1. Introduction

Solitons are ubiquitous in nature, occurring in a variety of physical, chemical, and biological systems [1-3]. The terminology soliton was invented to describe nonlinear solitary wave solutions of integrable equations [3-5]. Optical solitons in conservative systems result from the balance between the nonlinearity of the material and the dispersion (or diffraction).

In contrast to solitons in conservative systems, dissipative solitons (DSs) exist in non-conservative systems, which are far from equilibrium [2]. They are produced by dissipative systems and hence their dynamics is markedly different from that of conventional solitons. Since the energy of dissipative solutions has to be dissipated in the medium, their most prominent feature is that they exist only when there is a continuous energy supply from an external source [6]. The gain and loss coexist in the dissipative system and play an essential role in the generation of DSs [6-8]. Therefore, DSs have characteristics that drastically depend on the system parameters and are unique for the given external conditions.

A laser with purely positive group-velocity-dispersion (GVD) (or large positive GVD together with slight negative GVD) would presumably have to exploit dissipative processes to a greater degree in the steady-state pulse-shaping [8]. Recently, it has been demonstrated that pulse shaping in normal GVD fibers is favourable for the generation of DSs in fiber oscillators [8-11]. These demonstrations have attracted much interest in the development of all-normal-dispersion fiber lasers since this type of lasers has the capacity of significantly improving the deliverable energy per pulse, approaching or even exceeding 100 nJ [10, 11].

Although various regimes involving multiple pulses have been observed in the net-anomalous-dispersion fiber lasers [12-14], experimental observations of three (especially more than three) pulses operating on the all- or net-normal-dispersion lasers are rare. Zhao et al. reported an experimental observation of two pulses in an Er-doped fiber laser made of all-normal- dispersion fibers [15]. Haboucha and Martel et al. proposed that the spectral filtering and/or the overdrive of nonlinear polarization (NP) effect play essential roles in the multiple pulse formation in the all-normal-dispersion fiber lasers [16, 17]. In this paper, we establish an experimental setting to investigate a class of multiple DSs. The dynamic evolution of solitons as a function of the pump power has been observed. There exist multistability and hysteresis phenomena with respect to the pump power. Our experimental observations are quite different from the results reported on [15-17]. Our experiments suggest that the multi-pulse operation behavior of DSs tend to be managed by the NP effect and the accumulation of excessive pulse chirps.

2. Experimental setup

The fiber oscillator for DSs is shown schematically in Fig. 1. It consists of a polarization-sensitive isolator (PS-ISO), two sets of polarization controllers (PCs), a wavelength-division-multiplexed (WDM) coupler, an Er-doped fiber, and a fused coupler (10% output). The Er-doped fiber has a length of 19.9 m with a dispersion of about -30 ps/nm/km and a nonlinear coefficient of about 3 /W/km at 1550 nm. The other fibers in cavity are standard single-mode fiber (SMF) with a GVD of about 17 ps/nm/km and a length of 3.7 m. The total length of cavity is 23.6 m and the net cavity GVD is then +0.68 ps². The polarization-sensitive isolator provides unidirectional operation and polarization selectivity in a ring-cavity configuration. The polarization state of lightwaves in the cavity can be controlled by adjusting two polarization controllers. The NP rotation technique is used for locking the laser. A 977-nm laser diode (LD) can provide the pump power of up to 500 mW. An autocorrelator, an optical spectrum analyzer (OSA), and an 11-GHz oscilloscope (LeCroy SDA) together with a 12-GHz photodetector are used to simultaneously monitor the laser output.

 

Fig. 1. Schematic diagram of DS laser cavity.

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3. Results and analysis

With appropriate orientation and pressure settings of the polarization controllers, self-started mode locking can be achieved when the pump power P is beyond a threshold value. For P≈45 mW (i.e., the threshold power), the proposed fiber laser operates on the mode-locked (ML) state from the CW state. In this stage, the measured output average power is about 0.45 mW. Figures 2(a)-2(c) show the optical spectrum of the pulses, the corresponding autocorrelation and oscilloscope traces, respectively. We can observe from Figs. 2(a)-2(c) that (1) the optical spectrum has the typical characteristic of all-normal-dispersion pulses (i.e., steep spectral edges); (2) the 3-dB spectral width Δλ of solitons is about 10.25 nm; (3) the autocorrelation trace has a full width at half maximum (FWHM) of about 62.7 ps; (4) the fundamental frequency of cavity is 8.73 MHz, corresponding to the pulse separation of about 114.55 ns; (5) the center wavelength of spectra is about 1564 nm. If a Gaussian pulse profile is assumed, the pulse width Δτ is about 44.5 ps, which gives a time-bandwidth product (Δτ·Δν) of ~55.9. Therefore the pulses are strongly chirped.

 

Fig. 2. (a). Optical spectra of pulses; (b) Autocorrelation trace; (c) Oscilloscope trace at pump power P=45 mW; (d) Oscilloscope trace at P=406 mW

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When the pump power P is increased from 45 to 52 mW, the laser still emits single pulse although both the pulse duration Δτ and the spectral width Δλ are broadened. However, the laser changes to the unstable operation when P is beyond 52 mW. If P is further increased to about 71 mW, an interesting phenomenon happens, i.e., the number of pulses is doubled (Fig. 4). As P is more than 71 mW, the above evolution repeats again. As a result, the number of pulses increases one by one for increasing pumping. The multistability evolution of DSs in terms of P is shown in Fig. 3(a) (Media 1) in detail. We can observe from Fig. 3(a) (Media 1) that, for P=406 mW, the laser produces 10 solitons in the cavity. The corresponding oscilloscope trace is shown in Fig. 2(d), where there are 10 solitons coexisting in the cavity round-trip time of 114.55 ns. Comparing Fig. 2(c) with 2(d), we can see that pulse height in the oscilloscope trace for the single-soliton state is approximately the same as that for the 10-soliton state (their relative height difference is less than 6%). The experiments also show that, with increasing pump power, the pulse duration is broadened slightly at a certain N-pulse operation regime and the energy per pulse is increased slightly. However, with the increase of pulse number N, the energy per pulse is decreased, e.g., the pulse energy is about 51 pJ in the single-pulse operation regime and about 45 pJ in the dual-pulse regime.

However, the dynamics of DSs is very different when the pump power is decreased. The number of solitons decreases one by one for decreasing pumping. This occurs until the laser becomes continuous again at P≈34 mW. By comparing Media 1 to Media 2, we can see that, when the laser works in the N-pulse ML regime, the range and value of the pump power on the ascending procedure are different from those on the descending procedure. The detailed evolution process is shown in Fig. 4, where the blue and red step lines show the ascending and descending procedure, respectively. The dashed and solid lines denote that the laser operates on the unstable state and the stable N-soliton ML state (N=1, 2,… 10), respectively. We can observe from Fig. 4 that (1) the formation and annihilation of each pulse show the pump power hysteresis, (2) the threshold power of multistability is low, (3) the unstable and stable ML states alternately evolve along the pump power P, and (4) the threshold power of CW state on the ascending and descending procedure is approximately equal to about 9.6mW.

 

Fig. 3. Videos showing multistability evolution of DSs in terms of pump power P. The left and right videos show the operation state of increasing pump power (Media 1) and decreasing power (Media 2), respectively. P represents the pump power and N denotes the number of solitons in the cavity.

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Fig. 4. Relationship between the soliton number N and the pump power P. The thin dashed segments denote the unstable state of laser. The thick solid segments represent the stable N-soliton ML state. The blue and red step lines show the ascending and descending procedures, respectively.

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Fig. 5. Relationship of (a) the 3-dB spectral width Δλ of pulses and (b) the corresponding time-bandwidth product Δτ·Δν versus the pump power P.

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The relationships of the spectral width Δλ of pulses and the corresponding time-bandwidth product Δτ·Δν versus the pump power P are shown in Figs. 5(a) and 5(b), respectively. The blue solid curves show the stable ML state of the laser whereas the red dashed lines are the unstable state. We can observe from Fig. 5(a) that Δλ approximately periodically varies in terms of P and linearly increases in the stable N-pulse ML regime. Our experimental observations are very different from the experimental reports in [10], where Δλ was monotonically increased along P. From Fig. 5(b), one can see that the time-bandwidth product Δτ·Δν of the laser fluctuates from ~50 to ~65. These values are much larger than the reported results (e.g., Δτ·Δν≈2.4 in [11], ≈5.5 in [10], ≈3–17 in [8], and 21.8 in [5]).

In comparison with the multi-soliton fiber laser reported on [15], the threshold power for multi-soliton ML state of our laser is extremely low, e.g., our laser produces 10 pulses in the cavity for P≈400 mW whereas the laser in [15] only creates 2 pulses for P≈1500 mW. The cavity pulse peak clamping effect proposed by Tang et al. [13] successfully explains the formation of multiple solitons in the net-anomalous-dispersion fiber lasers. This theory even successfully explains the multiple gain-guided soliton generation in a fiber laser made of purely normal-dispersion fibers [15]. Wise et al. had proposed that the generation of ultrashort optical pulses tend to be limited by the accumulation of excessive nonlinear phase shifts [8]. A high-energy pulse with large peak intensity accumulates a high nonlinear phase shift that may cause the pulse to break up. Obviously, the above-mentioned theories may fail to explain our experimental observations because the peak power of pulses in our experiments is far lower than that in [8, 13, 15].

Haboucha and Martel et al. proposed that the spectral filtering effect of gain plays an essential role in the generation and evolution of multiple pulses [16,17]. We can observe from Fig. 5(a) that the spectral width Δλ of pulses is about 9–10.5 nm, which is much less than the gain bandwidth Δλ_g of the Er-doped fiber (Δλ_g is about 35 nm in our experiments). According to the theory in [18], the spectral filtering effect has slight influence on the pulse in our reports. Using the NP technique, Komarov et al. successfully explained the multi-stability and hysteresis phenomena in passively ML fiber lasers operating in the normal dispersion regime [12]. Martel et al. successively proposed that the overdrive effect of NP evolution plays a key role in the multi-pulse evolution [17]. However, our experimental observations are very different from the reports in [12, 17]. For example, as long as the laser is mode locked, new pulses can be generated one by one when the pump parameter is increased [12]. Contrarily, the multistability operation in our observations is unstable at some special pump power region (e.g., 144 to 130 mW in the descending pump procedure and 135 to 152 mW in the ascending pump procedure). The destabilization of the multi-pulse ML regimes towards CW state often occur through the generation of an unstable state in [17], whereas this phenomenon is never observed in our experiments except that the single-pulse regime towards CW state is destabilized by the generation of an unstable state (an example is shown in Fig. 4). Therefore, besides the effect of NP evolution, a new mechanism may cover our experimental observations.

Because the proposed laser has very long fiber length and very high net-normal cavity dispersion (+0.68 ps²), the GVD significantly induces the pulse broadening according to the theory in [1]. Furthermore, since the length of the Er-doped fiber in the cavity is very long with a large nonlinear coefficient, the combined effects of self-phase modulation (i.e., Kerr effect) and GVD contribute on the strong chirp of pulses. It is seen from Fig. 5(b) that, when Δτ·Δν is beyond a value (e.g., ~64), the laser operates unstably until it produces an additional soliton. We estimate that it is the accumulation of excessive pulse chirps that results in unstable operation state of the multiple soliton in the DS fiber lasers. The detailed explanation is as follows. According to the theory of the NP technique, the nonlinear transmission acts as the positive feedback at great pump intensity, whereas it acts as the negative feedback at higher pump intensity [12, 17]. As a result, the pulses appear one by one when the pump is increased [12]. However, when Δτ·Δν is very large in the N-soliton ML regime, the pulses are very strong chirped. The different frequency components of a pulse thus travel at larger different speeds along the fiber. Red components travel much faster than blue components. When their difference is beyond a threshold (e.g., the chirped pulses have Δτ·Δν of ~64), the pulse does not sustain the extremely strong chirp and hence the laser is unstable. If P is further increased and the peak intensity of pulses becomes greater than a threshold, the feedback becomes negative. An additional pulse therefore arises from the background instability [12]. At this stage, the peak intensities of the steady-state pulses become less than the threshold because of energy balance. As a result, the laser will stably operate in the (N+1)-soliton ML regime with lower frequency difference of red and blue components (i.e., Δτ·Δν of the chirped pulses is less than about 64). On the other hand, when P is decreased to a certain value (e.g., P≈144 mW), the pumping energy may not drive N-soliton ML operation because the balance between the energy being supplied and lost has to be exact in the dissipative system. Then the laser changes to the unstable operation. If P is further decreased (e.g., P≈130 mW), the appropriate pumping energy can support the (N-1)-soliton operation rather than the N-soliton operation with a lower pulse chirp. As a result, the laser works on the (N-1)-soliton state.

Based on a rigorous model [19], Olivier et al. investigated the role of the chirp on the bound-state (two-pulse) formation in a stretched-pulse fiber laser in detail. However, the temporal separation among the pulses in our reports is far larger than that in [19]. According to the theory in [19], such a large separation in this paper can not support the bound state of pulses despite the strong chirp. Because of longer fiber length and larger net-normal GVD in our experiment, the accumulation of excessive pulse chirps associated with the NP effect plays an essential role in the appearance and evolution of multiple DSs. As a result, the self-starting pump power threshold, as well as the incremental power to generate an additional pulse is extremely low (less than 45 mW).

4. Conclusions

The experimental observations of multiple DSs are reported in a passively ML fiber laser with large normal cavity dispersion (the net cavity GVD is as large as ~ 0.68 ps²). The dynamics of the number of solitons is demonstrated for increasing and decreasing pump power [Fig. 3 (Media 1) and (Media 2)]. Our laser produces multiple DSs from N=1 to 10 during the pump power range of 45 mW to 406 mW (Fig. 4). The spectral width Δλ and the time-bandwidth product Δτ·Δν approximately periodically vary as a function of the pump power P (Fig. 5), respectively. Multistability and hysteresis phenomena observed in our laser are qualitatively distinct from those observed in the net-anomalous-dispersion conventional-soliton fiber lasers. The experimental results show that the laser operates unstably when Δτ·Δν is beyond ~64. As a result, we estimate that the stable multi-soliton evolution and the unstable operation of DSs may tend to be managed by the accumulation of excessive pulse chirps together with the NP effect. To our best knowledge, it is the first time to report on multiple DS evolution, which evolves on the stable and unstable states alternately, in a fiber laser with very large normal GVD at a very low pump power.