A Wavelength-Multiplexed Dual-Frequency Mode-Locked Fiber Laser with Tunable Repetition Frequency Difference

Abstract

We present a wavelength-multiplexed dual-frequency mode-locked fiber laser capable of generating asynchronous dual-frequency pulses with a widely tunable repetition frequency difference. The laser achieves the center wavelength separation adjustable from 1.02 to 26.7 nm, while the repetition frequency difference can be tuned from 147 Hz to ~3 kHz. Such wide tunability is realized within a compact cavity by leveraging the low-threshold saturable absorption of carbon nanotubes (CNTs) and the wavelength selectivity introduced by nonlinear polarization rotation (NPR). The tunability of repetition frequency difference not only broadens the scope of wavelength-multiplexed asynchronous mode-locking but also provides a versatile and practical dual-frequency source.

1. Introduction

Fiber lasers, with their excellent beam quality, high stability, compact design, and ease of operation, have become important tools in optical communications [1], nonlinear optics [2], and bio-photonics [3]. Passive mode-locking is a common method to generate ultrashort pulses in fiber lasers. It can be realized by artificial saturable absorbers utilizing nonlinear polarization rotation (NPR) [4] and nonlinear optical loop mirrors (NOLM) [5], and real saturable absorbers like semiconductor saturable absorber mirrors (SESAM) [6,7] and saturable absorbers based on graphene [8], carbon nanotubes [9,10], or other nano materials [11]. Beyond single-pulse generation, mode-locked fiber lasers also provide a practical route for optical dual-frequency combs, which are essential for dual-frequency sources, where two pulse trains with slightly different repetition frequency interfere to produce a powerful tool for broadband, high-resolution spectroscopy and fast data acquisition [12,13,14]. In conventional dual-comb systems, two independent mode-locked ultrafast lasers are typically employed. Their repetition frequencies must be precisely synchronized to be nearly identical, while maintaining a slight and well-controlled repetition frequency difference. This small repetition frequency offset introduces a temporal walk-off between the pulse trains. However, because the two pulse trains are generated by separate mode-locked lasers with independent cavities, any cavity-length drift or environmental perturbation will degrade the mutual coherence and frequency stability of the two combs [15]. This imposes stringent requirements on active stabilization and synchronization systems, significantly increasing system complexity. An alternative solution leverages cavity-multiplexing schemes within a single fiber laser resonator, enabling dual-frequency mode-locking in a common cavity. In such single-cavity configurations, both pulse trains share the same optical path and environmental fluctuations, which inherently improves mutual coherence and ensures a more stable repetition rate difference. Consequently, single-cavity dual-frequency sources offer a compact, passively stable platform that overcomes the coherence and stability limitations of conventional two-laser systems. Several approaches have been explored to generate dual-frequency pulses using fiber lasers, including wavelength multiplexing, bidirectional mode-locking [16] and polarization multiplexing [17,18]. Among these methods, wavelength multiplexing has emerged as a particularly attractive and straightforward method, as it allows two spectrally distinct pulse trains to coexist in a single cavity, naturally enabling asynchronous dual-frequency pulse generation.

Wavelength-multiplexed fiber lasers can be realized using both real saturable absorbers [19,20,21] and artificial saturable absorbers [7,22,23]. Early demonstrations of dual-wavelength mode-locking generally produced fixed central wavelength separation without tunability. In 2016, Zhao et al. [10] employed single-walled carbon nanotubes as a saturable absorber and achieved dual-wavelength mode-locking with central wavelengths at approximately 1533 nm and 1544 nm. The repetition frequencies of the two outputs are 52.743118 MHz and 52.744368 MHz, respectively, with a repetition frequency difference of 1250 Hz and a central wavelength separation of 11 nm. The outputs were further spectrally broadened for dual-comb spectroscopy applications, but tunability of the repetition frequency difference was not realized. Zhao et al. [24] later realized dual-frequency pulses in a single-ring cavity based on nonlinear multimode interference, corresponding to a repetition frequency difference of 633 Hz. This scheme did not rely on additional filters, yet the repetition frequency difference was fixed. Several works have focused on introducing tunability on repetition frequency differences. In 2019, Luo et al. [25] constructed an all-fiber linear-cavity mode-locked laser using a SESAM as the saturable absorber. By incorporating a Lyot filter, they realized dual-frequency pulses with a tunable repetition frequency difference in several hundred hertz. However, the achievable tuning range of repetition frequency difference was rather limited. More recently, in 2020, Liu et al. [26] demonstrated NPR-based dual-wavelength mode-locking, emitting two pulse trains at different repetition frequency. These dual-frequency pulses with different central wavelength enable the switching of repetition frequency difference of 460 Hz and 635 Hz, which can be attributed to the spectral filtering effect introduced by NPR, thereby allowing a certain degree of tunability of repetition frequency difference. Luo et al. [22] reported an all-fiber switchable dual-wavelength mode-locked laser employing a Sagnac loop mirror. In this configuration, the Sagnac loop functions as a tunable periodic filter, with the wavelength separation determined by the length of the polarization-maintaining fiber (PMF). However, the system only enabled tunability of the central wavelength for a single wavelength operation. To adjust the repetition frequency difference, the length of the polarization-maintaining fiber within the Sagnac loop must be modified, making the tuning process highly inconvenient. Hybrid mode-locking has been extensively explored for the generation of dual-frequency pulse trains, leveraging the complementary advantages of real and artificial saturable absorbers. In 2022, Lin et al. [27] demonstrated a hybrid mode-locked Tm/Ho-doped fiber laser based on NPR and SESAM and realized switching and coexistence between dissipative solitons and stretched pulses with the repetition frequency of 10.502056 MHz and 10.502142 MHz, respectively, corresponding to a repetition frequency difference of 86 Hz. However, the two central wavelengths could not be tuned simultaneously, resulting that the repetition frequency difference in dual-frequency pulses could not be tuned. In 2024, Lu et al. [28] introduced a filtering mechanism using spun fiber (SPF) filter, which modifies the birefringence of the fiber and introduces a spectral filtering effect. While maintaining a repetition frequency difference of ∼1.5 kHz, the repetition frequency varied from 30 MHz to 20 MHz by increasing the cavity length. With repetition frequency of 20 MHz, the difference in repetition frequency changes from 0.5 kHz to 1.4 kHz by decreasing SPF length. However, the length of the SPF must be precisely controlled.

The tunability of repetition frequency difference has been explored using filters, NPR-based spectral filter and Sagnac loop mirrors. While these methods can adjust the repetition frequency differences, they are often limited by narrow tuning ranges or cumbersome filtering operation (need precisely control of the length of PMF or other fiber). A wide tuning range of repetition frequency difference is very important for dual-frequency pulse applications. Larger repetition frequency differences support broadband measurements, whereas smaller repetition frequency differences enhance the applications in high-resolution sampling. Thus, tunable repetition frequency difference in dual-frequency pulse provides a versatile and efficient platform for spectroscopy, metrology, and optical communications, with remarkable advantage over those dual-comb sources with fixed repetition frequency difference. The choice of saturable absorber and filter plays a critical role in determining both stability of dual-frequency pulse and tunability of repetition frequency difference. Real saturable absorbers, with their broadband operating range and low mode-locking threshold, enable stable dual-frequency operation, but there is typically no spectral filtering effect. Artificial saturable absorbers, by contrast, provide stronger spectral filtering, though at the expense of higher mode-locking thresholds. These limitations highlight the need for more efficient and versatile schemes. In light of these considerations, we implement a wavelength-multiplexed dual-frequency mode-locked fiber laser capable of generating asynchronous pulses with a widely tunable repetition frequency difference. By integrating the complementary advantages of different saturable absorbers, the hybrid scheme of carbon nanotubes-based SA and NPR overcomes the limitations of single-mode-locking approaches and facilitates efficient generation of dual-frequency pulses with widely tunable parameters. The repetition rate difference is tunable over a broad range, from as low as 147 Hz to nearly 3 kHz. More importantly, this dual-frequency mode-locking is achieved in a simple cavity design without external filtering components, and with straightforward control of central wavelength.

2. Experimental Setup

2.1. Preparation and Characterization of CNT-Based Saturable Absorber

Carbon nanotubes (CNTs) typically possess diameters of a few nanometers and lengths of several micrometers, exhibiting excellent mechanical strength and flexibility. Depending on the number of rolled graphene layers, CNTs can be classified into single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). In this experiment, commercially available SWCNT powder (Merck Life Science, Shanghai, China) was employed. At the initial stage, 12 mg of SWCNT powder was uniformly dispersed into 40 g of deionized water. The mixture was then subjected to intensive ultrasonic treatment for 6 h in order to obtain a stable aqueous dispersion of SWCNTs with superior optical absorption performance. After ultrasonication, the dispersion was centrifuged at 9500 rpm for 20 min. During the preparation of SWCNT-based saturable absorbers, the aggregation of nanotubes into clusters is a common issue, which leads to increased non-saturable losses. Centrifugation facilitates the sedimentation of aggregated SWCNT clusters, thereby improving the uniformity of the dispersion. Following centrifugation, the supernatant was carefully collected. A film-forming agent was then introduced by mixing the supernatant with a 6 wt% polyvinyl alcohol (PVA) aqueous solution. The mixing ratio between the SWCNT dispersion and the PVA solution was 2:1. The resulting SWCNT–PVA composite mixture was subsequently placed in a high-temperature environment at 80 °C and subjected to ultrasonic stirring for an additional 5 h. After this treatment, the SWCNT–PVA dispersion was poured into a Petri dish and placed in an oven for solvent evaporation. The oven temperature was strictly controlled and maintained at 40 °C throughout the evaporation process. Once the solvent had completely evaporated, a solid SWCNT–PVA composite film was obtained. The film was then cut into small pieces of 1 mm × 1 mm. Each film piece was positioned at the interface between two fiber patch cords, ensuring coverage of the fiber core. Finally, the film was tightly secured between the two fiber patch cords using a flange, thus forming a CNT-based saturable absorber (CNT-SA) device ready for subsequent experimental characterization and analysis.

In this experiment, a self-developed balanced dual-detector system was used to measure the nonlinear absorption characteristics of the CNT-SA, as shown as Figure 1a. The core light source is a ring-cavity mode-locked fiber laser, with a high erbium-doped fiber as the gain medium and a 976 nm semiconductor laser as the pump source. The mode-locked mechanism is based on NPR. The output wavelength of this light source is 1565 nm, with a pulse width of approximately 1 ps, a repetition frequency of ~50 MHz, and an initial power of around 1 mW. After amplification by a laboratory-built amplifier, the output power is adjusted using an optical attenuator. Subsequently, the light signal is split into two paths by a 50:50 coupler: one path serves as the reference light directly connected to a high-precision power meter (measuring power P0), and the other path acts as the probing light, illuminating the sample under test and being received by another high-precision power probe (measuring power P1). By adjusting the pump power of the amplifier to vary the incident light intensity and recording the relationship between the ratio of P1/P0 and the reference light power (P0), the nonlinear absorption characteristics of the SWCNT-SA are obtained.

The intensity-dependent transmission curve of the CNT-SA is a key parameter in determining its saturable absorption characteristics. In general, the transmission curve of a saturable absorber can be fitted using the following equation:

$$\mathrm{T}\left(I\right)=1-∆\mathrm{T}\cdot \exp \left(-{\displaystyle \frac{I}{{\mathrm{I}}_{\mathrm{sat}}}}\right)-{\mathrm{T}}_{\mathrm{n}\mathrm{s}}$$

(1)

where $∆\mathrm{T}$ represents the modulation depth, ${\mathrm{T}}_{\mathrm{n}\mathrm{s}}$ denotes the non-saturable loss, $\mathrm{I}$ is the input intensity, ${\mathrm{I}}_{\mathrm{sat}}$ is the saturation intensity, and $\mathrm{T}\left(I\right)$ is the transmission ratio.

From Figure 1b, it can be observed that the transmittance exhibits a nonlinear increase at low incident light flux, indicating operation in the nonlinear absorption region. This behavior is primarily attributed to the saturation effect of the SWCNT-SA. As the incident light flux gradually increases, the light absorption of the CNT-SA reaches saturation, entering the linear absorption region. By fitting the curve using the aforementioned formula, the modulation depth of the saturable absorber is determined to be 21.2%. The magnitude of the nonlinear absorption of CNT-SA reflects its role in modulating the mode-locked pulses.

2.2. The Principle of NPR Mechanism

The nonlinear polarization rotation (NPR) technique relies on the intensity-dependent polarization evolution of light propagating through a birefringent fiber segment, which, when combined with a polarization-dependent element such as a polarizer or isolator, forms an effective saturable absorber. The transmission of the NPR element can be rigorously derived using the Jones matrix formalism [29].

Assuming that the input electric field is linearly polarized at an angle α with respect to the fast axis of the birefringent fiber, the Jones vector of the input field can be expressed as

$${\mathrm{E}}_{\mathrm{i}\mathrm{n}}=\left({\displaystyle \genfrac{}{}{0pt}{}{\cos \alpha }{\sin \alpha }}\right)$$

(2)

where α is the angle between the input polarization and the reference axis.

After propagating through a birefringent fiber segment of length L, the field experiences different phase delays along the fast and slow axes, which can be represented by

$$\mathrm{J}=\left({\displaystyle \genfrac{}{}{0pt}{}{{\mathrm{e}}^{\mathrm{i}{\phi }_x}}{0}}{\displaystyle \genfrac{}{}{0pt}{}{0}{{\mathrm{e}}^{\mathrm{i}{\phi }_y}}}\right)$$

(3)

where φ_x and φ_y denote the phase delays along the two principal axes. Defining the phase difference as $\Delta \phi ={\phi }_x-{\phi }_y$, and the average phase ${\phi }_{\mathrm{avg}}=\left({\phi }_x+{\phi }_y\right)/2$, the output field can be written as

$${\mathrm{E}}_{\mathrm{o}\mathrm{u}\mathrm{t}}={\mathrm{e}}^{\mathrm{i}{\phi }_{avg}}\left({\displaystyle \genfrac{}{}{0pt}{}{\cos \alpha {\mathrm{e}}^{\frac{\mathrm{i}\Delta \phi }{2}}}{\sin \alpha {\mathrm{e}}^{-\frac{\mathrm{i}\Delta \phi }{2}}}}\right)$$

(4)

After passing through a polarization analyzer (or polarizer) oriented at an angle θ, the transmitted field amplitude becomes

$$\mathrm{A}={\mathrm{e}}^{\mathrm{i}{\phi }_{\mathrm{a}\mathrm{v}\mathrm{g}}}(\cos \theta \cos \alpha {\mathrm{e}}^{{\displaystyle \frac{\mathrm{i}\Delta \phi }{2}}}+\sin \theta \sin \alpha {\mathrm{e}}^{-{\displaystyle \frac{\mathrm{i}\Delta \phi }{2}}})$$

(5)

The normalized transmission intensity, given by $\mathrm{T}={\left|\mathrm{A}\right|}^2$, is obtained as

$$\mathrm{T}\left(\Delta \phi ;\alpha ,\theta \right)={\displaystyle \frac{1}{2}}\left[1+\cos 2\theta \cos 2\alpha +\sin 2\theta \sin 2\alpha \cos ∆\phi \right]$$

(6)

The total phase delay ∆φ consists of a linear (wavelength-dependent) component and an intensity-dependent nonlinear component:

$$\Delta \phi \left(\lambda ,I\right)=\Delta {\phi }_0\left(\lambda \right)+\Delta {\phi }_{\mathit{NL}}\left(I\right)$$

(7)

The linear birefringent phase delay is

$$\Delta {\phi }_0\left(\lambda \right)={\displaystyle \frac{2\pi \Delta n\left(\lambda \right) {\mathrm{L}}_{\mathrm{e}\mathrm{f}\mathrm{f}}}{\lambda }}$$

(8)

where Δn(λ) is birefringence (wavelength-dependent), L_eff is the equivalent birefringence length. the nonlinear phase shift due to the Kerr effect is expressed as $\Delta {\phi }_{\mathit{NL}}\left(I\right)=\kappa I$ or $\Delta {\phi }_{\mathit{NL}}=\gamma {\mathrm{L}}_{\mathrm{eff}}\mathrm{P}$, with κ (or γL_eff) being the nonlinear phase coefficient, and I (or P) the optical intensity (or power).

Since the term 1/λ is included in Δφ₀(λ), the transmission oscillates periodically with wavelength. The nonlinear phase φ_NL(I) changes with intensity, causing a phase shift in the interference fringes on the spectrum, thereby forming an intensity dependence for the transmission at a certain wavelength. This is the fundamental reason why NPR can act as an equivalent saturable absorber. It can be seen that NPR exhibits an adjustable multi-peak (comb-like) filter in the frequency domain. By adjusting α, θ and nonlinear phase, the passband position and shape can be changed, thereby achieving wavelength selection or multimode soliton formation. Changing PC is equivalent to changing α, θ. In summary, the NPR transmission spectrum exhibits periodic modulation determined by the birefringence and cavity parameters, while its intensity dependence originates from the Kerr-induced nonlinear phase. This property allows NPR to act as an artificial saturable absorber, enabling passive mode-locking and wavelength-selective operation in fiber lasers.

2.3. Experimental Setup of the Laser

In this study, a CNT-SA was employed as one of the mode-locking elements, in combination with the NPR mechanism, to construct a hybrid mode-locked erbium-doped fiber laser (see schematic in Figure 2). The laser adopted a ring cavity configuration, whose core components included: a 976 nm pump laser source, a wavelength-division multiplexer (WDM), erbium-doped fiber (EDF), a polarization-dependent isolator (PD-ISO), the custom-fabricated CNT-SA, polarization controllers (PCs), and a 30:70 fiber coupler. The system was pumped by a single-mode semiconductor laser diode operating at 976 nm with a maximum output power of 500 mW. The pump light was coupled into the cavity through a 980/1550 WDM. A 0.5 m-long EDF (Er110-4/125, LIEKKI, Camas, WA, USA) with a dispersion of approximately −20 ps/(nm·km) at 1550 nm was spliced after the WDM and served as the gain medium. Backward pumping was adopted to improve conversion efficiency, while the PD-ISO ensured unidirectional light propagation in the cavity. Together with the PCs, the PD-ISO formed the key elements of the NPR mode-locking mechanism, enabling the polarization state of the intracavity light to be controlled. The homemade CNT-SA film was mounted between two fiber connectors and integrated into the cavity as an additional saturable absorber. A 30:70 fiber coupler extracted 30% of the intracavity power for output and diagnostics. All other fiber segments in the cavity were Corning SMF-28e+ (dispersion ≤ 18 ps/(nm·km) at 1550 nm, Corning, NY, USA), resulting in a total cavity length of ~4.5 m. The laser output was characterized using a variety of instruments: an optical spectrum analyzer (OSA, AQ6370B, YOKOGAWA, Tokyo, Japan) for optical spectra; a photodetector (DET08CFC/M, 500 MHz, Thorlabs, Newton, NJ, USA) and an oscilloscope (DS4054, 4 GHz, RIGOL, Suzhou, China) for temporal pulse measurements; a radio-frequency spectrum analyzer (DSA815, RIGOL, Suzhou, China) for RF spectra; an optical autocorrelator (FR-103XL, FEMTOCHROME, Berkeley, CA, USA) for pulse width determination; and an optical power meter (UT692G, UNI-T, Dongguan, China) for output power measurements.

3. Experimental Results and Analysis

3.1. Single-Wavelength Mode-Locking

We first demonstrate and investigate the operation of the laser in the single-wavelength mode-locking regime. When the pump power was increased to 50 mW, continuous-wave (CW) emission centered around 1550 nm was observed at the output port of the coupler. Keeping the intracavity polarization unchanged and further increasing the pump power, stable single-wavelength mode-locked pulses were obtained when the pump power reached 100 mW. By adjusting the polarization controller while keeping other parameters fixed, the intracavity polarization state could be modified, leading to mode-locked pulses at different central wavelengths and 3 dB bandwidth. Figure 3a–c shows examples of three single-wavelength mode-locked pulses with different central wavelengths under the same pump power (pump power is 150 mW). The central wavelengths are 1563 nm 1557.9 nm and 1558.2 nm with different 3 dB bandwidths of 10.9 nm, 9 nm and 2.2 nm, respectively. In the single-pulse regime, according to Figure 3d,e, the pulse train exhibited a repetition frequency of ~45.539799 MHz, which is consistent with the cavity round-trip time of 22 ns.

3.2. Dual-Wavelength Dual-Frequency Mode-Locking

As the pump power was gradually increased from zero, the fiber laser initially exhibited continuous-wave (CW) output at 1532 nm and 1556 nm. When the pump power reached ~150 mW, narrow CW peaks appeared near 1530 nm and 1556 nm in the output spectrum, indicating the potential for dual-wavelength mode-locking. By carefully adjusting PC1 and PC2, stable dual-wavelength mode-locking was achieved. Figure 4a shows the measured mode-locked spectrum. The laser generated two pulse trains at distinct central wavelengths of ~1530.9 nm and ~1557.6 nm, with a wavelength separation of 26.7 nm. Notably, the mode-locked spectrum around 1557.6 nm exhibited pronounced Kelly sidebands, suggesting traditional soliton operation, whereas the sharp and asymmetric peak at ~1530.9 nm is likely a narrow-bandwidth mode-locked pulse. The total output power was measured to be 2.17 mW, comparable to that in the single-wavelength case. The corresponding RF spectrum of the pulse trains is shown in Figure 4b. The RF trace exhibited a distinct dual-peak structure, arising from the group velocity dispersion of the cavity, which leads to a slight difference in repetition frequencies between the two wavelength pulse trains. Since the laser operated in the anomalous dispersion regime, the longer-wavelength (1557.6 nm) pulse corresponded to a slightly lower repetition frequency, while the shorter-wavelength (1530.9 nm) pulse corresponded to a higher repetition frequency. Specifically, the repetition frequency of the 1557.6 nm pulse was ~45.536098 MHz, whereas that of the 1530.9 nm pulse was ~45.539157 MHz, with a repetition frequency difference of ~3.059 kHz. Moreover, the pulse centered at 1557.6 nm exhibited a broader mode-locked spectrum, and its RF peak was higher than that of the 1530.9 nm pulse, further confirming dual-frequency mode-locking operation. On a larger time scale on the oscilloscope, the oscilloscope trace revealed temporal interference fringes with a period of ~320 μs (Figure 4c), which is consistent with the inverse of the repetition frequency difference, thereby confirming the stability of the dual-comb operation. Additionally, time-domain measurements demonstrated the relative drift of the two pulse trains (Figure 4d,e). Due to their distinct group velocities, the two pulse trains produced periodic scanning beat notes on the oscilloscope, further validating that the dual-frequency mode-locked pulses were operating in an asynchronous mode-locking regime.

Figure 4f shows the autocorrelation trace of the dual-frequency mode-locked pulses, where the measured profile results from the combined contribution of the two pulse trains. Gaussian fitting yielded a pulse width of 1.064 ps. To further investigate their individual characteristics, the dual-wavelength pulses were separated using an external 1530/1550 coarse wavelength-division multiplexer (CWDM). The measured spectra after separation are presented in Figure 5a,b. As shown, the CWDM successfully divided the original dual-wavelength spectrum into two parts: 1520–1540 nm and 1540–1560 nm. Although some spectral deformation occurred, the two pulse trains centered at 1530.9 nm and 1557.6 nm could be well distinguished.

After separation, the output powers of the 1530.9 nm and 1557.6 nm pulses were 0.225 mW and 0.524 mW, respectively. Both were comparable in magnitude but lower than the total output power measured directly from the laser cavity, mainly due to spectral losses introduced by the CWDM filtering process. To accurately evaluate the temporal and spectral features of the two ports, RF spectra and autocorrelation traces were measured using a spectrum analyzer and an optical autocorrelator, as shown in Figure 5c,d. The RF spectrum of the 1550 nm port exhibited a clear single peak, further confirming successful pulse separation. In contrast, the 1530 nm port RF spectrum contained a small residual contribution from the 1557.6 nm pulse, owing to the broad spectral bandwidth of the 1557.6 nm component, which extended into the 1530 nm region and could not be fully filtered by the CWDM. From the autocorrelation traces in Figure 5e,f, noticeable temporal broadening was observed in both separated pulse trains, with the pulse widths increasing to 1.477 ps and 1.441 ps for the 1530 nm and 1550 nm ports, respectively.

By appropriately adjusting PC1 and PC2 without changing the pump power, dual-wavelength dual-frequency pulses with different repetition frequency differences were obtained. Figure 6a shows the measured mode-locked spectrum, with central wavelengths at ~1430.76 nm and ~1555.92 nm, corresponding to a wavelength separation of 25.16 nm, indicating a certain degree of spectral tunability. The total output power was measured to be 2.1 mW. The RF spectrum of the pulse trains is presented in Figure 6b. A distinct dual-frequency peak structure was again observed, with repetition frequencies of ~45.539262 MHz and ~45.536485 MHz, corresponding to a repetition frequency difference of ~2.777 kHz. Compared with the previously reported dual-frequency pulses, the longer-wavelength mode-locked pulse at 1555.92 nm exhibited a narrower optical spectrum, which was also reflected in the RF spectrum as a lower repetition frequency peak relative to the 1430.9 nm component. On a large time scale, the oscilloscope trace revealed clear temporal interference fringes (Figure 6c), with a measured period of ~380 μs, consistent with the repetition frequency difference. Furthermore, time-domain measurements in Figure 6d,e once again showed the relative drift of the two pulse trains, producing two periodically scanning pulse sequences on the oscilloscope, with one sequence exhibiting a lower intensity compared to the other.

The dual-frequency pulses were subsequently separated using an external 1530/1550 CWDM. The measured spectra after separation are shown in Figure 7a,b. The separated spectral widths were consistent with the original dual-wavelength spectrum, and the RF signals were more cleanly separated (Figure 6c,d). After separation, the output powers of the 1430.76 nm and 1555.92 nm pulses were 0.3 mW and 0.8 mW, respectively. Compared to the previous case, the power difference between the two pulses was more pronounced.

These observations confirm the stable generation of asynchronous dual-frequency pulses. Importantly, the ability to obtain different spectral spacings and repetition frequency offsets further demonstrates the tunability of the hybrid mode-locking scheme. In the following, we present another dual-frequency state with distinct spectral and temporal characteristics.

With the pump power kept constant, further adjustment of PC1 and PC2 to control the intracavity polarization and loss led to the emergence of a dual-frequency pulse state with an extremely narrow optical spectrum. Figure 8a shows the narrow-band mode-locked spectrum, where a sharp and narrow peak was observed around 1557 nm, while no significant peaks were detected at other wavelengths. A magnified view of the spectrum (Figure 8b) reveals that this sharp feature actually consisted of two closely spaced wavelength components at 1556.62 nm and 1557.64 nm, with a spectral separation of only 1.02 nm, representing a remarkably small wavelength separation. The corresponding RF spectrum, measured with a spectrum analyzer (Figure 8c), displayed a dual-frequency peak structure similar to the previous results, confirming that the cavity was operating in a dual-frequency state. The two pulse trains exhibited repetition frequencies of 45.536076 MHz and 45.536213 MHz, with a frequency difference of 137 Hz. The output pulse sequences were further measured using a photodetector and oscilloscope, as shown in Figure 8. In a large time scale, clear temporal interference fringes with a period of ~7 ms were observed, in great fitting with the measured repetition frequency difference. Figure 8e,f illustrates the relative temporal drift of the two pulse trains, again confirming asynchronous mode-locking. Although no intracavity polarization beam splitter (PBS) or PMF was used, additional polarization diagnostics were performed at the output port by inserting an external PBS and a PC to analyze the vector characteristics and polarization states of the emitted light. The results showed that the two RF peaks did not exhibit relative intensity variation when the external PC was adjusted; instead, the intensities of both decreased or increased simultaneously. This behavior indicates that the two asynchronous pulses shared nearly identical polarization states, consistent with the characteristics of wavelength-multiplexing dual-frequency mode-locking.

4. Discussion

The hybrid mode-locking scheme combining CNTs and NPR enables stable dual-frequency operation due to the complementary roles of the two mechanisms. CNTs, as broadband saturable absorbers, provide low-threshold, fast saturable absorption across a wide spectral range, allowing simultaneous mode-locking at multiple wavelengths. NPR, on the other hand, introduces intracavity spectral filtering and polarization-dependent loss, which selectively favors the formation of two distinct wavelength components and helps to suppress undesired modes. Importantly, the combination of CNTs and NPR also facilitates a balance of gain competition: the broadband absorption of CNTs supports simultaneous amplification of both wavelengths, while the polarization-dependent filtering of NPR adjusts the relative intracavity losses, preventing one wavelength from dominating the other. This synergy stabilizes dual-frequency mode-locking and enables tunable repetition frequency differences without the need for external filters. Due to the presence of cavity dispersion, different central wavelengths lead to different group velocities. According to the work of [25], the relationship between the repetition frequency difference and the wavelength difference can be expressed as follows. In high-repetition-frequency pulses, the repetition frequency difference between the two pulses is negligible compared to the overall repetition frequency, which allows further simplification of the formula:

$$\Delta \mathrm{f}={\displaystyle \frac{\mathrm{L}\mathrm{D}\Delta \lambda }{{\mathrm{T}}_1{\mathrm{T}}_2}}=\mathrm{L}\mathrm{D}\Delta \lambda {\mathrm{f}}_1{\mathrm{f}}_2\approx \mathrm{L}\mathrm{D}\Delta \lambda {\mathrm{f}}^2$$

(9)

where $\Delta \mathrm{f}$ and $\Delta \lambda$ represent the repetition frequency difference and the corresponding wavelength difference, respectively. $\mathrm{L}$ and $\mathrm{D}$ are the cavity length and average cavity dispersion, $\mathrm{f}$ is the fundamental repetition frequency of the mode-locked pulses. Using this formula, we can roughly estimate the corresponding values: the wavelength separation of 26.7 nm corresponds to a repetition rate difference of approximately 3.4 kHz (3.059 kHz in our experiment), while a spacing of 1.02 nm corresponds to about 130 Hz (147 Hz in this work). These values are in good agreement with our experimental results. Both the theoretical fitting and experimental data demonstrate that small variations in central wavelength separation can induce substantial changes in repetition rate difference, confirming the reliability of the experimentally achieved wide tunability in repetition frequency difference. The observed tunability of both central wavelength separation and repetition frequency difference represents a substantial advancement over previous reports. Our laser can generate dual-frequency mode-locking with large repetition frequency differences on the order of kilohertz, as well as narrowband outputs with repetition frequency differences as small as hundreds of hertz, corresponding to central wavelength separations ranging from 26.7 nm down to 1.02 nm. Such a wide tuning capability enables seamless switching between broadband and narrowband dual-wavelength operation.

Table 1. Comparison of adjustable wavelength separation and repeat frequency difference in different studies. $\lambda$ represents the exemplary center wavelengths achievable in dual-wavelength mode-locking, $∆\lambda$ denotes the tunable center wavelength separation, and $∆f$ indicates the tunable repetition frequency difference.

Reference	Gain	Mode Locked Mechanism	λ (nm)	Δλ (nm)	f (MHz)	Δf (Hz)
[30]	EDF	SWCNT-SA+NPE	1533.1, 1555.0	17.6–22	~28.29	1130–1370
[23]	PM-EDF	NALM	1558, 1581	23–25	~13.00	976–1164
[18]	EDF	BPQDs-SA+TF	1533.86, 1535.24	0.8–1.38	~23.3	188–200
[28]	PM-EDF	CNT-SA+SPF	1554, 1580	14.1–28.2	~19.7	507–1438
[20]	PM-EDF	GO-COOH-SA	1529, 1564	5–35	~34.82	444–2933
This work	EDF	SWCNT-SA+NPR	1530.9, 1557.6	1.02–26.7	~45.53	147~3059

compares the tunability of wavelength multiplexing dual-frequency mode-locking in erbium-doped fiber lasers as reported in recent studies alongside the results of this work. It can be observed that spectral filtering structures—such as NPR, tunable filters (TF), and spun fiber filter (SPF) or functional equivalents are commonly incorporated into the laser cavities. Notably, the use of polarization-maintaining erbium-doped fiber (PM-EDF) effectively introduces a Lyot filter effect into the cavity. A comparison of the tunable wavelength separation across studies reveals that our system achieves a notably wide tuning range of up to approximately 25 nm, while also capable of reaching a narrow separation of as low as 1 nm. The consistency of our minimal separation value with that reported in Ref. [18] further supports the validity of our findings. Although Ref. [20] demonstrates a broader maximum wavelength separation of 30 nm, our work offers a significantly wider range of repetition frequency difference tuning. Compared with recently reported dual-frequency mode-locked fiber lasers, our hybrid architecture provides a structurally simple, single-cavity implementation that avoids precisely manufactured filters or modification of fiber lengths. This combination enables tunable Δλ without insertion of bulky tunable filters or PMF. In summary, the exceptionally wide tunability of the dual-frequency mode-locking arises from two main factors. First, the intracavity saturable absorption provides broadband support and spectral filtering, enabling stable dual-wavelength operation with flexible control over the central wavelengths and spectral bandwidths. Second, the relatively large average cavity dispersion D allows for a substantial repetition frequency difference between the dual wavelengths around 1530 nm and 1550 nm. This high dispersion results from the combination of SMF with inherently large dispersion and a short segment of EDF that provides limited compensation. Consequently, a high net cavity dispersion facilitates larger repetition frequency differences at wider wavelength separations, while smaller wavelength separation allows correspondingly smaller repetition frequency differences, significantly expanding the accessible tuning range.

5. Conclusions

In this work, a hybrid mode-locking scheme using CNT and NPR is employed to achieve asynchronous mode-locked pulse with wavelength multiplexing. The laser can operate in a wide spectral range, achieving dual-frequency pulse with tunable different repetition frequency differences. Additionally, narrow-band dual-wavelength output with a wavelength separation of approximately 1 nm was obtained near 1556 nm. The repetition frequency difference can be tunable from 147 Hz to 3 kHz, significantly expanding the mode-locking range of the dual-frequency operation. It provides valuable insights for the study of wavelength-multiplexed mode-locked fiber lasers and dual-frequency applications with multiple repetition frequency differences. The flexibility in tuning repetition frequency difference provides reference for advancements in a variety of fields, including high-precision optical frequency metrology, broadband and narrowband dual-comb spectroscopy and other dual-comb applications.