**4. Nonlinear fiber amplifier**

262

port was measured to be 2.84 ps. Additional PM-1550 fiber was used to dechirp the preamplified pulse to 72 fs (shown in **Figure 6(b)**). Therefore, considering a repetition rate of 60 MHz, the pulse peak power achieved as high as 34.7 kW. Three types of HNLFs, such as NL 1550-ZERO, PM-HNLF, and Zero-slope HNLF, were applied to generate the supercontin‐

As shown in **Figure 6(c)**, 20-cm-long PM-HNLF with nonlinearity of 10.5 W−1 km−1 achieved the broadest spectrum, covering from 950 to 2200 nm, which is sufficient broad to produce *f*ceo signal. The HNLF type should be taken into consideration as it influences the SC generation. We used three kinds of HNLFs: 25-cm-long NL 1550-ZERO with nonlinear coefficient of 10.4-

sponding SC was depicted in **Figure 6(c)**. Obviously, PM-HNLF produces broader spectrum

As shown in **Figure 7**, a collinear setup was established for the detection of *f*ceo signal. The SC generated by 20-cm-long PM-HNLF was coupled into free space via a lens (L1) with adjustable focal length. An inline f-2f interferometer, including a PPLN, several wave plates and lens, and a PM-fiber delay line, is used to produce the temporal overlapped components at 1.0 μm. The long-wavelength component of SC at 2092 nm was frequency doubled to match with the shortwavelength component at 1046 nm. After the PPLN, two lenses, L3 and L4, were used to couple the two components at 1046 nm back to PM-980 fiber. A half-wave plate, HWP2, is used to adjust the energy ratio on the fast and slow axes of PM-980 fiber. The pulse transmitted along the slow axis experiences a delay relative to the pulse on fast axis. With an optimized fiber length of 3.4 m, the differential delay between the fast and slow axes could be fully compen‐ sated [49]. Subsequently, a half-wave plate, HWP3, as well as a PBS were used to selected pulses to generate *f*ceo signal on APD. Finally, with 28-dB signal-to-noise ratio was generated

**Figure 7.** Setup for *f*ceo detection. Amp: fiber amplifier; L1, L4, and L5: optical lens with adjustable focal length; L2, L3, and L6: optical lens with focal length of 50 mm; HWP1, HWP2, and HWP3: half-wave plates; PPLN: periodically poled

lithium niobate; PBS: polarization beam splitter; APD: avalanche photodiode.

, 20-cm-long PM-HNLF with nonlinear coefficient

, 25-cm-long Zero-slope HNLF with

, and the corre‐

uum by splicing the HNLFs to the dechirping fiber directly.

of 10.5-W−1 km−1 and effective mode area of 12.7-μm2

nonlinear coefficient of 10.8-W−1 km−1 and effective mode area of 12.4-μm2

W−1 km−1 and effective mode area of 13-μm2

than other two HNLFs.

by using this setup.

With the increasing applications in frequency metrology, THz generation, and in cataract surgery, the development for high-energy transform-limited pulse generation around 1.55 μm is still a fascinating area [50–53]. Owing to the limited available output power from laser oscillator, erbium-doped fiber amplifiers (EDFAs) are commonly used. Nevertheless, highpower amplification in EDFA is inevitably accompanied by several unwanted effects, such as SRS and amplified spontaneous emission (ASE), which would significantly deteriorate the temporal and spectral duration of pulse [54]. Chirped-pulse amplification (CPA) provides an effective way to decrease pulse peak power and avoid the nonlinearity in optical fibers [55– 58]. In CPA, strong stretching and compression occurs to extract more energy and avoid nonlinear distortion as well as damage. However, CPA is inevitably accompanied by the gainnarrowing effect and therefore hardly produces pulse with temporal duration less than 400 fs [59].

Even though CPA has many advantages over the other techniques in amplifying pulses around 1.55 μm, ~100-fs pulse duration with above 10-nJ pulse energy is still a challenge because of the spectral gain-narrowing effects and nonlinear phase accumulation. More‐ over, due to the involvement of bulk media, CPA is not suited for applications that require compact size and alignment-free laser source. A recent developed technique, divided-pulse amplification (DPA), opens up a new way for high-power laser pulse amplification [60–62]. In the configuration of DPA, the initial pulse is divided into a sequence of lower-intensity pulse with orthogonal polarization for successive replicas, and subsequently, the lowenergy pulse is amplified and then recombined to create a high-energy pulse [61, 63].

In this section, we mainly focus on DPA at 1.56 μm where pulse amplification and compression can be simultaneous carried out so that a separate compressor is no longer necessary. The

**Figure 8.** (a) Schematic diagram for laser system. F-PBS: fiber-coupled polarization beam splitter; DCF: dispersion compensation fiber; FRM: Faraday rotation mirror; LD1, LD2: pump diodes at 976 nm; WDM: wavelength division multiplexer; EDF: erbium-doped single-mode fiber; Col1, Col2: high-power collimators; HWP: half-wave plate; PBS: polarization beam splitter cube; DCFA: double-clad fiber amplifier. (b) Schematic of the pulse divider. The left-hand three cylinders (21, 22, and 23) represent YVO4-based dividers with given direction of crystal optical axes (OAs) shown as the red dash dot lines. The right-hand three parts (24, 25, and 26) represent PBS-based dividers with p-polar‐ ized direction shown as red dash dot lines. The red dot lines represent the horizontal plane, which is the same direc‐ tion as the OA of 21 and 23 and the p-polarized direction of 24 and 26. (c) The measured autocorrelation trace of the divided replicas.

schematic diagram of DPA is shown in **Figure 8(a)**. The experimental setup is composed of a mode-locked fiber laser, a fiber stretcher, a single-mode fiber amplifier for preamplifying, and a pulse-divider as well as a double-clad fiber amplifier for main amplification. The Er-doped fiber laser with 80-MHz repetition rate shared the same configuration as **Figure 6(a)**, which takes the advantage of EPC to actively control the mode-locking. A photodiode and an electric loop were applied to monitor and feedback control the EPC for long-term stable operation. The fiber oscillator consisted of 1.74-m SMF28-e fiber with dispersion parameter of 19 ps/nm/ km and 0.82-m Er-doped fiber with dispersion parameter of −51 ps/nm/km. There, the laser operated in the stretched-pulse regime and produced positively chirped pulses. As a result, with 200-mW pumping power at 976 nm, the laser oscillator produces 5-mW output average power with 1.5-ps pulse duration and 28-nm spectral bandwidth, corresponding to a timebandwidth product of 5.2.

A fiber stretcher is spliced to the output of the fiber oscillator to stretch laser pulse and control the quantity of frequency up-chirp. However, over-long fiber could inevitably introduce too much high-order dispersion which is hardly compensated by the pulse-compressing stage. For the current configuration, a double-pass fiber stretcher consisting of a fiber-coupled PBS with PM fiber at input/output port and non-PM fiber at common port, a segment of non-PM dispersion compensation fiber with 6.0-μm-mode field diameter and −38 ps/nm/km dispersion at 1550 nm, and a FRM is used to reduce the environmental perturbation.

In our experiment, 6-m dispersion compensation fiber was applied to stretch pulses from the fiber oscillator. A dual-pass bidirectionally pumped single-mode fiber preamplifier was used to boost the average power to more than 100 mW to ensure efficient operation of the subsequent amplifiers. A FRM reflected the incident pulse to suppress ASE noise and rotated the polari‐ zation of the pulses by 90° to cancel all birefringence effects in the dual-pass amplifier. A fiberbased polarization beam splitter (F-PBS) was used to couple the seed laser to the preamplifier and reflected preamplified pulses to subsequent components. The output characters of the preamplifier were shown as the blue curves in **Figure 9(a)** and **(b)**. The FWHM temporal duration and spectrum bandwidth of the preamplified pulses is 4 ps and 15 nm, respectively, generating a time-bandwidth product of 7.4. Dramatic decrease in spectral bandwidth was

**Figure 9.** The temporal duration (a) and spectral bandwidth (b) of laser pulses from laser oscillator (red curves) and SMFA (blue curves).

observed due to the limited transmission bandwidth of WDM and FRM as well as spectralnarrowing effect in fiber amplifier.

264

bandwidth product of 5.2.

SMFA (blue curves).

schematic diagram of DPA is shown in **Figure 8(a)**. The experimental setup is composed of a mode-locked fiber laser, a fiber stretcher, a single-mode fiber amplifier for preamplifying, and a pulse-divider as well as a double-clad fiber amplifier for main amplification. The Er-doped fiber laser with 80-MHz repetition rate shared the same configuration as **Figure 6(a)**, which takes the advantage of EPC to actively control the mode-locking. A photodiode and an electric loop were applied to monitor and feedback control the EPC for long-term stable operation. The fiber oscillator consisted of 1.74-m SMF28-e fiber with dispersion parameter of 19 ps/nm/ km and 0.82-m Er-doped fiber with dispersion parameter of −51 ps/nm/km. There, the laser operated in the stretched-pulse regime and produced positively chirped pulses. As a result, with 200-mW pumping power at 976 nm, the laser oscillator produces 5-mW output average power with 1.5-ps pulse duration and 28-nm spectral bandwidth, corresponding to a time-

A fiber stretcher is spliced to the output of the fiber oscillator to stretch laser pulse and control the quantity of frequency up-chirp. However, over-long fiber could inevitably introduce too much high-order dispersion which is hardly compensated by the pulse-compressing stage. For the current configuration, a double-pass fiber stretcher consisting of a fiber-coupled PBS with PM fiber at input/output port and non-PM fiber at common port, a segment of non-PM dispersion compensation fiber with 6.0-μm-mode field diameter and −38 ps/nm/km dispersion

In our experiment, 6-m dispersion compensation fiber was applied to stretch pulses from the fiber oscillator. A dual-pass bidirectionally pumped single-mode fiber preamplifier was used to boost the average power to more than 100 mW to ensure efficient operation of the subsequent amplifiers. A FRM reflected the incident pulse to suppress ASE noise and rotated the polari‐ zation of the pulses by 90° to cancel all birefringence effects in the dual-pass amplifier. A fiberbased polarization beam splitter (F-PBS) was used to couple the seed laser to the preamplifier and reflected preamplified pulses to subsequent components. The output characters of the preamplifier were shown as the blue curves in **Figure 9(a)** and **(b)**. The FWHM temporal duration and spectrum bandwidth of the preamplified pulses is 4 ps and 15 nm, respectively, generating a time-bandwidth product of 7.4. Dramatic decrease in spectral bandwidth was

**Figure 9.** The temporal duration (a) and spectral bandwidth (b) of laser pulses from laser oscillator (red curves) and

at 1550 nm, and a FRM is used to reduce the environmental perturbation.

Then, the concept of DPA was carried out to boost the laser to Watt-level average power. The preamplified laser is coupled into free space by collimator C1 and rotated to horizontal polarization to reach maximum transmission on PBS. The pulse division and combination were achieved by applying cascaded YVO4-based and PBS-based dividers with the help of a FRM to reflect the replicas passing through the same divider but in the opposite direction. Each divider (YVO4-based or PBS-based) can divide a single pulse into two cross-polarized replicas; hence, a single seed pulse could be temporally divided into 2*<sup>N</sup>* (where *N* is the stage number of the divider) replicas. Ideally, each replica has identical pulse energy after division. As depicted in **Figure 8(b)**, three YVO4 crystals with lengths of 10, 20, and 40 mm divided the initial pulse into *N* = 8 replicas. A half-wave plate (HWP) was used to produce the desired polarization of input pulses. The first (21 ) and third (23 ) YVO4 crystals had their crystal optical axes (OA) oriented in the same direction as the horizontal plane, while the OA of the second (22 ) YVO4 crystal oriented at a 45° angle to the horizontal plane. The polarization-mode delay between ordinary and extraordinary waves in YVO4 is 0.7 ps/mm at 1560 nm. The shortest crystal length for our system was chosen to split the input pulse into replicas with 7 ps separation, about twice of the seed pulse duration.

To mitigate the nonlinearity in main amplifier, the string of pulse (*N* = 8) was further divided by three PBS-based dividers, resulting in a final pulse number of 64. For PBS-based divider, each incoming pulse was divided into an s-polarized beam and a p-polarized beam. All ppolarized components were directly transmitted the PBS, while the s-polarized components were reflected to the folded delay line. For the sake of simplicity, the second PBS-based divider (25 ) had its p-polarized direction oriented 45° to the direction of the horizontal plane, while the first (24 ) and third (26 ) PBS-based oriented in the same direction as the horizontal plane, such that separate half-wave plates were no longer necessary.

Owing to the delay length of 10, 20, 40, 26.8, 53.6, and 107.2 mm, the 21 , 22 , 23 , 24 , 25 , and 26 stages approximately provided time delay of 7, 14, 28, 130, 260, and 520 ps, respectively. **Figure 8(c)** shows the measured autocorrelation trace of the pulse string which matches well with the designed time delays. The 7-ps interval between adjacent peaks in the same envelope was consistent with the expected time delay with 10-mm increment length of YVO4, and the ~140 ps spacing between two adjacent envelopes was consistent with the expected time delay introduced by the PBS-based divider.

Intuitively, for simultaneous pulse amplification and compression in EDFAs, a positively prechirping seed pulse is desired. Numerical simulations show that there exists an equilibrium position that can not only restrict excessive nonlinear effects to ensure high-quality temporal integrity but also produce sufficient optical nonlinearity to broaden the spectrum around the wavelength of 1.55 μm. The generalized nonlinear Schrödinger equation (7) with the split-step Fourier method was used to carry out the simulation [24].

$$\frac{\partial A}{\partial z} = \frac{a}{2}A + \sum\_{n \ge 2} \frac{i^{n+1}}{n!} \beta\_n \frac{\partial^n A}{\partial T^n} + i\gamma A \int\_{-\alpha}^{\alpha} R(T') \left| A(z, T - T') \right|^2 dT' \tag{7}$$

where *A* = *A*(*z*, *t*) is the complex amplitude of the pulse envelope of pulses, *α* is the laser gain coefficient, *βn* is the dispersion parameter at *ω*0 (1560 nm), and *γ* (3 W−1 km−1) is the nonlinear coefficient. The right-hand side of Eq. (7) models laser gain, dispersion, and nonlinearity. Pulse of 2.5-ps temporal duration and 19.9-nm spectral width (corresponding to 0.16 ps2 prechirping on 180-fs transform-limited pulse) and a pulse energy of 0.05 nJ were applied in the simulation.

**Figure 10.** Amplified output pulse duration versus propagation length for the cases of different *α* (a) and *β*2 (b). (c) Pulse duration versus the total gain provided by 6.5-m fiber. (d) Time-bandwidth product at different position of gain fiber.

The interplay of the SPM and group-velocity dispersion (GVD) as well as laser gain can lead to a qualitatively different behavior compared with that expected from them alone. SPM broadened the spectrum with increase in pulse energy, and simultaneously, the anomalous dispersion of the fiber compressed the new spectral components resulting in temporal shortening. **Figure 10(a)** compares the simulation results with different *α* but a fixed *β*<sup>2</sup> (−22 fs<sup>2</sup> /mm). It is clear that pulse compression operates in linear regime when the laser gain is low, then it enters in nonlinear regime when the laser gain gradually increased. The shortest transform-limited pulse duration decreased from 180 fs at 7.0 m (*α* = 0 dB/m) to 60 fs at 4.3 m (*α* = 3 dB/m). **Figure 10(b)** compares the pulse compression with different *β*<sup>2</sup> but a fixed *α* (3 dB/m). The maximum pulse energies with respect to fiber length of 4.65, 5.07, and 5.53 m reached 1.24, 1.66, and 2.28 nJ, respectively. Therefore, higher *α* and smaller |*β*2| are benefit to overcome spectral bandwidth limitation for high-energy pulse amplification. For refer‐ ence, the blue curves in **Figure 10(a)** and **(b)** present pulse evolution with the same parame‐ ters.

Next, we focus on pulse amplification and compression in a fixed fiber length by way of guiding the subsequent experiment. About 5.0-m-long fiber with *β*2 = −22 fs2 /mm was intro‐ duced to simulate the output pulse duration and the time-bandwidth product (TBP) at different position along the fiber. As shown in **Figure 10(c)**, when the total gain is smaller than 16 dB, the output pulse duration deceases linearly owing to the GVD and insufficient nonli‐ nearity. As the total gain is greater than 24 dB, the output pulse duration dramatically decreases owing to strong nonlinear compression. Theoretically, pulse as short as 80-fs duration can be achieved with a total gain of nearly 28 dB. Although the pulse duration could be further decreased to 20 fs with 32-dB gain, considerable pedestal as well as wave breaking appears due to excessive nonlinearity. Meanwhile, the TBP of the pulse along the fiber gradually decreases from 4.1 at the input port to 0.5 at the output port.

266

fiber.

(−22 fs<sup>2</sup>

ters.

<sup>1</sup> <sup>2</sup>

=+ + - ¢ ¢¢ ¶ ¶ å ò ! (7)

<sup>2</sup> ( ) (, ) <sup>2</sup>

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where *A* = *A*(*z*, *t*) is the complex amplitude of the pulse envelope of pulses, *α* is the laser gain coefficient, *βn* is the dispersion parameter at *ω*0 (1560 nm), and *γ* (3 W−1 km−1) is the nonlinear coefficient. The right-hand side of Eq. (7) models laser gain, dispersion, and nonlinearity. Pulse of 2.5-ps temporal duration and 19.9-nm spectral width (corresponding to 0.16 ps2 prechirping on 180-fs transform-limited pulse) and a pulse energy of 0.05 nJ were applied in the simulation.

**Figure 10.** Amplified output pulse duration versus propagation length for the cases of different *α* (a) and *β*2 (b). (c) Pulse duration versus the total gain provided by 6.5-m fiber. (d) Time-bandwidth product at different position of gain

The interplay of the SPM and group-velocity dispersion (GVD) as well as laser gain can lead to a qualitatively different behavior compared with that expected from them alone. SPM broadened the spectrum with increase in pulse energy, and simultaneously, the anomalous dispersion of the fiber compressed the new spectral components resulting in temporal shortening. **Figure 10(a)** compares the simulation results with different *α* but a fixed *β*<sup>2</sup>

/mm). It is clear that pulse compression operates in linear regime when the laser gain is low, then it enters in nonlinear regime when the laser gain gradually increased. The shortest transform-limited pulse duration decreased from 180 fs at 7.0 m (*α* = 0 dB/m) to 60 fs at 4.3 m (*α* = 3 dB/m). **Figure 10(b)** compares the pulse compression with different *β*<sup>2</sup> but a fixed *α* (3 dB/m). The maximum pulse energies with respect to fiber length of 4.65, 5.07, and 5.53 m reached 1.24, 1.66, and 2.28 nJ, respectively. Therefore, higher *α* and smaller |*β*2| are benefit to overcome spectral bandwidth limitation for high-energy pulse amplification. For refer‐ ence, the blue curves in **Figure 10(a)** and **(b)** present pulse evolution with the same parame‐

*A iA <sup>A</sup> i A R T A z T T dT*

<sup>+</sup> ¥ ³ -¥

*n n n n n*

b

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> Experimentally, the divider first operated with ×8 replicas. A dual-pass double-cladding fiber amplifier (DCFA) was used to boost the divided pulses. The DCFA consisted of 1.2-m 12/130 Er-doped double-clad fiber and 2.5-m SMF fiber. With a pump power of 4.3 W at 976 nm, the DCFA delivered 600 mW output power at 1560 nm, as measured at PBS output port. Along the first pass of DCFA, pulse evolution worked in the near-linear regime. Subsequently, as the pulse reflected by the FRM and passed through DCFA again, SPM and anomalous dispersion brought the pulse amplification into the moderate nonlinearity regime. **Figure 11(a)** and **(b)** shows pulse duration and spectral bandwidth of the combined pulse measured by an FROG (15-100-USB, Swamp Optics). The linear spectral phase indicates a nearly transform-limited pulse with the FWHM duration of 97 fs if a Gaussian pulse shape is assumed. The output power limitation was resulted from splicing losses of different type fibers and the insertion loss of FRM.

**Figure 11.** The measured pulse duration (a) and spectral width (b) of pulses from the nonlinear fiber amplifier.

Furthermore, a PPLN with 20.9-μm poling period and 0.3-mm length was used for frequencydoubling the amplified laser and checking the available peak power at 1560 nm. A pair of lens was used to focus and collimate the input and output beam on the PPLN, respectively. The output average power at 1560 nm and the corresponding SHG is shown in **Figure 12**. The highest SHG conversion efficiency was obtained as 56.3% with 302 mW incident power at 1560 nm. Further increasing the power at 1560 nm induced decay of conversion efficiency, shown as the black squares in **Figure 12**.

**Figure 12.** The output average power of 1560 nm (blue triangles) and 780 nm (red circles), and the SHG conversion efficiency.

To extract more energy from the double-pass amplifier, we will increase the number of the replicas from 8 to 16 and 32. The results for ×16 and ×32 replicas are still under investigation.

In conclusion, a divided fiber laser fiber amplifier delivering 500 mW average power at 1560 nm by the interplay between divided prechirped pulse amplification and nonlinear pulse com‐ pression. A small core double-clad erbium-doped fiber with anomalous dispersion carries out the pulse amplification and simultaneously compresses the laser pulses such that a separate compressor is no longer necessary. A numeric simulation reveals the existence of an optimum fiber length for producing a transform-limited pulse. Furthermore, frequency doubling to 780 nm with 170-mW average power is realized by using a PPLN at room temperature.

### **5. Repetition rate stabilization**

Fiber-based frequency comb is recognized as the key breakthrough in the field of optics for it brings high accuracy in frequency domain as well as low jitter in time domain [49, 64–67]. Principally, as a frequency comb, two RF frequencies, *f*ceo and *f*rep, are required to be stabilized to external references. Therefore, the optical frequencies can be written as *ν* = *m* × *f*rep + *f*ceo, where *m* is a large integer of order 106 that indexes the comb line. Nevertheless, recent developments in adaptive dual-comb spectroscopy successfully employed free-running mode-locked lasers where the *f*ceo instabilities could be compensated by data acquisition and electronic signal processing [68, 69]. Therefore, high accuracy *f*rep stabilization of passively ML lasers is of great importance.

268

efficiency.

shown as the black squares in **Figure 12**.

**5. Repetition rate stabilization**

where *m* is a large integer of order 106

highest SHG conversion efficiency was obtained as 56.3% with 302 mW incident power at 1560 nm. Further increasing the power at 1560 nm induced decay of conversion efficiency,

**Figure 12.** The output average power of 1560 nm (blue triangles) and 780 nm (red circles), and the SHG conversion

To extract more energy from the double-pass amplifier, we will increase the number of the replicas from 8 to 16 and 32. The results for ×16 and ×32 replicas are still under investigation. In conclusion, a divided fiber laser fiber amplifier delivering 500 mW average power at 1560 nm by the interplay between divided prechirped pulse amplification and nonlinear pulse com‐ pression. A small core double-clad erbium-doped fiber with anomalous dispersion carries out the pulse amplification and simultaneously compresses the laser pulses such that a separate compressor is no longer necessary. A numeric simulation reveals the existence of an optimum fiber length for producing a transform-limited pulse. Furthermore, frequency doubling to 780 nm with 170-mW average power is realized by using a PPLN at room temperature.

Fiber-based frequency comb is recognized as the key breakthrough in the field of optics for it brings high accuracy in frequency domain as well as low jitter in time domain [49, 64–67]. Principally, as a frequency comb, two RF frequencies, *f*ceo and *f*rep, are required to be stabilized to external references. Therefore, the optical frequencies can be written as *ν* = *m* × *f*rep + *f*ceo,

developments in adaptive dual-comb spectroscopy successfully employed free-running mode-locked lasers where the *f*ceo instabilities could be compensated by data acquisition and

that indexes the comb line. Nevertheless, recent

The relatively mature method for *f*rep locking is to use a piezoelectric ceramic transducer (PZT) to control the geometrical length L of the laser cavity, and the best locking accuracy is in the range of ±0.5 mHz with the corresponding SD of 220 μHz [70]. However, the PZT-based stabilization encounters many limitations, such as significant positioning errors, hysteresis effect, bulky-design, and the need for time-consuming alignment.

In this section, we focus on the *f*rep stabilization by using optical pumping scheme which can be achieved via resonantly enhanced optical nonlinearity or so-called pump-induced refractive index change (RIC) in doped fibers. In optical pumping scheme, the *f*rep is stabilized by modulating the refractive index *n*, while keeping the geometrical cavity length *L* fixed. In the past, this method has been successfully applied in fiber switch where a low pump power and a short length doped fiber are sufficient for the switching [71]. Moreover, the validity of this concept has also been achieved in coherent combining and adaptive interferometry [72]. In 2013, Rieger et al. reported all-optical stabilized repetition rate by using the RIC-based method. With the help of thermos-electric element, over 12-h long-term stabilization was achieved in an NPE-mode-locked Er-doped fiber laser, while the SD of repetition rate drift was measured

**Figure 13.** Experimental setup. LD1, LD2, and LD3: pump diodes at 976 nm; WDM1, WDM2, and WDM3: 980/1550 nm wavelength division multiplexers; EDF1, EDF2: erbium-doped fiber; BP: 2-nm bandpass filter centered at 1550 nm; CP1, CP2, and CP3: 1550 nm couplers with splitting ratio of 45:55, 30:70, and 50:50, respectively; DCF: dispersion com‐ pensation fiber; PD: photodiode detector; Rb: Rubidium clock; LFP: low-pass filter; A: electronic amplifier; PID: pro‐ portional–integral–derivative controller; Driver: precision current source.

to be 22 mHz. A recent experiment extends this concept to Yb-fiber laser and achieves 1.39 mHz SD of residual fluctuation in an hour measurement [73].

As reported in Ref. [74], a commercial available pump current supply can provide a minimum resolution of pump power as 1.5 μW and thus achieve a controlling accuracy of 0.05 Hz, which is more than two orders of magnitude than PZT-based method. Therefore, an interesting experiment worth to do is to use RIC-method to achieve high-precision *f*rep stabilization. So far, the RIC method has been fully investigated in NPR mode-locked lasers, which applied non-PM fibers and components [73–75], and the locking accuracy limited to ~mHz. Considering the environmental perturbation on non-PM fiber, a straightforward idea is to implement RIC method in a PM fiber laser. Therefore, the following part will discuss high-precision repetition rate stabilization by using RIC method in a PM figure-eight laser cavity.

The laser setup shown in **Figure 13** is same as **Figure 4(a)**, except the net dispersion of laser cavity. In the current experiment setup for all-optical repetition stabilization, a 56-cm-long Er3+-doped fiber (EDF2) is spliced asymmetrically in the NALM to act as a frequency controller, while the LD3, which is controlled by the error signal from frequency mixer, provides the feedback modulating pump power via WDM3 on EDF2. Besides, a segment of DCF38 is used to compensate the anomalous dispersion of PM1550 fiber. The dispersion of linear loop and the NALM was estimated numerically to be −0.208 and 0.025 ps2 , producing −0.183 ps2 net dispersion for the whole cavity. Self-started mode-locking in multiple-pulse regime can be achieved by over-pumping method, and stable single-pulse operation can be obtained by decreasing the pump power of LD1 and LD2. At fundamental repetition rate of 11.9 MHz, the figure-eight laser cavity delivers 1.5-mW average power via CP2.

The repetition rate was detected by PD3 and successively compared with standard reference (Rb clock) in a frequency mixer to produce the error signal. Subsequently, the error signal was filtered and amplified by low-noise voltage preamplifier with frequency cutoff at 1 MHz and a maximum voltage gain of 5 × 104 and further processed by a proportional-integral-derivative (PID) controller.

The long-term stabilization was depicted in **Figure 14**. As low as 27-μHz accuracy is achieved within 16-h measurement. The inset of **Figure 14** magnifies the measured dates from 30,000 to 31,000 s and shows fluctuation range within ±0.1 mHz. Typically, thermal effect, Kerr nonlinear effect, pump-induced nonlinear effect, and random acoustic perturbations contribute to the precision of *f*rep stabilization. For our experiment, a temperature-controlled incubator with a ripple of 0.2°C was used to take the laser cavity to isolate environmental perturbation. As for Kerr-nonlinearity, the RIC is proportional to the traveling power of resonant laser. Assuming 5-mW traveling power in NALM, the Kerr-induced RIC is estimated as 1.2 × 10–7/mW, having the same order of magnitude of the pump-induced RIC (2.1 × 10−7/mW). However, when the pump power of LD3 increased from 30 to 205 mW, only 1.6% of output power change was observed, which means little change on the dynamic process of pulse evolution in NALM. Thus, the Kerr-induced RIC is near ~1% of the RIC by pump-induced nonlinearity. Therefore, we postulate that the nonlinearity on the RIC of fibers owes to pump-induced nonlinear effect and thermal effect rather than Kerr effect.

**Figure 14.** The long-time stabilization of repetition rate.
