1. Introduction

Via the advances in fiber laser technology and ultrafast optics, passively mode-locked fiber lasers (PMLFLs) have been established as stable ultrafast sources for a variety of scientific and industrial applications [1–6]. From a theoretical point of view, PMLFLs can be regarded as nonlinear dissipative systems, incorporating saturable absorption, self-phase modulation, dispersion, amplification, spectral filtering, and higher-order nonlinearities [7]. Depending on how significant the impact of each effect is, various types of ultrashort pulses can be generated, i.e. solitons, similaritons, and dissipative solitons [8]. In contrast to such PMLFLs generating a coherent single pulse per roundtrip (RT), if the cavity conditions become extraordinary, i.e., the cavity is excessively pumped or amplitude-modulated, PMLFLs also start to operate in various quasi-mode-locked (QML) regimes, generating complex, multiple pulses per RT [9–26]. For example, Komarov et al. theoretically investigated the periodically-spaced stable multi-pulse operation in a fiber ring cavity [9], where they showed that the cavity could operate in a QML regime based on the nonlinear polarization rotation technique, depending on the angles of the polarization controllers. Soto-Crespo et al. reported the chaotic evolution of multiple pulses in a fiber ring cavity [11], where they showed that the cavity operation could be switched from a single-pulse regime to a randomly fluctuating multi-pulse regime, depending on the level of pump power. In particular, they also showed that with the pump power higher than a certain critical level, the multiple pulses turned into forming a bunched wave packet, operating in the so-called noise-like pulse (NLP) regime. In fact, NLPs have been observed experimentally in a variety of different types of PMLFLs [12–25]. In recent years, there have been a handful of reports that were aimed to resolve the complex dynamics of the NLP formation and their shot-to-shot characteristics, utilizing the dispersive Fourier transform technique or the single-shot measurement technique [20,23,24]. The authors have also extensively investigated the formation of NLPs in various cavity conditions, including the degree of polarization rotation, the level of pump power, and the net dispersion of the cavity [17].

Despite the aforementioned reports, the overall dynamics and origins of the various QML regimes have yet to be clarified: First, it is unclear what cavity conditions crucially lead to the distinctions in terms of the operation regimes, whilst some previous works tried to identify them from a limited perspective of the level of pump power [11,17] or of the transmission characteristics of the saturable absorber (SA) [14,17,20,25]. This remains an issue for the so-called symbiotic regime as well, in which NLPs and solitary pulses coexist [18–22]. Second, the phase stability of multiple pulses in QML regimes has not been investigated extensively, whilst for some specific cavity conditions the degrees of phase coherence between consecutive shots in NLP [17,23,24] or multi-soliton [26] regimes were separately discussed.

To fill these insufficiencies, we here present a phenomenological numerical study on the formation of multiple pulses and the interplays of various cavity parameters for all the constitutively different QML regimes. In our study, we adjust both the saturation power of the SA element and the length of the single-mode fiber (SMF) section in the laser cavity under the condition of high pump power and high output coupling ratio (OCR), in order to alter the type of the QML regime. This eventually leads to the changes in the amounts of nonlinear phase shift (NPS) and net cavity dispersion (NCD) per RT. We also analyze the phase stability of the output signal for the individual QML regimes via quantifying the modulus of the complex degree of first-order coherence (MDOC) of the output signal and present a contour map for it as a function of the cavity parameters. Relying on the MDOC contour map, we discuss the classification of the constitutively different QML regimes and their criteria, and finally draw our conclusion.

2. Numerical model

The fiber ring laser cavity we consider for our study is depicted in Fig. 1, which is conceptual but general enough for investigating the evolution of multiple pulses in QML regimes. The cavity consists of an erbium-doped fiber (EDF), an SMF section, and a fast SA element. In addition, a fixed portion of the intracavity power is extracted at each RT through the output coupler (OC) located after the SA element. The unidirectional propagation of the signal in the cavity is obtained by means of inserting an optical isolator (ISO) in the cavity. The theoretical modeling of each element of the laser cavity is given as follows. The nonlinear propagation of the signal through the EDF and SMF sections are modeled by the Ginzburg-Landau equation, expressed as [27]

 

Fig. 1 Schematic of the PMLFL under consideration. EDF: erbium-doped fiber, SMF: single mode fiber, ISO: isolator, SA: saturable absorber, and OC: output coupler.

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Table 1. Simulation parameters.

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As for the initial condition of the numerical simulation, we apply a 20-ps time-bandwidth-limited sech²-shaped pulse with 1 μW of peak power rather than applying a noise signal with random amplitude and phase. This method is to accelerate the convergence of the simulation without loss of generality [25]. After a cavity RT, the resulting signal is then used as a new initial value for the next RT calculation. The final convergence to the quasi-steady-state (QSS) solution is determined by the criterion such that the relative deviation of the total energy stored inside the cavity over a single iteration remains below 0.01, i.e., $\left|E_{tot,n}-E_{tot,n-1}\right|/E_{tot,n-1}<0.01$, for at least 50 RTs. It is worth noting that whilst we repeat the iteration for 1000 RTs, the QSS solution is normally obtained well before the maximum number of the RT times.

In addition, the phase stability of the output signal is quantified by the shot-to-shot spectral coherence via the MDOC calculated in the spectral-domain as [8]

3. Numerical results

3.1 QML cavity operation: NLP evolution

With the given parameters specified in Table 1, we numerically characterize the dynamics of the laser cavity, initially paying our attention to the formation of NLPs. It is worth noting that the given values for the gain per RT (20 dB) and the OC (70%) are substantially high, in contrast with the case supporting a stable single-soliton regime [17]. This means that the energy built up inside the cavity becomes to undergo large amounts of accumulation and dissipation in turn repeatedly during RTs. In such a condition of extreme gain and loss, the laser cavity tends to become unstable and will eventually be switched to operate in the so-called QML regimes.

As a typical example, we present in Fig. 2 the temporal and spectral behaviors of the optical signal generated in the laser cavity operating in a QML regime, where the saturation power and the length of the SMF section are given by P_sat = 1 kW and L_SMF = 100 m, respectively. One can see that the optical signal forms a packet of bunched ultrashort pulses randomly evolving in terms of amplitude and width at every RT. A distinct feature of the optical signal generated under these conditions, is that a packet of pulses remains isolated from all other parasitic pulses. In fact, the parasitic pulses tend to drift and fade away from the main packet or merge again into it after several RTs. These phenomena are highlighted with the white dashed arrows in Fig. 2(a) that shows the evolution of the optical signal in the time-domain with respect to the cavity RT number. It is worth noting that the evolution of the optical signal becomes to satisfy the condition for the QSS solution after 100 RTs. The packet of pulses consists of multiple sub-picosecond pulses with random heights and widths. A quantitative measure of the complex signal intensity is given by the RT-averaged auto-correlation (RTA-AC) trace shown in Fig. 2(d). The RTA-AC trace presents a coherent-spike on top of a broader pedestal, the time durations of which are 900 fs and 120 ps, respectively. The former indicates the duration of the individual sub-pulses constituting the packet, whereas the latter corresponds to the overall packet duration. The shape of the given RTA-AC trace is a very typical form of QML pulses, frequently observed in their experimental multi-shot AC measurements [12–14,16,17].

 

Fig. 2 (a) Temporal and (b) spectral evolutions of a time-BW-limited sech²-shaped input pulse with P_sat = 1 kW and L_SMF = 100 m with respect to the number of RTs. (c) Intracavity evolution of bunched sub-pulses shown in (a) for 5 RTs. (d) RTA-AC trace of the pulses shown in (a).

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The complex dynamics seen in the temporal domain subsequently gives rise to a highly structured spectrum in the spectral-domain, which also evolves stochastically at every RT as shown in Fig. 2(b). Nevertheless, the spectrum becomes to have a 20-dB BW of 14.8 nm after the first 100 RTs. We classify this operating regime into the NLP regime on account of its exact similarity with the characteristic features of the NLPs observed in many experimental reports [12–19]. On the other hand, it is worth noting that, besides the aforementioned NLP regime, different combinations of P_sat and L_SMF can also result in other types of QML regimes, in which the multiple pulses are generated in the civility but fail to form any isolated packet. To distinguish these types of regimes from the NLP regime, we classify them into multi-soliton (MS) regimes. Since an RTA-AC trace itself cannot provide detailed information on the complex intensity structure of the bunched multiple pulses and its temporal evolution, we take a zoomed-in snapshot of the random temporal evolution of the multiple pulses within a few RTs as shown in Fig. 2(c). The noticeable horizontal lines crossing the figure correspond to the significant changes in the optical signal while passing through the SA and OC elements shown in Fig. 1. A close investigation of the result shown in this figure is important in that it clearly reveals various distinct features of the temporal evolution of the multiple pulses in the QML regime. In particular, one can notice that some pulses propagate without undergoing strong fluctuations but others do not if one limits the number of RTs to a small number, as depicted in Fig. 2(c). For example, pulse a, located at t = −23 ps, propagates as a soliton without undergoing substantial fluctuations for at least 5 RTs, whilst pulse b splits and merges repeatedly for 4 RTs, but finally splits into two pulses owing to the interaction with pulse c between RT# 403 and RT# 404. On the contrary, pulse d is dispersed after 4 RTs, and a part of the pulse merges into pulse e. After having a phenomenological observation on the various distinct pulse evolutions for a small number of RTs, a “question” that arises is that if a QML regime embodies a fraction of such pulses that can propagate without undergoing significant fluctuations within a few RTs, can it contribute to enhancing the degree of spectral coherence of the output signal? Thus, in the next section, we will find the answer to it, investigating the degree of spectral coherence in conjunction with the distinct behaviors of the multiple pulses generated in various QML regimes.

3.2 Multi-pulse dynamics in the five constitutive QML regimes

In this section, we identify constitutively different QML regimes obtainable from the laser cavity specified in Fig. 1 and Table 1, while varying P_sat and L_SMF, and investigate their overall characteristics in conjunction with their wavelength-dependent MDOC characteristics. In fact, the whole simulation result that will be discussed in detail in the following indicates that one can classify them into five constitutive regimes: (1) the incoherent NLP regime, (2) the partially-coherent NLP regime, (3) the symbiotic regime, (4) the partially-coherent MS regime, and (5) the coherent MS regime, which are named according to their multi-pulse dynamics and spectral coherence properties, as represented in Fig. 3, where the temporal-evolution characteristics, RTA-AC traces, spectral characteristics, and wavelength-dependent MDOCs of the generated multiple pulses are illustrated individually for the five constitutive regimes. It is worth noting that the RTA-ACs, spectra and MDOCs are obtained via ensemble-averaging the corresponding parameters over the last 500 RTs.

 

Fig. 3 Temporal evolution (left), AC trace (middle), spectrum (right; lower), and MDOC (right; upper) in (a) the incoherent NLP regime with P_sat = 700 W and L_SMF = 125 m, (b) the partially-coherent NLP regime with P_sat = 700 W and L_SMF = 12.5 m, (c) the symbiotic regime with P_sat = 1200 W and L_SMF = 50 m, (d) the partially-coherent MS regime with P_sat = 1700 W and L_SMF = 125 m, and (e) the coherent MS regime with P_sat = 1700 W and L_SMF = 12.5 m.

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Let us start with the simulation results with relatively low saturation power combined with different lengths of the SMF section (P_sat = 700 W; L_SMF = 125 or 12.5 m), which are shown in Fig. 3(a) and 3(b), respectively. As can be seen in the first column of the figure, it is clear that in both cases the multiple pulses tend to form a packet of bunched ultrashort pulses and parasitic pulses isolated from the packet are not allowed to grow. The RTA-AC traces in the second column show the double-scaled structure that is one of the characteristic features of NLPs as shown in Fig. 2(d). Thus, the QML regimes for the multiple pulses illustrated in Fig. 3(a) and 3(b) are classified into the NLP regimes. Although these two regimes have similar temporal characteristics, there is a noticeable difference in terms of shot-to-shot stability. The insets of the first column of Fig. 3(a) and 3(b) show the magnified images illustrating the evolutions of the multiple pulses for 100 RTs. For the long cavity length case (i.e., L_SMF = 125 m) shown in Fig. 3(a), the evolution of the sub-pulses located inside the packet is significantly more chaotic and complex than those for the short cavity length case (i.e., L_SMF = 25 m) shown in Fig. 3(b). Whilst we admit that it is not straightforward to justify the origin of the chaotic nature at this stage, we would like to explain it from the following two perspectives, focusing on the consequence by increasing L_SMF: First, the increase of the cavity length implies the increase of the NPS per RT, which potentially tends to give rise to the collapse of solitons [13,29,30]. Thus, the long cavity length provides a harsh environment for the sub-pulses to survive as a stationary single entity. The complex sub-pulse dynamics incurred by the collapsed solitons can eventually lead to the chaotic evolution of the multiple pulses within the NLP packet, as having been verified in the previous reports [13,17].

Second, the increase of the cavity length provides more chances for sub-pulses to interact with other sub-pulses. Depending on the phase differences between the sub-pulses, they can attract or repel each other [28], and the amounts of the interactions depend on the amplitudes of the sub-pulses. Since the phases and amplitudes of the sub-pulses within the NLP packet are nearly random, the interactions with the neighboring pulses tend to result in irregular temporal shifts and amplitude fluctuations. The consequence of such dynamics has already been observed in Fig. 2(c). The increase of L_SMF means that the longer interaction length is provided before most of the pulse energy is lost by the SA and OC elements. Thus, for each RT, the temporal locations of the sub-pulses and their strengths tend to evolve as irregularly as shown in Fig. 3(a). We note that this cavity-length-dependent, characteristic feature is in accordance with those verified in the previous report [26].

The aforementioned two kinds of sub-pulse dynamics subsequently degrade the shot-to-shot stability of the NLPs. As a result, in the third column of Fig. 3(a), the MDOC becomes close to zero within the 20-dB BW, which means that in this QML regime, two consecutive NLPs have nearly no phase correlation, thereby leading to a considerable level of shot-to-shot fluctuations. In particular, this incoherent NLP regime is in accordance with the experimental result reported in Ref. 23. On the other hand, if L_SMF is significantly shortened as in the case of L_SMF = 25 m, the NPS per RT and the interaction between sub-pulses are correspondingly reduced, resulting in producing relatively more regular multiple pulses, as shown in the first column of Fig. 3(b). In particular, from the inset, one can see that some of the sub-pulses propagate even for a few tens of RTs without undergoing a considerable level of disruption. As a result, the MDOC of the multiple pulses in this regime is noticeably enhanced, in comparison with the incoherent NLP regime. Thus, this phenomenological evidence indicates that one should not completely exclude the existence of an NLP regime having a partial-coherence feature. We think this is an important numerical confirmation that can justify the different spectral coherence properties reported in Refs. 17 and 23. Interestingly, the calculated MDOC shows that the partial coherence mainly arises from the spectral range slightly deviated from the center wavelength, whereas the spectral contents near the center wavelength still bear a low MDOC. We emphasize that this spectrally structured, partial-coherence characteristic of QML multiple pulses has also been analyzed experimentally [20].

Next, when we increase the saturation power of the SA element a bit more and take an intermediate length of the SMF section such that P_sat = 1200 W and L_SMF = 50 m, respectively, we encounter a very intriguing QML regime as shown in Fig. 3(c), which is the so-called symbiotic regime [18–22]. In this regime, the sub-pulses bunched in a packet like NLPs and the coherent pulses isolated from the packet can coexist. We note that the moderate increase in P_sat in comparison with that for the previous NLP regimes eventually reduces the tendency of grouping of sub-pulses, so that as shown in the first column of Fig. 3(c), a few of sub-pulses can escape from the main NLP packet and form individual pulses that are temporally stable enough to coexist with the NLP packet. In contrast, if we increase P_sat even higher such as shown in Fig. 3(d) and 3(e), the tendency of grouping of sub-pulses into a packet completely disappears, and thus, the sub-pulses gradually spread out in the time domain. In fact, the laser cavity becomes to operate in the MS regimes. In this sense, the symbiotic regime can be regarded as the intermediate, transitional regime between the NLP regimes and the MS regimes, such that the sub-pulses bunched in the main packet behave like NLPs whereas the isolated sub-pulses behave like coherent MSs. This consequence is also reflected into its MDOC property shown in the third column of Fig. 3(c), where one can see a considerably high partial coherence, higher than that of the partially-coherent NLP regime. We attribute this increase to the coexistence of the individual coherent pulses.

When we increase the saturation power of the SA element even higher to P_sat = 1700 W, we observe that the QML operation is switched into MS regimes as already shown in Fig. 3(d) and 3(e), in which the lengths of the SMF section are set by L_SMF = 125 m and L_SMF = 12.5 m, respectively. It is worth noting that the main distinct feature of the MS regimes from the NLP regimes is the fact that sub-pulses tend to keep spreading out in the time domain without forming any stationary packet of bunched sup-pulses. In fact, one can justify that the extinction of a stationary packet of sub-pulses in the laser cavity is owing to the substantial increase in P_sat, because as for the lengths of the SMF section there are no significant differences between the MS regimes and the NLP regimes. Since the sub-pulses are not tightly packed in the cavity, the interactions between them become less significant for the MS regimes than for the NLP or symbiotic regimes, thereby leading to the significant increase in the MDOC as shown in the third column of Fig. 3(d) and 3(e). In addition to the reduction of the sub-pulse interactions, the substantial increase in P_sat can drastically alter the amount of the NPS that all the sub-pulses undergo in the cavity per RT. We note that the increase of P_sat means the increase of the cavity loss, thereby resulting in the decreases of the peak power levels of the individual sub-pulses. Thus, the NPSs that all the sub-pulses in the MS regimes undergo per RT tend to decrease significantly, so that the soliton-collapse effect cannot be apparent [29], as verified in Fig. 3(d) and 3(e). Thus the lower NPS per RT via the alleviation of the soliton collapse effect can be another reason for the increase of the MDOCs in the MS regimes. Actually, we think the amount of the NPS per RT is closely correlated with the shot-to-shot stability of multiple pulses. This issue will be discussed in more detail in Section 4.

In addition, the difference between the partially-coherent MS regime and the coherent MS regime is similar to the relationship between the incoherent NLP regime and the partially-coherent NLP regime. For the long SMF section (i.e., L_SMF = 125 m), as shown in Fig. 3(d), the evolution of the sub-pulses becomes relatively more irregular and chaotic, thereby resulting in the partially-coherent MS regime. In contrast, for the short SMF section (i.e., L_SMF = 12.5 m), the interactions between the sub-pulses are substantially diminished, forming more stable and coherent multiple pulses, as shown in Fig. 3(e).

As discussed above, we have verified that the two cavity parameters of P_sat and L_SMF play a pivotal role in determining the type of the QML regime and also the shot-to-shot stability of the corresponding multiple pulses. In order to visualize their characteristic relations, we present a contour map in Fig. 4(a), plotting the averaged MDOC with respect to P_sat and L_SMF and also indicating the whereabouts of the five constitutive QML regimes on it. It is worth noting that the averaged MDOC is quantified via the following equation:

 

Fig. 4 (a) Contour map of the averaged MDOC for 1 RT difference as a function of P_sat and L_SMF in conjunction with the specified locations of the five constitutive QML regimes. (b) Averaged MDOC as a function of the RT difference number between pulses for the five constitutive QML regimes. IC: incoherent; PC: partially-coherent; and C: coherent.

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In fact, we justify on the contour map five different regions for the five different constitutive QML regimes that include the incoherent NLP, partially-coherent NLP, symbiotic, partially-coherent MS, and coherent MS regimes, considering the characteristic features of the multiple pulses obtainable at the given location of P_sat and L_SMF, whilst we cannot distinctly classify every location into a specific regime, because the transitional and intermediate status of the multiple pulses soon becomes apparent in the border areas across the different regimes. Nevertheless, we can still say that the incoherent NLP regime, in which the averaged MDOC is below 0.2, is obtained with P_sat < 1000 W and L_SMF > 50 m. While either decreasing L_SMF or increasing P_sat from the incoherent NLP regime, we can expect the improvement of the shot-to-shot stability into the three partially-coherent regimes that include the partially-coherent NLP, symbiotic, and partially-coherent MS regimes. The partially-coherent NLP regime is obtained by decreasing the L_SMF below 25 m, in which the averaged MDOC varies from 0.3 to 0.5. The symbiotic regime is obtained with the conditions of 1100 W < P_sat < 1500 W and 40 m < L_SMF < 90 m, in which the averaged MDOC varies from 0.3 to 0.7, depending on the strengths of the isolated coherent pulses relative to that of the NLP packet. The partially-coherent MS regime is obtained with the conditions of P_sat > 1500 W and L_SMF > 80 m, in which the averaged MDOC varies from 0.5 to 0.8. By increasing P_sat higher than 1500 W and decreasing L_SMF shorter than 60 m, the QML regime finally turns into the coherent MS regime. As having been shown in Fig. 3(e), the shot-to-shot stabilities of the multiple pulses are well maintained, thereby resulting in the averaged MDOC higher than 0.8 in this regime.

In addition, we illustrate in Fig. 4(b) the averaged MDOC with respect to the RT difference number between the shots. It is worth noting that $\tau$ in Eq. (4) can be given by n/f_rep with an integer n, and that all the previous results shown in Fig. 3 and Fig. 4(a) were obtained with n = 1, which means the MDOC was evaluated between the two consecutive pulses. In contrast, we pay our attention to how the averaged MDOC evolves if n increases, i.e., the RT difference number increases. In the incoherent NLP regime, the averaged MDOC for n = 1 is about 0.12, and decreases further down below 0.1 for n = 2. In contrast, in the three partially-coherent regimes that include the partially-coherent NLP, symbiotic, and partially-coherent MS regimes, the partial coherence natures are maintained to a considerable level until n become 5 to 10, although they decrease to the similar level as in the incoherent NLP regime afterwards. In this regard, the coherence length of the multiple pulses is around the cavity length for the incoherent NLP regime and about 5 to10 times of the cavity length for the three partially-coherent regimes. In further contrast, the decreasing rate of the averaged MDOC in the coherent MS regime is much slower than the others. We have verified that the averaged MDOC eventually decreases to below 0.1 when n = 194, which means the coherence length of the multiple pulses operating in the coherent MS regime is exceptionally long relative to the other regimes.

4. Discussion

As shown in Fig. 3 and 4, the shot-to-shot coherence of the QML multiple pulses is critically affected by the cavity conditions that are parameterized by P_sat and L_SMF. We believe that the coherence property is closely linked with the amount of the NPS that the QML multiple pulses undergo per RT, because the multi-pulse dynamics is presumably determined by the soliton collapse effect and the sub-pulse interactions inside a packet if any. Thus, we rework the contour plot of the shot-to-shot coherence shown in Fig. 4(a) in terms of the amount of the NPS per RT and illustrate it in Fig. 5(a), which is quantified by the following equation:

 

Fig. 5 (a) Contour map of the NPS per RT as a function of P_sat and L_SMF in conjunction with the specified locations of the 5 constitutive QML regimes. (b) Averaged MDOC as a function of the NPS per RT. The blue dots and the green curve represent the original data set and the fitted curve with a Lorentzian distribution function, respectively. IC: incoherent; PC: partially-coherent; and C: coherent

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In fact, we emphasize that the reworked contour plot shown in Fig. 5(a) is greatly analogous to the one shown in Fig. 4(a). In other words, the averaged MDOC is presumably closely related with the amount of the NPS per RT. Then, we can reinterpret the mechanism of how the type of the QML regime is eventually determined in terms of the NPS per RT as the following: The NPS per RT below 0.2π assures the coherent MS regime, resulting in a high degree of shot-to-shot coherence. We note that the contour lines for φ_NL = 0.2π and φ_NL = 0.4π are nearly perfectly matched with those for ${\left|g^{(1)}\right|}_{ave}$ = 0.9 and ${\left|g^{(1)}\right|}_{ave}$ = 0.8, respectively. The φ_NL ranging from 0.4π to 2π corresponds to the averaged MDOC ranging from 0.3 to 0.8. We note that in this range the three partially-coherent regimes are obtained. For φ_NL > 2π, the QML regime turns into the incoherent NLP regime, in which the averaged MDOC becomes lower than 0.2. In particular, in the case of the incoherent NLP regime shown in Fig. 3(a), the NPS per RT has reached 6.5π, thereby exhibiting a very typical chaotic feature of NLPs. In fact, the close correlation between the averaged MDOC and the NPS per RT is well verified in Fig. 5(b). Actually, the data points can be fitted by

In addition to the NLP per RT, we here consider another critical parameter that determines the type of the QML regime in the given condition, which is the NCD. Since the NCD is simply given by the following relation: NCD = β_2,EDFL_EDF + β_2,SMFL_SMF, the y-axis of Fig. 5(a) can be reinterpreted into the amount of the NCD, because the first term of the relation is fixed in our case. Then, one can say that the incoherent NLP and coherent MS regimes are mainly determined by the amount of the NPS per RT. In contrast, in the cases of the other three partially-coherent regimes, i.e., the partially-coherent NLP, symbiotic, and partially-coherent MS regimes, the amount of the NCD plays a critical role in precisely determining the type of the QML regime in addition to the amount of the NPS per RT. In other words, the five constitutive QML regimes are now determined by the two general cavity parameters of the NPS per RT and the NCD, instead of P_sat and L_SMF. In a quantitative manner, the cavity operation can be classified into three cases in terms of the amount of the NCD: (1) If the NCD is over −1 ps², the QML regime turns into either the partially-coherent NLP regime or the coherent MS regime. (2) If the NCD is below −2 ps², the QML regime turns into either the incoherent NLP regime or the partially-coherent MS regime. (3) If the NCD is between −1 ps² and −2 ps², various QML regimes, including the symbiotic regime, can appear, depending on the detailed conditions of the NPS per RT and the NCD. On this foundation, we would like to add two more points on the alternation of the QML regime: First, the symbiotic regime that is located approximately at the center of the contour map has very intermediate characteristic features that can also be quantified in the middle of both the NPS per RT and the NCD, so that it can evolve into any of the other four regimes, depending on the amounts of effective nonlinearity and dispersion of the laser cavity. In this regard, the symbiotic regime can be thought as a four-way saddle point. Second, the MS regimes can somehow turn into the NLP regimes by increasing the NPS per RT regardless of the amount of the NCD. In other words, the formation of an NLP packet is critically related with the amount of the nonlinearity in the given laser cavity, regardless of the amount of the NCD as having been verified experimentally in Refs. 11, 17, and 20.

5. Conclusion

We have numerically studied the multi-pulse dynamics in a fiber ring laser cavity. By applying a large amount of gain and loss per RT to the laser cavity, we could make it operate in QML regimes. Adjusting two key cavity parameters of P_sat and L_SMF, we could trigger five distinct constitutive QML regimes, which include the incoherent NLP, partially-coherent NLP, symbiotic, partially-coherent MS, and coherent MS regimes. We paid particular attention to analyzing the shot-to-shot stabilities of the QML regimes, in order to figure out the justification and identification of the dynamics and origins of the individual regimes. We found that the generated NLPs could be either incoherent or partially coherent, and that the generated MSs could be either coherent or partially coherent, depending on the cavity conditions parameterized by P_sat and L_SMF. In particular, we numerically clarified and confirmed that the existence of the symbiotic regime recently observed experimentally [18–22], in which NLPs and MSs could coexist stably at the same time.

Based on the whole numerical investigations extensively carried out here, we quantified the shot-to-shot coherence characteristics of all the constitutive QML regimes in terms of the averaged MDOC, and presented them in a contour map with respect to P_sat and L_SMF. Reinterpreting the contour map, we drew the relationship between the shot-to-shot coherence and the amount of NPS per RT, and verified that both characteristics are strongly correlated. In addition, we justified the origins of the individual QML regimes in terms of the NPS per RT and the NCD, instead of P_sat and L_SMF, and showed that the NCD plays a very critical role in determining which QML regime the laser cavity would operate in, particularly when the NPS per RT is in an intermediate level, whereas the NPS per RT itself mainly determines the shot-to-shot coherence characteristics. These results are all in consistent with the previous, experimental observations [17]. Furthermore, we verified that the symbiotic regime is a four-way saddle point, on which the QML regimes could evolve into any of the other four constitutive QML regimes, depending on the degree of the modification of the cavity parameters. Whilst our discussion is currently up to the QML multi-pulse dynamics in the anomalous dispersion regime, we believe that it can be extended to other regimes, including the net-normal or all-normal regimes [14,25] in due course.