1. Introduction

Near transform limited picosecond (ps) pulses have important applications in many fields due to their advantages in both high spectral purity and high peak power [1]. To date, various ps pulse sources have been proposed and developed [2–4]. Particularly, it has found that the all-normal dispersion (ANDi) passively mode-locked fiber laser can generate a near transform-limited ps pulse if the bandwidth of its intracavity filter is less than 1 nm. Moreover, such an optical filter can be easily implemented by common FBG, and the pulse width can be adjusted within a range of 1∼200 ps by simply changing the FBG bandwidth [5]. The ps fiber laser pulse not only can offer time-frequency performance comparable to that generated from solid-state laser, but also have the advantages of high beam quality, compact structure, and low cost, and has thus gradually become an important ps pulse seed source. Three types of narrowband dissipative soliton passively mode-locked fiber lasers that can produce near transform-limited ps pulses, based on semiconductor saturable absorption mirror (SESAM), nonlinear polarization rotation and nonlinear amplifying loop mirror (NALM) [6–8], have been developed. Among them, the Figure-9 fiber laser is a desirable ps pulse source due to its excellent self-start mode-locking mechanism and lifetime unlimited NALM based saturable absorber (SA) [8].

In many applications such as ablation micromachining [9], medical imaging [10], and nonlinear frequency conversion [11–13], near transform-limited ps pulses with a repetition rate of less than 1 MHz are required. Yet, the repetition rates for all ps-pulsed fiber lasers are higher than ∼10 MHz, implying that pulse selectors may have to be used to reduce the repetition rate, which has led to complex structure and high cost in laser designing and fabrication. This has led to the development of low repetition rate ps-pulsed fiber laser oscillators. Unfortunately, all the above-mentioned ps-pulsed fiber lasers encounter the same problem when the intracavity fiber length is increased to reduce the repetition rate, i.e., pulse formation in fiber cavity is dominated by self-phase modulation (SPM), which is caused by weak group velocity dispersion (GVD) experienced by the ps pulses [14]. To avoid wave breaking and multi-pulse generation, the nonlinear phase shift (NPS) accumulated in the intracavity fiber must be kept low, typically ∼1 rad [5,15], meaning that in order to reduce the repetition rate by increasing the intracavity fiber length, the pulse peak power (or energy) in cavity has to be adjusted small enough to suppress extra nonlinearity caused by this increase in fiber length. When the pulse energy is too low, however, the SA may be incompletely bleached and Q-switching instability may be triggered [16], which in turn restricts the reduction of repetition rate for the fiber laser. To overcome this difficulty, Agnesi et al. reported a method using an SA with saturation energy low enough to avoid the Q-switching instability [17]. We have also proposed a method to reduce repetition rate, i.e., first a fiber coupler is inserted into the cavity to extract the pulse energy, then the length of the fiber behind the coupler is increased [18]. Nevertheless, both methods are demonstrated in SESAM based mode-locked fiber lasers, which have monotonic transmittance curves [19]. For NALM-based SA, on the other hand, the transmittance varies with incident pulse power periodically [20]. Therefore, whether the repetition rate of ps-pulsed Figure-9 fiber laser with periodic SA can still be decreased, namely by suppressing Q-switching instability through reduction in saturation power of the NALM-based SA, and whether reduction in repetition rate through increasing intracavity fiber length is limited by other factors, deserves further investigation.

In this paper, we propose and demonstrate a method for reducing the repetition rate of a Figure-9 fiber laser. We study the restricting factors of decreasing the repetition rate for the Figure-9 fiber laser with periodically SA, and find that by asymmetrically increasing the passive fiber length of NALM, SA saturation power can be lowered and thus Q-switching instability can be avoided. We demonstrate our method with a typical Figure-9 fiber laser, and by increasing the intracavity fiber length outside the SA, we further overcome noise-like pulses from excessive reduction in SA saturation power. We achieve effective reduction in repetition rate of ps pulses, and in particular rates for ∼10 and ∼20 ps pulses are reduced down to 1.7 MHz and 848 kHz, respectively.

2. Method analysis

Let us consider the typical ANDi passively mode-locked Figure-9 fiber laser shown in Fig. 1. The gain fiber is a polarization maintaining (PM) single-mode fiber with length of ${L_1}$, and its two ends are accordingly fused with PM single-mode fibers with length of ${L_2}$ and ${L_3}$. They are then twisted by 90 degrees and respectively coupled with the fast and slow axes of a polarization beam splitter (PBS1) to form a PM fiber loop [21]. A nonreciprocal linear phase shifter (PS), composed of a half-wave plate (HWP), ${45^ \circ }$ Faraday rotator (FR), and quarter-wave plate (QWP), is placed between PBS1 and PBS2. The transmission port of PBS2 is connected to a fiber Bragg grating (FBG) with a PM fiber of length $L_{4}$ The fiber loop, PS, PBS1, and PBS2 constitute an NALM based SA. The transmittance of the SA can be expressed as [22]

 

Fig. 1. Schematic diagram of the program controlled all-PM Figure-9 mode-locked fiber laser. LD: laser diode; GF: gain fiber; WDM: wavelength-division-multiplexer; PS: phase shifter; HWP: half-wave plate; QWP: quarter-wave plate; FR: Faraday rotator; PBS: polarization beam splitter; FBG: fiber Bragg grating; ISO: isolator.

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From Eq. (1) the SA transmittance varies periodically with $P_{\textrm{in}}^0$. In addition, it depends not only on the parameters of passive fiber loop, but also on the gain fiber and its saturation characteristics. For an incident pulse on the SA, if $P_{\textrm{in}}^0 \ge {P_{\textrm{sat,A}}}$, the transmittance of SA can reach its maximum value G to compensate the cavity loss, resulting the laser to operate in a mode-locked state. If the pump power then decreases gradually, $P_{\textrm{in}}^0$ must also decrease due to reductions in both ${g_0}$ and ${E_{\textrm{sat,G}}}$. When $P_{\textrm{in}}^0$ is reduced to less than ${P_{\textrm{sat,A}}}$, then $T < G$. Once the pump is so low that the cavity gain cannot automatically adjust to compensate for the loss through the gain saturation characteristic, the SA will no longer be bleached completely, and the laser may not be able to oscillate. Then, the population inversion of gain fiber may have to be re-accumulated with a ∼μs dynamic time [25], which is much longer than a round-trip time in the cavity, resulting the manifestation of Q-switching instability or Q-switched mode-locking [16]. Therefore, to avoid Q-switching instability for the Figure-9 fiber laser, the peak power of incident pulse of SA should satisfy

Clearly, the allowed $P_{\textrm{in}}^0$ that does not trigger Q-switching instability can be very low by suitably designing the NALM to reduce ${P_{\textrm{sat,A}}}$, so the fiber length of cavity can be increased to reduce the repetition rate without accumulating excessive extra nonlinearity. Note that since the saturation characteristic of gain fiber is determined by the pulse energy, even if the parameters of SA and pump condition are identical, ${P_{\textrm{sat,A}}}$ for the pulses with different durations are different and the pulse with narrower duration has lower ${P_{\textrm{sat,A}}}$, according to Eq. (2) combined with the gain saturation property, $g = {{{g_0}} / {({1 + {{{T_0}P_{\textrm{in}}^0} / {{E_{\textrm{sat,G}}}}}} )}}$. Moreover, considering that the second term in the denominator on the right side of Eq. (2) is always negative and ${L_1}$ is generally around several meters, ${P_{\textrm{sat,A}}}$ can be reduced by lengthening ${L_3}$while shortening ${L_2}$ as much as possible. The increase of ${L_3}$ just corresponds to the decrease of repetition rate for the laser.

Meanwhile, if it is assumed that the SA is not overdriven, the wave breaking should also be avoided for the pulses in cavity to ensure the laser operates in the single pulse mode locking. Without loss of generality, consider the case of $\rho > 0.5$, where the NPS accumulated by the pulse along cw direction in fiber loop is always larger than that of ccw direction. Thus, wave breaking must first take place in fiber ${L_3}$caused by the NPS accumulated for the pulse along the cw direction. The NPS accumulated by the pulse transmitting along fiber ${L_4}$ can be extremely low if ${L_4}$ is short enough, due to the nonreciprocal output coupling of PBS2. Thus, to avoid wave breaking, ${L_3}$ must be less than the distance at which the wave breaking occurs in this fiber section, ${L_{\textrm{WB}}}$ [14]. Assuming the ps pulse propagating along ${L_3}$ has a Gaussian shape, the allowable incident pulse peak power of SA that can avoid wave breaking should meet

Equations (3) and 4 give the lower and upper bounds to the allowable peak power for the incident pulse of SA for operating the Figure-9 fiber laser in single pulse mode-locking, respectively corresponding to the Q-switching instability and the wave breaking effects. Since the lower and upper bounds are inversely proportional to ${L_3}$ and $L_3^2$, respectively, as ${L_3}$ increases, they tend to coincide, giving a maximally acceptable ${L_3}$, which corresponds to the lowest repetition rate achievable for the laser.

3. Experimental results and discussion

We use the configuration of the ps-pulsed Figure-9 fiber laser shown in Fig. 1 to verify our method for reducing the repetition rate by increasing ${L_3}$. A 1.5 m PM single-mode Yb-doped fiber (YDF, Nufern, PM-YSF-6/125-HI) with absorption of 250 dB/m at 975 nm is used as the gain fiber, which is pumped by a 975-nm laser diode through PM wavelength division multiplexer (WDM). The center wavelength, bandwidth, and reflectivity of the FBG are 1064.2 nm, 0.2 nm, and 99%, respectively. The lengths of ${L_2},\,{L_3}$ and ${L_4}$ are initially set at 0.6, 9, and 0.6 m, respectively, all of which are PM 980 fibers. ${\theta _1}$ and ${\theta _2}$ are set to ${20^ \circ }$ and ${10^ \circ }$, respectively, which corresponds to ${\phi _\textrm{L}} = 1.5\pi$ and $\rho=0.5$ [23]. When the pump power is increased to 312 mW, the laser self-starts mode-locking to a multi-pulse state, and as the power is decreased to 270 and 196 mW, the outputs are switched to single pulse mode-locked and Q-switching mode-locked states, respectively. Thus, the pump range of single pulse mode-locked operation for the laser is [196 mW, 270 mW], with its upper and lower bounds limited by wave breaking and Q-switching instability, respectively.

Figure 2 shows the output pulse trains, autocorrelation trace, optical and RF spectra measured at pump of 270 mW. As shown, the repetition rate is 14.7 MHz, corresponding to the cavity length. The 3-dB spectral width, pulse duration and the time-bandwidth product are 0.16 nm, 19 ps and 0.79, respectively, which are almost unchanged in the single pulse mode-locked operation region, since they are determined by the FBG bandwidth [5]. When the 0.2 nm FBG is replaced by a 0.3 nm FBG for generating ∼10-ps pulses, it was observed that the laser exhibits the same self-start single-pulse mode-locking process. When ${\theta _1}$ and ${\theta _2}$ are adjusted to vary ${\phi _\textrm{L}}$ and $\rho$, similar single pulse operation region restricted by wave breaking and Q switching instability was also observed for the laser as long as it can self-start mode-locking.

 

Fig. 2. Measured output pulse properties when the fiber laser with 0.2 nm FBG is pumped at 270 mW. Here, ${L_3}$is 9 m, ${\phi _\textrm{L}}$ is $1.5\pi$, and $\rho$ is 0.5. (a) Output pulse trains. (b) Autocorrelation trace (blue) and its Gaussian fitting curve (red). (c) Optical spectrum. (d) RF spectrum with a resolution bandwidth (RBW) of 100 kHz.

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Figure 3 shows the pulse energy regions of single-pulse mode-locking as functions of ${L_3}$ for the fiber lasers with 0.2 and 0.3 nm FBGs. As expected, as ${L_3}$ increases from 9 m to reduce the repetition rate, the upper and lower bounds of the pulse energy gradually decrease, but the decreasing speed of the upper bound is faster than that of the lower bound, resulting in the energy range of single pulse operation decreasing with the increase of ${L_3}$. This is consistent with our theoretical prediction. For the 0.2 nm FBG case, as ${L_3}$ is increased to 90 m, the repetition rate is reduced to 2 MHz, and the pulse energy regions are reduced from [0.45 nJ, 0.95 nJ] to [0.059 nJ, 0.073 nJ], though still restricted by wave breaking and Q-switching instability. Moreover, the single-pulse operating range eventually disappears when ${L_3}$ is increased to 90 m. Since it is impossible for the over-driving of SA to cause the disappearance of single-pulse operation range [26], this indicates that it is reasonable to assume that the multi-pulse oscillation caused by the SA overdriving is negligible when the single-pulse operating range evolution is theoretically analyzed. For the 0.3 nm FBG case, similar results are observed, i.e., as ${L_3}$ is increased to 37 m, the repetition rate are reduced to 5 MHz, with pulse energy regions dropping from [0.15nJ, 0.28nJ] to [0.03nJ, 0.042nJ].

 

Fig. 3. Pulse energy regions of single-pulse mode-locking as functions of ${L_3}$ for the fiber lasers with FBG bandwidths of 0.2 nm (a) and 0.3 nm (b).

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When ${L_3}$ is increased beyond 90 m, the laser with 0.2 nm FBG can still self-start to the multi-pulse mode-locking state, but single pulse mode locking was no longer achievable. For the 0.3 nm FBG laser, however, when ${L_3}$ is increased beyond 37 m, different self-starting mode-locked behaviors are exhibited, i.e., the lasers self-start to single pulse mode-locking directly once the pumps reach the mode-locking thresholds. Although the measured output pulse spectrum is still within the FBG bandwidth of 0.3 nm, the spectral shape is different from that shown in Fig. 2(b), and small burrs are superimposed on the spectral curve (see Fig. 4 (b) below). Reading the results of a high-speed detector with a 500 MHz oscilloscope, it is observed that the measured output pulse is composed of a relatively narrow peak (about 5.6 ps measured with the autocorrelator) sitting on the center of a ns pulse pedestal. Moreover, with the increase of pump power, the spectral shape and the duration of the peak on the pulse pedestal remain basically steady. Only the ns pedestal of the pulse is widened, resulting in the increase of the pulse energy. Figures 4 (a) and (b) show the measured output pulse waveforms and normalized spectra when the laser with ${L_3} = 40$m is pumped at 234 and 317 mW, respectively. It can be seen that the pulse pedestals for the pump powers of 234 and 317 mW are widened to ∼1.5 and ∼2.3 ns, giving the single pulse energies of 0.96 and 1.6 nJ (though the peak powers of the pulse basically remain unchanged), respectively. These results show clearly the characteristics of noise-like pulse (NLP) [27,28]. This is not surprising because ${L_3}$ has been increased to be long enough to give very small SA saturation power and period of SA transmittance curve, according to Eq. (4). Thus, the pulse peak power clamped by inverse SA effect strengthened, which causes the NLPs to occur. The reason for NLP to appear in ∼10-ps pulse laser but not in ∼20-ps laser is that, since the pulse with narrower width has lower ${P_{\textrm{sat,A}}}$, the SA transmittance curve period will be shorter and the clamping effect on the pulse peak power will be stronger, so that it is easier for NLP to take place in the ∼10-ps laser. Thus, NLP may occur in narrowband dissipative soliton ps-pulsed Figure-9 fiber laser due to the low saturation power of periodically SA. Although the repetition rate can still be decreased by increasing ${L_3}$, the output is not the ps pulse we expected (see the inset of Fig. 4 (a), which shows the measured typical autocorrelation trace without NLP effect). Therefore, the NLP restricts the further reduction of repetition rate by increasing ${L_3}$ for the ∼10 ps pulse laser.

 

Fig. 4. Measured temporal pulse waveforms (a) and normalized spectra (b) when the 0.3 nm FBG laser with ${L_3}$ of 40 m is pumped at 234 (red curve) and 317 mW (blue curve), respectively. The inset shows the measured typical autocorrelation trace (blue) without NLP effect and its Gaussian fitting (red), for convenience of the comparison with the measured pulse shown in Fig. 4(a).

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According to the results of Fig. 3, the lasers with 0.2 and 0.3 nm FBGs respectively do not allow further reduction in ps pulse repetition rate through increasing ${L_3}$ beyond 90 and 37 m because of multi-pulse state and the NLP effect. The repetition rates for both lasers can be decreased further by increasing the length of fiber outside SA, since the single pulse operation regions are still sufficiently wide, i.e., [0.059 nJ, 0.073 nJ] and [0.03 nJ, 0.042 nJ] for the lasers with 0.2 and 0.3 nm FBGs, respectively (see Fig. 3). In particular, the NPS accumulated in ${L_4}$ is weak due to the non-reciprocal output coupling of PBS2. Results show that under the conditions of Fig. 3, the repetition rate for ∼10 and ∼20 ps pulses can be reduced to 2.5 MHz and 1.6 MHz by using PM single mode fiber to increase $L_4 $. In fact, they can be further lowered to 1.7 MHz and 848 kHz when the low nonlinear large-mode field PM fibers (PLMA-GDF-10/125) with lengths of 40 and 75 m are used to increase $L_4 $, with the lower and upper bounds of single pulse energy regions for the ∼10-ps and ∼20-ps lasers increased from [0.03 nJ, 0.04 nJ] and [0.059 nJ, 0.073 nJ] to [0.12 nJ, 0.19 nJ] and [0.09 nJ, 0.1 nJ], respectively. The reason may be that the upper and lower pump power bounds may have to be increased for the single pulse mode locking because the fusion loss between PLMA and PM980 is slightly increased, which results in larger output pulse energies. For ∼20-ps 848 kHz laser, the peak power range of SA input pulse corresponding to single pulse operation is estimated to be [7 W,10 W] by measuring the average output power at the FBG port, implying that the saturation power of SA should be slightly less than 7 W. If taken the value of $\gamma $ of $3.5 \times {10^{ - 3}}$ W^-1m^-1 [18] and the cavity loss of about 5 dB into accounted, the value of $P_{sat,A} $ obtained from Eq. (2) is 5 W, which is close to the result estimated by the experimental measurement. Figure 5 shows the autocorrelation trace, optical spectra, RF spectra and the pulse profile, which is measured with a 2 GHz photodetector and a 4 GHz oscilloscope (Agilent, DSO9404A), of the ∼20-ps pulses generated by the 848-kHz laser pumped at 100 mW. From the baseband RF signal spectrum for the pulse train measured with 1-Hz resolution and 1-MHz range, the signal-to-noise ratio is 64 dB, showing clearly the stable single pulse mode-locking for the laser. The time-bandwidth product is about 1.02, indicating that the NPS accumulated in the cavity increases slightly by compared with the results in Fig. 2 [18]. Notably, 1.7 MHz and 848 kHz are, to the best of our knowledge, the lowest repetition rates for ∼10 and 20 ps narrowband dissipative solitons Figure-9 fiber lasers reported thus far.

 

Fig. 5. Measured output pulse properties of the laser with FBG bandwidths of 0.2 nm, ${L_3}$of 90 m and ${L_4}$of 75 m pumped at 100 mW. (a) Autocorrelation trace (blue) and its Gaussian fitting (red). (b) Optical spectrum. (c) RF spectrum with a resolution bandwidth (RBW) of 1 Hz, with inset showing the higher harmonics with a RBW of 10 kHz. (d) Pulse profile, with inset showing the pulse train.

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4. Conclusion

In summary, we have investigated how to decrease the repetition rate of narrowband dissipative soliton ps-pulsed Figure-9 fiber laser based on periodic SA. The repetition rate reduction by increasing intracavity fiber length is limited by Q-switching instability and wave breaking. By asymmetrically increasing the passive fiber length of NALM to lower SA saturation power, the repetition rate can be reduced effectively. However, the excessive reduction of SA saturation power may lead to NLP, which restricts further reduction in repetition rate. Thanks to the non-reciprocal output characteristics of periodic SA, the repetition rate is allowed to be further reduced by increasing the intracavity fiber length outside the SA. With a large-mode field PM fiber, we successfully reduce the repetition rates of ∼10 and ∼20 ps pulsed Figure-9 fiber lasers to 1.7 MHz and 848 kHz, respectively, which are the lowest repetition rates of Figure-9 laser so far to the best of our knowledge.