Dissipative solitons emitted in a mode-locked fiber laser are characterized by particle-like dynamics [1,2]. Beyond individual soliton singlets, self-assembling soliton molecules are allowed within the much longer timescales of a mode-locked fiber laser [3]. Attractive and repulsive interplays induced by the binding mechanisms of nonlinear saturation loss and gain, spectral filtering, and cross-phase modulation (XPM) determine the behaviors of soliton molecules [4,5]. Soliton dynamics involving multiple pulse patterns such as collision, vibration, and rogue wave are revealed, emphasizing the resultant “light–matter” analogy [6–8]. Given the increasing temporal complexity, soliton molecules permit in extended encoding formats of optical bits [9,10]. In addition to the traditional binary digits “0” or “1” that depend on the presence or absence of a single pulse, a multi-pulse bound state could be assigned an advanced digit “2” or higher encoding formats [11,12]. The synthesis and dissociation mechanisms of soliton complexes and their potential applications in all-optical signal processing have attracted increasing attention [3,5–7,10,13,14]. Particularly, manipulation of soliton molecules plays an important role, which has been intensively studied recently [3,10,15–18]. M. Pang et al. have proposed control of bound pulses through optomechanical interactions in a 2.5-m photonic crystal fiber (PCF) together with a nonlinear-polarization-evolution (NPE)-based pulse erasing scheme [10]. A similar mechanism is also used to stimulate the trapping potential for bound pulse control by a 2-m PCF and an externally addressing pulse source [3]. In addition, modulating pump lasers and polarization controllers have also been demonstrated to directly control the pattern of soliton molecules [15–17].

However, these works are based on non-polarization-maintaining (non-PM) configurations, which always suffer from random polarization perturbations and are thus less robust and controllable compared to general all-PM setups. Therefore, a simple yet reliable solution that is compatible with PM fiber (PMF) lasers is favorable. Meanwhile, preferably it is passive.

Limitation in time complexity shown in the previous studies becomes another issue that could hinder further applications. However, drawing concepts from multiplexing communications, a wavelength-multiplexing soliton complex that consists of bound solitons centered at different wavelengths can expand operable degrees of freedom [19]. Thus, available encoding formats are extended from the time to the frequency domain. Particularly, independent control of different wavelengths can be realized by introducing appropriate wavelength selection and tuning mechanisms. Recently, a synchronous emission of soliton complexes and singlets in separated wavebands has been observed [19–21]. However, pulse patterns with higher complexity, such as dichromatic soliton pairs, have not yet been reported.

In this Letter, an all-PM fiber laser based on a twistable tapered-fiber filter is developed, which delivers controllable pulse patterns of dichromatic soliton complexes. Depending on the twist angle, dual-wavelength bound states are switched between different patterns. The assembly format of pulses at each wavelength can also be manipulated. Moreover, long-term stability is improved by 2.2 times due to the temperature feedback package design. This work demonstrates a simple and reliable PM solution for dichromatic coexisting bound state manipulation.

The all-PM fiber laser is depicted in Fig. 1(a), where a 980-nm pump laser, a hybrid component consisting of a 980/1550-nm wavelength division multiplexer (WDM) and a 10% extracting tap, a fast-axis blocked optical isolator (ISO) and 1.4-m PM erbium-doped fiber (EDF) are employed. Passive mode-locking is realized by a CNT-coated tapered fiber. As shown in Fig. 1(b), its waist is 12 µm in diameter and 3 cm in length; the two symmetrical tapers are 4 cm long with a gradient of 0.0019 rad. Its pigtails are 45° cross-spliced. The insertion losses before and after coating are 0.47 and 6.70 dB, respectively. The tapered fiber is installed in a polymethyl methacrylate (PMMA) package that consists of two layers of coaxial hollow tubes. Thus, a hybrid device combining a saturable absorber (SA) and a Lyot filter is realized, which allows birefringence management by twisting. The total cavity length is ∼7.37 m with a net dispersion of −0.158 ps². The output signal is filtered by a wave shaper and analyzed by an optical spectrum analyzer (OSA, AQ6375), an electrical spectrum analyzer (EAS, E4440A), an oscilloscope (OSC, DS2202), and an autocorrelator (AUT, FR-103XL).

 

Fig. 1. Laser schematic diagram: (a) laser setup, (b) twistable tapered-fiber filter and package. Single-wavelength performance at 45-mW pump power: (c) optical spectrum, (d) autocorrelation trace, and (e) RF spectrum. Dual-wavelength performance at 34-mW pump power: (f) optical spectra and (g) pulse trains before and after filtering, and (h) RF spectrum.

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When the pump power is increased to 45 mW, self-started mode-locked pulses are emitted, the single-wavelength performance of which is summarized in Figs. 1(c)–1(e). The optical spectrum is centered at 1562 nm with a 3-dB bandwidth of 3.3 nm. The pulse duration is ∼859 fs fitting sech² profile, coming to a time-bandwidth product (TBP) of 0.348. The fundamental repetition rate is ∼27.545 MHz with over 70-dB signal-to-noise ratio (SNR). Furthermore, dual-wavelength mode-locking is obtained via twisting the tapered fiber. Due to the pump power hysteresis [22], the pump power can be carefully reduced to 34 mW to maintain single-pulse generations at both wavelengths respectively centered at 1552.5 nm (λ₁) and 1568.6 nm (λ₂). Their performances including spectra and oscilloscope traces before and after filtering and radio frequency (RF) spectrum are illustrated in Figs. 1(f)–1(h), respectively. The group velocity difference induces two nanosecond-scale wandering single-pulse trains as the oscilloscope snapshot, which correspond to two separate RF signals [23]. The 1.5-kHz interval on the RF spectrum fits the 16.1-nm span on the optical spectrum.

Further increasing the pump power to 192 mW and twisting the tapered PM fiber, the laser mode-locking state can be switched between different dichromatic states. Although here the tapered fiber is only twisted 50° in total, it can withstand 120° of twist without breaking. The optical spectra of the switching process are depicted in Fig. 2(a), and autocorrelation traces of short- and long-wavelengths are illustrated in Figs. 2(b) and 2(c), respectively. Intuitively, dichromatic encoding formats are introduced as follows to determine different assembly patterns: monochromatic pulse singlets and pairs “10”, “01”, “20”, “02”; dichromatic coexisting pulse singlets and pairs “11”, “21”, “12”, “22”.

 

Fig. 2. Switchable dual-wavelength pulse pattern: (a) optical spectra, (b) autocorrelation traces of the short and the (c) long wavelengths.

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In addition, opposite twist directions could cause asymmetric deflection of polarization and result in asymmetric denseness of ripples. This allows independent manipulation in each wave band. Then the number of pulses involved in a wave band can be manipulated via controlling the pump power while the twist angle remains unchanged. As depicted in Fig. 3(a), when the twist direction is clockwise (positive angle), the coexisting pulse number centered at a short wavelength (λ₁) grows as the pump power increases. Similar dependence is observed at long wavelength (λ₂) with negative angles. The corresponding temporal distributions are illustrated in Fig. 3(b), where the pulse pattern upgrades from “3 + 2” to “6 + 2” and “2 + 3” to “2 + 6”. Meanwhile, the pulses become closer with the increasement of pump power, indicating tighter inter-pulse interactions. The observed up-limit pattern is shown in Figs. 3(c)–3(e), where a 10-pulse soliton molecule centered at 1552.5 nm and an 8-pulse soliton molecule at 1567.5 nm coexist.

 

Fig. 3. Independent manipulation of separate wavebands: (a) spectra, (b) autocorrelation traces (inset: time-domain traces). Observed up-limit state: (c) spectrum, (d) autocorrelation traces at short- and (e) long-wavelengths.

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Moreover, the power characteristics are presented in Fig. 4. The pulse number grows linearly as the pump power increases in a quantized manner (red and blue solid lines). The same pulse pattern can be maintained with a lower pump power due to pump power hysteresis during the backward adjustment (gray and yellow dash lines). Similar linearity and hysteresis in terms of pulse energy and average output power are also unveiled in Fig. 4(b).

 

Fig. 4. Relationships between the pump power and the (a) number of bound pulses, (b) single-pulse energy and average output power.

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Benefiting from the PM-cavity configuration, the laser has good operational stability and repeatability. Since the birefringence of a PMF can be affected by the fluctuations of ambient temperature and thus causes small filtering shifts, an adaptative temperature feedback that relies on temperature-sensitive PMMA is applied within our filter design. As the temperature climbs, the fiber strain is released; while the PMMA package expands simultaneously, which reversely strengthens the strain. Consequently, impacts of ambient thermal fluctuations are passively compensated. Figure 5 demonstrates the enhancement of long-term robustness through comparative experiments of a stationary soliton pair that freely evolves for 50 min. PMMA and glass holders are respectively used, and the laser is adjusted to have the same initial state shown in Fig. 5(a). Every sampling wavelength at each time moment shown in Fig. 5(b) is compared to the expected value to calculate the 1-σ standard deviation (SD). Coefficients of variation (CVs) are further calculated to indicate the overall instability as the red and blue dots shown in Fig. 5(c). The average SDs for the PMMA and glass are 2.20% and 4.89%, and the corresponding CVs are 4.06% and 8.92% respectively. Their occurrence probability is also measured, where the probabilities of lower CVs are much higher than those of higher ones when using the PMMA. This indicates that the instability is constrained to a lower level. In contrast, the probability distribution of using glass is flat, which cannot guarantee improved robustness. Thus, 2.2-times improvement is secured under the average temperature fluctuation of ∼0.47°C/h measured in the lab.

 

Fig. 5. PMMA enhances long-term robustness: (a) initial optical spectrum, (b) 50-min spectral monitoring, (c) distribution of average coefficient of variations and corresponding statistical probabilities.

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To understand the possible mechanism regarding the impacts of twistable filtering, the filter transmission is calculated and applied in a simulated laser (see Supplement 1). Figure 6(a) shows the filter diagram consisting of the input light (E_in), the input/output pigtails (L₁, L₅), the down/up tapers (L₂, L₄), and the waist (L₃) of the tapered fiber, the polarizer (P), and its pigtail fiber (L₆). The twist angle (θ) is step-split accumulated, and the overall transmission is calculated by the Jones calculus. The results are illustrated as Fig. 6(b), which demonstrate enhanced spectral modulations. The larger the applied twist angle, the denser the resulting spectral ripples. The envelopes shown in Fig. 6(c) denote that the ripples could be approximately recognized as the result of cosine-shaped carrier and modulation. Two indicators of free spectral range (FSR) and amplitude are studied as dependent variables in the laser simulation to reveal possible mechanism. The laser simulation is based on the coupled nonlinear Schrödinger equation (CNLSE) that can be analytically solved via the split-step Fourier transform method (SSFM) [24] and the CNT coating function according to a slow SA model [25] (see Supplement 1). The steady state, determined as the root mean square error (RMSE) of convergence results, is less than 0.1%. As the modulation depths measured in Fig. 6(d), the initial setting value is 2.60% (required threshold), while the actual value (degraded one) is found to be reduced to 0.52% when the calculation finally converges to a steady state. Figure 6(e) analyzes the requirements to maintain the same single-pulse steady state in the presence of ripples of varying density and strength. As the ripples enhance, the required threshold climbs, which indicates greater degradation from the setting to the actual SA effect. Moreover, the degraded modulation depth and emitted pulse number at steady states are measured under the impact of ripples. The relationships between the SA degradation, number of simultaneously emitted pulses, and optical gain level are depicted in Fig. 6(f). The laser tends to operate in a multi-pulse region until reaching chaos when the SA effect is degraded or the gain level is increased with higher small-signal gain or saturation energy. Thus, twistable filtering enhances ripples, and ripples could suppress the SA effect; while degradation of the SA effect stimulates multi-pulse generation—causes more opportunities for pulse interaction and higher possible temporal complexity.

 

Fig. 6. (a) Simulation diagram of twistable tapered-PMF filter. (b) Spectral ripples versus the twist angle. (c) Ripples at ±25- and 0-degree twists. (d) Pulse before and after passing through the SA, and corresponding SA losses. (e) Requirements to maintain a single-pulse steady state. (f) Relationship between the number of simultaneously emitted pulses, gain level, and SA degradation (see Supplement 1).

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However, some ignoration and assumptions are made to simplify the calculation. The simulation accuracy mainly depends on the errors from the twist-taper modeling, fiber parameters, and laser modeling and analysis methods. The assessment of simulation accuracy is explained in Supplement 1. Intuitively, the simulation confirms the generation of dichromatic multiple pulses at 1552 and 1568 nm. Similar results on bound state changes according to the Lyot filter polarization angle have also been reported recently [18].

To summarize, dichromatic pulse pattern manipulation is studied in an all-PM fiber laser based on a twistable tapered-fiber filter. To our knowledge, it is the first demonstration of bound states that simultaneously coexist in two wave bands. Particularly, bound pulses are switchable between different pulse patterns, from monochromatic soliton singlets “01” or “10” to dichromatic soliton pairs “22”, depending on the twist angle. The twist direction also determines the dominant wavelength, which allows independent control at each wavelength. Higher order pulse encodings are thus realized by increasing the pump power, following a linear quantized relationship. Up to 18-pulse pattern is observed, which is the combination of a 10-pulse bunch at a short wavelength and an 8-pulse bunch at a long wavelength. Besides, 2.2-times improved robustness is secured by the PMMA package. In the simulation, the counteraction between twist ripples and the actual SA effect causes more pulse-participation to generate complex pulse patterns. This work proposes a solution of complex pulse pattern manipulation that is compatible with PM fiber configurations while being passive.