Soliton molecules in a fiber laser mode-locked by a graphene-based saturable absorber

Erbium-doped fiber (EDF) lasers have attracted a great deal of research interest due to their wide applications in fiber sensors and communications [1–9]. They can deliver a continuous wave (e.g. dual-wavelength lasers [5, 6], multi-wavelength lasers [7–10], ultra-narrow linewidth lasers [11, 12]) or generate ultrafast pulses based on the mode-locking technique [13–22]. The passively mode-locked fiber lasers can operate on various states, such as hyperbolic-secant profile soliton [23–25], stretched pulses [26], self-similar pulses [27] and dissipative solitons [28–31]. Various phenomena are experimentally observed, such as multiple pulses [32], vector solitons [33], ultra-broadband pulses [34, 35], wave-breaking-free pulses [36, 37], dissipative soliton resonance [1], coexisting pulses [4, 38–43], bright-dark pulse pair [44], two different output states [45, 46], high power femto-second pulse in a long cavity [47]. Especially, Liu et al first proposed a distributed ultrafast fiber laser that is dependent on pulse wavelength [25]. Choi et al reported a dissipative-soliton laser by using an evanescent field interaction with graphene [31]. Cui et al reported conventional and dissipative solitons in a CFBG-based fiber laser [43]. Gao et al demonstrated a bright-dark laser pair in a fiber ring cavity, mode-locked by graphene [44]. Huang et al obtained two different applied output states from an all-normal dispersion ytterbium-doped fiber laser [45, 46]. He et al reported a long-cavity pulse generation fiber laser by using graphene saturable absorber-based multi-mode fiber [47].

In conventional soliton lasers, the pulse energy is limited to 0.1 nJ because of the cavity peak power clamping effect [48, 49] and multiple pulses would be formed under an excessive pump [49–51]. In practice, different multiple pulsing regimes can exist due to the soliton interaction [49]. In a bound state, strong phase locking can appear between pulses with comparatively fixed pulse–pulse separation. The bound-state soliton would not be easily destroyed by external perturbations [50]. In contrast, the binding force between pulses in bunch-state soliton is relatively weak. Each pulse may vibrate on a certain scale with the ruleless pulse–pulse separation. The bound- and bunch-state soliton can be regarded as special cases for soliton molecules [52–57]. Usually soliton molecules are characterized by the peak-to-peak separation (ρ) and the phase difference (Ψ) between pulses [52, 57]. The stability as well as amplitude of ρ and Ψ will strongly affect the modulation of output optical spectrum [53]. In previous work, fully modulated spectra have been reported when pulses in soliton molecules exhibit fixed pulse separation and invariant phase relation [54]. Simultaneously, vibrating and rotating phase molecules that perform a blurred modulated or an unmodulated spectrum have also been reported in [55] and [56], respectively.

Soliton molecules have been extensively investigated for fiber lasers operating in normal or anomalous dispersion regime [57, 58]. In a large normal dispersion regime, dissipative soliton molecules (DSMs) with an independently evolving phase, flipping phase, invariant phase, and a rotating phase have been demonstrated [59]. Liu has reported DSMs with an independently evolving phase in numerical simulation and experimental observation, and the pulse duration T₀ is about 20 ps with the pulse–pulse separation of about 8T₀ [57]. All of the above reports focus on bound-state molecules with a fixed peak separation and symmetrical autocorrelation trace. However, the operation of a bunch-state molecule with a flipping peak separation has not, to our best knowledge, been addressed yet.

In this paper, we report the experimental observation of a bunch-state molecule in a passively mode-locked erbium-doped fiber laser with anomalous dispersion. The pulse molecule acts as a unit and exhibits rectangular profiles on an oscilloscope while shows randomly distributed peaks on an autocorrelator. The optical spectrum of a pulse molecule shows random modulations that may be attributed to the interaction of the intra-molecule solitons. Three pairs of typical sidebands appear on the spectrum profile, which are characteristic for solitons operating at a negative dispersion regime [26]. These results demonstrate that solitons in the pulse molecule oscillate slightly in temporal domain and move as a unit at a fundamental cavity repetition rate, which is quite different from that of the multi-solitons or bound-state solitons.

The fiber laser configurations used in our experiment are schematically shown in figure 1. It has a ring cavity of a segment of 5.5 m erbium-doped fiber (EDF) with an absorption of 6 dB m⁻¹ at 980 nm, a polarization controllers (PC), a polarization-insensitive isolator (PI-ISO) to ensure unidirectional operation, a fused optical couplers (OC) with 10% output, and a 980/1550 nm wavelength-division multiplexed (WDM) couplers. The D-shaped-fiber saturable absorber (DF-SA) is deposited with CNTs by using an optical-deposition technique. The other fiber in the cavity is the standard single-mode fiber (SMF). The total length of laser cavity is 14 m. The dispersion parameter D for the EDF and SMF are about −42 and 17 ps nm⁻¹ km⁻¹ at 1550 nm, respectively. The net cavity dispersion and fundamental cavity repetition rate of the ring laser are estimated to be −0.03 ps² and 10.35 MHz, respectively. The laser cavity is pumped by a 980 nm laser diode (LD) with the output power up to 550 mW. An optical spectrum analyzer, an auto-correlator, and a 6 GHz oscilloscope with a 10 GHz photodetector are used to monitor the laser output simultaneously.

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A Nonlinear polarization rotation technique is widely used to achieve passive mode-locking in fiber laser [60–63]. In this report, however, we use a graphene-based saturable absorber to generate the solitons. In the experiments, we can achieve several kinds of soliton emission including the single pulse and bunch-state molecules. When the pump power is about 9 mW, the cavity operates at stable bunch-state molecule emission, as shown in figure 2(a). The separation between adjacent molecules is ~96 ns, which agrees well with the fundamental cavity repetition rate of 10.4 MHz. The molecule profile in figure 2(b) exhibits rectangular shape with a nanosecond duration. We suspect that each pulse in a molecule can not be directly observed due to the limited resolution of the oscilloscope (~200 ps).

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There are three peaks on the autocorrelation trace as shown in figure 3, which indicates that the rectangular molecule on the oscilloscope is not a single pulse. Figures 3(a)–(c) are recorded at a different time while under the same operation state. The separations between the neighboring peaks are ~75, ~65 and ~35 ps for figures 3(a) and (c), respectively. Furthermore, the experimental observation shows that the peak separation changes randomly. We can see that the state depicted in figure 3(c) has a different peak separation comparing with those of figures 3(a) and (b). The results here are quite different from that of other experimental results where the peak separations on an autocorrelation trace are identical [53, 57, 58]. Since the peak separation of autocorrelation trace represents the pulse–pulse separation, we can conclude from the above observation that the intra-molecule pulses have unstable relative positions. However, experimental results on the oscilloscope reveal that the soliton bunch could still move as a unit at the fundamental cavity repetition rate, which forms the state of pulse molecules. The inset of figure 3(c) shows that the full width at half maximum (FWHM) of the highest peak is 0.473 ps. Note that the blue and red curves in the inset are the experimental data and the fit curve, respectively. If a sech²—shaped temporal profile is assumed, the pulse duration is estimated as 0.307 ps. So the pulse–pulse separation is hundreds times that of the pulse duration.

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The theoretical and experimental results show that the peak-to-peak separation of pulses in the molecule vibrates randomly [57]. The experimental observations here demonstrate that the pulse vibration in the molecules is much larger than that in [57]. On the other hand, two or multiple solitons interact in the oscillator whereas they never form a molecule [32, 41, 49]. Obviously, our experimental results are different from the observations reported by Liu [25, 49].

As shown in figure 4, the spectrum exhibits characteristic profiles of conventional soliton with three pairs of sidebands. Based on the contour of the spectrum, we roughly estimate the 3 dB bandwidth to be ~8 nm and the corresponding time-bandwidth product to be 0.33. A notable feature is that ruleless modulation occurs on the whole spectral profile, which is different from the regular modulation of the bound-state soliton. The spectral modulation can be attributed to the pulse–pulse interaction. Because of the random distributed in the bunch-state molecule, the optical spectral exhibit ruleless modulation. The maximal modulation depth appears at the center of spectrum, which is about 7 dB. In addition, the modulation intensity decreases from the center to the verge.

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In summary, the pulse–pulse interaction depends on the ratio (γ) between the pulse separation and pulse duration. The smaller the parameter of γ, the stronger the interaction [49].

For the typical bound-state soliton, pulses are closely spaced in temporal domain, so the parameter γ is usually small and the pulse–pulse interaction is very strong. In the frequency domain, the corresponding spectrum exhibits clear modulation for a certain period. For the typical bunch-state soliton, the pulse separation is from a few to several-hundred times of the pulse duration with the comparably weak pulse–pulse interaction. As a result, the corresponding spectrum performs blurred, or even no modulation and pulses oscillate in temporal domain. For the proposed pulse molecules, the interaction is stronger than that of traditional bunch-state soliton, but weaker than that of a bound-state soliton, so the molecule operation state can be regarded as the transitional state between the bound- and bunch-state soliton.

At the same time, we achieved a conventional single soliton pulse emission in the same cavity, which also operates at the fundamental cavity repetition rate and has a similar pulse train to the bunch-state molecule as shown in figure 2(a). The spectrum of conventional single soliton as shown in figure 5 only exhibits clear sidebands without any modulation. Moreover, the corresponding autocorrelation trace (the inset of figure 5) performs only one peak. If a sech² temporal profile is assumed, the pulse duration is estimated to be 1.186 ps, which is three times more than that of the intra-molecule pulse. It is obvious that the characteristics of pulse molecules achieved in our experiment are obviously different from that of conventional solitons.

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In this paper, we have experimentally observed a kind of soliton molecules in a passively mode-locked erbium-doped fiber laser with anomalous cavity dispersion. The solitons in the molecule oscillate randomly around a stationary state in temporal domain and there exists only one molecule in the cavity. Meanwhile, the spectrum shows blurred modulation over the whole profile induced by the interaction of the intra-molecule solitons. Therefore, the proposed soliton molecule is a very novel operation state for fiber lasers and can be regarded as the transitional state between bound- and bunch-state soliton.