1. Introduction

Compared with traditional solid-state lasers, fiber lasers are highly regarded as an alternative option due to their exceptional beam quality, high energy conversion efficiency, compact structure, and reliable operation [1,2]. In addition to an ultrashort pulse source, the passively mode-locked fiber laser is also a fantastic platform for investigating the nonlinear dynamics of optical solitons [3]. Generally, multiple solitons can be generated in fiber lasers with increasing pump power [4,5]. To date, the formation of multi-soliton patterns through short-range [6–9] and long-range [10–12] interactions have been intensively investigated in nonlinear optical systems owing to their fruitful dynamics. Through soliton interactions, multiple solitons propagating in the laser cavity are able to form different patterns such as soliton molecules [13,14], supramolecular structures [3], or soliton crystals [15,16]. Several mechanisms for the soliton interactions have been proposed, possibly resulting from multiple or individual actions mediated by Casimir-like forces [17], perturbation [18,19], thermal effect [20], and photoacoustic effect [10,12,21]. However, the temporal spacing of the generated multiple solitons is difficult to manipulate in fiber lasers, because the soliton interactions are complex and somewhat random.

Recently, a breakthrough was made in the performance improvement of passively mode-locked fiber lasers, namely intelligent control of operation regimes. Relying on the electronic polarization controller (EPC) or liquid crystal wave plate, the automatic mode-locking as well as precise control of mode-locked states can be obtained in ultrafast fiber lasers based on machine-learning strategies [22,23]. Aiming at accessing specific mode-locked states (i.e., harmonic mode-locking and soliton pulsation), versatile intelligent control algorithms have been proposed to drive the polarization searching through EPC in the laser cavity, such as human-like algorithms [24], depth-first search algorithms [25], evolutionary algorithms [26,27], and so on. On the other hand, the temporal spacing manipulation of multiple solitons is meaningful from fundamental research to practical applications, i.e., multi-soliton interaction [28,29], high-repetition-rate pulse train [30], and laser micromachining [31,32]. The common approach to tune the temporal spacing of multiple solitons is to precisely adjust the cavity parameters, such as pump power and polarization states [33]. However, because of the lack of a clearly determinate relationship between the temporal spacing of multiple solitons and cavity parameters, a long trial-and-error process is required to obtain the desirable mode-locked state. Considering the significance of desirable multi-soliton pattern generation, it would be meaningful to develop an intelligent identification and control of multi-soliton temporal spacing in ultrafast fiber lasers.

In this work, we demonstrated the auto-setting temporal spacing of multiple solitons in a nonlinear polarization rotation (NPR) mode-locked fiber laser by a hybrid genetic algorithm-particle swarm optimization (GA-PSO) algorithm. By combining the strong diversity of the GA with the strong convergence of the PSO algorithm, the operation regime of the mode-locked fiber laser can be intelligently controlled through the automatic polarization tuning of an EPC. In particular, the temporal spacing of double-soliton mode-locking can be effectively adjusted. We believe that the proposed intelligent control of the double-soliton spacing would find potential applications in the exploration of nonlinear soliton dynamics and the development of ultrafast laser technology.

2. Experimental setup and operation principle

2.1 Experimental setup

Figure 1(a) depicts the flowchart of the hybrid GA-PSO algorithm that is used to search and control the double-soliton operation in the fiber laser. Firstly, the computer randomly generates an initial population consisting of multiple individuals with positions (representing three voltages applied to the EPC) and velocities (referring to the search step). After applying voltages to the EPC, the data acquisition module records the data from the output of the laser and inputs it to the computer. Note that the data acquisition is set to wait for 3 seconds after the adjustment of EPC to avoid the influence of the chaotic signal recorded in the process of laser buildup. After the evaluation is done, local best values so far are obtained (denoted as pbest), out of which a global best value can be acquired (denoted as gbest) [34]. The velocity of each individual is related to the following terms: the distance between the current position and pbest, and the distance between the current position and gbest [35]. Using the principle of “survival of the fittest”, two steps of crossover and mutation are performed to update the positions of individuals with fitness values higher than the average one. The crossover process involves the pairwise selection of individuals in order, with their fitness values exceeding the population's mean fitness value. A random number between 0 and 1 is generated and compared with the crossover probability (denoted as P_c= 0.5). If the generated number is greater than or equal to P_c, the parameters within the same dimensions of the two individuals are exchanged; otherwise, both individuals remain unchanged. Each pairwise crossover operation is conducted three times, until all dimension parameters of the individuals have undergone crossover, with parameters in different dimensions remaining independent of each other. Similarly, the mutation process follows a similar pattern, using a mutation probability (denoted as P_m), which is equivalent to the crossover probability P_c. When the random number is greater than or equal to P_m, the individual's parameters are replaced by new ones within the specified range. Then these individuals enter into the next generation with the remaining individuals. Finally, stop and exit the algorithm when individuals are meeting the requirements in the population or when the maximum number of iterations is reached. The hybrid GA-PSO algorithm combines the advantages of the swarm intelligence of the PSO algorithm and the natural selection mechanism of the genetic algorithm. This strikes a balance between high diversity and fast convergence, which greatly improve search efficiency [36]. The EPC is composed of three cascaded variable fiber waveplates, and each of them is controlled by applying different analog voltages. It enables the manipulation of all possible polarization states on the Poincaré sphere, with each set of voltages corresponding to a specific polarization state. Therefore, employing only one EPC inside the NPR-based laser setup provides the efficient manipulation of polarization state for the intra-cavity propagation light, thus achieving control over the nonlinear transfer function [37]. Moreover, the EPC usually has a response time on the order of milliseconds, allowing quick adjustment of laser operation state, and enabling the efficient implementation of the GA-PSO algorithm to search the double-soliton regime with user-defined spacing in an acceptable time scale. The hybrid algorithm is initialized and iterated with a population of 80 individuals. Evaluating the properties of all individuals in an entire generation typically takes about 4 minutes.

 

Fig. 1. (a) Flowchart of hybrid GA-PSO algorithm searching for the double-soliton state. (b) Schematic of an intelligently NPR-based mode-locked fiber laser with computer-controlled feedback control. WDM: wavelength division multiplexing; EDF: erbium-doped fiber; PD-ISO: polarization-dependent isolator; EPC: electronic polarization controller; PC: personal computer, DAC: digital to analog converter; OSA: optical spectrum analyzer; PD: photodetector; OC: optical coupler.

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The configuration of the proposed automatic mode-locked fiber laser is shown in Fig. 1(b). The laser cavity comprises a 9-m long erbium-doped fiber (EDF) with a dispersion parameter D ≈ -15 ps/nm/km, a total length of 2.1-m single-mode fiber (SMF) with a dispersion parameter D = 17 ps/nm/km, and a polarization-dependent isolator (PD-ISO). Thus, the fiber laser is operating in net-normal dispersion regime. An EPC (Phoenix Photonics, EPC-15-1-1-2) is used to adjust laser operation by applying different DC voltages to its three channels. In this case, the polarization state of propagating light in the cavity can be adjusted automatically. The intelligent feedback system of the fiber laser is composed of an oscilloscope, a personal computer (PC), a DAC (DAC8563) module, and an EPC. The oscilloscope (Tektronix DSA70804) acts as a high-speed data acquisition card to collect the output signal of the laser. The PC serves as the computational core and control unit of the feedback system, which is responsible for data processing and the execution of the hybrid algorithm. After receiving and processing the data sent by the oscilloscope, the PC outputs corresponding commands to the DAC module, and thus, to apply voltages to the EPC. Two couplers (20% and 10%) are used to extract part of the light from the cavity for characterization and feedback control of the laser performance. The output laser is detected with an ultrafast photodetector (Alphalas, UPD-15-IR2-FC). The optical spectrum is measured by an optical spectrum analyzer (OSA, Yokogawa AQ6317C).

2.2 Discrimination of the double-soliton regime

The signal acquired by the oscilloscope is the instantaneous voltage value after conversion by a photodetector, as shown in Fig. 2. The black line represents the actual waveform, which consists of a large number of discrete voltage points. For the double-soliton discriminant diagram depicted in Fig. 2(a), it should be emphasized that sample values surpassing the pulse count threshold are designated as effective pulses, whereas those falling below are classified as noise [24]. The pulse count threshold can usually be set to an appropriate value according to the preset pump power. Then the average pulse amplitude can be expressed as [25]:

 

Fig. 2. Schematic diagram of acquisition of double-pulse mode-locked regime with different spacing. (a) Acquisition diagram of double-pulse mode-locking state. The solid black line represents the actual outline of the pulse train. The red dashed line represents the effective pulse count threshold, and T represents the sampling time of the oscilloscope. (b)-(c) Two situations for judging the spacing of double pulses. T₀ is the period of single pulse mode-locking.

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Here, Max(A₁, A₁, $\ldots $,A_N) represents the maximum amplitude of the effective pulses, while Min(A₁, A₁, $\ldots $,A_N) indicates the minimum one. The magnitude of pulse amplitude jitter serves as an indicator of the stability of the mode-locking state. A smaller pulse amplitude jitter corresponds to a higher probability of a mode-locking state. However, it is insufficient to accurately discriminate the double-soliton state only by the pulse amplitude jitter. Because this criterion can also be fulfilled by other types of mode-locking states, including single-soliton and triple-soliton regimes. When the pulse amplitude jitter is small and acceptable, the soliton spacing and the repetition rate should be considered to check whether the fiber laser works in the desirable double-soliton regime. The pulse repetition rate can be obtained by calculating the number of effective pulses in a fixed sampling time. If the double-soliton state is confirmed, the spacing of the double solitons is further checked.

When calculating the spacing between the double solitons, two cases need to be considered, as shown in Figs. 2(b) and 2(c). Commencing from the first identified temporal position of a valid pulse, a waveform segment lasting twice the cavity roundtrip time is extracted for subsequent analysis. Thus, the double-soliton spacing can be given as:

Here, the variables x₁, x₂, and x₃ correspond to the positions of the solitons within the sampled data. Δt is the temporal resolution of the oscilloscope. In order to explore the double-soliton state with a specific spacing, a composite fitness function is defined, which integrates pulse amplitude jitter, pulse count, and pulse spacing:

Regarding the three terms on the right-hand side of Eq. (5), the first term represents the uniformity of the pulse sequence, the second term indicates the degree to which the number of valid pulses approximates the predetermined value, and the third term characterizes the proximity of the actual double-soliton spacing to the target one. Each term ranges from 0 to 1. Hence, it is imperative to establish predefined conditions to ensure that the values remain within the designated range. For instance, in the case where A_jitter is much smaller than A_average, the fitness value of pulse amplitude jitter approaches 1, indicating that the fiber laser operates in the mode-locking state. So, in the case where A_jitter exceeds A_average, the value of the first term should be negative. Nevertheless, given the constrained range of values from 0 to 1, the first term is enforced to be 0. C_{ideal_count} denotes the ideal count of pulses at a fixed sampling time of the oscilloscope, determined by the fundamental repetition frequency of the pulse train. For the pulse count, as the measured pulse count approaches the ideal value, the fitness of the pulse count tends to approach 1. If three or more pulses simultaneously exist in the laser cavity, the second term is constrained to 0 to prevent it from taking negative values. Similarly, when T_spacing is greater than T_set, the value of the third item is 0. Therefore, the ideal fitness value is 3. Considering the effects of laser noise and oscilloscope sampling errors in practical operation, the fitness value of the double-soliton state with the desirable spacing is set to 90% of the ideal one. Depending on the aforementioned criteria, the hybrid GA-PSO algorithm is utilized to drive the adjustment of EPC voltages, thereby controlling the cavity parameters and achieving a double-soliton state with different temporal spacing.

3. Results and discussion

Firstly, the pump power is set to 200 mW, enabling the laser to achieve the self-starting mode-locking with single-pulse regime. Figure 3(a) illustrates the corresponding mode-locked spectrum, revealing a center wavelength of 1558.4 nm and a 3-dB bandwidth of 29.61 nm. It is noteworthy that the mode-locked spectrum manifests itself as a rectangular profile, which represents a typical feature of dissipative soliton mode-locking [38,39]. The pulse train is depicted in Fig. 3(b), demonstrating a soliton spacing of 53.62 ns. The fundamental repetition frequency of the fiber laser is determined by the cavity length, which is measured to be 18.65 MHz. The autocorrelation trace of the mode-locked dissipative soliton, displayed in Fig. 3(c), yields a duration of 9.8 ps assuming a Gaussian profile. Furthermore, the radio frequency (RF) spectrum of the output laser, as depicted in Fig. 3(d), showcases a remarkable signal-to-noise ratio (SNR) of up to 61-dB located at the fundamental repetition frequency of 18.65 MHz. This further confirms the stable operation of the fiber laser.

 

Fig. 3. Experimental results of the fundamental mode-locked regime. (a) Spectrum; (b) pulse train; (c) autocorrelation trace; (d) RF spectrum.

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The generation of double solitons is attributed to the peak power clamping effect in the NPR -based mode-locked fiber laser operating in normal dispersion regime [4,40]. The research findings indicate that the manipulation of the polarization controller (PC) can effectively adjust the cavity loss in an NPR-based fiber laser [41]. In this scenario, the settings of the intracavity pulse phase are influenced by the PC and the pump power, resulting in changes to the transfer function. When the peak power of the mode-locked soliton reaches the threshold at negative feedback condition, increasing the laser gain does not amplify the pulse parameters, but amplifies the dispersive waves [41,42]. In this way, a new soliton is generated. The newly emerging soliton engages in mutual competition and influence with the original pulse. The preceding soliton induces phonons through the electrostriction effect [43,21], resulting in effective refractive index modulation that exerts an attractive/repulsive force on the subsequent soliton. Furthermore, the dispersive waves emitted by the preceding pulse exhibits faster propagation than the soliton itself, eventually reaching the second pulse and inducing perturbations through cross-phase modulation [44,45]. Therefore, the stability of the double-soliton structure stems from the equilibrium between the attractive interaction among solitons induced by the acousto-optic effect and the repulsive interaction between solitons caused by dispersive wave perturbations [46]. In this way, the variation of soliton energy, which can be achieved by adjusting the EPC, will lead to the variation of soliton spacing.

At the beginning of the experiment, the polarization state of EPC is initialized as a random state. Then the target values for the double-soliton spacing are preset at 2 ns, 7 ns, 12 ns, 18 ns, 22 ns, and 26.8 ns, respectively. The pump power is initially increased to 400 mW to effectively achieve the double-soliton regime. By employing the hybrid algorithm incorporating the composite fitness function above, the automatic configuration of double-pulse mode-locking with varying temporal spacing is successfully achieved in the laser. Figures 4(a)-4(f) present the convergence curves of the hybrid GA-PSO algorithm computed in 25 generations, depicting the historical best values and average fitness values of the populations corresponding to different temporal spacings of the double-soliton regime, respectively. As can be seen here, the evolution of the gbest fitness (red round points) and average fitness (black squared points) aligns with the expected pattern as the number of generations increases. It should be noted that occasional populations exhibit fluctuations in average fitness during the process of algorithm iteration. This can be mainly attributed to the existence of randomness during the mutation process, leading to lower fitness values for the laser control under the specific set of voltage values. The consecutive preservation of the gbest fitness value across multiple generations occurs when there is a lack of offspring with better fitness. In such instances, the diversification of the search is enhanced by appropriately adjusting the pump power.

 

Fig. 4. The convergence curve of the GA-PSO algorithm with setting different double-soliton spacing containing gbest (historical optimal individual of the population) and mean fitness values. (a) 2 ns; (b) 7 ns; (c) 12 ns; (d) 18 ns; (e) 22 ns; (f) 26.8 ns.

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Figure 5 illustrates the double-soliton operation with different temporal spacings enabled by the hybrid algorithm. It can be observed that the temporal spacing of the double-soliton pattern is not exactly the same as those preset initially. It is important to clarify that slight deviation from the preset value of double-soliton spacing in the actual results are reasonable. Because the fitness is influenced not only by the soliton spacing but also by the amplitude jitter and pulse counts. Furthermore, it can also be observed that the pulse-train suffers slight intensity fluctuations, which may be attributed to the imperfect adjustment of EPC that is driven by the evaluation terms. Correspondingly, the mode-locked spectra of double-soliton regime with different temporal spacings exhibit a high degree of consistency, with slight variations observed in their central wavelengths and 3-dB bandwidths, as depicted in Fig. 6. The occurrence of this phenomenon can be attributed to the modification of the intracavity polarization state which leads to variations in the effective gain bandwidth and cavity loss [4,39]. The results discussed above indicate that the proposed hybrid GA-PSO algorithm is an effective way for auto-setting multi-soliton temporal spacing in a fiber laser, which will find important applications in manipulating the soliton nonlinear behavior and improving the ultrafast laser performance. In recent years, dispersive Fourier transform (DFT) technology has been applied for intelligent mode-locking of fiber lasers [47,48]. Thus, future efforts can be devoted to combine DFT with the hybrid GA-PSO algorithm, aiming to intelligently explore other complex soliton nonlinear phenomena, such as soliton molecules [8], multi-period pulsations [3], and rogue waves [48].

 

Fig. 5. The double-soliton sequences with various temporal spacings are obtained using a hybrid GA-PSO algorithm. (a) 2.01 ns; (b) 7.10 ns; (c) 12.36 ns; (d) 17.89 ns; (e) 22.41 ns; (f) 26.81 ns.

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Fig. 6. Spectra of the double-soliton state with different temporal spacings.

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4. Conclusion

In summary, we have investigated the automatic manipulation of multi-soliton temporal spacing in a fiber laser by using a hybrid GA-PSO algorithm in conjunction with EPC intelligent control. Specifically, the automatic setting of double-soliton state with different spacings could be realized by using a composite fitness function. This approach exemplifies an intelligent control of multi-soliton temporal behavior in fiber laser systems through the establishment of suitable fitness functions. Our findings enable the effective exploration of diverse and desirable multi-soliton patterns, holding significant promise for the study of soliton nonlinear dynamics and laser performance improvement.