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Abstract
In this paper, we propose a method for narrowing the spectrum in high-power narrow-linewidth polarization-maintaining (PM) fiber amplifiers and investigate its potential for suppressing the stimulated Brillouin scattering (SBS). In this method, in addition to common phase modulation to suppress SBS, precisely designed amplitude modulation is induced to generate self-phase modulation in a high-power PM fiber amplifier. In this co-modulation way, the spectrum can be gradually compressed along the fiber. Compared to phase modulation alone or fiber-Bragg-gratings (FBGs) based narrow-linewidth fiber oscillator schemes, in which the spectrum remains the same or broadens, this scheme can achieve a higher SBS threshold for the same output spectral linewidth. Experiments on a ∼ 3 kW peak power quasi-continuous wave (QCW) fiber amplifier show that the co-modulation scheme can compress the spectrum from 0.25 nm to 0.084 nm as output peak power increases from 13 W to 3.2 kW and enhances the SBS threshold by ∼1.7 times compared to traditional FBGs-based fiber oscillator schemes, and by ∼1.4 times compared to common phase modulation schemes. This co-modulation scheme has the potential for mitigating SBS in high-power fiber amplifiers.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
For high-power coherent beam combining (CBC) and spectral beam combining (SBC), enhancing the power of an all-fiber polarization-maintaining (PM) narrow-linewidth amplifier is now a focus of the current research [1–4], which is mainly limited by stimulated Brillouin scattering (SBS). Other than optimizing the amplifier itself, lots of efforts have been made on the injected seeds to mitigate the SBS in the amplifier. The common seeds used in this way can be divided into two categories: single-frequency seeds with phase-modulation and fiber-Bragg-gratings (FBGs)-based narrow-linewidth fiber oscillators. For the former one, the spectrum of output remains the same as the phase-modulated seed [5]. In general, with a wider phase-modulated linewidth, the SBS threshold can get increased [2,6,7]. For example, with the white noise signal (WNS) modulated seed injection, in 2017 Su. R et. al successfully realized a 1.49 kW, 45 GHz linearly polarized output [2]. In 2018, M. D et.al achieved a 0.82 kW,1.8 GHz output [6]. Besides, in 2022, Ren. S et.al reported a 3.96 kW, 164 GHz output [7]. Compared to the conventional WNS modulation, even for the same linewidth, fine-optimizing phase modulation can also increase the SBS thresholds [8–10]. In 2022, M. Shi et.al reported that by precisely adjusting the parameters of high-order modulation such as frequency, amplitude, spectral interval, and spectral envelope, the SBS thresholds under the same linewidth were increased by 1.3 times [8] and 1.5 times [9], respectively. In the same year, our group also reported a 1.6 times enhancement of the SBS threshold by choosing a proper recurring random signal [10]. What is more, with a changing linewidth, e.g. the FBGs-based narrow-linewidth fiber oscillator whose spectrum broadens due to four-wave mixing (FWM) [11–13], the SBS threshold cannot be directly related to the output linewidth.
To investigate the SBS threshold with a changing linewidth along the fiber, we induced the parameter of total Brillouin gain GB, which donates the accumulation of Brillouin gain along the fiber in an amplifier. The total Brillouin gain GB can be calculated as exp(∫0LgBeff(z)IL(z)dz), where the gBeff(z) denotes the effective Brillouin gain and IL(z) represents the laser intensity along the fiber [14]. For an amplifier with changing linewidth, we can estimate the effective Brillouin gain gBeff(z) as gp/Δωeff(z) where gp denotes the Brillouin gain peak and Δωeff denotes the effective spectral linewidth (FWHM). If the spectrum gradually broadens along the amplifier, which is the case for FBGs-based fiber oscillator injection, gBeff(z) should gradually decrease. In Fig. 1, we show the process to evaluate the GB in this case and to compare that with a linewidth unchanged, which is the case for phase-modulated seed injection. In Fig. 1(a), gBeff1 is the effective Brillouin gain with a linearly increased linewidth. To compare, we also draw a constant gBeff2 line with the same output linewidth as gBeff1. In Fig. 1(b), the red line with the right axis shows the process of signal power increasing alone the gain fiber in a typical amplifier and remaining the same in the passive fiber after the amplifier. With the gBeff curves in Fig. 1(a) and power IL curve in Fig. 1(b), we then can calculate the product of them, which are shown as the dashed line and black line with the left axis in Fig. 2(b). According to the definition of GB, the area below these two lines can represent the stimulated Brillouin total gain. Then in Fig. 1(b), it is clear that GB1 > GB2 for the same output power and the same output linewidth. That indicates for the same output linewidth, the SBS threshold in a broadened spectrum case is lower than that in a constant spectrum case. That explains, at the same output linewidth, the SBS threshold should be higher in the phase-modulation scheme than that in the FBGs-based narrow-linewidth fiber oscillator scheme.
Fig. 1. The accumulation of SBS gain with a comparison of the constant spectrum and broadened spectrum.
Fig. 2. The accumulation of SBS gain with a comparison of the constant spectrum and narrowed spectrum.
From the above discussion, we can easily think that a narrowing spectrum case may achieve an even higher SBS threshold. As shown in Fig. 2, for the same output power and output linewidth, the total SBS gain is lower when the spectrum is narrowed compared to a constant spectrum, GB2 > GB3. Therefore, compared to keeping the spectrum constant, gradually compressing the spectrum along the fiber would further increase the SBS threshold.
To achieve that gradual spectral compression, we introduce a nonlinear spectral narrowing method based on Self-Phase Modulation (SPM). SPM-based spectral compression is widely employed for negatively chirped picosecond pulses [15,16]. In the continuous wave (CW) domain, previous work by Goodno et al. demonstrated that SPM-induced nonlinear spectral compression can be achieved by applying correlated amplitude modulation (AM) and phase modulation (equivalent to frequency modulation (FM)) to the input field [17–19]. Their research successfully achieved up to a 2.5× narrowing of SBS-limited linewidths from a kW-class fiber amplifier in 2019 [17]. Later, in 2021, they reported a 7× narrowing factor using this spectral narrowing method [18].
In their work, after common phase modulation of a single frequency seed broadening to several GHz levels, they imposed 32 GHz FM to shift the central frequency to several Bessel order lines. Then, a synchronous 32 GHz AM was imposed. By establishing a relationship between AM and FM, the amplifier works as a phase demodulator, which can cancel the FM via SPM at target B-integral (determined by the output power and the effective fiber length). At last, the power in several Bessel order lines transfers back to the central frequency, resulting in spectral compression. They used three modulations with an appropriate electrical driver, which is a 32 GHz Mach-Zehnder electro-optical modulator (EOM) with three independent drive inputs for each leg [20].
In our study, with a single commercial dual-channel (both slow axis and fast axis work) 5 GHz EOM and one RF amplifier, we achieve amplitude and phase co-modulation to the single frequency seed. In this way, nonlinear spectral compression is also achieved via SPM in our fiber amplifier. A detailed description of this seed structure and a detailed description of the mechanism and simulation for achieving spectral compression are presented in Part 2. The experimental setup of this scheme is described in Part 3. The validation of the achieved spectral compression is presented in Part 4.
To evaluate the potential of this gradual spectral compression scheme for enhancing the SBS threshold, we take two other seeds for comparison experiments: a phase-modulated seed and an FBGs-based narrow-linewidth fiber oscillator. These two types of seeds correspond to spectrum unchanged and spectrum broadening, respectively. Based on the master oscillator power amplifier (MOPA) configuration, we build a ∼3 kW peak-power quasi-continuous wave (QCW) fiber amplifier for the experiment. By injecting different seeds separately into the same fiber amplifier, SBS thresholds of the amplifier under these three types of seed are experimentally investigated in Part 4.
2. Spectral narrowing scheme
One of the key techniques to achieve this spectral narrowing induced by SPM is an amplitude and phase co-modulated seed. As for the typical phase-modulated single-frequency seed, the intensity fluctuation can normally be ignored. That means, as the SPM can also be ignored during a high-power amplification, the spectrum almost maintains the same as the seed. To shape the spectrum, we need to deliberately induce amplitude modulation. In that way, it is possible that by designing the amplitude modulation, we may achieve spectral narrowing via SPM in a high-power amplification. That then results in SBS enhancement for a specific fiber amplifier.
The phase and amplitude co-modulation can be realized by a dual-channel electro-optical phase modulator (EOPM) with different half voltages on the fast and slow axes. If two single-frequency lights with orthogonal polarization passed through this dual-channel EOPM with the same modulation voltages, two lights with different phase modulation can be realized. Then with a rotatable polarizer, arbitrary interference of these two lights can generate that amplitude and phase co-modulation. Figure 3 illustrates the setup where a single-frequency fiber laser (SFFL) is injected into a dual-channel (both fast-axis and slow-axis work) EOPM. This EOPM with different half voltages on the slow and fast axes is used to modulate the phase of the injected light. Afterward, this phase-modulated light transmits through a PM isolator. WNS is added to this dual-channel EOPM. In this setup, the PM isolator works as the polarizer, and by rotating the fiber to a certain angle during fusing, the arbitrary interference of these two signals can be achieved.
Fig. 3. A single frequency fiber laser modulated by a dual-channel (both fast axis and slow axis work) EOPM passed through a PM isolator. SFFL, single frequency fiber laser, WNS, white noise signal, EOPM, electro-optical phase modulator.
If the fast axis light is taken as the primary polarization output, then the output electric field can be expressed as:
Based on the above analysis, we take some practical parameters into account to numerically analyze how this amplitude and phase co-modulation works in spectral narrowing.
First, in our simulation, we model a co-modulated seed with a commercial dual-channel EOPM driven by a WNS. The half-wave voltage of that EOPM on the fast axis and slow axis is ∼2 V and ∼6 V, respectively. That means α in our model is 0.33. The typical time difference value of PM980 is about 1.3 ps/m. In this EOPM setup, the total length of input and output fiber of this EOPM is usually less than 0.5 m, which means τ is less than 0.65 ps. In our setup, the modulation depth of the WNS is ∼10$\pi $ level. As the highest frequency is 5 GHz, the maximum phase shift within 0.65 ps can be calculated as 0.65 ps/0.2 ns*10$\pi $ ≈ 0.03$\pi $ <<$\pi $. Therefore, τ can be ignored as not so much phase can be changed within 0.65 ps.
Then we can inject this modulated seed into an ideal fiber amplifier and analyze the changes in the input-output spectrum. Since the white noise signal is randomly generated, we averaged the spectrum over 1000 sets of WNS data and calculated the final spectrum. In terms of the simulation parameters for the amplifier, the non-linear refractive index ${n_2}$ is 2.7 × 10−20 m2/W, the effective mode area Aeff is 2.5 × 10−10 m2, and the center wavelength is 1064 nm, thus the Kerr-nonlinearity parameter γ=2πn2/(λAeff is about 6.37 × 10−4 W/m. As for a typical fiber amplifier at 3 kW output with < 20 m effective fiber length, the B integral (γPLeff) is determined to be less than 30 rads. Therefore, we calculate the spectral shaping process with B integral ranges from 0-30 rads.
In our simulation, we use the broadening factor Δλout/Δλin to describe the change of the spectrum with varying PER01 and B integral. Δλin is defined as the input spectral linewidth with pure phase modulation on the fast axis. Δλout is defined as the output spectral linewidth with amplitude modulation. The results are presented in Fig. 4. From Fig. 4(a), it can be observed that as the B integral increases, by setting a certain value of PER01, the broadening factor would first decrease to the minimum value and then get increased. If the value of PER01 is smaller, the decreasing process would be faster. The spectral shaping process can be described as firstly narrowing and then broadening. What is more, if the PER01 is too small (5 dB), the initial output spectral linewidth would be narrowed even without the B integral, which is not conducive to mitigating the SBS during the spectral narrowing process. By comparing Fig. 4(a) with Fig. 4(b), we can find that different initial linewidth will lead to almost the same spectrum-shaping process.
Fig. 4. Broadening factor with different PER01 and B integrals. When (a) initial FWHM = 0.25 nm. (b) initial FWHM = 0.4 nm.
Therefore, the key to effectively compressing the spectrum is to choose a proper rotation angle θ0 ($PE{R_i} = 10{\log _{10}}({1/{{\tan }^2}{\theta_i}} ),\; PE{R_{01}} = ({PE{R_0} + PE{R_1}} )/2$, dB, θ1 is set to 5°) for matching the desired B integral in the experiment. In our experiment, the designed active fiber length is 10 m, and the passive fiber length is 5 m. B integral be calculated by Eq. (7)
The desired B integral in our setup is determined by the output power. Then, we can choose the appropriate θ0 by matching the desired output power. We draw Fig. 5 to show how this angle changes output spectral linewidth as the output power increases. The co-modulated seed in Fig. 5 is the same as in Fig. 4(a). By adjusting the angle of θ0 from 10° to 35°, corresponding to PER01 decreasing from 18.1 dB to 12.1 dB, the FHWM of the output spectrum can be narrowed as the output power increases. Simulation results are presented in Fig. 5, which shows that FWHM can be compressed from 0.25 nm to 0.08 nm at ∼3.7 kW (B = 18.7 rads) when θ0 = 30°. When the output power is 3.2 kW (B = 16.2 rads), FWHM can be compressed to 0.09 nm, 0.08 nm, and 0.075 nm with setting θ0 to 25°, 30°, and 35°, respectively.Fig. 5. (a) Detailed simulation results of Fig. 4(a), show the process of linewidth being compressed gradually as the power increases with different θ0. (b) Simulation results of spectral compression at ∼3.7 kW when θ0 = 30°. The FWHM can be compressed from 0.25 nm to 0.08 nm.
3. Experimental setup
The SBS effect in a fiber amplifier operated in QCW mode with a certain peak output power can be consistent with that amplifier for the same output power in CW mode. Due to the thermal loading and safety concerns, we choose QCW mode for our proof-of-principle experiment of nonlinear spectral compression, which facilitates us to extend our future work in CW mode. Figure 6 shows our QCW fiber amplifier system. The signal is generated from a single-frequency CW diode laser which worked at 1064 nm with 30 mW output power. After passing through a dual-channel EOPM, driven by a WNS source covering a band of 0.01-5 GHz, the spectrum was broadened to 0.25 nm (FWHM). The rotation angles before and after this EOPM are set to 5° and 30°. The measured half-wave voltage of the fast and slow axes of this dual-channel EOPM is 2.1 V and 6.2 V. The total length of input and output fiber of this EOPM is 0.4 m. The PM isolator allows only the primary polarization light to pass.
Fig. 6. Schematic of the QCW fiber amplifier with amplitude and phase co-modulated seed injection. CPS, cladding power strippers, YDF, Yb-doped fiber, PBS, Polarizing Beam Splitter, M1, total internal reflection mirrors, M2, high reflectivity mirror, M3, 50% reflectivity mirror, OSA, Optical Spectrum Analyzer, PD, photodiode.
Then, this seed can be pre-amplified to ∼13 W by PM fiber pre-amplifiers. To monitor the backward signal, a ∼1‰ tap is set on the PM isolator before the PM main fiber amplifier. The PM main fiber amplifier is based on a counter-pumped configuration, made up of a piece of 10 m PM large mode area (∼250 µm2) YDF, a PM counter signal-pump combiner, two PM cladding power strippers (CPS), and a PM endcap. The total length of passive fiber is about 5 m. The pump sources of the main amplifier are six high-power laser diodes (LDs) capable of providing a maximum power of 600 W. We operate them in quasi-continuous mode with a repetition rate of 1 kHz and an output pulse width of 42 µs. Two high-power PM CPS are used to eliminate the redundant light in the cladding.
A Polarizing Beam Splitter (PBS) with a half-wave plate and two power meters are used to measure the polarization extinction ratio (PER). To measure the output spectrum, we use the transmission direction of the high reflectivity (99.9%) mirror (M2) to pick up the signal from the primary polarizing beam. To measure the output time temporal trace, a photodiode (PD) with an oscilloscope is used. Power meter 1 is used to record the backward laser power. The output power is recorded by power meter 2 and power meter 3.
4. Results and discussion
4.1 Spectral narrowing validation
To validate the above spectral narrowing method, we conduct a series of experiments. Firstly, we record the spectrum at different output power, ranging from 13 W to 147.5 W, and plot them in Fig. 7(a). The seed is operating in CW mode and the pump of the amplifier is operating with a pulse width of 42 µs and a repeated frequency of 1kHz. The peak power of the output can increase from 13 W to 3.2 kW with a pulse width of 42 µs at the maximum power of 147.5 W (temporal trace measured from PD is inserted in Fig. 7(a)). The measured PER ranges from 17∼18 dB. As the measured results shown in Fig. 7(a), the FWHM of the spectrum exhibits a significant compression from 0.25 nm to 0.084 nm as the peak power increases from 13 W to 3.2 kW. The experimental results are consistent with the simulation results (from 0.25 nm to 0.08 nm), in Fig. 5(a) and Fig. 5(b).
Fig. 7. Measured Spectrum and time temporal traces of output. (a) shows the spectrum narrowing process with the power increase, insert, the output time temporal trace (−50∼80 µs) at 3.2 kW peak-power output. (b) shows the spectrum with different angles of θ0 when the peak power is 3.2 kW.
Subsequently, we measure the spectrum at the maximum output with different θ0 settings, which are shown in Fig. 7(b). The FWHM is 0.129 nm, 0.084 nm, and 0.074 nm respectively when θ0 is set to 25°,30°, and 35° (corresponding to PER01 values of 13.8 dB, 12.9 dB, and 12.1 dB). These experimental results also validate the simulation results in Fig. 5(a) (0.09 nm, 0.08 nm, 0.075 nm). Thus, it verifies that to compress a spectrum at the same B integral, a smaller value of PER01 corresponds to a narrower spectral linewidth at the initial compression process.
4.2 Comparison of SBS thresholds at the same output spectral linewidth
To evaluate the SBS suppression capability of this gradual spectral compression scheme, we measure the SBS threshold. In this work, the backward reflectivity of 0.1‰ [21,22] is defined as the SBS threshold. A plot of the backward peak power versus the output peak power is presented in Fig. 8. The initial linewidth is 0.25 nm. As laser power increases, the output linewidth compresses to 0.195 nm at 1.3 kW and 0.084 nm at 3.2 kW. The backward power in the latter case is 330 mW, representing 0.103‰ of the output power. The SBS threshold for this PM fiber amplifier with amplitude and phase co-modulated seed injection is ∼3.2 kW at 0.084 nm output linewidth.
Fig. 8. Backward peak-power versus the laser peak-power with amplitude and phase co-modulated seed injection.
Then we take two other seeds to measure the SBS threshold of this fiber amplifier for comparison experiments. The first one is a phase-modulated seed. In constructing this seed, we replace the dual-channel EOPM in the above co-modulated seed with a single-channel EOPM and adjust the WNS to cover a band with 0-2.8 GHz. In this way, the initial FWHM of this phase-modulated seed is set to 0.084 nm. We also record the backward peak-power versus the output peak-power with this seed injection and present the results in Fig. 9. The results show that the output linewidth remains constant at 0.084 nm at ∼1.3 kW, and it is still constant at ∼2.3 kW. The backward power in the latter case is 240 mW, representing 0.105‰ of the output power. Thus, the SBS threshold for this PM fiber amplifier with this phase-modulated seed injection is ∼2.3 kW at 0.084 nm output linewidth.
Another is an FBGs-based fiber oscillator. This oscillator works at 1064 nm with a maximum output power of 13W. We use a ∼1‰ tap to extract the light from this oscillator as a seed with a power of 13 mW for amplification. By continuously adjusting the cavity length and the output power of this oscillator, we set the initial linewidth of this seed to 0.031 nm. We also record the backward peak-power versus the laser power with this seed injection and present the results in Fig. 10. Results show that as power increases, the output linewidth is 0.04 nm at ∼0.8 kW, and it is 0.083 nm at ∼1.9 kW. The backward power in the latter case is 206 mW, representing 0.109‰ of the output laser power. Therefore, the SBS threshold for this PM fiber amplifier with this seed injection is ∼1.9 kW at 0.083 nm output linewidth.
Fig. 10. Backward peak-power versus the laser peak-power with FBGs-based fiber oscillator injection.
Based on our experimental results, the amplitude and phase co-modulation scheme shows an advantage in SBS mitigation among these three schemes (amplitude and phase co-modulation, phase modulation, FBGs-based fiber oscillator). Compared to phase modulation, our scheme achieves a 40% improvement in the SBS threshold at the same output linewidth (∼0.08 nm), increasing it from 2.3 kW to 3.2 kW. Furthermore, compared to the traditional FBGs-based fiber oscillator scheme, our proposed scheme demonstrates a 70% increase in the SBS threshold, rising from 1.9 kW to 3.2 kW at the same output linewidth (∼0.08 nm).
Besides, we also investigate the peak power intensity by comparing the final output spectra via co-modulation and the initial input spectra via pure phase modulation on the fast axis. The normalized intensity distribution is plotted in Fig. 11.
Fig. 11. The comparison of the final output spectra via co-modulation with the initial input spectra via pure phase modulation on the fast axis.
From Fig. 11, it can be seen that, compared to the initial pure phase modulation case, the peak power intensity for the co-modulation scheme gets increased by ∼1.7 times, which allows us to realize a higher SBS-limited power spectral density (PSD). In 2021, P. Yan et.al realized an output power of 3010 W, a spectral 3-dB bandwidth of 103pm, and a PSD of 29.2 W/pm based on an FBGs-based scheme by suppressing the nonlinear effects to reduce the spectral broadening [23]. In our study, the three schemes above realize a PSD of ∼38 W/pm (co-modulation), ∼27W/pm (phase modulation), and ∼23 W/pm (FBGs-based), respectively. Their study also indicated that compared to the pure phase modulation scheme, if the spectrum broadening is not suppressed, the SBS-limited PSD of the FBGs-Based scheme is relatively low. The comparison of these two schemes in our study also verifies their points. Moreover, the experimental results show that by employing our co-modulation (spectral-narrowing) scheme, the SBS-limited PSD of fiber amplifiers can be further increased.
5. Conclusion
In conclusion, this paper proposes a nonlinear spectrum-narrowing method that employs an amplitude and phase co-modulation technique to suppress SBS. Experimental results show that by precisely controlling the modulation parameters, the output linewidth can be narrowed from ∼0.25 nm to ∼0.084 nm with output power increasing. In particular, compared to the commonly used phase modulation, simply by replacing the fast-axis blocked EOPM with a dual-channel EOPM, our method achieves a 40% improvement in the SBS threshold. This demonstrates the conciseness of our approach and its potential applicability to other wavelengths. Compared to the traditional FBGs-based fiber oscillator scheme, this method exhibits a 70% increase in the SBS threshold. Therefore, by employing this spectrum-narrowing method, we have successfully enhanced the SBS threshold of this fiber amplifier. In the future, we plan to investigate the SBS suppression performance of this approach in high-power fiber CW lasers.
Funding
the Innovation Development Fund of CAEP (C-2021-CX20210047).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. P. Ma, R. Tao, R. Su, X. Wang, P. Zhou, and Z. Liu, “1.89 kW all-fiberized and polarization-maintained amplifiers with narrow linewidth and near-diffraction-limited beam quality,” Opt. Express 24(4), 4187–4195 (2016). [CrossRef]
2. D. Meng, W. Lai, X. He, P. Ma, R. Su, P. Zhou, and L. Yang, “Kilowatt-level, mode-instability-free, all-fiber and polarization-maintained amplifier with spectral linewidth of 1.8 GHz,” Laser Phys. 29(3), 035103 (2019). [CrossRef]
3. N. Platonov, R. Yagodkin, J. De La Cruz, A. Yusim, and V. Gapontsev, “1.5 kW linear polarized on PM fiber and 2 kW on non-PM fiber narrow linewidth CW diffraction-limited fiber amplifier,” Proc. SPIE 10085, 100850M (2017). [CrossRef]
4. R. Su, R. Tao, X. Wang, H. Zhang, P. Ma, P. Zhou, and X. Xu, “2.43 kW narrow linewidth linearly polarized all-fiber amplifier based on mode instability suppression,” Laser Phys. Lett. 14(8), 085102 (2017). [CrossRef]
5. Y. Wang, W. Peng, W. Ke, Y. Sun, Z. Chang, Y. Ma, R. Zhu, and C. Tang, “Influence of seed instability on the stimulated Raman scattering of high power narrow linewidth fiber amplifier,” Opt. Quantum Electron. 52(4), 193 (2020). [CrossRef]
6. R. Su, Y. Liu, B. Yang, P. Ma, X. Wang, P. Zhou, and X. Xu, “Active polarization control of a 1.43 kW narrow linewidth fiber ampli fi er based on SPGD algorithm,” J. Opt. 19(4), 045802 (2017). [CrossRef]
7. S. Ren, P. Ma, W. Li, G. Wang, Y. Chen, J. Song J, W. Liu, and P. Zhou, “3.96 kW All-Fiberized Linearly Polarized and Narrow Linewidth Fiber Laser with Near-Diffraction-Limited Beam Quality,” Nanomaterials 12(15), 2541 (2022). [CrossRef]
8. M. Shi, M. Yu, Z. Fang, Y. Wu, J. Li, J. Wang, H. Mu, W. Hu, and L. Yi, “Real-time definite sequence modulation based spectral broadening scheme for high-power narrow-linewidth fiber laser,” J. Lightwave Technol. 40(18), 6222–6229 (2022). [CrossRef]
9. M. Shi, Z. Wu, J. Li, Y. Wu, M. Yu, Y. Sun, W. Hu, and L. Yi, “High-power narrow-linewidth fiber lasers using optical spectrum broadening based on high-order phase modulation of inversion probability-tuning sequence,” Opt. Express 30(6), 8448–8460 (2022). [CrossRef]
10. Y. Wang, Y. Sun, W. Peng, J. Wang, Y. Feng, Y. Ma, Q. Gao, R. Zhu, and C. Tang, “Effect of the recurring random signal waveform on SBS and self-pulsing in a phase-modulated narrow-linewidth linearly polarized fiber amplifier,” Opt. Commun. 523, 128683 (2022). [CrossRef]
11. W. Shi, Q. Fang, J. Fan, T. Qu, and X. Meng, “High power monolithic linearly polarized narrow linewidth single mode fiber laser at 1064 nm,” in Conference on Lasers and Electro-Optics/Pacific Rim (2015) p. 26P_16.
12. J. Man, P. Ma, H. Long, J. Xu, P. Zhou, and X. Gu, “kW-level, narrow-linewidth linearly polarized fiber laser with excellent beam quality through compact one-stage amplification scheme,” High Power Laser Sci. Eng. 5, e30 (2017). [CrossRef]
13. Y. Wang, W. Ke, W. Peng, Z. Chang, Y. Feng, Y. Sun, Q. Gao, Y. Ma, R. Zhu, and C. Tang, “3 kW, 0.2 nm narrow linewidth linearly polarized all-fiber laser based on a compact MOPA structure,” Laser Phys. Lett. 17(7), 075101 (2020). [CrossRef]
14. G. P. Agrawal and G. P. Agrawal, Nonlinear fiber optics, (Springer, 2000), pp. 195–211.
15. R. H. Stolen and C. Lin, “Self-phase-modulation in silica optical fibers,” Phys. Rev. A 17(4), 1448–1453 (1978). [CrossRef]
16. E. R. Andresen, J. M. Dudley, D. Oron, C. Finot, and H. Rigneault, “Transform-limited spectral compression by self-phase modulation of amplitude-shaped pulses with negative chirp,” Opt. Lett. 36(5), 707–709 (2011). [CrossRef]
17. G. D. Goodno and J. E. Rothenberg, “Suppression of stimulated Brillouin scattering in high power fibers using nonlinear phase demodulation,” Opt. Express 27(9), 13129–13141 (2019). [CrossRef]
18. G. D. Goodno, “Linewidth Narrowing of a High Power Polarization Maintaining Fiber Amplifier using Nonlinear Phase Demodulation,” in CLEO: Science and Innovations (2021), pp. SM4K. 1.
19. G. D. Goodno and J. Rothenberg, “Spectral brightness scaling of kW fiber amplifiers via nonlinear linewidth narrowing (Conference Presentation),” Proc. SPIE. 11260, 1126004 (2020).
20. G. D. Goodno, “AM/FM seed for nonlinear spectrally compressed fiber amplifier,” U.S. Patent No. 10,811,837. 20 Oct. 2020.
21. B. Anderson, A. Flores, R. Holten, and I. Dajani, “Comparison of phase modulation schemes for coherently combined fiber amplifiers,” Opt. Express 23(21), 27046–27060 (2015). [CrossRef]
22. C. Robin, I. Dajani, C. Zernigue, A. Flores, B. Pulford, A. Lanari, and S. Naderi, “Pseudo-random binary sequency phase modulation in high power Yb-doped fiber amplifiers,” Proc. SPIE 8601, 86010Z (2013). [CrossRef]
23. Y. Huang, Q. Xiao, D. Li, J. Xin, Z. Wang, J. Tian, Y. Wu, M. Gong, L. Zhu, and P. Yan, “3 kW narrow linewidth high spectral density continuous wave fiber laser based on fiber Bragg grating,” Opt. Laser Technol. 133, 106538 (2021). [CrossRef]
