We demonstrate the fast-axis instability in mode-locked fiber lasers numerically for the first time. We find that the energy of the fast mode will be transferred to the slow mode when the strong pump strength makes the soliton period short. A nearly linearly polarized vector soliton along the slow-axis could be generated under certain cavity parameters. The final polarization of the vector soliton is related to the initial polarization of the seed pulse. Two regimes of energy exchanging between the slow mode and the fast mode are explored and the direction of the energy flow between two modes depends on the phase difference. The dip-type sidebands are found to be intrinsic characteristics of the mode-locked fiber lasers under high pulse energy.
© 2017 Optical Society of America
Vector solitons are fascinating subjects in nonlinear optics and mode-locked fiber lasers. The modes with orthogonal polarization in a perfectly circular fiber are degenerate and one can use a scalar nonlinear Schrodinger equation (NLSE)  to describe the propagation and evolution of a light pulse in such a fiber. However, the fiber is not perfectly circular in most causes because of the random pressure, bending and defects during production. The two modes is not degenerate and we should consider the difference and interactions between the two modes and regard the pulse in a birefringent fiber as a vector pulse and use the coupled NLSE  to describe propagation of the vector pulse. The polarization evolution of a soliton in a birefringent fiber has been extensively investigated in the past years [3–5]. The soliton in the mode-locked fiber lasers also have polarization evolution in the cavity and one can control the states of the polarization through tuning the energy and averaged beat length of the cavity. Vector solitons can been formed in fiber lasers when there is no polarization-sensitive components in the cavity. Many satuarable absorbers such as semiconductor satuarable mirrors [6,7], carbon-nanotube [8,9], graphene [10–13] and two-dimension matetials  are polarization-insensitive and have been used to generate vector solitons. Different kinds of vector solitons in mode-locked fiber lasers have been researched such as group velocity-locked vector solitons (GVLVSs) [15–18], phase-locked vector solitons (PLVSs) [19–21], polarization-rotation locked vector solitons (PRLVSs) [22,23] and bright-dark vector solitons (BDVSs) . It is useful to control the polarization of solitons in lasers since it has potential applications in polarization multiplexing  and coherent optical detectors  in optical communication systems. Both GVLVS and PLVS have elliptical polarization. GVLVSs have a rotational polarization along the cavity because of the large linear birefringence  which cannot be balanced by the nonlinear birefringence. The two orthogonal components shift their central wavelength to compensate the group-velocity difference caused by the linear birefringence and move with the same group-velocity as a GVLVS.While a PLVS has a fixed polarization across the cavity because the nonlinear birefringence compensates the weak linear birefringence, what’s more, the phase difference between the orthogonal components is  so that there is no energy transfer between them. Under these conditions, both phase and amplitude of the two component are locked resulting in the polarization-locked vector solitons. It is interesting to know if the vector soliton can become linearly polarized without polarization-sensitive components.
For a soliton polarized along the slow axis of the fiber, nonlinear birefringence induced by self-phase modulation (SPM) and cross-phase modulation (XPM) will enhance the birefringence and small deviation from the slow-axis wouldn’t destroy the stable pulse propagation. However, a pulse polarized along the fast-axis would be unstable if the nonlinear birefringence compensate the linear birefringence [3–5]. It has been proposed that a soliton propagating in a birefringent fiber will transfer the energy of fast component to its slow component because of the fast-axis instability when the linear beat length is comparable with or longer than the nonlinear beat length [3–5]. Theoretical and experimental works also confirmed the fast-axis instability in a birefringent fiber . It seems interesting to use the fast-axis instability to generate a linearly polarized soliton and control the polarization of vector solitons in a mode-locked fiber laser. Different from the birefringent fiber, the fiber laser is a non-conserved system because the soliton experiences gain and loss in the cavity and the evolution of the soliton should satisfy the boundary conditions of the laser cavity, however, a soliton propagating in a fiber can be regard as a conserved one when assuming the loss of fiber is weak. S. T. Cuddiff et. al found a kind of linear polarization-locked solitons in a mode-locked fiber laser . They also confirmed that the linear polarization-locked pulse polarized along the slow-axis of the fiber . This is the first and only experimental demonstration of the fast-axis instability in a mode-locked fiber laser. To our best knowledge, no numerical simulations of the fast-axis instability in mode-locked fiber lasers have been reported so far. A simulation could give an insight into the fast-axis instability in a mode-locked fiber laser and it is useful for one to select parameters to realize the fast-axis instability in fiber lasers.
In this paper, we present a comprehensive numerical study of the fast-axis instability in a mode-locked fiber laser. We find that the energy of the fast and slow modes increase until their total energy stabilized. After their total energy stabilized, the energy of the fast mode will be transferred into the slow mode when the linear beat length is larger than the soliton period. Under certain conditions, the fast-axis pulse would evolve into a very tiny pulse compared with the slow component and the vector soliton is nearly linearly polarized. Because the order of the vector soliton in the anomalous dispersion is nearly 1, so a soliton with high intensity has narrow width and short soliton period. When the beat length is longer than the soliton period, fast mode is not stable. Under the circumstances of high intensity and large beat length, it is easy to get the fast-axis instability, which is confirmed by our simulations. We think our simulations would give an insight into the dynamics of the vector soliton and can greatly help others to research the fast-axis instability in mode-locked fiber lasers experimentally.
2. Model and methods of the simulation
The model of simulation we use is shown in Fig. 1. The main parts of the cavity include 15m single-mode fiber(SMF), 5m Er-doped fiber (EDF), a lumped polarization-insensitive saturable absorber (SA) and a 10:90 output coupler (OC). Our simulation starts from an arbitrary weak signal and the pulse runs from the EDF to SMF in each round. We use the coupled Ginzburg-Landau equations to describe the vector pulse propagating in the laser cavity:
Where, u and v are the envelopes of the lightwave polarized along the slow and fast axes, respectively. is half of the wave-number difference between the slow and fast modes and is the difference of the effective refractive index between the slow and fast modes. is the inverse group-velocity difference between two modes. is the second order dispersion which is −21ps^2/km in EDF and SMF. is nonlinear coefficient of the fiber which is 1.3W−1km−1 in EDF and SMF. represents the gain bandwidth of 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 lumped SA is represented by:
Where P is the instantaneous power of the light pulse. In the simulation, we choose different pump strength Es while other parameters are fixed. We use the symmetric split-step Fourier method to implement our simulations. We run the programs for 10000 rounds for each case.
3.1 Instability under different initial polarization states
A vector soliton polarized along the slow or fast axis will propagate stablely and the polarization wouldn’t be changed. If a soliton is polarized along the slow axis, small deviation from the axis of the initial polarization can’t destroy the stable state because the nonlinear refractive index and linear index enhance the birefringence of the fiber. However, a soliton polarized along the fast axis wouldn’t be stable if small deviation happens because the nonlinear birefringence is opposite to the linear birefringence and the polarization will evolve far from the fast axis when the pulse power is larger than the threshold . Generally one can say that when the nonlinear beat length or soliton period is shorter than the linear beat length, the fast-axis instability will happen [3–5], however, no strict defination of the nonlinear beat length has been proposed. Unstable polarization evolution in the birefringent fiber depends on the initial polarization states and initial soliton paramenters, solitons with rotating-phase, oscillating-phase or oscillating to rotating phase might be excited depending on initial conditions . It is known to all that the final states of a pulse such as the pulse energy, peak power, width, spectral width and chirp in the scalar model of a mode-locked laser depend on the cavity parameters such as length of cavity, net-dispersion, gain, loss and filtering . However, whether the parameters of final stable vector solitons depend on the initial polarization or not still remains a question. We use three different initial linear polarization states which are polarized near the fast axis, slow axis and 45° respect to both axes, respectively. We should point here that near fast axis is not perfectly polarized along fast-axis, but means that a small deviation from the fast axis. The pulse polarized perfectly along the principle axes will propagate without polarization changing according to the Ginzburg-Landau equation. Fast-axis instability was demonstrated experimentally in fibers  and fiber lasers . In fibers, the instability is confirmed by the crowding of output polarization near the slow axis when different linear polarization is injected in the fiber. In fiber lasers, the instability is confirmed by the linearly polarized pulse along slow-axis . According to the experimental facts, it might be simple and clear for us to observe the energy evolutions of slow and fast modes instead of polarization points on the Poincare sphere to catch the fast-axis instability in the simulation.
First, we set the initial polarization of the seed pulse polarized 45° to both principle axes. We fix the beat length at 7m and increase the pump from 10pJ to 100pJ before the pulse splitting. The energy evolutions of the fast and slow modes with different circulating rounds are shown in Fig. 2. Figures 2(a)-(f) correspond to different pump strength of 10pJ, 30pJ, 40pJ, 50pJ, 70pJ and 100pJ, respectively. As we can note from Fig. 2 that the component polarized along the fast axis is not stable. After the total energy of pulse is stable, the fast mode transfers its energy to the slow one as a result of the fast-axis instability. When the pump strength is 10pJ as shown in Fig. 2(a), the energy of pulse is low, so the pulse has wider pulse duration according to soliton-area theory  and thus has long soliton period compared to the linear beat length, so the axis instability is weak and the rate of energy transfer is very slow. The GVLVS is formed in Fig. 2(b) as there is no axis-instability happens and the phase and amplitude of the slow and fast modes are not fixed but evolving across the cavity. Figures 2(c)-2(f) correspond to the high energy pulse with short duration and short soliton period, so the beat length is larger than the soliton period and the fast-axis instability is relatively strong. The pulse at the pump of 40pJ is almost linearly polarized as the the energy of fast mode is very weak after 4000 rounds. These are qualitatively agree with the experimental results of the linearly polarized pulse in . In Figs. 2(e) and 2(f), after the fast mode transferring its energy to the slow mode after thousands of rounds, the energy curves verse rounds for the slow and fast modes are oscillating with anti-phase. The energy of pulse we calculate and plot is at the end of the laser cavity and this means that the energy of the slow and fast modes at a fixed position in the laser cavity is oscillating slightly round to round. In a word, the fast-axis instability exists in the mode-locked laser demonstrated by the energy transfer from the fast mode to slow mode in Figs. 2(c)-2(f).
The above simulation has an initial polarization which is 45° to both principle axes. Next, we set the initial seed pulse linearly polarized near the fast axis. The energy evolutions of both modes are shown in Fig. 3. Figures 3(a)-3(f) correspond to different pump strength of 10pJ, 30pJ, 40pJ, 50pJ, 70pJ and 100pJ, respectively. The linear beat length is also fixed at 7m. When the pump strength is 10pJ as shown in Fig. 3(a), the final energy is all relax onto the fast mode and the slow mode is completely suppressed. This is because the pulse has low energy and peak power, the soliton period is long compared with the beat length, so the fast-axis instability is weak and the slow mode is also suppressed because of the gain competition with fast mode, thus vector solitons linearly polarized along the slow axis is obtained. However, things become different when pump strength is high as shown in Figs. 3(b)-3(f), the fast-axis instability is strong and the energy transfer from the fast mode to slow mode makes the slow mode gradually grow up . The fast mode is completely suppressed and the vector soliton is polarized along slow axis in Figs. 3(c) and 3(e). The energy curves are oscillating in Figs. 3(b) and 3(f) similar to Figs. 2(e) and 2(f). Compared with Fig. 2 and Fig. 3, we can conclude that the initial polarization has effects on the final polarization of vector pulse in a mode-locked fiber laser, however, it has no effects on the fast-axis instability as we can see in Fig. 2 and Fig. 3 that all the fast modes under high pump strength would transfer their energy to the slow modes. The fast-axis instability in our simulation manifests as the energy transferring from the fast mode to slow mode. The energy evolutions under initial polarization state of near slow axis are shown in Fig. 4. The fast-axis instability is similar with Fig. 2 and Fig. 3. However, the suppression of fast mode is stronger than those in Fig. 2 and Fig. 3. When the initial slow mode is strong than the fast mode, with the help of fast-axis instability and gain competition, the weak initial fast mode has no chance to be amplified which is different from Fig. 3(a). In Fig. 3(a), the initial weak slow mode could be amplified and finally suppresses the fast mode with the help of fast-axis instability.
From the above simulations, we can conclude that the final polarization of the vector soliton is related to the initial polarization of the weak seed pulse. When the polarizations of vector solitons are different, dispersive-wave caused by the periodic perturbation would be different , thus the sidebands, energy and width of the vector solitons might be different, too. Figure 5 shows the pulse profiles in time and wavelength domains related to the cases of Fig. 2(b), Fig. 3(b) and Fig. 4(b) as the final three vector solitons have obviously different polarization. The pump strength is 30pJ and beat length is 7m. As we can see from Fig. 5(a) that pulses with different polarizations have different pulse power and energy because of different strength of dispersive wave radiation. The total energy of both pulse and dispersive wave in three cases are 162.1pJ, 162.2pJ and 162.2pJ, respectively. The energy of the three vector solitons are 159.1pJ, 157.5pJ and 155.5pJ, respectively. Assuming the difference between the total energy and pulse energy is the energy of the dispersive wave, the strength of the dispersive wave in three cases are found to be 3pJ, 4.7pJ and 6.7pJ, respectively. We can also obtain from Fig. 5(b) that the locations and strength of sidebands are also different, which means the three vector solitons have different pulse width , however, such difference is not big. When the final vectors have similar linear polarizations such as shown in Fig. 2(c), Fig. 3(c) and Fig. 4(c), they should have the same strength of dispersive wave, pulse energy and pulse width. As we can see from Fig. 6 that the three solitons have nearly the same intensity profiles and spectra. In a word, the parameters of a vector soliton circulating stably in a mode-locked fiber laser are not only determined by the cavity parameters but also by the initial polarization of the seed pulse.
3.2 Phase difference and in-cavity energy exchanging
As we know that two components of a vector soliton have coherent energy exchanging (CEC) caused by the four-wave mixing . The CEC depends on the phase difference of the two components, CEC is zero when the phase difference is N*90°, where N is an arbitrary integer. It is difficult and unclear for us to calculate the nonlinear beat length analytically, however, we can estimate the average beat length from the in-cavity phase difference evolution. We choose the phase difference between the slow mode and fast mode at the point of the peak power of the vector soliton. We select two examples, one is the case in Fig. 4(e) and the other is the case in Fig. 4(f). Figure 7 shows the energy evolution of the fast mode and phase difference verse position in the cavity. We should state that the slow mode has a energy evolution like the fast mode but with anti-phase and the energy modulation depth is very small compared with the large energy of slow mode, so we just plot the evolution of the fast mode. It is clear in Figs. 7(a) and 7(c) that the energy exchanging exists in the laser cavity. We can see from Figs. 7(b) and 7(d) that the phase difference is not locked but evolving in the cavity. We note that when phase difference is N*90° the strength of energy exchanging is zero. When the phase difference locates in N*180°~N*180° + 90°, the energy exchanging is from the fast mode to the slow mode while the energy exchanging is from the slow mode to the fast mode when the phase difference locates in (N-1/2)*180°~N*180°. The increasing of the phase difference is not linear with the distance but oscillating around its linear fitting curve, as shown in Figs. 7(b) and 7(d). From the linear fitting curves, we can estimate the average beat length in the laser cavity is 4.6087m and 3.9696m in Figs. 7(b) and 7(d), respectively. Because the total birefringence is enhanced by the nonlinear birefingence when the power of the slow mode is stronger than that of the fast mode, so the average beat length is shorter than the linear beat length.
3.3 Dip-type sidebands under high pump strength
In our simulations, dip-type sidebands are generated when the pump strength is strong and the soliton has high peak power and energy. As has been explored that the dip-type sidebands are caused by the parametric process of the energy exchanging between soliton and dispersive-wave through four-wave mixing , the sidebands would be the dip-type when energy flow is from the soliton to dispersive-wave or be the peak-type when the direction of the energy flow is from the dispersive-wave to soliton. The spectra of the slow and fast modes under different pump strength when the initial polarization is near the slow axis and linear beat length of 7m is shown in Figs. 8(a)-8(f). We state that the spectra are all at the end of the laser cavity (starting from the 5m EDF and ending at the 15m SMF), we have also checked that no peak to dip altering from position to position in the cavity as that in .
We find that the dip-type sidebands are strong on the spectra of the slow modes while the peak-sidebands are strong on the fast mode as is shown in Fig. 8. The positions of the dip-type sidebands of the slow modes are not equal to the positions of the peak sidebans on fast modes, so it is not the case of the direct coherent energy exchanging between the fast and slow modes. So we can conclude that there are two ways for the energy exchanging between the slow and fast modes, one is the direct CEC by four-wave mixing between the two modes, the other is that the slow mode transfers its energy to the dispersive-wave and then the dispersive-wave transfers its energy to the fast mode. Even the fast modes are very weak, the components polarized along the fast axis is not noise but a weak soliton with obvious sidebands, so the “linearly polarized soliton” is not exactly a perfectly linear one, but with very weak orthogonal component. Some other kinds of sidebands such as peak-dip sidebands shown in  are also found in our simulation. The authors of  thought the dip-type sidebands were related with the polarization-selective effects of the taper covered by the WS2 film, however, no components are polarization-sensitive in our model, so we think the dip-type sidebands are intrinsic characteristics of the mode-locked fiber laser.
We have presented a clear simulation of the fast-axis instability in the mode-locked fiber lasers. The fast-axis instability is independent of the initial polarization of the seed pulse while the parameters of the final vector soliton depend on the initial polarization of seed pulse. The fast-axis instability manifests as the fast mode transferring its energy to the slow mode while the final polarization approaching to the slow axis, which qualitatively agrees well with the experimental results reported in the past [20, 28]. If the final vector pulses have similar polarizations (for the same cavity but different initial seeds), the parameters of the final pulses would be similar and their sidebands on the spectra are also similar. Two components will exchange their energy through direct CEC or indirect energy exchange with dispersive-wave, which is shown as the dip-type sidebands on the spectra of the slow modes and peak sidebands on the spectra of the fast modes. The direction of the energy flow is related to the phase difference between two components. We also show that the dip-type sidebands are independent of polarization selective effects and are the intrinsic properties of the mode-locked fiber laser. We hope our work might give an insight into the dynamics of the vector solitons in a mode-locked fiber laser. Future work will be focus on the experimental evidences of the fast-axis instability and whether the fast-axis instability would happen in some other pulse regimes such as the stretched pulse, the dissipative soliton or the similariton.
Director Fund of WNLO; National 1000 Young Talents Program, China.
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