1. Introduction

Passively mode-locked fiber lasers have attracted much attention because they can be used to construct compact, robust, and versatile ultrashort pulsed sources. When the pumping power exceeds a certain value, more than one pulse will emerge in an anomalously dispersive cavity because of quantization of the soliton energy. Due to the interactions among solitons, dispersive waves and continuous waves (CW), these multiple-solitons will form various dynamic patterns. Since Malomed first predicted the existence of bound state of two solitons in 1991 [1], several groups have experimentally demonstrated or theoretically studied the soliton bound states of two or a few solitons [2–6]. Apart from several solitons coexisting in a cavity, recently, the coexistence of hundreds of solitons in a cavity has also been studied theoretically [7] or experimentally observed under high pumping power [8]. Komarov and Amrani have also reported high order harmonic mode locking in fiber lasers, in which several hundred pulses are uniformly distributed along the cavity [7, 9]. In addition to ordered patterns of multiple-solitons, pulse bunching states have also been observed by several groups, in which several identical soliton pulses group themselves in a tight packet whose duration is much smaller than the cavity roundtrip time. Haboucha et al have theoretically analyzed spontaneous periodic pattern formation in fiber lasers [10]. In addition, Amrani et al have experimentally investigated the formation of ordered and disordered patterns of solitons in a double-clad fiber laser, and have identified patterns that they characterize as a gas, a supersonic gas flow, a liquid, a polycrystal and a crystal of solitons [11, 12]. Chouli et al have also observed soliton rains in which new soliton pulses form spontaneously from the background fluctuations and drift until they reach the condensed soliton phase [13, 14]. Since all the above multiple-soliton states form in ring cavities that are passively mode-locked through nonlinear polarization rotation (NPR), in order to confirm that the pulse bunching states are an intrinsic feature of high power fiber lasers independent of the exact mode locking mechanism, Amrani et al have investigate multiple-soliton pattern formation in a figure-of-eight passively mode locked fiber laser where mode locking is achieved with a nonlinear amplifying loop mirror (NALM) [15].

Since the NPR and the NALM mode-locked laser are environmentally unstable, in order to sidestep this drawback, the use of real passive SA is usually preferred in the development of commercial mode-locked sources. One of the possible SAs is the semiconductor saturable absorber mirror (SESAM), which has a narrow tuning range (tens of nanometers) and requires complex fabrication and packaging. Another SA is made of single-walled carbon nanotubes (SWNT). Although its fabrication is simpler and cost effective, the SWNT always results in non-saturable loss. Compared with SWNT, a new SA, graphene, has a higher optical damage threshold and lower losses, and is wavelength independent. After Hasan et al. firstly demonstrated a graphene-based modelocker [16], a variety of graphene mode-locked fiber lasers, capable of producing a variety of outputs such as ultrashort soliton pulses [17, 18], high energy pulses [19], and wideband tunable pulses [20, 21] have all been achieved. However, to our knowledge, there has been no report concerning the formation of multiple pulses in a graphene mode-locked fiber laser.

More recently, Hendry et al. have demonstrated that graphene has an exceptionally high nonlinear response, described by the effective nonlinear susceptibility $\left|{\chi }^{(3)}\right|\sim {10}^{-7}$esu (electrostatic units), which is 8 orders of magnitude larger than the nonlinearities observed for glasses [22]. Subsequently, Luo et al. proposed and experimentally demonstrated that highly nonlinear graphene was helpful for mitigating the mode competition of EDF lasers [23]. The fact that the graphene shows a strongly nonlinear effect suggests that this character may be benefit for forming multiple pulses in the cavity at relative lower pumping power. This was the initial motivation of our work. In this paper, we have investigated experimentally multiple-soliton dynamic patterns in a graphene mode-locked fiber laser, and have found that, not only can multiple-soliton patterns including disordered multiple-solitons, bunched solitons and high order harmonic mode locking be formed with relatively low pumping power, but flow of solitons can also be obtained with high pumping power. All these phenomena are reported here for the first time for the case of a graphene mode-locked fiber laser.

2. Experimental setup

The experimental setup of the fiber ring laser is shown in Fig. 1 . It consists of a 2 m long, high concentration Erbium-doped fiber (EDF) with group velocity dispersion (GVD) of 21 ps²/km. A fiber-pigtailed 980-nm laser diode with a maximum pumping power of 300 mw was used to pump the EDF through a 980/1550 wavelength-division multiplexer (WDM). A polarization-independent isolator (PI-ISO) was employed to force unidirectional operation of the cavity. A polarization controller (PC) was used to adjust the state of polarization in the cavity. A 10% fiber output coupler (OC) was used to extract the signal. The cavity length was 18.76 m, including 16.76 m standard telecommunications single mode fibers (SMF) with GVD of −22 ps²/km. The net cavity dispersion was about −0.326 ps². An optical spectrum analyzer (OSA) with a minimum resolution of 0.01 nm (AQ6317C), a 1-GHz digital sampling oscilloscope (DL9140) and an autocorrelator (FP-103XL) were used to observe the optical spectrum and temporal pulse shape.

 

Fig. 1 Experimental setup. WDM: wavelength division multiplexer. EDF: Erbium-doped fiber. PC: polarization controller. PI-ISO: polarization-independent isolator. OC: output coupler.

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3. Production and characterization of graphene mode locker

The graphene mode locker was made by transferring graphene film on nickel onto a fiber pigtail. Production of the SA included the following steps: chemical oxidation of graphene wafers in FeCl₃ solution to remove the nickel (see Fig. 2(a) ); floating of the graphene wafers on the surface of deionized water to clean the graphene film and make it accessible to the fiber end face (see Fig. 2(b)); careful positioning of the fiber end face to contact the graphene film (see Fig. 2(c)), and finally, adherence of the graphene film to the fiber end face through mutual Van Der Waals forces (see Fig. 2(d)). Figure 2(e) shows optical image of fiber pinhole coated with graphene. Figures 2(f) and 2(g) show AFM image of graphene on the fiber core and Raman spectra of the graphene film respectively. The transmittance of our SA was 85%, the modulation depth of our SA was 18.6%. 3−5 layers of graphene were placed on the fiber core.

 

Fig. 2 Production and characterization of graphene mode locker. (a) Graphene wafers in FeCl₃ solution. (b) Graphene film floating on the surface of deionized water. (c) Adjustment of the position of the fiber end face to contact the graphene film. (d) Graphene film coating on the pigtail cross-section. (e) Optical image of fiber pinhole coated with graphene. (f) AFM image of graphene on the fiber core. (g) Raman spectra of the graphene film.

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4. Experimental results and discussion

4.1 Disordered multiple-solitons

Our fiber laser has a very low mode locking threshold of about 25 mW. When the pumping power exceeds this threshold, self-starting mode locking can be easily achieved. The solitons observed in our laser have a pulse width (FWHM) of 0.98 ps, as confirmed by the autocorrelation trace shown in Fig. 3(a) . Repetition rate of the pulse is 10.66 MHz, which corresponds to the cavity length. On increasing the pumping power to 40 mW, multiple solitons appeared in the cavity. By tuning the orientation of the polarization controllers and the pumping power, different multiple-soliton operation patterns could be formed. The most easily observed state was that of disordered multiple solitons, in which solitons were randomly distributed in the cavity. In this state, solitons occupied either all or part of the available space along the cavity. As an example, Figs. 3(b) and 3(c) show two different oscilloscope traces of such states. The output power was 3.2 mW in this case. In order to understand the origin of this state, we observed the corresponding optical spectrum which is shown in Fig. 3(d). In contrast to the generic soliton optical spectrum, there is strong and unstable CW lasing coexisting with the soliton operation. The existence of the unstable CW lasing plays an important role in the formation of the disordered multiple-solitons. Since the unstable CW component can introduce a kind of global soliton interaction mechanism linking all the solitons, if the solitons have different central wavelengths, they will have different shift velocities because of dispersion. The distribution of these solitons along the cavity becomes disordered and the solitons are in constant motion with respect to each other.

 

Fig. 3 (a) Autocorrelation trace of soliton. Disordered multiple solitons occupy all the cavity (b) and part of the cavity (c). (d) Optical spectrum of the disordered multiple solitons.

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4.2 Soliton bunching

We found that the strength of the CW lasing could be controlled by adjusting the pumping power and the PC. As the strength of the CW lasing decreased, it became more stable. When the CW became stable, we obtained another multiple-soliton dynamic pattern — soliton bunching. As mentioned in the introduction, in this case, many identical soliton pulses group themselves in a tight packet. Figures 4(a) and 4(b) show the optical spectrum and the oscilloscope trace of this state, respectively. The output power was 3.6 mW with the pump power of 44 mW in this case. Though the time resolution of our oscilloscope is not sufficient to calculate accurately the number of solitons, we can still see that the separations between adjacent solitons are not equal, and decrease with time. In this operation state, we also found two noticeable phenomena. The first was that although bunching of an increased number of solitons becomes easier at high pumping power, we observed bistability in this state. That is to say, when the soliton bunches had formed, changing only the pumping power in a certain range resulted in no change in the shapes of the bunch. However, by changing the pumping power and simultaneously adjusting the PC, the shapes of the bunch could be significantly changed. The second phenomenon was that harmonic mode locking of bunches could be achieved by adjusting the PC. Figure 4(c) shows 2nd harmonic mode locking of bunches. The harmonic repetition frequency of the bunch was 21.32 MHz, twice that of the fundamental cavity repetition rate. The above two phenomena further confirm that although there exist random soliton interactions in the bunching, under the right conditions, soliton bunches can operate stably and function as a unit with the same intrinsic characteristics as a single soliton.

 

Fig. 4 (a) Spectrum of bunched solitons. (b) Soliton bunching with many solitons. (c) 2nd harmonic mode locking of soliton bunch.

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Comparing with the generic soliton spectrum of the nonlinear Schrödinger equation (NLSE), we find that there exist anisomerous dispersive waves in the bunching spectrum (see Fig. 4(a)), leading to the conclusion that the soliton interactions mediated through the radiative dispersive waves play more important roles in the formation of soliton bunching.

4.3 High order harmonic mode locking

When the CW lasing was settled in an appropriate position in the soliton spectrum as shown in Fig. 5(a) , carefully adjusting the PC a little further from the conditions for operation with disordered multiple solitons in which all solitons occupy all the available space along the cavity, we obtained the third multiple-soliton pattern — high order harmonic mode locking, in which all solitons distribute along the cavity with equal spacing. When the pumping power was 65.3 mW, we obtained the 15th harmonic with a repetition rate of 0.16 GHz as shown in Fig. 5(b). The output power was 5.3 mW in this case. With a further increase in the pumping power and with adjustment of the PC, higher order harmonics could be observed. Figure 5(c) shows the 46th harmonic (repetition frequency is 0.49 GHz) with the pumping power of 162.4 mW. The output power reached 13.5 mW in this case. At higher pumping powers, we could obtain higher order harmonics. However, the stability of the solitons decreased as the order increased. Observing the optical spectrum and the oscilloscope trace of this state, we deduced that the unstable CW lasing and its position in the soliton spectrum may play an important role for the formation of harmonic mode locking. As mentioned in disordered multiple-solitons, unstable CW lasing would cause all solitons in the cavity to move. When the CW lasing was adjusted appropriately, phase locking between one of the dynamical modes of the solitons and the CW lasing would occur automatically. In this way, the phases of all the solitons in the laser cavity could be synchronized to those of the CW lasing except for an arbitrary phase constant. Under these conditions, harmonic mode locking appeared.

 

Fig. 5 (a) Spectrum of high order harmonic mode locking. (b) The15th harmonic (repetition frequency is 0.16 GHz) with a pumping power of 65.3 mW. (c) The 46th harmonic (repetition frequency is 0.49 GHz) with a pumping power of 162.4 mW.

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4.4 Flow of solitons

On further increasing the pumping power and adjusting the PC a little from the soliton bunching state, we observed the last pattern —flow of solitons in a diphasic system. Although the flow of solitons looks similar to soliton rain, which has been studied in detail by Chouli et al [14]. In distinction from the work of Chouli, in which they observed the soliton pulses either forming or disappearing all in the background fluctuations, in our experiment, solitons occupied all the available space along the cavity, and we observed the pulses appearing to arise from one condensed phase and drifting to another one with a fixed width for the condensed phase. Figure 6(a) and Media 1 show this process. In our experiment we also found that, although with a higher pump power it was easier to obtain higher drift speeds of the solitons, once flow of solitons was formed, if the polarization state in the cavity remained fixed, increasing or decreasing the pumping power did not change the drift speed of the solitons. However, by changing the pumping power and simultaneously adjusting the PC, the drift speed of the solitons could be significantly changed. This indicates that the drift speed of the solitons is related to both the pump power and the orientation of the PC.

 

Fig. 6 (a) Video recording of the flow of solitons (Media 1). (b) Oscilloscope trace of the condensed phase with a pump power of 170 mW. (c) Spectrum of the condensed solitons phase. (d) Autocorrelation trace of the condensed phase.

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Further adjusting the PC, we have also observed the cessation of the flow of solitons. In this case, a large number of solitons approached each other and formed a condensed phase. Because these solitons were very closely spaced, our detection system could not resolve them, so only a single very large pulse appeared on the oscilloscope. Figure 6(b) shows oscilloscope trace of the condensed phase with a pump power of 170 mW, the corresponding output power was 14.1 mW. The width of the bunch can be a few nanosecond or less than 1 ns, as determined by the number and the separation of the solitons in the condensed phase. Figure 6(c) shows the spectrum of the condensed solitons phase. Due to the phase of the solitons are not entirely locked, weaker modulations can only be observed in the spectrum. No obvious CW or dispersive waves in the spectrum indicates that direct soliton interactions play a main role in the case. Figure 6(d) shows the autocorrelation trace of the condensed phase. Higher intensity on both sides of the main peak further indicates that a large number of solitons whose phase are not entirely locked exist in the condensed phase.

In our experiment, the pumping power mainly affects the soliton number, and the dynamic regimes are sensitive to the orientation of the PC. We do not fully understand the effect of the orientation of the PC on the form of the dynamic regimes, one possible is that, there exists birefringence in the laser cavity, this birefringence may produce a linear, wavelength-dependent polarization rotation [24], hence adjusting the PC in the cavity may effectively induce intensity- and wavelength-dependent loss [25]. Then tuning the PC is indeed able to change the position, the intensity and the stability of the CW and the dispersive waves in the soliton spectrum. As mentioned above, these factors play an important role in forming different dynamical regimes.

5. Conclusion

Using graphene film as a saturable absorber, we have experimentally investigated multiple-soliton dynamic patterns in a ring fiber laser. The multiple-soliton dynamic patterns are determined by the continuous wave lasing, radiative dispersive waves and direct soliton interactions. Unstable continuous wave lasing gives rise to disordered multiple-solitons. With a weak continuous wave, the radiative dispersive waves play a more important role in the form of soliton bunching. When the CW lasing position is settled in an appropriate position in the soliton spectrum, multiple-solitons will distribute along the cavity with equal spacing to form high-order harmonic mode locking. Due to the strong nonlinearity of graphene and the all-fiber cavity of the laser, flow of solitons and condensed phase could also be observed with significantly lower pumping power compared with previous experiments.