Multiple dissipative solitons are numerically studied in the normal-cavity-dispersion Yb-doped fiber laser. Soliton pairs and triplets of different types are found in parameter domain where single soliton mode-locking is unstable. Similar scenario is found for soliton pair and triplet: an additional weak soliton supplements already existing solitons, then with increasing gain the solitons start oscillating and at higher gain the soliton complex becomes stable. Spectral profiles of the solitons dynamically change taking either parabolic-like or Π-like or M-like shape similar to those described in the literature for individual dissipative soliton at different pump level.
©2009 Optical Society of America
Mode-locked fiber lasers are becoming excellent alternatives to solid-state lasers owing to their greater stability, weaker alignment sensitivity, and reduced cost . These attributes triggered substantial development efforts to achieve laser pulse characteristics comparable or better than the solid-state counterparts. Fiber lasers with net anomalous cavity dispersion operating in the soliton regime cannot produce ultra-short energetic pulses due to limitations imposed by the soliton area theorem. Therefore, a lot of attention has been recently paid to fiber lasers with normal group velocity dispersion (GVD) where either self-similar evolution of the laser pulse [2, 3] or formation of so-called gain-guided solitons [4, 5] or dissipative solitons [6, 7] inside the cavity is possible. The linear chirp accumulated across the pulse makes the pulse robust against optical wave breaking and allows compression of the pulse outside the laser cavity. Recently, generation of sub-100 fs pulses with energies above 30 nJ from normal GVD fiber laser without external amplification has been reported .
Generation of a single ultra-short energetic pulse is hindered by the harmonic mode-locking when multiple pulses are formed inside the laser cavity with increasing external pump level. Interaction between multiple solitons has been extensively studied both theoretically and experimentally. It has been shown that due to the cavity pulse peak clamping effect, number of pulses increases with gain level with a hysteresis transition from one multiple soliton state to another [9,10]. Another study has demonstrated that solitons may form pairs with predetermined separation and phase difference between two pulses [11, 12]. Most recent work show that both gain-guided solitons  and dissipative solitons  may demonstrate even more sophisticated interaction dynamics including vibrating, shaking and mixed soliton pairs.
In our previous paper  stability limits of the single mode-locked pulse operation have been studied for a quite generic model of the normal-cavity-dispersion Yb-doped fiber laser. It has been shown that when pump level increases (or the cavity dispersion decreases) the single dissipative soliton state loses its stability versus low-frequency perturbations. In this paper we numerically study multiple soliton states which arise in the domain of instability of the single dissipative soliton. We find a variety of dynamical regimes with properties distinctive from those reported earlier in the literature. As the gain increases we find oscillating soliton pairs and triplets which become stable and symmetric at some gain level. Asymmetric multiple soliton states are found at the boundaries between different attractors. Spectrum profiles of the solitons change their shapes from M-like to Π-like and parabolic during oscillations that correspond to the spectra of the single dissipative soliton at different pump levels. We study the dynamics of the Yb-doped fiber laser in the framework of extended Ginzburg-Landau equation and compare our finding with that reported earlier in the literature.
2. Laser cavity model
The laser cavity model used in the current work, illustrated in Fig. 1 is the same as that introduced before in . Here, we briefly describe the model: the Yb-doped fiber with a length of 60 cm is considered as the main element in the laser cavity. The model includes also few centimeters of single mode fiber (SMF) as a part of WDM coupler to provide external pumping but the presence of the SMF is not essential for the obtained results. Polarization beam splitter (PBS) is assumed to output light from the cavity as well as to initiate mode-locking through nonlinear polarization evolution (NPE). A wave plate (WP) is used for controlling the polarization state of the propagating light, and the isolator is used to provide unidirectional propagation.
The pulse evolution in the Yb-doped fiber is governed by the Ginzburg-Landau equation for the complex amplitude A:
where z is the propagation coordinate, t is the retarding time, A(z, t) is the complex field amplitude, α is the linear loss taken as 0.04 m-1, β2 is the GVD parameter taken as 24000 fs2/m that corresponds to the normal cavity dispersion, γ is the nonlinear parameter taken as 0.005 W-1m-1, go is the small signal gain parameter with parabolic frequency dependence and a bandwidth Ωg taken as 35 nm, Epulse is the pulse energy and Esat is the gain saturation energy (5 nJ). Light transmission through the polarizer has been modeled by a transfer function T=1-lo/[1+ Ppulse/Psat], where lo=0.5 is the unsaturated loss, Ppulse is the pulse power and Psat=12 kW is the saturation power. The total loss of the cavity has been taken as 10 dB corresponding to losses at all interfaces as well as the useful loss due to light output from the cavity. The numerical method used is based on the split-step Fourier transform algorithm. As an initial condition, white noise has been used. The accuracy of the numerical results has been verified by reproducing them at different spatial and temporal grid parameters.
3. Results and discussion
Recently, it has been shown in several studies [6, 15–17] that stable single dissipative soliton regime can be obtained in a limited area of the fiber laser parameters space. When one of the parameters (cavity dispersion, pump strength or gain bandwidth) extremely changes then the balance necessary for the single soliton state formation gets broken and either a cw state, chaotic dynamics or multiple soliton state establishes. In a previous work  we have observed in numerical simulations that when the pump is switched on for the laser parameters corresponding to the unstable area, initially the single soliton is formed from noise. Later, this soliton evolves to low-frequency fluctuations and is then transformed to the quasi-cw state. When following the quasi-cw state for several thousands of roundtrips we observe the formation of the multiple soliton states that will be described below in this section. For the sake of concreteness we fix all the laser parameters and vary the gain parameter go only.
3.1. Soliton pairs: characteristics and classification.
When gain parameter go is chosen above the level of 45dB/m, we observe formation of the oscillating soliton pair (OSP) in the laser cavity with the characteristics presented in Figs. 2 and 4. Temporal profiles of the OSP at different moments are shown in Fig. 2. The profiles are presented both in linear, Fig. 2 (a) and logarithmic, Fig. 2(b) scales to enable us to see their interaction through the tails. The three curves in Fig. 2 correspond to three instants in time which either one of the soliton, left (L) or right (R) achieves peak energy or the energy is split nearly equally between them. Apparently, the solitons are oscillating at fixed positions; hence the distance between them is fixed. To illustrate evolution of the OSP in the cavity within one roundtrip, in Fig. 3 we present normalized profiles which appear to be almost the same at the different cavity locations. This is attributed to the high normal GVD value of the cavity which makes the breathing ratio of the single pulse almost unity [3, 6]. It can be seen that solitons exchange energy but their tails do not overlap so that the individual pulses can be separated and its spectrum can be calculated. The spectra of the individual solitons, L and R with maximal and minimal energy are presented in Fig. 4 (a) while the plot in Fig. 4(b) shows the total spectrum of the OSP. It is seen from Fig 4(a) that the spectrum profile changes from the parabolic-like shape at the energy minimum to the M-like shape at maximum. It is worth mentioning that previous studies showed such a spectra of the stable dissipative soliton when pump level increases.
When analyzing the spectrum of the soliton pair in Fig. 4(b) and comparing it with the spectra of Fig. 4(a), it becomes clear that the spectral width is determined by the more energetic narrower soliton of the pair while the weaker soliton creates modulation on the spectrum pedestal.
It should be noted that in our case there is no direct interaction between solitons through their tails which is different from the situation of the vibrating solitons described by Soto-Crespo et al . We find some resemblance between our oscillating solitons and the nonstationary behavior of the multiple gain-guided solitons reported by Zhao et al  in the study of normal-dispersion erbium doped fiber laser. We do not find any correlations between the phases of the oscillating solitons what follows from the fact that solitons do not interact through their tails. We see that phase difference between soliton peaks changes quite chaotically in time which can be explained by the fact that electric field decays significantly (in other words, goes through numerical zero) between the pulses as it is shown in Fig. 2(b). We assume that random change of the phase difference is associated with numerical noise at the points of field singularities.
With increasing pump level we see that the amplitude of oscillations decreases. Eventually, beyond gain parameter go value of 49dB/m we observe formation of the stable soliton pair (SSP) with characteristics presented in Figs. 5 and 6. It is seen that the solitons, L and R in the SSP are identical and the spectrum has Π-like shape similar to what has been reported for individual dissipative solitons in our system .
To give more general picture of system dynamics, the transient behavior of the total normalized energy Q with respect to the Esat of the light field in the cavity versus number of roundtrip for different values of the gain parameters is presented in Fig. 7. It can be seen that for the gain parameter go below 49 dB/m, Q demonstrates oscillating behavior with the period and amplitude of oscillations decreasing for higher go. On the other hand, we observe formation of the stable soliton pair in the range of the gain parameter greater than 49 dB/m and up to 54 dB/m. It is worth noticing that the transient time to achieve SSP is becoming larger for higher gain. It is an indication that we are approaching new attractor in the infinite-dimensional phase space of our system. The next subsection describes new states that are observed in the region of higher gain.
3.3. Soliton triplets: characteristics and classification.
When the pump parameter go is taken above 55 dB/m, as a final state we observe formation of the oscillating soliton triplet (OST) whose characteristics is presented in Figs. 8–10. In Fig. 8 temporal intensity profiles of the OST are shown in linear (a) and logarithmic scale (b) at go=60 dB/m. Two snap-shots correspond to the moments of time separated by 75 roundtrips when either the single soliton on the left (L) or the other two nearly identical solitons on center (C) and on right (R) reach maximum power. We observe that the soliton triplet consists of a soliton pair and a separate soliton: The solitons in the pair are nearly identical and oscillate in phase while the individual soliton oscillates in the opposite phase. Although the solitons are oscillating, the peaks positions do not change; hence the distances among the solitons are fixed. Moreover, the triplet profiles at the different positions of the cavity are almost the same, like in the case of the soliton pair Again, from the analysis of the profiles in the logarithmic scale we see that the solitons do not interact directly through their tails. Another interesting feature is that while the shape of the C and R pulses in the pair does not change much with the number of roundtrip, the width of the separate soliton L decreases when its energy grows.
The spectra of the individual pulses as well as that of the whole OST are presented in Fig. 9 and 10. Three profiles in Fig. 9(a) are the spectra of the separated pulses of Fig 8 when the individual soliton L reaches its minimum energy. The spectrum of this individual soliton has M-like shape and is different from the Π-like profiles of the other two solitons C and R. The Π-like spectra are wider and they determine the spectral width of the whole OST in Fig. 9(b) while the individual soliton L contribution is the modulation in the spectrum center.
As time (the roundtrip number) progresses, the energy of the individual soliton L grows and its M-like spectrum widens. Fig. 10(a) presents spectra at the instant of time when the separate pulse L reaches maximum energy (the pair, C and R pulses is at minimum energy). It is seen that the M-type spectrum is now wider than the Π-type spectrum of the pulses of the pair. The spectrum width of the whole triplet in Fig. 10(b) is determined now by the individual soliton and the signature of the pair is the modulation.
When simulations are performed at higher levels of the gain parameter go we observe that oscillations decay at approximately go=60 dB/m and stable soliton triplet (SST) is formed in the cavity. In Fig. 11, the intensity profiles of the pulses are shown for go=61 dB/m. The pulses are almost identical, i.e. they have nearly the same width and peak power. The spectra of the solitons look identical. An example of the individual soliton spectrum is presented in Fig. 12(a), it is seen that the spectrum profile seems to be intermediate between the M-type and the Π-type shapes. Fig. 12(b) illustrates the spectrum of the whole triplet.
The transient dynamics of the normalized energy inside the cavity for the case of soliton triplet is presented in Fig. 13. It is seen that after transient process, periodic oscillations settle down. The period of the oscillations decreases with growing gain while the oscillation amplitude reaches maximum at g0=57 dB/m and then decreases. In the oscillation shape one can note two scales: a slow change from one maximum to another and then a sudden drop after which the process is repeated. When we follow individual pulse dynamics we see that slow scale is associated with the change of the energy of the soliton pair while sharp drop is due to sudden change of the separate soliton energy. For gain parameter above g0=60 dB/m we observe decay of the oscillations and the steady soliton triplet previously described in Fig. 12 establishes. One can see that dynamics presented for soliton triplets in Fig. 13 is similar to the case of the soliton pair in Fig. 7. With increasing gain we see that oscillation behavior vanishes and the stable multiple solitons appear. It should be noted that stable soliton triplets as well as collision between soliton pair and the single soliton have been found as solutions of the cubic-quintic complex Ginzburg–Landau equation [12, 18]. In our case, interaction between solitons leading to specific type of oscillatory behavior is different from that described earlier in the literature.
3.3. Asymmetric border states.
At some values of the gain parameter we observe extremely long transient dynamics which corresponds to the boundaries between different attractors of our infinite-dimensional dynamical system. Examples of such dynamics are presented in Fig. 14. It is seen that the transient time to reach stable attractor here is more than two times longer as compared to Figs. 7 and 13. As a result of extremely long transient we observe formation of stable asymmetric soliton pairs and triplets presented in Figs. 15 and 16 respectively. Stable asymmetric soliton pair in Fig. 14 obtained at the gain parameter go=44 dB/m is a border state which corresponds to emergence of the soliton pair. Below go=42 dB/m we observe formation of the single dissipative soliton in the laser cavity.
Formation of the soliton pair shown in Fig. 15(a), with one soliton being much weaker than another, can be explained by the cavity pulse peak clamping mechanism. This is generally considered as a cause of harmonic mode-locking. With increasing gain, the weaker soliton grows and gets closer to the more powerful one. Later we observe oscillatory behavior described above of the OSP. The spectrum of the asymmetric soliton pair is presented in Fig. 15(b). The total spectrum width is determined by more energetic soliton while the weaker one creates modulation in the spectrum center.
Stable asymmetric soliton triplet obtained at go=55 dB/m is presented in Fig. 16. It is seen from Fig. 16(a) that third pulse has quite low amplitude compared to other two. Two powerful solitons determine spectrum envelope in Fig. 16(b), whereas the weakest one is responsible for narrow modulation in the spectrum center.
In the present paper, multiple dissipative soliton states are found in the model of the mode-locked normal-cavity-dispersion Yb-doped fiber laser. Soliton pairs and triplets arise in the parameter domain where single dissipative soliton is unstable. The following sequence has been observed. At the border of single soliton stability, formation of the asymmetric soliton pair is found with one soliton being much weaker than another. When gain parameter is increased the solitons form symmetric oscillating soliton pair, and for higher gain the oscillations decay and symmetric soliton pair becomes stable. It should be noted that solitons do not interact directly through their tails so that no correlation between phases of the solitons has been observed.
Similar scenario has been found for soliton triplets for higher value of the gain parameter. Beyond stability border of the soliton pair, third weak soliton is added to the pair so that the soliton triplet emerges. As the gain increases further, the triplet starts oscillating in such a way that the two identical solitons oscillate in the same phase while the third one oscillates in the opposite phase. Finally at higher gain values, all three solitons become stable. Spectral profiles of the solitons dynamically change taking either parabolic-like or Π-like or M-like shape similar to those observed for individual solitons at different pump level.
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