1. Introduction

Passively mode-locked fiber lasers have attracted considerable interest due to their high efficiency, environmental stability, low cost, and compactness, features that are advantageous for many applications in scientific and industrial fields [1–3]. Compared with the traditional solitons [4, 5] and dispersion-managed solitons [6, 7] that are employed in other cavity dispersion arrangements, the dissipative solitons [2] that are used in all-normal dispersion (ANDi) fiber lasers are capable of generating higher-energy compressible pulses. The pulse formation of dissipative solitons is governed not only by the cavity nonlinearity and dispersion, but also by the gain and loss [2].

In passively mode-locked fiber lasers, saturable absorbers, which may be either physical or artificial, are essential for stable mode-locking. Physical saturable absorbers [8–11] lack long-term reliability and high power capacities. However, artificial saturable absorbers such as those use nonlinear polarization rotation [12], as well as nonlinear optical loop mirrors (NOLMs) and their variants [13–15], can withstand high peak power and can be utilized to construct all-fiber mode-locked lasers. Compared with fiber lasers mode-locked by nonlinear polarization rotation, NOLM-based lasers can be fabricated using polarization-maintaining fibers and therefore are more environmentally stable and reliable.

However, NOLM-based ANDi fiber lasers with dissipative solitons have some limitations. For example, multi-pulsing (MLP) [16–18] or dissipative soliton resonance (DSR) [14, 17, 19] usually occurs when the pump power is increased to a certain level. In the MLP regime, as the peak power increases, the pulse breaks into several pulses with lesser energies and lower peak powers [17, 20]. In the DSR regime, rectangular-shaped pulses can be generated with high energies [14]. Li et al. demonstrated that DSR pulses can be efficiently compressed. In their experiment, a 63 ps DSR pulse was dechirped to 760 fs. However, the pulse compressibility deteriorated due to its nonlinear frequency chirp, and the narrow spectral width of the DSR pulse limited the dechirped pulse duration. Apart from the MLP and DSR regimes, passively mode-locked lasers can operate in a regime characterized by unstable picosecond pulse bundles consisting of stochastic femtosecond subpulses [21, 22]. Such noise-like pulses are incompressible and thus inappropriate for ultrafast pulse generation.

Notably, dissipative solitons do not only occur in ANDi cavities. They can also exist in other types of cavities, because they are defined as localized solutions in systems in which there is substantial energy exchange with the environment [2]. Strong dissipation is the key to pulse formation and stability in ANDi fiber lasers. Thus, the DSR pulses formed in ANDi cavities can be regarded as constituting a sub-category of dissipative solitons [19]. MLP is a common phenomenon in both traditional solitons [5] and dissipative solitons. There are also stable, non-pulse-breaking, bell-shaped dissipative soliton pulses in ANDi cavities. In the remainder of this manuscript, we specifically use the term typical dissipative soliton (TDS) to refer to bell-shaped dissipative soliton pulses. Hence, three general states can be identified for the dissipative solitons formed in ANDi fiber lasers: TDS, MLP, and DSR states.

Although it is possible to optimize laser cavities to avoid destructive effects and improve the pulse properties, the experimental optimization method is cumbersome, which might result from a lack of understanding of the physical processes involved. Consequently, numerical methods are needed to reveal the connections between the pulse dynamics and laser structural parameters. Moreover, numerical analysis would play a constructive and significant role in the development of new laser designs. For instance, to overcome the power limitations induced by MLP, excessive nonlinear spectral broadening should be alleviated. It has been demonstrated that large-mode-area fibers with small nonlinear coefficient could be used to increase the MLP threshold and, consequently, the pulse energy [23].

Numerical simulations of dissipative soliton generation have been performed in numerous studies. Nevertheless, the properties of the pulsing states (i.e., the TDS, DSR, MLP, and noise-like pulses) and dynamics of transitions between them have only partially been reported [17, 21, 22, 24–26]. In those studies, a single-control-variable method that involves leaving the other laser structural parameters fixed has commonly been applied. In that method, a specific pulse state is chosen as the starting point, and one parameter is then varied to investigate the pulse property evolution. However, due to the coupling among the laser structural parameters, the characteristics of the transitions between pulse states that are determined by utilizing the single-variable-control method may change if other parameters are altered. To overcome this limitation and reveal the relationships between parameters, we used a two-dimensional control variable method, which is called a two-dimensional parameter space method in this paper.

Two-dimensional parameter spaces have been employed in numerous previous works, especially in the framework of the complex cubic-quintic Ginzburg-Landau equation. Stable pulse state distribution in the normal dispersion regime was discussed in [27]. The distribution of unstable solutions such as pulsating solitons, chaotic solitons, solutions exhibiting period doubling, etc. was investigated in [28]. However, the Ginzburg-Landau equation neglects the importance of intracavity pulse evolution, since it includes the key physical processes of mode-locked lasers in a distributed manner. Thus, Li et al. [17] applied the discrete model instead and presented the distributions of the MLP and DSR states in saturation-power–bandwidth parameter space in their report, yet the pulse parameters (peak power, pulse energy, etc.) were not quantitatively characterized. In [29], Bale et al. reported on the use of reduced equations to calculate the pulse parameters directly, which saved computational time. However, the pulse envelope and spectrum were not simulated, leaving the pulse state distributions unknown and unstable pulse states neglected. In our work, a comprehensive investigation of the pulse state distributions was conducted using a more visual approach by strictly solving the related equations in the discrete model and simultaneously obtaining the pulse parameters and the corresponding pulse states.

To date, no systematic numerical analysis of the state distributions in ANDi fiber laser setups has been performed. By using the numerical model and parameter space method described in this paper, we report for the first time the pulse state distributions in a wide range of stable and unstable regions in NOLM-based mode-locked ANDi fiber lasers. Since we are interested in utilizing large-mode-area fibers to obtain high pulse energies, an all-10/125-fiber laser model was employed, which is different from the models used in many other mode-locked fiber laser simulations. Section 2 describes this numerical model in detail. Then, the simulation results are presented and discussed in Section 3. Finally, Section 4 contains our conclusions.

2. Numerical model

The concept of a typical NOLM-based ANDi fiber ring laser, also known as a figure-of-eight laser, is schematically illustrated in Fig. 1. In our study, each part of the laser cavity was simulated separately. The bandpass filter width (BPFW) was varied, and a typical seven-order super-Gaussian profile was applied to model the rectangular-shaped transmission spectrum of the filter. All of the parameters used for each part are listed in Table 1 and are either typical or calculated values.

 

Fig. 1 Schematic diagram of the laser cavity used in the simulation. SMF = single-mode fiber; OC = output couple.

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Table 1. Parameters of the numerical model

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As shown in Table 1, the laser components such as the filter, passive fibers, active fiber, pump, and coupler are described by numerous parameters. Among these, we selected four parameters, specifically, the BPFW, length of the second SMF L_SMF2, gain saturation power P_sat, and NLOM loop length L_Loop, which contribute to the spectral filtering loss, spectral broadening, gain, and saturable absorption, respectively, in ANDi fiber lasers. Length of the first SMF L_SMF2 and length of active fiber L_AF are constants. In our simulation, we focused on these factors and researched the pulse state distributions in two-dimensional parameter spaces, which is similar to drawing a map by varying two parameters simultaneously. Specifically, we used BPFW–L_SMF2, BPFW–P_sat, and BPFW–L_Loop parameter spaces.

We did not calculate the state distributions in parameter spaces that did not include the BPFW mainly because the BPFW prominently influences the state distributions. Also, varying the BPFW does not require additional computing resources, while varying the other parameters would substantially increase the simulation time and required memory space. Moreover, when calculated using different BPFWs, the state distributions in the excluded spaces could change significantly, which would make it difficult to select an appropriate BPFW. Thus, it is reasonable to focus primarily on the most influential factor, and we believe that the selected parameter spaces are adequate to investigate the coupling between the chosen parameters.

Optical pulse propagation in optical fibers can be described by using the generalized nonlinear Schrodinger equation (GNLSE). Since the pulses simulated in our model were comparatively wide (>10 ps), self-sharpening and intrapulse Raman scattering were ignored [30]. Due to the relatively high Raman thresholds of large-mode-area fibers and the short cavity length, we did not consider the stimulated Raman scattering effect. This effect contributed little to the pulses generated in our simulation, especially after they become stable. Considering second- and third-order fiber dispersion, the GNLSE can be written as follows [30]:

We numerically solved Eq. (1) by utilizing the interaction picture method in the frequency domain, which was demonstrated in [31] to be faster and more precise than the commonly used split-step Fourier method. To accelerate the calculation process further, we used a conservation quantity error algorithm [32, 33] to optimize the step size automatically and to limit the computing error. By applying this method, we were able to control the computing error threshold and calculation speed explicitly. A small error threshold usually results in small optimized step sizes and thus a long computing time.

Gain saturation was also considered in this work. As a consequence of the comparatively long excited state lifetime of Yb³⁺, the pulses were treated as a continuous wave signal with the same average power. Thus, the gain saturation of the amplifier was modeled according to

In our method, the gain saturation model and interaction picture method were applied in a split-step manner. The iterative process in an active fiber consists of the following steps: 1. determination of the length of the current step; 2. calculation of the gain coefficient using Eqs. (2) and (3) with the envelope of the pulse injected; 3. solution of the GNLSE using the interaction picture method to calculate the pulse for the next step; 4. termination of the iteration process if the output end of the fiber is reached, and return to step 1 otherwise.

To simulate the saturable absorption of an NOLM, which consists of a coupler and a passive fiber section, the incident light field was firstly divided into clockwise and counter-clockwise light fields, using the method described in [13, 14]. During this stage, the coupler in the NOLM was considered to be a lumped device with $A_{CW}=\sqrt{\rho }{A}_{in}$ and$A_{CCW}=i\sqrt{1-\rho }{A}_{in}$, where $A_{CW}$ and $A_{CCW}$are the two light fields divided by the coupler and $\rho$is the coupling ratio. Next, the propagation of each of the fields was simulated individually according to Eq. (1). The interference between the two anti-propagating fields was then simulated to obtain the transmitted light field $A_t\text{=}\sqrt{\rho }{A}_{CW}^{\text{'}}+i\sqrt{1-\rho }{A}_{CCW}^{\text{'}}$, where $A_{CW}^{\text{'}}$and $A_{CCW}^{\text{'}}$ are the estimated amplitudes of the two light fields after propagating through the fiber loop. In this way, nonlinear spectral evolution inside the fiber segment of the NOLM was included. Simply modeling the NOLM as a lumped element with an ideal sinusoidal saturated absorption profile would not have been sufficiently precise, because non-negligible disturbances occur on saturated absorption curves incubated by complicated pulse spectra under high pump powers.

The simulation began with Gaussian white noise windowed by a ~30 ps sech-shaped profile, to accelerate the calculation process. After several tens of roundtrips, the temporal and spectral outputs stabilized under specific parameters, indicating that convergence was achieved and the pulse evolution inside the cavity was self-consistent. To enable large numbers of pulse states in the parameter spaces to be calculated automatically, a convergence assessment method was implemented. In brief, we kept only two sequential spectral output shots, whose cross-correlation was calculated later to estimate their similarity. Once the cross-correlation coefficient exceeded a certain threshold, the simulation was terminated. This concept could be applied to temporal pulse shots in a similar manner. Considering that the shapes of pulses vary less than their spectra near convergence, we used the spectral cross-correlation method in our simulation.

3. Simulation results and discussion

3.1 State distribution overviews

To obtain overviews of the state distributions, we first simulated the pulse formation by using L_Loop = 1.5 m and P_sat = 23.5 dBm and varying the BPFW (from 8 nm to 20 nm) as well as L_SMF2 (from 1 m to 30 m). Then, we estimated the spectral width, peak power, pulse energy, and pulse width based on the simulation output, which yielded the mosaic-liked results for the BPFW–L_SMF2 space that are presented in Fig. 2. The coordinates of the lower left points of the colored rectangles correspond to specific parameter-pair values. To associate the pulse states with different regions in the parameter space, we refer to the regions by using their labels (capital letters in Fig. 2) in this subsection. In the remaining subsections, we directly use the pulse state abbreviations for clarity and concision, except for the unstable and non-mode-locked regions.

 

Fig. 2 State distributions in BPFW–L_SMF2 parameter space when L_Loop = 1.5 m and P_sat = 23.5 dBm. (a) Spectral width. (b) Peak power. (c) Pulse energy. (d) Pulse width. The dark blue areas indicating zero values correspond to unstable regions that failed to converge. Four state regions are labeled with capital letters (A: DSR, B: TDS, C: unstable non-vanishing solution including pulsation and noise-like pulses, D: non-mode-locked states). Four points, which are labeled with numbers, were chosen and used to obtain the temporal and spectral pulse output examples displayed in Fig. 3.

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Figure 2 displays state distributions typical of the ANDi fiber laser simulation model. The levels of the four laser output metrics are presented using colors ranging from blue to yellow. The values of the laser output metrics at the points at which the simulation could not converge appear as zero (dark blue). Four state regions are clearly distinguishable in Fig. 2(a): the upper area (Region A, with a narrow spectrum), the middle yellow area (Region B, with a wide spectrum), the middle-left dark blue area (Region C), and the lower-right dark blue area (Region D). As discussed in the following paragraphs, Regions A and B exhibit the properties of the DSR and TDS states, respectively.

There is a distinct boundary (BPFW ≈15 nm) between Regions A and B (DSR and TDS) in Fig. 2(a), which was simply caused by the method we used to estimate the spectral width. Since we located the half-power point by searching from both sides of the frequency domain, when the pulse evolved to the DSR state, the quick disappearance of the sharp spectral edges of the TDS state leads to a sudden decrease in the calculated spectral width.

To identify the pulse characteristics in the stable regions (Regions A and B), we selected four points in whose vicinities the most common pulsing characteristics were apparent. Those points are indicated in Fig. 2(a) and correspond to the conditions L_SMF2 = 10 m or 25 m and BPFW = 11 nm or 16 nm. The simulated output pulse and spectral profiles obtained at each of the selected points are presented in Fig. 3. It is obvious that Points 1 and 2 (blue and yellow) correspond to the TDS state because the laser output profiles acquired at those points exhibit the typical steep edges, while their spectral profiles are cat-ear-like. Also, the pulses are almost linearly chirped. In Fig. 2, as L_SMF2 increases from 10 m to 25 m, the spectral width in the TDS state levels off at ~30 nm; the pulse energy and width increase from ~20 nJ to ~60 nJ and ~20 ps to ~40 ps, respectively; and the peak pulse power decreases slightly. Meanwhile, the output pulse profiles obtained at Points 3 and 4 (orange and purple) exhibit several characteristics typical of DSR. Notably, the introduction of greater dispersion by lengthening SMF2 led to broader pulses and higher energies increasing almost to infinity in Figs. 2(c) and 2(d), which are key properties of DSR [19]. Furthermore, the pulse profiles have flat tops, and the slope ratios of the frequency chirps are small near the center and large near the edges.

 

Fig. 3 Output pulse and spectrum profiles. (a) Normalized output spectra. (b) Output pulses. (c) Output pulse chirp profiles. The four state points are marked and numbered in Fig. 2 using the same colors as the lines: Point 1, blue, L_SMF2 = 10 m, BPFW = 11 nm; Point 2, yellow, L_SMF2 = 25 m, BPFW = 11 nm; Point 3, orange, L_SMF2 = 10 m, BPFW = 16 nm; Point 4, purple, L_SMF2 = 25 m, BPFW = 16 nm. Points 1 and 2 are typical dissipative solitons (Region B); Points 3 and 4 accord with the properties of DSRs (Region A).

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There are unstable but non-vanishing solutions in the middle-left area (Region C) in Fig. 2. The simulated pulses could not converge to stationary solutions in this region, but rather either exhibit so-called short-period pulsations [35] or approach stochastic solutions. One example of pulsation (BPFW = 12 nm, L_SMF2 = 1 m) is provided in Fig. 4, in which the spectrum output after each cycle is depicted. Considering the clear four-roundtrip periodicity, we attribute this state to the short-period pulsation described in [35]. Furthermore, it is in agreement with [2] that solutions with such few-cycle periodicity were only found in parts of the unstable region (Region C) and that these parts are located near the boundary between Regions B and C. In terms of stochastic solutions, the temporal waveform exhibits oscillatory structures along the top, while the pulse as a whole has an approximately flat top. Thus, this state may be a transition stage of DSR and noise-like pulse states, as discussed by Cheng et al. in [21]. In the light of their work, by increasing P_sat to 30 dBm, which is much higher than the value we used to obtain Fig. 2, the noise-like state was eventually obtained in this area. A spike appeared in the pulse autocorrelation trace, and the pulse chirp became much messier, indicating a high degree of randomness.

 

Fig. 4 Output spectrum evolution of short-period pulsation state over 100 roundtrips.

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Last, in the lower-right area (Region D) in Fig. 2, pulses could not stably form, which occurs in the so-called non-mode-locked state. We observed the pulses repeatedly rise and then decay in several roundtrips. The cause of this behavior is the fact that the net gain was too small to support a pulse. In this case, the pulses exhibit more nonlinear loss since the cavity length is increased and the BPFW is narrower with respect to the length and BPFW in Region B. Once the pulses rose above the noise, their spectra broadened mainly through self-phase modulation (SPM) in the SMF and were then absorbed by the bandpass filter, while the background noise existed in the middle of spectrum and therefore suffered less loss. The pulse loss increases as the SMF length increases and the BPFW decreases. This competing effect formed the border separating Regions B and D, along which the pulse gain and loss are nearly comparable.

3.2 NOLM-induced mode-locking pulsation

In this subsection, we discuss in detail the short-period pulsation point in Region C by presenting the intracavity spectral evolution recorded in the last two consecutive roundtrips, as displayed in Figs. 5(a) and 5(b). The pulse spectrum obtained in the 101st roundtrip [Fig. 5(a)] exhibits two intensified sidebands in the transmitted NOLM port [Fig. 6(a)]. After that, the sidebands exhibit attenuation due to the filter, and the central band appears to have been built up again in the active fiber in the next roundtrip [Fig. 5(b)]. The pulse spectrum evolved similarly in the following two roundtrips, except that the spectral bands were broader as displayed in Fig. 5. Due to this process, the laser output repeated itself every four roundtrips. Having observed the intensified sidebands in the NOLM results, we determined that the spectral properties of the NOLM play an essential role in this case.

 

Fig. 5 In-cavity spectral evolution of the pulsation state in the (a) 101st and (b) 102nd roundtrips. Only the propagation of the clockwise light field in the NOLM, which corresponds to the high-tap-ratio side in our simulation, is displayed.

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Fig. 6 Pulse interference at the transmittance port of the NOLM in the 101st roundtrip. (a) Pulse spectra after propagation inside the NOLM loop and their interferential spectrum. (b) Amplitudes of clockwise, counter-clockwise, and transmitted pulses.

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To validate this conclusion, we calculated the saturated absorption curve depicted in Fig. 7 based on the input and output pulses of the NOLM. The input pulse was assumed to have an intensity equal to the sum of the intensities of the clockwise and counter-clockwise light fields. Dividing the transmitted pulse amplitude by the input pulse amplitude yielded the transmittances corresponding to the instantaneous pulse powers in different parts of the pulse. In this step, we calculated the saturated absorption curves for the 101st and 103rd roundtrips. For comparison, a theoretical curve was also calculated according to the method described in [36].

 

Fig. 7 Virtual saturable absorption curves (Sim1 and Sim2) calculated using the clockwise, counter-clockwise, and transmitted beams of the NOLM in the 101st and 103rd roundtrips and determined theoretically.

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Compared with the theoretical curve (the yellow line in Fig. 7), the simulated curves (the blue and red lines in Fig. 7) exhibit severe disturbances, and the saturated power corresponding to the maximum transmittance is around 2500 W. Given that the pulse is nearly linearly chirped, the central band of the spectrum corresponds to the center of the pulse, where the instantaneous power exceeds 2500 W. Hence, there are dips in the centers of both the spectral [Fig. 6(a)] and temporal [Fig. 6(b)] domains due to the anti-saturable absorption.

In our simulation, the input light field of the NOLM consisted of chirped pulses with complicated spectral components instead of monochromatic continuous light. Different parts of the pulse corresponded to frequency components with distinct intensities, causing the nonlinear SPM-induced phase shift to fluctuate and resulting in disturbances in the interferential transmittance of the NOLM. Apart from that, the SPM-induced spectral broadening also influenced the transmittance curve. In Fig. 7, a platform is observable in the instantaneous pulse power from 300 W to 1300 W. Judging from the pulse chirps, this region corresponds to the oscillatory side-band of the pulse spectrum. Because of the power imbalance triggered by the coupler ratio, the high-power clockwise pulse spectrum is wider than the counter-clockwise one. This side-band mismatch is evident in Fig. 6(a) and results in a platform in the transmittance curve. The disturbances in the transmittance curve are responsible for the instability of DSR, as demonstrated in [21]. Consequently, the flat tops of DSR pulses oscillate stochastically over many roundtrips and cannot converge to stationary solutions.

3.3 NOLM loop length variation

To investigate the impacts of the NOLM on the state distributions, we focused on the BPFW–L_Loop space. The length of SMF2 was set to 2.5 m in order to include Regions A, B, and C. Keeping the other parameters fixed, BPFW and L_Loop were varied from 5 nm to 20 nm and from 1 m to 4.5 m, respectively. All of the pulse states mentioned in Subsection 3.1 were included. The same labels are used to indicate these states in this subsection. Similarly to in Fig. 2, the spectral width, peak power, pulse energy, and pulse width of the simulation output are depicted in Fig. 8. The area containing blue points represents to the MLP region. The maximum number of subpulses in an MLP bundle is five, corresponding to the bottom-right corner of Fig. 8(a). Since the distribution of the exact subpulse numbers is complicated, we merged the MLP points in the whole region.

 

Fig. 8 State distributions in BPFW–L_Loop parameter space when L_SMF2 = 2.5 m and P_sat = 23.5 dBm. (a) Spectral width. (b) Peak power. (c) Pulse energy. (d) Pulse width. The dark blue areas indicating zero values represent unstable regions, and the area containing blue dots in (a) corresponds to the MLP state. The MLP region is labeled with the capital letter E. In Region E, the total energy of each pulse bundle, instead of the energies of the individual pulses, was calculated.

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As shown in Fig. 8(a), when L_Loop ≥ 1.5 m, there are two critical BPFW values corresponding to specific L_Loop values. For instance, when L_Loop = 3 m and the BPFW is greater than 11.5 nm, the pulses are in the DSR state. When the BPFW becomes less than 11.5 nm, the output immediately exhibits instability. Mode-locking is achieved again in the MLP state when the BPFW is ≤ 7.5 nm. Then, as the BPFW decreased further, the cavity evolved to an unstable state because the net gain was reduced and was not able to support a pulse.

The two critical BPFW values decrease as L_Loop increases, which corresponds to a decrease in the saturated power of the NOLM. As L_Loop decreases to less than 1.5 m, the unstable region (Region C) does not appear in Fig. 8. In addition, as depicted in Fig. 2, Region C also disappears as the length of SMF2 increases because of the lower peak power. In other words, using either a short NOLM loop or a long SMF2 can cause the pulse to be transformed continuously from the TDS state to the DSR state as the BPFW increases. The distribution of the continuous transition is quite similar to that reported in [17]. However, only one critical BPFW value was revealed in that work, since the Region C was excluded. A possible reason that the unstable region was excluded is that the spectral disturbance of the saturated absorption curve was neglected, making unstable non-vanishing solutions difficult to achieve.

Meanwhile, the transitions between the TDS, DSR, and MLP states are critically influenced by both the BPFW and L_Loop. As shown in Fig. 6(b), if the BPFW is sufficiently small (<10 nm), the TDS state will be transformed into the MLP state as L_Loop increases, given that other parameters are fixed. This characteristic indicates that the MLP power threshold decreases, since the pump power remains the same. When BPFW < 10 nm, a rather large value of L_Loop (usually tens of meters) is required to transform MLP to DSR according to our simulation. When BPFW > 10 nm, as L_Loop increases, the TDS state could become rather unstable or turn into a TDS–DSR mixed state states (BPFW > 14.5 nm) and eventually evolve into DSR, bypassing the MLP state.

The peak-power-clamping effect caused by anti-saturable absorption is the main factor in the formation of DSR and MLP, as is manifested clearly in Fig. 8(b). The peak powers in both the DSR and MLP regions smoothly decrease as L_Loop increases. At the same value of L_Loop, the peak powers in both regions are almost the same, which indicates that the NOLM imposes the peak-power-clamping effect on both states equally. Meanwhile, the BPFW determines whether DSR or MLP is favored when anti-saturable absorption is activated. In terms of pulse energy, the pulse bundle energy distribution in the MLP region in Fig. 8(c) exhibits many oscillatory patterns. As L_Loop increases, the pulse bundle energy tends to increase. Because of the intensified power-clamping, more subpulses emerge to join the pulse bundle. Thus, the pulse bundle energy increases due to the increase in the total pulse width of the bundle, which is similar to that in the DSR region. Regarding the pulse width in Fig. 8(d), the subpulse duration remains almost the same in the MLP region. As the peak power decreases, the subpulse energy decreases. On the contrary, the corresponding pulse bundle energy increases, indicating that the number of subpulses and the total width of the pulse bundle both increase.

3.4 Pump power variation

Here we present the simulation results in BPFW–P_sat parameter space. L_Loop was set to 2 m to reduce the MLP pump power threshold. As depicted in Fig. 9(a), the TDS (yellow area, Region B), DSR (blue area, Region A), MLP (dotted area, Region E) and unstable (deep blue area, Region C) regions are all included.

 

Fig. 9 State distributions in BPFW–P_sat parameter space when L_SMF2 = 2.5 m and L_Loop = 2 m. (a) Spectral width. (b) Peak power. (c) Pulse Energy. (d) Pulse Width. The dark blue area (Region C), corresponding to a value of zero represents the unstable region. The area containing blue dots in (a), Region E, corresponds to MLP operation. In Region E, the total energy of a pulse bundle was calculated instead of the energy of each individual pulse.

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Near the bottom of Fig. 9(a) (BPFW < 10 nm), as P_sat increases, the SPM effect intensifies, broadening the TDS spectrum. After P_sat exceeds a critical value that corresponds to the MLP pump power threshold, MLP is eventually achieved. As soon as MLP occurs, the peak output pulse power drops, as illustrated in Fig. 9(b). When BPFW ≥ 7 nm, the calculated pulse energy of the MLP state is increased by the formation of a two-pulse bundle [Fig. 9(c)], since there are only slight decreases in the peak power [Fig. 9(b)] and subpulse duration [Fig. 9(d)]. When BPFW < 7 nm, the peak power decreases substantially, while the calculated pulse energy increases slightly due to the formation of the two-pulse bundle. As a result of the weakening of the SPM effect, the spectral width of the MLP region decreases. Since the MLP region (Region E) in Fig. 9(a) entirely corresponds to the two-pulse bundle state, which means that the pump power remains below the next-order MLP threshold, the MLP spectrum is similarly broadened as P_sat increases continuously. For other BPFWs, MLP does not occur immediately after P_sat exceeds the same value. On the contrary, the critical P_sat value varies with the BPFW, as will be discussed at the end of this section.

In the area with BPFW >14 nm, the laser operates in the DSR state. When the pump power increases, the pulse spectrum becomes narrower. As demonstrated in [21], the typically narrow spectral width in the DSR state is governed by the filtering of the gain spectrum. With increasing pump power, this effect intensifies, thus decreasing the spectral width of the DSR pulses. Meanwhile, as P_sat increases, the temporal widths and energies of the DSR pulses increase, with the peak power reaching a plateau [Fig. 9(b)].

In Fig. 9(a), the critical BPFW point changes with P_sat. As P_sat increases, the upper limit corresponding to the boundary between the DSR region (Region A) and Region C increases, while the lower limit that splits the TDS (Region B) and MLP (Region E) regions from the unstable region (Region C) decreases. This finding indicates that a transition from a TDS–DSR mixed state to Region C occurs as the pump power increases. As P_sat increases, stable solutions near the critical points become unstable. In addition, the centers of unstable areas eventually evolve to noise-like states, as discussed above.

Last, another notable finding is that both the two-pulse power threshold and spectral width vary synchronously with the BPFW at a specific saturation power (21 dBm in Fig. 10). By increasing the saturation power and recording the spectral width immediately before pulse breaking, the maximum spectral width can be obtained. As illustrated in Fig. 10, the metrics crest at exactly the same BPFW. The optimum BPFW maximizes the MLP pump power threshold and spectral width simultaneously. Based on the effects of the fiber length and NOLM loop length that were described above, we determined that the highest MLP power threshold can be obtained by using a long cavity and a short NOLM loop and varying the BPFW to obtain the widest spectral width while keeping the laser operating in the TDS state.

 

Fig. 10 Spectral width (P_sat = 21 dBm), maximum spectral width, and MLP threshold variation with BPFW.

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4. Conclusion

In summary, the state distributions of an NOLM-based ANDi fiber laser were systematically investigated in two-dimensional parameter spaces. The evolution of stable solutions (in the TDS, DSR, and MLP states) and unstable solutions (non-vanishing solutions and non-mode-locked areas) depends on different laser parameters. Firstly, in the BPFW–L_SMF2 parameter space, different laser states were identified and their distributions were clarified. Next, we investigated the unstable non-vanishing region in detail. The pulsation soliton in this area was found to be triggered by the anti-saturable absorption of the NOLM. The stochastic solution was caused by the fluctuation of the NOLM transmittance curve. Then, by switching to the BPFW–L_Loop parameter space, we observed that the peak-power-clamping effect was imposed equally on the DSR and MLP states. Thus, optical filters play an essential role in the formation of DSR and MLP, and two critical BPFW points exist that separate the DSR, MLP, and unstable non-vanishing regions. Finally, in the BPFW–P_sat parameter space, when P_sat was sufficiently high, the anti-saturable absorption was found to be so strong that the laser tended to operate in the DSR or MLP state. With an optimum BPFW, we could achieve the highest peak power and broadest spectral width simultaneously at a proper pump power level. This result also reaches the limits of most of the available TDS output metrics.