8]. The interaction traces of the slow and fast soliton molecules in Fig. 2(i) show that the fast soliton molecule experiences stronger oscillation than that of the slow soliton molecule. The results in Fig. 2 demonstrate that the internal interaction dynamics of the slow soliton molecule is usually different from that of the fast soliton molecule. The soliton molecule in a fiber laser mode-locked by NPR is a scalar soliton molecule because the fast mode is proportional to the slow mode due to the polarizer, which is quite different from the vector soliton molecule in our case. We also find that the final state of the vector soliton molecule converges to the state in Figs. 2(a)-2(c) when the initial separation is less than 7ps while the final separation keeps almost fixed when the initial separation is larger than 8ps in our simulations under the beat length of 2m. We believe there is a threshold locating in the range of 7-8ps acting as the boundary of the two evolution behaviors, however, to find this threshold is beyond the scope of our paper.

 

Fig. 2 Soliton molecule evolution of the vector soliton molecule, peak separations (blue solid), phase difference (red solid) between leading and trailing pulses, interaction evolution traces of the slow (pink solid) and fast (black solid) soliton molecules: (a) slow soliton molecule with 800fs separation, (b) fast soliton molecule with 800fs separation, (c) interaction traces of the slow and fast soliton molecules with 800fs separation and 2000 rounds, (d) slow soliton molecule with 7.9ps separation, (e) fast soliton molecule with 7.9ps separation, (f) interaction traces of the slow and fast soliton molecules with 7.9ps separation and 2000 rounds, (g) slow soliton molecule with 17ps separation, (h) fast soliton molecule with 17ps separation, (i) interaction traces of the slow and fast soliton molecules with 17ps separation and 1000 rounds.

Download Full Size | PDF

Different evolution dynamics occurs if we change the linear beat length to be 5m and 7m, respectively. The results of the simulation under the linear beat lengths of 5m and 7m with the same initial conditions as the case in Figs. 2(a)-2(c) are shown in Fig. 3. The final separation of the vector soliton molecule is almost unchanged when the initial separation is large enough (>8ps), however, the final separation is sensitive to the initial conditions when the initial separation is less than 1ps. Although various solutions with different specific parameters can be obtained, all of them are of no difference from the dynamics presented in our paper. The slow and fast soliton molecules under 5m beat length shown in Figs. 3(a) and 3(b) are vibrating, however, the amplitudes of the peak separation and phase difference are so weak that they are almost locked on the interaction plane as shown in Fig. 3(c). Interestingly, the slow soliton molecule is locked to be in-phase while the fast one is locked to be out-of-phase as shown in Fig. 3(c). The inserts in Fig. 3(c) are the enlargement portions of the quasi-locked traces in Fig. 3(c). The small and closed interaction traces in Fig. 3(c) can be regarded as robust nonlinear attractors of the dissipative systems. We should state that the internal interaction traces in Figs. 3(c) and 3(f) are formed by the results of continuous 1000 round trips while only 200 and 300 round trips are shown in Figs. 3(a)-3(b) and Figs. 3(d)-3(e) for clarity, respectively. If the beat length is increased to 7m, the oscillations of the soliton molecule of both modes are even weaker as shown in Figs. 3(d)-3(f). The weak fine structures in Figs. 3(d) and 3(e) are due to the limited resolution in our simulation (~3fs) and the slight deviation of the separation calculation algorithm used in our simulation. The slow soliton molecule is locked to be out-of-phase while the fast soliton molecule is locked to be in-phase. Their interaction traces are limited in small regions as shown in Fig. 3(f) and the inserts. The initial conditions of the Fig. 3 are the same as those of Figs. 2(a)-2(c). We can speculate from Figs. 2(a)-2(c) and Fig. 3 that larger birefringence results in stronger oscillation of the soliton molecule. We should note that even the round-to-round dynamic of the soliton molecules is quasi-locked, their intra-cavity state is not locked everywhere in the laser cavity. Such a quasi-locked evolution is caused by the multiple 2π phase difference increasement in each round.

 

Fig. 3 Pulse interaction of the vector soliton molecule under beat lengths of 5m and 7m, peak separations (blue solid), phase difference between leading and trailing pulses (red solid), interaction traces of the slow soliton molecule (pink solid) and fast(black solid) soliton molecule: (a) soliton molecule of slow mode with 5m beat length, (b) soliton molecule of fast mode with 5m beat length, (c) interaction traces of the soliton molecules of slow and fast modes with 5m beat length and 1000 rounds, (d) soliton molecule of slow mode with 7m beat length, (e) soliton molecule of fast mode with 7m beat length, (f) interaction traces of the soliton molecules of slow and fast modes with 7m beat length and 1000 rounds.

Download Full Size | PDF

Figure 4 shows the interaction traces of the vector soliton molecules under a linear beat length of 10m. Chaotic evolution of the vector soliton molecules with the initial temporal separation of 400fs and the initial leading-to-trailing phase difference of -π/2 are shown in Figs. 4(a)-4(c). We can obtain from Fig. 4(a) that the slow soliton molecule has a chaotically oscillating separation and an independently evolving phase evolution. Such chaotically oscillating soliton molecule has been researched with a scalar model at normal-dispersion [18]. In our case, the model is vectorial and works at anomalous-dispersion, which demonstrates that independent evolution of the soliton molecule is an intrinsic solution of the dissipative systems under certain parameters. The interaction evolution of the fast soliton molecule in Fig. 4(b) is almost the same as that in Fig. 4(a), except the phase difference of the fast soliton molecule is slightly smaller than that of the slow mode. The evolution of the soliton molecules on the interaction plane in Fig. 4(c) are limited in the closed circles but with chaotic rotating directions. Vibrating soliton molecules can also be obtained through adjusting initial conditions. The internal interaction of the vector soliton molecule with the initial temporal separation of 790fs separation and the leading-to-trailing phase difference of π are shown in Figs. 4(d)-4(f). The period of the phase difference evolution in Fig. 4(d) and 4(e) is twice that of the separation evolution. The oscillation of the peak separation is weaker than that in Figs. 4(a)-4(c). The trajectories of the slow and fast modes in Fig. 4(f) are nearly symmetric to each other on the interaction plane. The oscillations of the soliton molecules are even weaker with the initial temporal separation of 10ps separation and the initial leading-to-trailing phase difference of π as shown in Figs. 4(g)-4(i), which is due to the weak interaction between the leading and trailing pulses under large temporal separation. The oscillating amplitude of the fast mode is larger than that of the slow mode. The trajectories of the fast and slow soliton molecules overlap with each other in Fig. 4(i). By comparing Fig. 4(f) with Fig. 4(i), we can conclude that the larger separation relates to the smaller oscillation of the phase difference evolution. The intensity dynamics of the vector soliton is not only caused by the balance between the leading and trailing pulses but also by the balance between two orthogonal modes considering the gain competition and coherent energy exchange by the FWM [33, 34]. Consequently, the phase difference between the leading and trailing pulse of a soliton molecule at polarized axis is not only determined by the self-phase-modulation (SPM) but also by the XPM and FWM induced by the soliton molecule at its orthogonal axis.

 

Fig. 4 Pulse interaction of the vector soliton molecule under 10m beat length, peak separations (blue solid), phase difference between leading and trailing pulses (red solid), interaction traces of the soliton molecule on the slow (pink solid) and fast (black solid) axes: (a) slow soliton molecule with 2.1ps separation, (b) fast soliton molecule with 2.1ps separation, (c) interaction traces of the soliton molecules with 2.1ps separation and 2000 rounds, (d) slow soliton molecule with 2.48ps separation, (e) fast soliton molecule with 2.48ps separation, (f) interaction traces of the soliton molecules of slow and fast modes with 2.48ps separation and 1000 rounds, (g) slow soliton molecule with 10.3ps separation, (h) fast soliton molecule with 10.3ps separation, (i) interaction traces of the slow and fast soliton molecules with 10.3ps separation and 1000 rounds.

Download Full Size | PDF

From the above results, we have revealed the internal interaction dynamics of the vector soliton molecules in a mode-locked fiber laser. Vibrating, sliding, quasi-phase-locked and chaotically oscillating vector solutions can be obtained with different beat lengths and initial conditions. The interaction trajectories of the slow and fast soliton molecules in a vector soliton molecule are different from each other. The slow and fast soliton molecules can interact with each other through XPM, FWM, gain competition and nonlinear loss, which will indirectly affect the phase difference and temporal separation between the leading and trailing pulses as well as the interaction trajectories of the soliton molecules on their orthogonal polarized axes. The leading and trailing pulses directly interact with each other through overlapping and DWs. The complex balance between gain and loss together with the complex interactions inside the soliton molecule results in various kinds of stable vector soliton molecules in the mode-locked fiber lases.

3.2 Polarization dynamics of vector L and vector T

It has been revealed numerically that the soliton in a NPR mode-locked fiber laser has a nearly uniform polarization across the pulse duration when the pulse reaches the stable state [37]. It is not clear whether the polarizations of the vector L and vector T in the vector soliton molecule are the same or not in a fiber laser with a polarization insensitive SA such as graphene or CNT. To investigate the polarization dynamics of vector L (T), we choose the normalized Stokes parameters at the peak point of vector L (T) to represent the polarization of vector L (T). To distinguish the phase difference between the slow and fast components from the phase difference between the leading and trailing pulses, we refer to the orthogonal phase difference as the phase difference between the slow and fast components in the subsequent contents. We also find that the polarization in the time slot of vector L(T) is nearly uniform, which is similar to the results in [37]. The uniformity of the polarization is shown in Fig. 5, which is under the pump strength of 250pJ with 10m beat length. We can obtain from Figs. 5(a) and 5(b) that the orthogonal amplitude ratio and phase difference between the slow and fast modes are nearly uniform in the central parts of vector L and vector T. It is known to all that the DWs are strong at the edges of the dissipative solitons, which results in the polarization non-uniformity at the edges of the vector soliton molecule. The polarization trajectories within the time slots of full width of quarter maximum (FWQM) of vector L and vector T are shown in Fig. 5(c). One can get from Fig. 5(c) that the polarization trajectories within the FWQM time slots of vector L and vector T are nearly uniform and manifest themselves as two points on the polarization Poincare sphere. From the simulation results in Fig. 5, we think it is suitable for us to use the polarization states at the peak points of vector L and vector T to describe their polarization evolution.

 

Fig. 5 (a)Intensity profile of the slow soliton molecule and the amplitude ratio between slow and fast modes across the time slot of the soliton molecule, (b)Intensity profile of the slow soliton molecule and the phase difference between slow and fast modes across the time slot of the soliton molecule, (c) polarization trajectories of vector L and vector T within their time slot of full width of quarter maximum.

Download Full Size | PDF

Detailed pulse characteristics in the case of Figs. 2(a)-2(c) are shown in Fig. 6. The amplitude evolutions of the four components of the vector soliton molecule, namely, the leading pulse of the slow soliton molecule, the tailing pulse of the slow soliton molecule, the leading pulse of the fast soliton molecule and the trailing pulse of the fast soliton molecule, are shown in Fig. 6(a). We can get from Fig. 6(a) that the fast soliton molecule is asymmetric with unequal leading and trailing amplitudes. Such asymmetric soliton molecule is similar to the so-called A-type soliton molecule in [10], which is caused by the slow response of the SA as well as the balance between gain and loss. We should note that such A-type soliton molecule in our case is with fast SA response, which is similar to the simulation results in [22]. We think the asymmetric soliton molecules are intrinsic solutions for the coupled CGLEs as well as the scalar CGLE with certain parameters. The evolutions of the unwrapped phase difference between the orthogonal components of vector L and vector T are shown in Fig. 6(b). From Figs. 6(a) and 6(b) we can get that vector L and vector T have different orthogonal amplitude ratio and phase difference, so the polarization dynamics of vector L and T are different as well. We should state that it is not suitable for one to calculate the total beat length from the unwrapped round-to-round phase difference evolution from Fig. 6(b) as the round-to-round phase difference evolution has neglected multiples of 2π phase difference inside each round during the unwrapping process. The polarization trajectories of vector L and vector T on the Poincare sphere are shown in Fig. 6(c). The trajectories of the vector L and vector T are polyline circles winding around the point of (0,1,0) (the slow linear polarized state), which can be regarded as the polarization attractors [25, 26]. Owing to the different amplitude ratio between the orthogonal components of vector L and vector T, their polarization attractors have different averaged radius as shown in Fig. 6(c). The spectra of the slow and fast soliton molecules in Fig. 6(d) show that the slow soliton molecule has different leading-to-trailing phase difference from that of the fast soliton molecule, which is in accordance with Fig. 6(a). Temporal intensity profiles of the slow and fast soliton molecules are shown in Figs. 6(e) and 6(f), respectively. We can see from Figs. 6(e) and 6(f) that the orthogonal amplitude ratio of vector L is larger than that of vector T, which results in the larger radius of vector T and smaller radius of vector L in Fig. 6(c). In conclusion, the vector L and vector T of the vector soliton molecule in this situation have different polarization as well as polarization dynamics. Such a kind of vector soliton molecule is formed by the complex balance between gain and loss together with the coupling between the slow and fast soliton molecules. One can use wave plates and polarizers at the output to control the relative strength of the leading and trailing pulses due to their different polarization, which has potential applications in optical signal processing and coding format.

 

Fig. 6 Polarization dynamics of the leading and trailing pulses with beat length of 2m and peak separation of 790fs: (a) the amplitude evolutions of the four components of the vector soliton molecule, (b) phase difference between the two polarization components of the leading and trailing pulses, (c) polarization trajectories on the normalized polarization Poincare sphere, (d) spectra of the soliton molecule of the slow(pink solid) and fast mode(black solid), (e) pulse intensity of the soliton molecule of the slow mode, (f) pulse intensity of the soliton molecule of the fast mode.

Download Full Size | PDF

In the next step, we increase the linear beat length to 5m and 7m while adjust the initial conditions to obtain different kinds of vector soliton molecules. In most cases of our simulations with 5m or 7m linear beat length, the polarization trajectories of the slow and fast soliton molecules rotate around (0,1,0) with the same averaged radius, which means their orthogonal amplitude ratio are the same. Typical results are shown in Fig. 7 and Fig. 8. The initial conditions for Figs. 7(a) and 7(c) are the same as those of Figs. 3(a) and 3(d), respectively. For Figs. 7(b) and 7(d), they have the same initial conditions of 8ps separation and zero leading-to-trailing phase difference. We can obtain from Figs. 7(a) and 7(b) that the trajectories of vector L and vector T under beat length of 5m are quickly winding around the point of (0,1,0) and manifest themselves as polylines strongly winding around circles. The related orthogonal phase difference between the slow and fast components are shown in Figs. 8(a) and 8(b). One can see from Fig. 8(a) that even the vector L and vector T have nearly the same polarization trajectories, their polarization are different from each other due to their different orthogonal phase difference as shown in the insert of Fig. 8(a). The difference between the two curves shown in the inserts of Figs. 8(a), 8(b), 8(c) and 8(d) are ~π rad, ~4.3 rad, ~π rad and 8.6 rad, respectively. Vector L and vector T have the same orthogonal phase difference increasement per round trip. The increasement of the orthogonal phase difference per round trip in Figs. 8(a)-8(d) are 3.3 rad, 3.5 rad, 1.1 rad and 2.1 rad, respectively. The large increasement of the orthogonal phase difference in Figs. 8(a) and 8(b) results in the strongly winding polylines of the trajectories in Figs. 7(a) and 7(b). The polarization trajectories of vector L and vector T under the beat length of 7m are shown in Figs. 7(c) and 7(d). The trajectories in Figs. 7(c) and 7(d) are weakly winding polylines rotating around (0,1, 0). We can obtain from Figs. 8(c) and 8(d) that the orthogonal phase difference increases nonlinearly, which is different from the linearly increasing phase difference in Figs. 8(a) and 8(b). The orthogonal intensity difference of vector L and vector T corresponding to Figs. 8(a)-8(d) are shown in Figs. 8(e)-8(h), respectively. Vector L and vector T have nearly the same oscillating evolution of the orthogonal intensity difference under 5m beat length and 820fs separation as we can see from Fig. 8(e). When the separation is increased to 7.95 ps, the orthogonal intensity difference of vector L is slightly different from that of vector T as one can obtain from Fig. 8(f). When the beat length is increased to 7m, the orthogonal intensity difference evolutions of vector L and vector T have dual-period oscillation behavior. We can get from Fig. 8(g) that vector L and vector T have the same evolution with a short period of 2 round trips and a long period of 33 round trips. Such dual-period oscillation is also shown in Fig. 8(h) with the same linear beat length but 7.95ps temporal separation. We think the dual-period behavior is due to the coherent energy exchange between the orthogonal components and the long-term gain dynamics. The coherent energy exchange caused by the FWM makes the intensity difference oscillate quickly in a few round trips while the long-term complex balance between the gain and loss makes the orthogonal intensity difference evolve slowly in dozens of round trips. We can also obtain from Figs. 8(e)-8(h) that the intensity difference evolution of vector L is asynchronous with vector T under large separation such as 7.95ps while it is synchronous with vector T under short separation such as 800fs and 820fs. We speculate that the strong interaction between vector L and vector T under short separation makes their orthogonal intensity difference oscillates synchronously, however, deeper mechanism still remains to be explored.

 

Fig. 7 Polarization dynamics of the vector L (blue solid) and vector T (red solid) with different beat lengths and peak separations: (a) 5m beat length with 820fs separation, (b) 5m beat length with 7.95ps separation, (c) 7m beat length with 800fs separation, (d) 7m beat length with 7.95ps separation.

Download Full Size | PDF

 

Fig. 8 Phase difference between the slow and fast components of vector L (blue solid) and vector T (red dash) and intensity difference between slow and fast components of vector L (pink solid) and vector T (black dash), the insert is the enlargement portion to show the difference between the vector L and T: (a) and (e) 5m beat length with ~820fs separation, (b) and (f) 5m beat length with ~7.95ps separation, (c) and (g) 7m beat length with ~800fs separation, (d) and (h) 7m beat length with ~7.95ps separation.

Download Full Size | PDF

Different polarization dynamics happens when we increase the linear beat length to 10m with different initial conditions. The results are shown in Fig. 9 and Fig. 10. The initial conditions of Figs. 9(a), 9(b) and 9(d) are the same as those of Figs. 4(a), 4(d) and 4(g), respectively. Figure 9(c) is with the initial temporal separation of 100fs separation and the initial leading-to-trailing phase difference of π. The polarization trajectories of vector L and vector T under separation of 1.94ps are shown in Fig. 9(a). We can obtain from Fig. 10(a) that the orthogonal phase differences of vector L and vector T oscillate chaotically around ~0.36rad and ~-2.81rad, respectively. The intensities of the four components of the vector soliton molecule in Fig. 10(e) also oscillate chaotically. The chaotically oscillating phase difference and amplitude ratio result in the chaotic polarization trajectories of vector L and vector T, which are limited in the areas on the Poincare sphere shown in Fig. 9(a). The chaotic polarization trajectories can be regarded as the chaotic polarization attractors. The polarization trajectories of vector L and vector T under the separation of 2.49ps are shown in Fig. 9(b). The trajectories of vector L and vector T rotate on the Poincare sphere with oscillating phase difference and different central points, as shown in Fig. 9(b). The orthogonal phase differences between the slow and fast components for vector L and vector T in Fig. 10(b) oscillate slightly around ~0.31rad and ~-2.82rad, respectively. The intensities of the slow and fast modes evolve with anti-phase as shown in Fig. 10(f), which is caused by the gain competition and coherent energy exchange between the orthogonal components. The polarization trajectories of vector L and vector T under the separation of 4.89ps are shown in Fig. 9(c). We can get from Fig. 9(c) that the polarization trajectory of vector L locates around the equator of the Poincare sphere, which means the polarization of vector L is linearly polarized. For vector T, the polarization trajectory is a strongly precessing polyline with wide-spread dynamic region on the Poincare sphere. The phase difference between slow and fast components and intensity evolution in 100 round trips are shown in Figs. 10(c) and 10(g), respectively. The orthogonal phase difference of vector L is almost locked to -π with slight oscillation as one can see from Fig. 10(c) and its inset while the phase difference of vector T decreases monotonically with round trips increasing. We can obtain from Fig. 10(g) that the intensities of the two orthogonal components of vector L oscillate slightly while the intensities of the two orthogonal components of vector T oscillate sharply. The decreasing phase difference as well as the sharply oscillating intensity of vector T result in the shaking polarization trajectories in Fig. 9(c). The polarization trajectories of vector T can be regarded as a shaking polarization attractor while the polarization trajectory of vector L can be regarded as a quasi-locked polarization attractor. The vector soliton molecule in Fig. 9(c) can be regarded as a composite vector soliton molecule whose leading pulse keeps a slightly oscillating polarization evolution while trailing pulse keeps a shaking polarization evolution. The shaking state of vector T is caused by the strong energy exchange between its orthogonal components through FWM under weak birefringence. When increasing the peak separation to 10.37ps as shown in Fig. 9(d), vector L and vector T have the same polarization trajectories. We can obtain from Figs. 10(d) and 10(h) that vector L and vector T have nearly the same orthogonal phase difference and amplitude ratio between the slow and fast components. The intensity profiles of the vector soliton molecules corresponding to Figs. 9(a)-9(d) are shown in Figs. 10(i)-10(l), respectively. We can get from Figs. 10(i)-10(l) that the slow soliton molecule has higher intensity than the fast soliton molecule, which is caused by the fast axis instability under weak linear birefringence [34–36]. In general, four kinds of polarization dynamics of the vector soliton molecules have been obtained under the beat length of 10m.

 

Fig. 9 Polarization trajectories of vector L (blue solid) and vector T (red solid) under different temporal separations: (a) ~1.94ps, (b) ~2.49ps, (c) 4.89ps, (d) 10.37ps.

Download Full Size | PDF

 

Fig. 10 Detailed evolution of the vector soliton molecules under beat length of 10m. Polarization phase difference between the orthogonal components of vector L and vector T with different separation: (a) 1.94ps separation, (b) 2.49ps separation, (c) 4.89ps separation, (d) 10.37ps separation. Intensity of leading and trailing pulses of slow mode and fast mode: (e) 1.94ps separation, (f) 2.49ps separation, (g) 4.89ps separation, (h)10.37ps separation. Intensity profiles of the vector soliton molecules with different separation: (i) 1.94ps separation, (j) 2.49ps separation, (k) 4.89ps separation, (l) 10.37ps separation. The inset in Fig. 10(c) is the enlargement portion of the vertical axis.

Download Full Size | PDF

The above simulations of the polarization dynamics of vector L and vector T in the vector soliton molecules under different beat lengths and peak separations reveal that the polarization across the whole vector soliton molecule is usually not uniform. The polarizations of vector L and vector T in a vector soliton molecule are different from each other. The slow and fast soliton molecules interact with each other through XPM, FWM, gain competition and saturable absorption. In each soliton molecule, the leading and trailing pulses interact with each other through direct overlapping or indirect interaction mediated by DW. The complex balance between the gain and loss of the dissipative system results in the complex dynamics of the internal interaction evolution of the slow and fast soliton molecules as well as the complex polarization dynamics of vector L and vector T.

3.3 Intra-cavity dynamics of the vector soliton molecule

The above dynamics in Sections 3.1 and 3.2 are the vector soliton molecule evolution at the end of the laser cavity versus round trips, the evolution inside each round can give a deep understanding of the intra-cavity dynamics as well as the round-to-round evolution. The intra-cavity dynamics corresponding to Figs. 2(a)-2(c) and Fig. 6 are shown in Fig. 11, which is with a beat length of 2m and a peak separation of ~790fs. The amplitude evolution and orthogonal phase difference of vector L are shown in Fig. 11(a). We can obtain from Fig. 11(a) that the amplitude of the slow component is larger than that of the fast component, as a result of which the dynamics of the fast component is dominated by the XPM and FWM due to the large intensity of the slow component while the evolution of the slow component is dominated by the balance between gain and loss. The oscillating amplitude of the fast component is caused by the FWM between two orthogonal components. The oscillating amplitude of the fast component has a fixed relationship with the orthogonal phase difference between the slow and fast components as we can obtain from Fig. 11(a). The amplitude of the fast component increases when the orthogonal phase difference locates in (N-1/2)π~Nπ while it decreases when the orthogonal phase difference locates in Nπ~(N + 1/2)π, where N is an arbitrary integer. When the phase difference is Nπ/2, there is no energy exchange between the slow and fast modes. Such a relationship confirms the amplitude oscillation is caused by the FWM as we can easily calculate from the FWM items in Eq. (1) to get such a relationship. The amplitude oscillation caused by the FWM for the slow component is not obvious in Fig. 6(a) because the slow soliton molecule is not dominated by the coupling with the fast mode but by the balance between gain and loss considering its large amplitude. The phase differences at the beginning and end of the cavity are 1.083rad and 6π + 2.094rad, respectively, which means the orthogonal phase difference at a fixed position of the laser cavity gradually increases from 1.083rad to “2.094rad” and thus the polarization trajectories rotates around the point (0,1,0), as shown in Fig. 6(c). The intra-cavity amplitude and orthogonal phase difference evolution of vector T in Fig. 11(b) is similar to those of vector L in Fig. 11(a). The intensity and phase difference between the leading and trailing pulses of the slow soliton molecule is shown in Fig. 11(c). In general, the leading-to-trailing phase difference is dominated by the leading-to-trailing intensity difference due to the SPM, as one can obtain from Fig. 11(c) that the leading-to-trailing phase difference increases when their intensity difference is negative while decreases when their intensity difference is positive even there is some deviation caused by the XPM and FWM. Things become different for the fast soliton molecule as we can see from Fig. 11(d). The phase difference between the leading and trailing pulses for the fast soliton molecule is not dominated by the SPM because the phase difference does not decrease monotonically when the leading-to-trailing intensity difference is negative. Because the intensity of the slow component is larger than that of the fast one, the XPM for the fast soliton molecule is stronger than the SPM, which means that the leading-to-trailing phase difference of the fast soliton molecule depends not only on the SPM but also on the XPM. What’s more, the SPM and leading-to-tailing phase difference of the fast soliton molecule can be affected by the XPM because the energy exchange between orthogonal components is sensitive to the XPM. From Figs. 11(a) and 11(b), we can get that the coupling between the slow and fast soliton molecules has important effects on the internal interaction dynamics and evolution. What’s more, the amplitude evolution of the leading and trailing pulses of the fast soliton molecule is anti-phase as shown in Figs. 11(a) and 11(b), which means there is a strong interaction between the leading and trailing pulses of the fast soliton molecule. From Figs. 11(c) and 11(d), we can conclude that the leading-to-trailing phase difference oscillates in the cavity while the overall round-to-round phase difference and separation evolution can exhibit vibrating, sliding, quasi-locked or chaotic behavior as shown in section 3.1. The coupling between the orthogonal components has strong effects on the internal interaction dynamics of the fast soliton molecule. The round-to-round internal interaction evolution depends on the composite effects of the dynamic balance between gain and loss, the coupling between the slow and fast components and the interaction between the leading and trailing pulses.

 

Fig. 11 Intra-cavity dynamics of the vector soliton molecule with beat length of 2m and 790fs temporal separation: (a) amplitude evolution of slow (blue solid) and fast (red solid) modes and phase difference between the slow and fast components of vector L (pink solid), (b) amplitude evolution of trailing pulses of slow (blue solid) and fast (red solid) modes and phase difference between the slow and fast components of vector T (pink solid), (c) intensity difference (blue dashed) and phase difference between the leading and trailing pulses of the slow soliton molecule, (d) intensity difference (blue dashed) and phase difference between the leading and trailing pulses of the fast soliton molecule.

Download Full Size | PDF

The intra-cavity dynamics of the soliton molecules corresponding to Figs. 9(c) and 9(d) are shown in Fig. 12. The amplitude evolution of the slow and fast components of vector L corresponding to Fig. 9(c) is shown in Fig. 12(a). Both the slow and fast components oscillate in Fig. 12(a) as a result of the coherent energy exchange caused by FWM. The relationship between the amplitude oscillation and phase difference is the same as that in Fig. 11. The orthogonal phase differences at the beginning and end of the cavity are π + 0.056rad and 3π + 0.004rad, respectively, which means that the round-to-round phase difference for a fixed position in the cavity is almost fixed as the phase difference increasement in each round is ~2π. The intra-cavity dynamics of vector T corresponding to Fig. 9(c) is shown in Fig. 12(b). Because the amplitudes of the slow and fast components of vector T have similar magnitude, the interaction between the two components mediated by XPM and FWM results in the strong amplitude oscillation of the orthogonal components. Also, the amplitude oscillation obeys the phase rules caused by the FWM. The phase differences at the beginning and end of the cavity are −2.82rad and 2π-4.53rad, respectively. The phase difference for the fixed position of the cavity decreases from −2.82rad to “-4.53rad” in one round trip and the amplitude evolution in each round is not self-consistant, which results in the shaking polarization evolution in Fig. 9(c). We can see from Fig. 12(c) that the spectra of the vector soliton molecule at the end of the cavity has blurring of the fringe contrast due to the unequal amplitudes of the leading and trailing pulses. The resonant sidebands in Fig. 12(c) means that the leading and trailing pulses can interact with each other mediated by the strong DW as well as the direct overlapping. The intra-cavity evolutions of vector L and vector T corresponding to Fig. 9(d) are shown in Figs. 12(d) and 12(e), respectively. We can obtain from Figs. 12(d) and 12(e) that vector L and vector T have nearly the same polarization everywhere inside the cavity. The same polarization properties of vector L and vector T are caused by the weak pulse interaction between them under large temporal separation of 10.37ps. The phase differences in Fig. 12(d) at the beginning and end are 0.311rad and 0.395 + 2π rad, respectively, while the phase differences in Fig. 12(e) at the beginning and end are 0.242rad and 0.3078 + 2π rad, respectively. We can obtain from Figs. 12(d) and 12(e) that the polarizations of vector L and vector T are nearly the same despite there is a deviation for the orthogonal phase difference. The slightly varying orthogonal phase difference and amplitude ratio in each round together with the balance between gain and loss result in the polarization trajectories with oscillating phase difference in Fig. 9(d). The resonant sidebands in Fig. 12(f) are weak, which means that there is no strong interaction between the leading and trailing pulses under large pulse separation. The weak interaction between the leading and trailing pulses results in the similar polarization evolution of vector L and vector T.

 

Fig. 12 Intra-cavity evolutions of the amplitudes of the slow component (blue solid), fast component (red solid) and phase difference (pink solid) between the slow and fast components under beat length of 10m: (a) vector L with separation of 4.89ps, (b) vector T with separation of 4.89ps, (c) total spectrum at the end of the cavity with separation of 4.89ps, (d) vector L with separation of 10.37ps, (e)vector T with separation of 10.37ps, (f) total spectrum at the end of the cavity with separation of 10.37ps.

Download Full Size | PDF

4. Conclusion

We have revealed the thorough physical pictures of the vector soliton molecule in a mode-locked fiber laser through numerical simulation for the first time. The slow and fast soliton molecules can have different internal interaction evolutions on the interaction plane. Vibrating, sliding, chaotically vibrating and quasi-locked vector soliton molecules can be obtained under certain conditions. Vector L and vector T can have different polarizations and polarization evolutions on the Poincare sphere, which have potential applications in optical communications and signal processing. Rotating, chaotically oscillating, oscillating, shaking and quasi-locked polarization evolutions can be obtained for vector L and vector T. The intra-cavity dynamics of the vector soliton molecule demonstrate that the direct coupling between the orthogonal components mediated by XPM and FWM plays an important role in the intra-cavity dynamics and round-to-round evolution of the vector soliton molecule. The overall complex balance between gain and loss together with the interaction between the orthogonal components mediated by XPM, SPM and FWM result in the complex polarization trajectories of vector L and vector T. The polarization difference between the vector L and vector T is caused by the pulse interaction between them through the direct overlapping, DW and gain competition. When their separation is large, they have similar polarization as a result of the weak interaction. Those vector-bound-state solutions are nonlinear attractors for the coupled CGLEs caused by the complex balance between gain and loss, the interaction between the orthogonal components and the internal interaction between the leading and trailing pulses. Simpler description and deeper understanding of the vector soliton molecules in ultrafast fiber lasers are still remaining to be researched. Our work gives a different and thorough view on the vectorial and molecular nature of the vector soliton molecules and can help one exploit the potential of the vector soliton molecules in optical communications and signal processing.