Wavelength-Tunable L-Band High Repetition Rate Erbium-Doped Fiber Laser Based on Dissipative Four-Wave Mixing

A wavelength-tunable high repetition rate (HRR) erbium-doped fiber laser in L-band based on dissipative four-wave mixing (DFWM) mechanism is demonstrated. The cavity can generate a single-soliton train and bound-soliton train with a fixed repetition rate of ~126 GHz, which is determined by the free spectral range of the intra-cavity Lyot filter. A wide wavelength-tuning operation can also be obtained by rotating the polarization controllers. The wavelength-tuning ranges of the HRR single-soliton state and HRR bound-soliton state are ~38.3 nm and ~22.6 nm, respectively. This laser provides useful references for the area of a wavelength-tunable fiber laser with high repetition rate. The laser may also find useful applications in high-speed communication, sensing, etc.


Introduction
In recent decades, mode-locked fiber lasers with wavelength tunability have attracted great attention for various applications, such as micromachining, sensing, spectroscopy, and optical communication [1][2][3]. In particular, for high-speed optical communication systems, the conventional C-band (1530-1565 nm) is unable to meet the increasing requirement of communication capacity. Therefore, it is of great significance to investigate L-band (1565-1625 nm) wavelength-tunable mode-locked fiber lasers. Experimentally, there are two major ways to achieve wavelength tunability. The first is incorporating a tunable spectral filter in the cavity, such as a W-shaped long-period fiber grating (LPFG) filter [4], chirped fiber Bragg grating [5], birefringence-induced fiber filter [6], and commercial tunable band-pass filter [7]. The other method is to control intra-cavity loss by using some special elements, such as a tunable-ratio optical coupler [8], fiber taper [9], mechanical attenuator [10], and LPFG-based narrow-band optical attenuator [11]. However, these conventional wavelength-tunable EDFLs mostly operate at the repetition rate of megahertz. In order to make full use of the L-band for optical communication, it is also necessary to improve the repetition rate of the EDFLs to gigahertz (GHz).
To achieve a high-repetition rate (HRR) pulse train, a variety of passive mode-locking mechanisms have been reported, such as harmonic mode locking [12][13][14], short cavity [15,16], and dissipative four-wave mixing (DFWM) [17,18]. Compared to harmonic mode locking and short cavity, DFWM is a simpler and more effective method to generate an HRR pulse train with a capability even up to terahertz. In order to achieve a DFWM modelocking operation, large nonlinearity is necessary, which contributes to phase matching between longitudinal modes in the cavity. Such intra-cavity nonlinearity can be enhanced by inserting a segment of highly nonlinear fiber, i.e., photonic crystal fiber [19]. In addition, Due to the existence of PD-ISO and PMF, a wavelength-dependent b gence-induced filter with a comb transmission spectrum is formed in the cavity This filter is also called a Lyot filter and its transmission function can be descr [36,37] cos cos sin sin 1 2 sin 2 sin 2 cos ∆ ∆ In Equation (1), and are azimuth angles of the two ends of the polariz tive to the fast axis of the PMF. ∆ 2 ∆ / is the linear phase delay, and ∆ 2 cos 2 / is the nonlinear phase delay. and ∆ are length an fringence intensity of PMF, respectively. is laser's operating wavelength, linear coefficient, is instantaneous power of optical signal, and is effecti of core of PMF. Additionally, the FSR of fiber-based Lyot filter can be calculated b

Δ ∆
The modal birefringence of PMF used in cavity is 4.32 × 10 −4 . Since the transm spectrum of the filter is only related to PMF, we disconnected the connection betw coupler and the PMF and spliced a polarizer after the PMF to measure the transm spectrum of the fiber-based Lyot fiber. As shown in Figure 2a, the solid line is perimental measurement result, and the FSR is ~126 GHz. The dashed line is the lated result, which is in good agreement with the measured results. We also me the transmission spectrum at different polarization states, which is illustrated in 2b. The depth of the transmission spectrum changes with the adjustment of the o tions of PCs. We also simulated the transmission spectrum under different angle and (dashed line in Figure 2(b-1,b-2)), which is in good agreement with the mental results. Due to the existence of PD-ISO and PMF, a wavelength-dependent birefringenceinduced filter with a comb transmission spectrum is formed in the cavity [34,35]. This filter is also called a Lyot filter and its transmission function can be described as [36,37] In Equation (1), θ 1 and θ 2 are azimuth angles of the two ends of the polarizer relative to the fast axis of the PMF. ∆ϕ L = 2πL∆n/λ is the linear phase delay, and ∆ϕ NL = 2πn 2 PL cos(2θ 1 )/ λA e f f is the nonlinear phase delay. L and ∆n are length and birefringence intensity of PMF, respectively. λ is laser's operating wavelength, n 2 is nonlinear coefficient, P is instantaneous power of optical signal, and A e f f is effective area of core of PMF. Additionally, the FSR of fiber-based Lyot filter can be calculated by [27] The modal birefringence of PMF used in cavity is 4.32 × 10 −4 . Since the transmission spectrum of the filter is only related to PMF, we disconnected the connection between the coupler and the PMF and spliced a polarizer after the PMF to measure the transmission spectrum of the fiber-based Lyot fiber. As shown in Figure 2a, the solid line is the experimental measurement result, and the FSR is~126 GHz. The dashed line is the simulated result, which is in good agreement with the measured results. We also measured the transmission spectrum at different polarization states, which is illustrated in Figure 2b. The depth of the transmission spectrum changes with the adjustment of the orientations of PCs. We also simulated the transmission spectrum under different angles of θ 1 and θ 2 (dashed line in Figure 2(b-1,b-2)), which is in good agreement with the experimental results.

HRR Output with the Repetition Rate of ~126 GHz
Due to the spectral filtering effect of the fiber-based Lyot filter, the DFWM mode-locking operation self-starts under the forward (LD1) and backward (LD2) pump power of 332 mW and 200 mW. Figure 3 shows a stable HRR single-soliton train when the pump power of LD1 and LD2 is 865 mW and 500 mW. The output power of the cavity is 13.69 mW. Figure 3a is the measured comb spectrum with a center wavelength of 1604.30 nm and a peak-to-peak interval of 1.08 nm, which corresponds to ~126 GHz. Here, we attribute this L-band oscillation to the in-band absorption of EDF [38,39]. The autocorrelation (AC) trace of the single-soliton train shown in Figure 3b has a pulse-to-pulse interval of 8.04 ps, which is in good agreement with the interval of spectral peaks. Additionally, the inset in Figure 3b is the measurement result across the entire scanning range. It is noted that the AC traces have a continuous wave background, and the pulse sequence is not flat. We speculate that this unflatness of the pulse intensity may be caused by the modulation instability of the two main spectral peaks [27].

HRR Output with the Repetition Rate of~126 GHz
Due to the spectral filtering effect of the fiber-based Lyot filter, the DFWM mode-locking operation self-starts under the forward (LD1) and backward (LD2) pump power of 332 mW and 200 mW. Figure 3 shows a stable HRR single-soliton train when the pump power of LD1 and LD2 is 865 mW and 500 mW. The output power of the cavity is 13.69 mW. Figure 3a is the measured comb spectrum with a center wavelength of 1604.30 nm and a peak-to-peak interval of 1.08 nm, which corresponds to~126 GHz. Here, we attribute this L-band oscillation to the in-band absorption of EDF [38,39]. The autocorrelation (AC) trace of the single-soliton train shown in Figure 3b has a pulse-to-pulse interval of 8.04 ps, which is in good agreement with the interval of spectral peaks. Additionally, the inset in Figure 3b is the measurement result across the entire scanning range. It is noted that the AC traces have a continuous wave background, and the pulse sequence is not flat. We speculate that this unflatness of the pulse intensity may be caused by the modulation instability of the two main spectral peaks [27].
When we keep the pump power of LD1 and LD2 fixed at 715 mW and 500 mW, the wavelength tuning operation can been achieved by carefully adjusting PCs. As illustrated in Figure 4a, the center wavelength of the HRR single-soliton train can be tuned from 1569.83 nm to 1608.21 nm, with a wavelength tuning range of~38.3 nm in the L-band. To our best knowledge, this is the widest spectral-tuning range from which the DFWM fiber laser works in the L-band so far. During the process of wavelength tuning, the HRR single-soliton train has been well maintained, which is demonstrated in Figure 4b. We believed that such wavelength tunability is attributed to broad transmission characteristics of the intra-cavity Lyot filter in the L-band. The orientations of PCs lead to the change of modulation depth of the Lyot filter and the position of maximum gain, which shift the center wavelength of output pulses. In particular, this spectral tuning process is reversible.
Keeping the pump power fixed and further adjusting the orientations of PCs, the bound-soliton state can be obtained with the output power of 8.275 mW and the repetition rate of a unit of bound solitons also matches the frequency spacing of the Lyot filter. We believe that modulation instability has played a role in the formation of the HRR bound-soliton train, altering the repetition unit of the cavity emission [27]. The measured results are shown in Figure 5. The middle of the spectrum is concave, and the peaks on two sides are the strongest in Figure 5a, which is a typical characteristic of the bound-soliton state. The peak-to-peak interval of the spectrum is 1.06 nm, corresponding to a repetition rate of 126 GHz. Figure 5b shows the AC traces with an interval of~8.03 ps, which is consistent with the result in the HRR single-soliton state. In addition, it seems that the pulse-to-pulse interval in a single bound soliton is related to the spectrum. The interval between two highest peaks is 3.18 nm, corresponding to a repetition rate of~380 GHz, which matches well with the side-pulse interval of 2.62 ps.

HRR Output with the Repetition Rate of ~126 GHz
Due to the spectral filtering effect of the fiber-based Lyot filter, the DFWM mode-locking operation self-starts under the forward (LD1) and backward (LD2) pump power of 332 mW and 200 mW. Figure 3 shows a stable HRR single-soliton train when the pump power of LD1 and LD2 is 865 mW and 500 mW. The output power of the cavity is 13.69 mW. Figure 3a is the measured comb spectrum with a center wavelength of 1604.30 nm and a peak-to-peak interval of 1.08 nm, which corresponds to ~126 GHz. Here, we attribute this L-band oscillation to the in-band absorption of EDF [38,39]. The autocorrelation (AC) trace of the single-soliton train shown in Figure 3b has a pulse-to-pulse interval of 8.04 ps, which is in good agreement with the interval of spectral peaks. Additionally, the inset in Figure 3b is the measurement result across the entire scanning range. It is noted that the AC traces have a continuous wave background, and the pulse sequence is not flat. We speculate that this unflatness of the pulse intensity may be caused by the modulation instability of the two main spectral peaks [27]. When we keep the pump power of LD1 and LD2 fixed at 715 mW and 500 mW, the wavelength tuning operation can been achieved by carefully adjusting PCs. As illustrated in Figure 4a, the center wavelength of the HRR single-soliton train can be tuned from 1569.83 nm to 1608.21 nm, with a wavelength tuning range of ~38.3 nm in the L-band. To our best knowledge, this is the widest spectral-tuning range from which the DFWM fiber laser works in the L-band so far. During the process of wavelength tuning, the HRR single-soliton train has been well maintained, which is demonstrated in Figure 4b. We believed that such wavelength tunability is attributed to broad transmission characteristics of the intra-cavity Lyot filter in the L-band. The orientations of PCs lead to the change of modulation depth of the Lyot filter and the position of maximum gain, which shift the center wavelength of output pulses. In particular, this spectral tuning process is reversible. Keeping the pump power fixed and further adjusting the orientations of PCs, the bound-soliton state can be obtained with the output power of 8.275 mW and the repetition rate of a unit of bound solitons also matches the frequency spacing of the Lyot filter. We believe that modulation instability has played a role in the formation of the HRR Similar to the situation of the HRR single-soliton state, the center wavelength of the spectrum in the bound-soliton state is also tunable. As illustrated in Figure 6a, the center wavelength has been shifted from 1578.98 nm to 1601.60 nm through adjusting PCs carefully and slowly, with a wavelength tuning range of ~22.6 nm in the L-band. Figure  6b exhibits the corresponding AC traces of the HRR bound-soliton state. There always is a strong background pedestal in measured AC traces regardless of HRR single-soliton or HRR bound-soliton emissions.This phenomenon may be caused by supermode noise in the cavity [20,26] and can be suppressed by inserting a sub-cavity structure for enhanced filtering [26] or using a high-finesse filter instead [40][41][42].  Similar to the situation of the HRR single-soliton state, the center wavelength of the spectrum in the bound-soliton state is also tunable. As illustrated in Figure 6a, the center wavelength has been shifted from 1578.98 nm to 1601.60 nm through adjusting PCs carefully and slowly, with a wavelength tuning range of~22.6 nm in the L-band. Figure 6b exhibits the corresponding AC traces of the HRR bound-soliton state. There always is a strong background pedestal in measured AC traces regardless of HRR single-soliton or HRR boundsoliton emissions.This phenomenon may be caused by supermode noise in the cavity [20,26] and can be suppressed by inserting a sub-cavity structure for enhanced filtering [26] or using a high-finesse filter instead [40][41][42]. Similar to the situation of the HRR single-soliton state, the center wavelength of the spectrum in the bound-soliton state is also tunable. As illustrated in Figure 6a, the center wavelength has been shifted from 1578.98 nm to 1601.60 nm through adjusting PCs carefully and slowly, with a wavelength tuning range of ~22.6 nm in the L-band. Figure  6b exhibits the corresponding AC traces of the HRR bound-soliton state. There always is a strong background pedestal in measured AC traces regardless of HRR single-soliton or HRR bound-soliton emissions.This phenomenon may be caused by supermode noise in the cavity [20,26] and can be suppressed by inserting a sub-cavity structure for enhanced filtering [26] or using a high-finesse filter instead [40][41][42].

HRR Output with the Repetition Rate of~71 GHz
In our experiment, HRR EDFLs with different repetition rates can be obtained by merely changing the length of PMF. This is because the FSR of the fiber-based Lyot filter is related to the length of PMF. For example, if we want to achieve a pulse train with a repetition rate of 70 GHz in the L-band, the calculated length of PMF is 10 m, and the corresponding FSR is 0.6 nm. In order to verify this, we insert a 10 m PMF into the cavity. When the pump power of LD1 and LD2 is 893 mW and 660 mW, the HRR single-soliton train can be generated. Figure 7a shows the corresponding optical spectrum with a peak-to-peak interval of 0.61 nm, which is consistent with the calculated result. Additionally, the interval of pulses is measured to be~14.39 ps. The experimental results strongly prove the efficiency of this technique. Keeping the pump power fixed, the above-mentioned wavelength-tuning operation can be obtained by adjusting PCs. As exhibited in Figure 8a, the central wavelength of the spectrum has been tuned from 1572.58 nm to 1608.12 nm, nearly 35.6 nm. Additionally, the corresponding AC traces are shown in Figure 8b. In addition, the HRR bound-soliton trains with the same repetition rate of~71 GHz and the corresponding wavelength-tuning operation can also be achieved, which are similar to the results in Figures 5 and 6.

HRR Output with the Repetition Rate of ~71 GHz
In our experiment, HRR EDFLs with different repetition rates can be obtained by merely changing the length of PMF. This is because the FSR of the fiber-based Lyot filter is related to the length of PMF. For example, if we want to achieve a pulse train with a repetition rate of ~70 GHz in the L-band, the calculated length of PMF is 10 m, and the corresponding FSR is ~0.6 nm. In order to verify this, we insert a 10 m PMF into the cavity. When the pump power of LD1 and LD2 is 893 mW and 660 mW, the HRR single-soliton train can be generated. Figure 7a shows the corresponding optical spectrum with a peak-to-peak interval of 0.61 nm, which is consistent with the calculated result. Additionally, the interval of pulses is measured to be ~14.39 ps. The experimental results strongly prove the efficiency of this technique. Keeping the pump power fixed, the above-mentioned wavelength-tuning operation can be obtained by adjusting PCs. As exhibited in Figure 8a, the central wavelength of the spectrum has been tuned from 1572.58 nm to 1608.12 nm, nearly 35.6 nm. Additionally, the corresponding AC traces are shown in Figure 8b. In addition, the HRR bound-soliton trains with the same repetition rate of ~71 GHz and the corresponding wavelength-tuning operation can also be achieved, which are similar to the results in Figures 5 and 6.

HRR Output with the Repetition Rate of ~71 GHz
In our experiment, HRR EDFLs with different repetition rates can be obtained by merely changing the length of PMF. This is because the FSR of the fiber-based Lyot filter is related to the length of PMF. For example, if we want to achieve a pulse train with a repetition rate of ~70 GHz in the L-band, the calculated length of PMF is 10 m, and the corresponding FSR is ~0.6 nm. In order to verify this, we insert a 10 m PMF into the cavity. When the pump power of LD1 and LD2 is 893 mW and 660 mW, the HRR single-soliton train can be generated. Figure 7a shows the corresponding optical spectrum with a peak-to-peak interval of 0.61 nm, which is consistent with the calculated result. Additionally, the interval of pulses is measured to be ~14.39 ps. The experimental results strongly prove the efficiency of this technique. Keeping the pump power fixed, the above-mentioned wavelength-tuning operation can be obtained by adjusting PCs. As exhibited in Figure 8a, the central wavelength of the spectrum has been tuned from 1572.58 nm to 1608.12 nm, nearly 35.6 nm. Additionally, the corresponding AC traces are shown in Figure 8b. In addition, the HRR bound-soliton trains with the same repetition rate of ~71 GHz and the corresponding wavelength-tuning operation can also be achieved, which are similar to the results in Figures 5 and 6.

Noise-like Pulse Generation
In addition to the above-mentioned HRR pulse train emission, we can also observe NLP in this laser regardless of whether the length of the intra-cavity PMF is 5.51 m or 10 m. When the orientations of PCs were adjusted to make the transmission coefficient very small, the comb filtering characteristic of the Lyot filter was suppressed. Therefore, the DFWM effect became weak and could not support the formation of the HRR pulse train, finally resulting in the generation of NLP. For example, when the length of PMF is 5.51 m, it is observed that the NLP is generated by adjusting PCs carefully under the total pump power of 1.215 W. The measurement results are shown in Figure 9. The center wavelength and full width at half height (FWHM) of the optical spectrum are 1607.07 nm and 28.34 nm in Figure 9a. Figure 9b exhibits a typical AC trace of the NLP, which is composed of a broad pedestal and a narrow spike. Additionally, the FWHM of this narrow spike is 151 fs (sech 2 assumed). The corresponding pulse train shown in Figure 9c is measured by the oscilloscope connecting with a high-speed photo-detector (Newport 818-BB-51F, 12.5 GHz). The pulse interval is 61.51 ns, which matches well with the cavity length. As presented in Figure 9d, the signal noise ratio (SNR) is 37.8 dB. Additionally, the RF spectrum with a 3.2 GHz span is shown in the illustration.

Noise-Like Pulse Generation
In addition to the above-mentioned HRR pulse train emission, we can also observe NLP in this laser regardless of whether the length of the intra-cavity PMF is 5.51 m or 10 m. When the orientations of PCs were adjusted to make the transmission coefficient very small, the comb filtering characteristic of the Lyot filter was suppressed. Therefore, the DFWM effect became weak and could not support the formation of the HRR pulse train, finally resulting in the generation of NLP. For example, when the length of PMF is 5.51 m, it is observed that the NLP is generated by adjusting PCs carefully under the total pump power of 1.215 W. The measurement results are shown in Figure 9. The center wavelength and full width at half height (FWHM) of the optical spectrum are 1607.07 nm and 28.34 nm in Figure 9a. Figure 9b exhibits a typical AC trace of the NLP, which is composed of a broad pedestal and a narrow spike. Additionally, the FWHM of this narrow spike is 151 fs (sech 2 assumed). The corresponding pulse train shown in Figure 9c is measured by the oscilloscope connecting with a high-speed photo-detector (Newport 818-BB-51F, 12.5 GHz). The pulse interval is 61.51 ns, which matches well with the cavity length. As presented in Figure 9d, the signal noise ratio (SNR) is 37.8 dB. Additionally, the RF spectrum with a 3.2 GHz span is shown in the illustration.

Stability Measurements
We also perform an experiment to verify the stability of this laser when the PMF is 5.51 m. Increasing the pump power of LD1 and LD2 to 893 mW and 660 mW and ensuring the cavity is working in an HRR single-soliton operation, the laser output is repeatedly recorded at an interval of 5 min for 45 min. As exhibited in Figure 10, the central wavelength drift and intensity fluctuation are 0.19 nm and 0.14 mW, respectively, indicating the good stability of this HRR EDFL.

Stability Measurements
We also perform an experiment to verify the stability of this laser when the PMF is 5.51 m. Increasing the pump power of LD1 and LD2 to 893 mW and 660 mW and ensuring the cavity is working in an HRR single-soliton operation, the laser output is repeatedly recorded at an interval of 5 min for 45 min. As exhibited in Figure 10, the central wavelength drift and intensity fluctuation are 0.19 nm and 0.14 mW, respectively, indicating the good stability of this HRR EDFL.

Conclusions
In conclusion, we experimentally demonstrate a wide wavelength-tunable L-band HRR EDFL based on a DFWM mode-locking mechanism. Due to the existence of an intra-cavity fiber-based Lyot filter, the repetition rate of the pulse train is fixed at ~126 GHz. Under the forward and backward pump power of 715 mW and 500 mW, two kinds of HRR pulse sequences, i.e., HRR single-soliton train and HRR bound-soliton train, can be obtained. With the broad transmission spectrum of this comb filter, the wavelength-tuning operation has been achieved by rotating the PCs in both pulse regimes. The center wavelength-tuning range in the HRR single-soliton state and HRR bound-soliton state is ~38.3 nm and ~22.6 nm, respectively. Compared to other filters, this technique especially has better flexibility. On the one hand, this fiber-based filter allows for an all-fiber structure, which has higher compactness and lower insertion loss. On the other hand, due to the FSR of this fiber-based Lyot filter being related to the length of PMF in the cavity, we can select the appropriate length to obtain a pulse train with the required repetition rate for different applications. Our research provides an easy and effective method to achieve a wavelength-tunable GHz HRR EDFL in the L-band, which will contribute to the development of wavelength-tunable HRR EDFLs. In addition, such high-repetition-rate comb fiber lasers can also work as seed sources for sensing, such as coherent ranging [43], gas detection at 1.5-1.6 μm [44], and temperature sensing [45].  Output power (mW)

Conclusions
In conclusion, we experimentally demonstrate a wide wavelength-tunable L-band HRR EDFL based on a DFWM mode-locking mechanism. Due to the existence of an intra-cavity fiber-based Lyot filter, the repetition rate of the pulse train is fixed at~126 GHz. Under the forward and backward pump power of 715 mW and 500 mW, two kinds of HRR pulse sequences, i.e., HRR single-soliton train and HRR bound-soliton train, can be obtained. With the broad transmission spectrum of this comb filter, the wavelengthtuning operation has been achieved by rotating the PCs in both pulse regimes. The center wavelength-tuning range in the HRR single-soliton state and HRR bound-soliton state is 38.3 nm and~22.6 nm, respectively. Compared to other filters, this technique especially has better flexibility. On the one hand, this fiber-based filter allows for an all-fiber structure, which has higher compactness and lower insertion loss. On the other hand, due to the FSR of this fiber-based Lyot filter being related to the length of PMF in the cavity, we can select the appropriate length to obtain a pulse train with the required repetition rate for different applications. Our research provides an easy and effective method to achieve a wavelengthtunable GHz HRR EDFL in the L-band, which will contribute to the development of wavelength-tunable HRR EDFLs. In addition, such high-repetition-rate comb fiber lasers can also work as seed sources for sensing, such as coherent ranging [43], gas detection at 1.5-1.6 µm [44], and temperature sensing [45].  Institutional Review Board Statement: Not applicable.