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# Stripe Soliton in All-Fiber Lasers

Qun Gao, Heze Zhang, Yueqing Du, Chao Zeng, Dong Mao, Philippe Grelu, Jianlin Zhao

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## To cite this version:

Qun Gao, Heze Zhang, Yueqing Du, Chao Zeng, Dong Mao, et al., Stripe Soliton in All-Fiber Lasers. Laser and Photonics Reviews, 2024, 10.1002/lpor.202400494, hal-04875141

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Distributed under a Creative Commons Attribution 4.0 International License
Stripe soliton in all-fiber lasers

QUN GAO,¹,† HEZE ZHANG,¹,† YUEQING DU,¹ CHAO ZENG,¹ DONG MAO,¹,*
PHILIPPE GRELU,² AND JIANLIN ZHAO¹

¹Key Laboratory of Light Field Manipulation and Information Acquisition, Ministry of Industry and Information Technology; Shaanxi Key Laboratory of Optical Information Technology; School of Physical Science and Technology, Northwestern Polytechnical University, Xi'an 710072, China

²Laboratoire ICB UMR 6303 CNRS, Université Bourgogne Franche-Comté, 9 avenue A. Savary, Dijon 21000, France

†These authors contributed equally to this work.

*Corresponding author: maodong@nwpu.edu.cn

Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; published XX Month XXXX

Stripe solitons (SSs), the special localized wavepackets exhibiting periodic modulations, have been identified in condensed matter physics and nonlinear science. Here, we demonstrate giant-chirp and chirp-free SSs in all-fiber lasers composed of single-mode fiber and polarization-maintaining fiber. The chirp property of SSs depends on the cavity dispersion while the stripe period relies on the fiber birefringence. Theoretical and numerical results coincide with experimental findings, revealing that the filtering and frequency shift of two vector modes are governed by the mode coupling related to birefringence and the nonlinear coupling induced by phase modulation, respectively, which disrupts their complementarity and ultimately leads to modulation of the overall spectrum. With the aid of the saturable absorption and cross-phase modulation effects, the multi-color pulses synergistically overlap, giving rise to the unique SSs. These results present a simple approach for generating chirp-controllable SSs in all-fiber lasers, which is crucial for applications in difference frequency generation and soliton physics.

©2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

http://dx.doi.org/10.1364/OPTICA.XXXXXX

# 1. INTRODUCTION

Solitons, as self-organized wavepackets against perturbations, have been discovered in various fields including optics [1, 2], hydrodynamics [3], plasma [4], and condensed matter physics [5]. Optical solitons are capable of maintaining their nature during propagation and interaction in nonlinear media due to the balance between diffraction and self-focusing effects in the spatial domain [6], or anomalous dispersion and self-phase modulation (SPM) in the time domain [7]. Owing to the flexible tunability of the dispersion and interaction length, fibers and fiber lasers serve as the workhorses for exploring novel solitons and elucidating their evolution dynamics [8-10]. Over the past few decades, scientists have demonstrated chirp-free solitons [11, 12] and stretched pulses [13, 14] by engineering the dispersion of cavities to anomalous and near-zero regimes. In normal-dispersion fiber lasers, giant-chirp self-similar pulses [15, 16] and dissipative solitons [17-19] can also be achieved, in which the former relies on the self-similar amplification in the gain fiber while the latter depends on the balance between gain and loss of the system. Strictly speaking, the mode-locked pulses in fiber lasers should not be defined as conservative solitons because the pulses exchange energy with the environment, and their properties vary along the cavity owing to gain, loss, dispersion, and nonlinear effects. Nonetheless, they are referred to as dissipative optical solitons in the general sense [20], which are attributed to self-consistent evolution per roundtrip in dissipative systems.
In addition to dispersion and nonlinearity, fiber birefringence significantly influences the properties of mode-locked pulses. For instance, low-birefringence single-mode fiber (SMF) lasers facilitate the formation of group-velocity-locked [21] and polarization-locked vector solitons [22, 23], whereas high-birefringence polarization-maintaining fiber (PMF) lasers enable the generation of extremely stable scalar solitons [24, 25]. These critical advancements, primarily focusing on the management of dispersion, nonlinearity, and birefringence in fiber lasers, have undoubtedly yielded profound insights into soliton physics and spurred numerous technological innovations, such as high-intensity pulse generation and compression [19, 26]. Limited by the gain competition effect, the aforementioned fiber lasers generally operate in a single-wavelength mode-locking regime, enabling $Sech^2$ or Gaussian pulse profiles. Thus, exploring new forms of solitons and elucidating their formation mechanisms continue to be an intriguing subject in the realm of nonlinear physics and mathematics.

Recent advancements in the modulation of cavity parameters and laser configuration have propelled the development of solitons to an unprecedented level, significantly enhancing their potential for practical applications. By implementing phase modulation through a programmable pulse shaper, fiber lasers can efficiently generate high-energy pure-quartic solitons [27] and chirp-free multi-color soliton molecules [28, 29]. The programmable pulse shaper is generally based on the diffraction grating and spatial light modulator, which destroys the all-fiber structure of the laser and introduces unexpected losses as well as dispersions. Currently, dual-color breathing soliton molecules were created in a synchronized two-path mode-locked fiber laser, where two pulses at 1550 and 1570 nm are amplified separately and then combined by a wavelength-division multiplexer [30, 31]. Such soliton molecules exhibit robust out-of-phase vibrations induced by the Q-switching instability. Similar to the stripe solitons (SSs) demonstrated in ultracold atomic systems [32-34], these soliton molecules exhibit stripe structures with periodic modulations and can be considered as a prototypical example of SSs in the temporal domain. Both types of SSs can be described by the similar nonlinear Schrödinger equation [35]. As such, the investigation of SSs in lasers has emerged as an exceptionally captivating research area, particularly for their important applications in Raman microscopy [36], terahertz generation [37], and timing distribution [38].

In this study, we demonstrate the chirp-controllable SSs in birefringence-managed all-fiber lasers composed of SMF and PMF, where the PMF-induced mode coupling in conjunction with self-phase modulation (SPM) and cross-phase modulation (XPM) dominate the evolution of two vector modes and results in the modulated structure on the overall spectrum. Due to the saturable absorption and XPM effects, the multi-color pulses are synchronized and overlapped in the time domain, giving birth to the unique SSs.

## 2. EXPERIMENT SETUP AND CONCEPT

The configuration of the SS fiber laser is depicted in Fig. 1(a), comprising a 6-m erbium-doped fiber (EDF), a 1-m polarization-maintaining fiber (PMF), a 90:10 output coupler, and a polarization-insensitive isolator. A 980 nm laser diode pumps the fiber laser through a 980/1550 nm wavelength-division multiplexer. Passive locking is ensured by a carbon nanotube saturable absorber, where the linear loss, modulation depth, saturation power, and response time are 51%, 8%, 9 W, and ~800 fs respectively. The birefringence coefficient of PMF is $4.2 \times 10^{-4}$, resulting in the differential group delay of 1.4 ps/m for the two vector modes. A polarization controller adjusted the polarization orientation of the light before the PMF. The pigtails of the fiber components are SMFs with a total length of 6.6 m. The cavity has a length of 13.6 m and a net dispersion value of $4.4 \times 10^{-3}$ ps$^2$. A polarization beam splitter is utilized to resolve the two orthogonally polarized components of SSs. The output lasers are monitored via an optical spectrum analyzer (YOKOGAWA: AQ6370), an autocorrelator (Pulsecheck: USB-150), a frequency-resolved optical gating (Femtoeasy: FS-600), a radio
frequency analyzer (Agilent: E4440A), and a 500-MHz oscilloscope (Tektronix: DPO 3054) simultaneously.

<figure><img src="image_4.png" /><figcaption>Fig. 1. Setup and principle of the SS fiber laser. (a) Configuration of the fiber laser. LD: laser diode; WDM: wavelength-division multiplexer; EDF: erbium-doped fiber; PC: polarization controller; SMF: single-mode fiber; PMF: polarization-maintaining fiber; OC: optical coupler; SA: saturable absorber; PI-ISO: polarization-insensitive isolator. (b) Coupling behavior of light in SMF-PMF-SMF structure. (c) Intensities of total pulse and two orthogonally polarized components versus wavelength for coupling angles $\theta = 0.1 \pi$ and $\alpha = 0.25 \pi$. (d) Modulation depth versus $\theta$ for $\alpha = 0.05 \pi$, $0.15 \pi$, and $0.25 \pi$ respectively.</figcaption></figure>

In comparison with other fiber lasers [39], the proposed fiber laser contains an SMF-PMF- SMF structure, which plays a dominant role in the formation of SSs. As depicted in Fig. 1(b), when a linearly polarized pulse ($u_0$) enters PMF from SMF, the coupling behavior is primarily influenced by the angle $\theta$ between the incident polarization and the slow axis of PMF. The two components along the slow ($u_s$) and fast axes ($u_f$) of the PMF are

$$ 
\begin{aligned}
u_s &= u_0 \cos \theta, \\
u_f &= u_0 \sin \theta.
\end{aligned} 
\qquad (1) 
$$

During propagation in the high-birefringent PMF, the four-wave mixing terms cancel on average, and $u_s$ and $u_f$ components accumulate a phase difference $\Delta\phi$ owing to the unequal refractive indexes and pulse intensities, which consists of the linear part $\Delta\phi_L$ and nonlinear part $\Delta\phi_{NL}$ [40]

$$ \Delta\phi = \Delta\phi_L + \Delta\phi_{NL} = 2\pi\Delta nL/\lambda + (\gamma L|u_0|^2/3)\cos(2\theta), \qquad (2) $$

$\Delta n$ and $L$ represent the birefringence coefficient and PMF length, respectively. $\lambda$ is the wavelength and $\gamma$ is the nonlinear coefficient of PMF. When the pulse propagates into the next section of the SMF, $u_s$ and $u_f$ components couple into $u_x$ and $u_y$ components and the coupling behavior depends on the angle $\alpha$ between the slow axes of PMF and SMF. According to Eq. (1), $u_x$ and $u_y$ components in the SMF after the PMF are

$$ 
\begin{aligned}
u_x &= |u_s| \cos \alpha e^{i\phi_s} - |u_f| \sin \alpha e^{i\phi_f} = |u_0| \cos \theta \cos \alpha e^{i\phi_s} - |u_0| \sin \theta \sin \alpha e^{i\phi_f}, \\
u_y &= |u_f| \cos \alpha e^{i\phi_f} + |u_s| \sin \alpha e^{i\phi_s} = |u_0| \sin \theta \cos \alpha e^{i\phi_f} + |u_0| \cos \theta \sin \alpha e^{i\phi_s},
\end{aligned} 
\qquad (3) 
$$

where $\phi_s$ and $\phi_f$ are the phases of $u_s$ and $u_f$ components, respectively. Taking the $u_x$ component as an example, the intensity is
$$I = |u_0|^2 \left( \cos^2 \theta \cos^2 \alpha + \sin^2 \theta \sin^2 \alpha - \frac{1}{2} \sin 2\theta \sin 2\alpha \cos \Delta\varphi \right). \quad (4)$$

For the fixed coupling angles $\alpha$ and $\theta$, the intensity of each component only relies on the phase difference $\Delta\varphi$, and changes with the wavelength at a period $\Delta\lambda = \lambda^2/\Delta nL$. Figure 1(c) shows the intensity evolution of $u_x$, $u_y$ components, and the total pulse versus wavelength for coupling angles $\theta$ of 0.1 $\pi$ and $\alpha$ of 0.25 $\pi$, where the periods are inversely proportional to the PMF length $L$. Due to the mode coupling between the two orthogonally polarized components, the intensities of $u_x$ and $u_y$ components exhibit complementary modulations whereas their total intensities remain constant. However, when the SPM and XPM effects are taken into account, the complementarity of two components may be destroyed due to their unequal intensities, which will be elaborated in the discussion section. The absence of polarizers in the cavity obviates any polarization-dependent loss or filtering effect on the total pulse, which is intrinsically different from that of Lyot filters [41]. From Eq. (4), the modulation depth on the spectrum of each component can be expressed as

$$T = \frac{I_{\max} - I_{\min}}{I_{\max} + I_{\min}} = \frac{|\sin 2\theta \sin 2\alpha|}{2(\cos^2 \theta \cos^2 \alpha + \sin^2 \theta \sin^2 \alpha)}. \quad (5)$$

The modulation depths of two components primarily depend on $\theta$ and $\alpha$, and vary periodically with $\theta$ at a period of $\pi$. Figure 1(d) shows the evolution of modulation depth with coupling angle $\theta$ for $\alpha$ of 0.05 $\pi$, 0.15 $\pi$, and 0.25 $\pi$ respectively. As the transmission property of each component relies on the modulation depth, the operation state of laser can be effectively controlled by the coupling angle. This assertion is supported by the observation that the peak-to-valley ratio of the spectrum can be adjusted by manipulating the polarization controller before the PMF (see Fig. S1 in Supplement 1).

### 3. RESULTS

The fiber laser initially emits a single-wavelength mode-locked pulse, characterized by a confined rectangular spectrum without any periodic modulation (Figs. S2 in Supplement 1). The pulse width is tens of picoseconds and the time-bandwidth product is calculated as 4.3, which is the typical fingerprint of the giant-chirp dissipative soliton in the normal-dispersion regime [18]. This state corresponds to the coupling angles of 0 or 0.5 $\pi$, that is, the pulse propagates along the slow or fast axis of the PMF. As one of the orthogonally polarized components approaches zero, the pulse can be treated as a scalar one, without considering the impact of vector mode coupling.

Multi-wavelength giant-chirp SSs are obtained from the same fiber laser when the incident polarization deviates from the principal axes of the PMF. Figure 2 shows the typical giant-chirp SS at a pump strength of 15 mW. The spectrum displays six discrete components, in which the two central peaks correspond to the mode-locked dissipative solitons, whereas the lateral peaks arise from the cascaded four-wave mixing [42]. In fact, the intensity difference between the central and lateral peaks is approximately 10 dB, implying that the lateral peaks are far below the formation threshold of dissipative solitons. The resonantly enhanced four-wave mixing process can effectively suppress the amplified spontaneous emission of the EDF by inhibiting photon excitation [43], thereby facilitating the formation of the stripe spectrum with a peak-to-valley ratio of up to 30 dB. Due to the filtering effect induced by linear coupling between two vector modes, the profiles of the central spectra are slightly asymmetric. The spectral spacing between the neighboring peaks is 5.54 nm, which matches the spectral modulation period using the 1-m PMF in Fig. 1(c). The experimental result slightly deviates from the theoretical value as the latter does not account for the birefringence of SMF. Similar to the vector solitons reported previously [12], the SS can be decomposed into two orthogonally polarized components with an intensity difference of ~8 dB. The trapping between two orthogonally polarized components is realized by the nonlinear
coupling induced by the XPM effect [44]. The autocorrelation trace of the SS [Fig. 2(b)] displays a stable stripe structure, indicating the synchronization and interference of pulses at these discrete wavelengths. The pulse profile is further measured with a second-harmonic generation FROG. As shown in Figs. 2(c) and 2(d), the retrieved FROG trace coincides well with its measured counterpart, confirming the reliability of the spectral-resolved measurement. Both the autocorrelation trace and FROG trace show that the modulation period of the SS is 1.5 ps, which is the inverse of the frequency difference between two neighboring wavelengths. By tuning the pump strength, the fiber laser can be switched between single-pulse and multiple-pulse states (see Fig. S3 in Supplement 1), while the spectral and temporal profiles remain nearly unchanged. The single-pulse energy of the SSs within the cavity ranges from 150 to 220 pJ, surpassing that of the chirp-free soliton [11]. The single-shot spectra suggest a consistent frequency-time relationship, which indicates that the pulses at these wavelengths are trapped in the temporal domain (see Fig. S4 in Supplement 1).

<figure><img src="image_5.png" /><figcaption>Fig. 2. Giant-chirp SS. (a) Optical spectra of SS and its two orthogonally polarized components. (b) Autocorrelation (AC) trace. (c) Measured (upper panel) and retrieved (lower panel) FROG spectrograms. (d) Retrieved pulse profile. (e) Spectra and (f) AC traces of filtered SS.</figcaption></figure>

The SS is further studied by filtering out each spectrum external to the cavity. As illustrated in Figs. 2(e) and 2(f), the durations of pulses at 1553.7 and 1559.2 nm are 34.9 and 35.2 ps, respectively, close to the meta-envelope of 35.1 ps observed in Fig. 2(b). Remarkably, both the SSs and filtered pulses have almost the same time-bandwidth product of 7.6, implying that they are strongly chirped [45]. In comparison with the chirp-free SSs [28, 41], the chirped SSs contain more stripes in the temporal domain due to the larger pulse duration. These findings validate that the meta-envelope of the SS is determined by that of its constituent dissipative solitons.

Interestingly, when we add a 0.2-m SMF to the same fiber laser, the cavity dispersion decreases from $4.4 \times 10^{-3}$ ps$^2$ to zero and the giant-chirp SSs transforms into a chirp-free SSs. As shown in Fig. 3(a), the two dominant spectra locate at 1555 and 1560.8 nm with 3-dB bandwidths of 1 and 1.06 nm, respectively. The spectral separation of the chirp-free SSs (5.80 nm) is close to that of the theoretical value (5.75 nm), reinforcing the conclusion that PMF-induced mode coupling dominates the formation of SSs. Here, the intensity ratio between the two orthogonally polarized components is approximately 10 dB, also validating the vector characteristics of such solitons. During the whole experiment, the intensity ratios always range from 7 to 11 dB, corresponding to spectral modulation depths of 66% to 85%, in accordance with Eq. (5), which indicates that the modulation on each orthogonally polarized component should be moderate to facilitate multi-wavelength lasing while ensuring sufficient bandwidth to support the mode-locked state.
<figure><img src="image_6.png" /><figcaption>Fig. 3. Chirp-free SS. (a) Optical spectra of SS and its two orthogonally polarized components. (b) AC trace. (c) Measured (upper panel) and retrieved (lower panel) FROG spectrograms. (d) Retrieved pulse profile. (e) Spectra and (f) AC traces of filtered SS.</figcaption></figure>

The envelope width of the chirp-free SS is calculated as 2.5 ps using the Sech² fitting [Fig. 3(b)]. Both the autocorrelation trace and retrieved pulse show that the separation between neighboring stripes is 1.4 ps [Figs. 3(c) and 3(d)], which corresponds to a pulse repetition rate of 0.71 THz. The durations of the two pulses at shorter and longer wavelengths are 2.5 and 2.6 ps respectively [Figs. 3(e) and 3(f)], giving a similar time-bandwidth product of approximately 0.32, which further validates the chirp-free property of the SSs.

<figure><img src="image_7.png" /><figcaption>Fig. 4. Evolution of SSs with the net cavity dispersion. (a)-(e) Spectra. (f)-(j) AC traces. Solid lines: experimental results; dash lines: fitting results.</figcaption></figure>

The pulses at different wavelengths are trapped by the XPM-induced binding force [46]. As the overlapped pulses have higher intensities, the fiber laser tends to generate SSs rather than multiple closely-trapped pulses owing to the saturable absorption effect, that is, this laser also follows the minimum-loss principle [47]. We investigate the evolution of the SSs with the net cavity dispersion by gradually reducing the SMF length from 7 to 6.6 m, as shown in Fig. 4. As the net cavity dispersion decreases from $4.4×10^{-3}$ to $-4.4×10^{-3}$ ps², the time-bandwidth product reduces from 7.6 to 0.35, indicating the flexible tunability of the pulse chirp. The transition from chirped to chirp-free SSs can be attributed to the change of the
lasing state, shifting from a normal-dispersion dissipative soliton to a dispersion-managed soliton. Beyond this range, the SSs become unstable and the laser evolves towards the single-wavelength regime owing to the dispersion-induced walk-off and gain-competition effect.

According to Eq. (4), the spectral modulation period depends solely on the birefringence of the PMF. We measure the separation between the neighboring spectra and compare it with the theoretical analysis. As depicted in Figs. 5(a) and 5(c), the experimental results agree well with the analytic solution, in which the spectral separation increases from 3.9 to 6.3 nm by reducing the PMF length from 1.4 to 0.9 m. For the difference-frequency wave generation, such pulses exhibit a higher conversion efficiency than that of the dual-wavelength continuous waves [48]. Furthermore, the cavity dispersion increases with the decrease of PMF length, causing the transition from chirp-free SSs to giant-chirp SSs. Figures 5(b) and 5(d) show the evolution of SSs with the increase of the pump strength. Taking the spectrum at 1560 nm as an example, the 3-dB bandwidth exhibits a nearly linear increase from 2.2 to 3.4 nm, while maintaining almost constant wavelength separation, which confirms that the spectral modulation period mainly relies on the fiber birefringence rather than nonlinear effects.

<figure><img src="image_8.png" /><figcaption>Fig. 5. Evolution of SSs with PMF length and pump strength. (a) Spectra and (c) spectral separation versus PMF length. (b) Spectra, (d) spectral separation and bandwidth versus pump strength.</figcaption></figure>

## 4. DISCUSSION

Based on a lumped propagation model that considers the transmission matrices of each component [40], we study the intracavity evolution of the SSs to reveal their formation mechanism. The propagation of pulses in the fibers is described by the coupled Ginzburg-Landau equations [12], in which the coupling behavior of pulse between the SMF and PMF is particularly considered. The simulation settings are the same as those of the experiments, and the coupling angles $\theta$ and $\alpha$ are set as 0.1 $\pi$ and 0.25 $\pi$, respectively. Simulation details and other parameters are elaborated in Section 5, Supplement 1.

Figure 6 displays the spectral evolution of SS over three consecutive roundtrips for a clear and demonstrative illustration. Because of the mode coupling and interference between the two vector modes, their spectra display modulating fringes with a period of 5.63 nm after traversing the PMF in the first roundtrip [II in Fig. 6(b)], which coincides with the theoretical analysis [Fig. 1(c)] and experiment observations [Fig. 2(a)]. As the two orthogonally polarized components are complementary, the overall spectrum maintains a smooth profile without any modulation. Then, the $u_x$ and $u_y$ components undergo cascaded mode coupling during propagation in PMF at the second roundtrip and additional modulation peaks appear on the spectra of each component. Simultaneously, a slight fluctuation emerges on the total
spectrum and the wavelength spacing between the two components decreased from 2.82 to 1.41 nm [IV in Fig. 6(b)]. As the pulse traverses through the PMF during the third roundtrip, the wavelength spacing further decreases to 1.02 nm and the multi-peak structure becomes much clearer on the total spectrum.

<figure><img src="image_9.png" /><figcaption>Fig. 6 (a) Simulated spectral evolution of SS and two orthogonally polarized components. (b) Optical spectra at I, II, III, IV, V, and VI positions in (a).</figcaption></figure>

In addition to the mode coupling and interference induced by the PMF, the SPM and XPM effects are essential for the formation of SSs. As illustrated in Fig. S5 of Supplement 1, the modulation of the overall spectrum vanished when the XPM and SPM effects are removed from the simulation model. The formation mechanism of SSs can be summarized as follows: when the pulse enters the SMF from the PMF, the two orthogonally polarized components possess different group velocities. The fast and slow components subsequently combine in the SMF, resulting in the spectral modulations caused by their interference. In the subsequent roundtrip, the spectral modulation period of each component is reduced due to the cascaded mode coupling and interference within the SMF-PMF-SMF structure. During this process, because the two components possess different intensities, they accumulate distinct nonlinear phase shifts during propagation in the fiber owing to the SPM and XPM effects, which destroys the complementarity and ultimately results in the modulation of the overall spectrum. As the cavity dispersion approaches zero, the saturable absorption and XPM effect automatically force the multi-color pulses into an overlapped state, giving rise to the unique SSs, despite the small residual group delay difference. This evolution behavior is reminiscent of SSs with a spatially oscillating density in condensed matter physics [49], which arises from the similar role of spin-orbit coupling between two hyperfine states of cold atoms.

Once the PMF is removed from the cavity, the SS operation switches to the single-wavelength mode-locked state. By replacing the polarization-insensitive isolator with a
polarization-sensitive isolator, the proposed scheme degenerates into a standard fiber laser including a Lyot filter. In both cases, the SSs cannot be achieved regardless of the pump strength and polarization state (see Fig. S6 in *Supplement 1*), which confirms the crucial role of vector-mode coupling in the SS formation.

## 5. CONCLUSION

We have demonstrated giant-chirp and chirp-free SSs in birefringence-managed fiber lasers operating in a near-zero dispersion regime. The cascaded mode coupling and interference between the two orthogonally polarized components result in the periodically modulated spectra. Due to the SPM and XPM effects, the overall spectrum evolves into the trapped stripe structure while the saturable absorption effect forces the multi-color pulses into an overlapped state. Compared to SSs based on a spatial light modulator or two-path laser, the proposed scheme features a simpler all-fiber configuration that enables effective manipulation of the spectral and temporal properties of pulses. Such unique SSs, featuring adjustable chirp and sub-terahertz pulse repetition, exhibit significant potential for applications in difference-frequency wave generation [37] and beat-wave plasma accelerators [50].

**Funding.** National Natural Science Foundation of China (12274344), Natural Science Basic Research Program of Shaanxi Province (2021JC-09), and Fundamental Research Funds for the Central Universities (3102019JC008).

**Disclosures.** The authors declare no conflicts of interest.

**Data Availability.** The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

**Supplemental document.** See Supplement 1 for supporting content.

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