Open Access
Add to BibSonomy
Abstract
In this paper, we study the power scaling in high power continuous-wave Raman fiber amplifier employing graded-index passive fiber. The maximum output power reaches 2.087 kW at 1130 nm with an optical conversion efficiency of 90.1% (the output signal power versus the depleted pump power). To the best of our knowledge, this is the highest power in the fields of Raman fiber lasers based merely on Stokes radiation. The beam quality parameter M2 improves from 15 to 8.9 during the power boosting process, then beam spot distortion appears at high power level. This is the first observation and analysis on erratic dynamic properties of the transverse modes in high power Raman fiber amplifier.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Over the years, high power fiber lasers have witnessed a striking progression with terrific output characteristics including high power level and excellent beam quality, which fulfill the desire in diverse application areas [1–4]. In particular, due to the wide and tunable lasing waveband, Raman fiber lasers (RFLs) that utilize the stimulated Raman scattering (SRS) effect in passive fibers as gain regime are prospective laser sources [5–9]. Unlike the rare-earth (RE) doped fiber lasers that generate radiation in fixed spectral range referring to the emitting band of specific ions, the broad Raman gain spectrum and various cascaded shifting number ensure that RFLs can attain the demands in spectral waveband beyond the RE-doped lasing [10–15]. Besides, some common advantages of RFLs, to cite for instance, relatively low quantum defect, the absence of photondarkening and parasitic lasing also make RFLs another potential candidate on power scaling rather than the RE-doped fiber lasers [16–18].
Power scaling of RFLs based on pure passive fiber has been the research interest and the power level is mounting continuously [19–25]. Particularly, thanks to the better output characteristics, RFLs with brightness enhancement (BE) have been paid more and more attention in recent years. To achieve this, one can employ cladding pumping which is similar to RE-doped fiber lasers. More specifically, pump laser with low beam quality is injected into the fiber cladding, and along with the Raman conversion, signal light with higher brightness is generated in the fiber core [20,22,25]. Nevertheless, to guarantee the efficient suppression on higher order Raman Stokes, the cladding-to-core area ratio needs to be less than 8, otherwise the multi-clad fiber with special designs on optical filtering at longer wavelength is required [26]. A relatively convenient alternative to acquire BE is utilizing commercial graded-index (GRIN) passive fibers directly, which is more economical and simplifies the implementation of RFL systems [23,24,27–33]. Along with the manufacturing development of fiber components and BE of pump source, much progress has been achieved based on SRS in continuous-wave (CW) GRIN fiber laser in the last few years. In 2004, Baek et al. reported a single mode RFL based on GRIN fiber in free space configuration and obtained beam parameter M2 of 1.66, while special fiber Bragg gratings (FBGs) inscribed in multimode (MM) GRIN fiber were applied, which had function of mode selection to support BE of laser [34]. In 2016, Kablukov et al. developed an all-fiber RFL employing FBGs to select single transverse mode and enhance the output brightness, then the research group progressively optimized the RFL and the latest maximum power of 62 W at 954 nm was reported with BE [9,29,31]. In 2017, Glick et al. realized a 154 W CW RFL at 1023 nm with an efficiency of 65% in free-space configuration, and reflective mirrors were employed to form laser cavity [30]. Then in 2018, FBGs were utilized and all-fiber RFL based on GRIN fiber was demonstrated with an output power of 135W [24]. Recently, Raman fiber amplifier (RFA) with BE and high power output was reported based on GRIN fiber [23]. The output power reached 528.8 W at 1060 nm with the conversion efficiency of 78.8%, then in Ref. [32] the lasing system was optimized and the power level was improved to 1002.3 W with the efficiency of 84%. In 2019, 1.6 kW lasing power at 1130 nm was reported in RFA based upon GRIN fiber and the optical-to-optical (o-o) efficiency was 87.3% (The efficiencies of the above mentioned RFAs equals the output signal power divided by the depleted pump power), which is so far the highest power from a RFA based on pure passive fiber [33]. Comparing with oscillator structure, the amplifier can couple more pump power and high power lasing is not limited by endurance capability of fiber components in cavity, which shows potential in power boosting.
Presented here is the power scaling of high power CW RFA. The gain scheme is based upon SRS in MM GRIN fiber, which is totally irrelevant with RE gain regime and offers power boosting as well as improvement of beam quality simultaneously. When 3.34 kW pump laser is injected, the output power reaches 2.087 kW at 1130 nm with an optical conversion efficiency of 90.1% (the output signal power versus the depleted pump power). To the best of our knowledge, this is the highest power in the fields of RFLs based merely on Stokes radiation. The beam quality parameter M2 improves from 15 to 8.9 during the power boosting process. After that, the distortion of spot at beam waist with instability of modes is monitored at high power level. In the fields of MM nonlinear dynamics, constant exploration has been made based on GRIN fiber, since that this fiber supports complex nonlinear phenomena, and the distribution of massive modes is controllable and measurable on account of the parameters of fiber [35–41]. This is usually in the domain of ultrafast fiber lasers and spatiotemporal solitons. However, the mode dynamics in the GRIN fiber with high power CW lasing is less discussed. Here is the first observation of erratic dynamics properties in 2 kW-level Raman fiber amplifier, and the corresponding analysis is presented.
2. Experimental setup
To achieve the laser generation at this record output power, an RFL in master oscillator power amplifier (MOPA) configuration is designed, which is composed of a fiber oscillator stage and a fiber amplifier stage, as illustrated in Fig. 1. In the oscillator stage, the resonant cavity is a Raman fiber oscillator pumped by an Ytterbium-doped fiber laser at 1080 nm, which contains one pair of FBGs and a piece of Germanium-doped fiber (GDF). The fiber length is 220 m and all the fiber components have the same core/cladding diameters of 20/400 µm. The numerical aperture (NA) of the core is 0.06. The FBGs centered at 1130 nm are used as wavelength selectors, which have reflectivity of 99% and 13%, respectively. Cladding pump stripper (CPS) is applied in the seed laser to ensure that the radiation emerges from fiber core. In the amplifier stage, a homemade 7×1 tapered fiber bundle (TFB) is utilized to combine the seed and pump lasers [42]. The pump lasers are Ytterbium-doped fiber lasers at 1080 nm and the total output power at maximum is 3.5 kW after combination. The core/cladding diameter of the output pigtailed fiber of the TFB are 50/360 µm. The amplification stage adopts a piece of 22 m-long GRIN fiber as the Raman gain medium which has core diameter of 100 µm and NA of 0.29. At the end of the GRIN fiber, one specifically manufactured end cap is spliced to deliver the amplified Raman laser into free space in security. The GDF and GRIN fiber are coiled round metal barrel with bending radius of 0.24 m, and water cooling boards are applied to the amplifier to moderate the thermal load.
In order to evaluate the output laser power and spatial parameters, a beam splitting system is established, containing one collimating mirror, dichroic mirrors and HR mirrors. Since the power levels of pump and signal radiation are too high to be completely separated, two dichroic mirrors are installed to ensure the laser segregation after the collimating mirror. Meanwhile, we employ several splitters in the part of HRs to reduce the power orders of output laser to a suitable magnitude for beam shape monitoring and beam quality measurement.
3. Experimental results
In the detecting system in Fig. 1, the outgoing laser power after the end cap is measured by a power meter that supports high power endurance. In the oscillator stage, the signal laser at 1130 nm is obtained through Raman conversion in the GDF, and the maximum output power is 142 W with corresponding laser at 1130 nm of 105 W and residual unconverted 1080 nm power of 37 W. Here the unconverted 1080 nm light is injected into the amplifier along with the 1130 nm seed laser, which can later be a portion of pump light in the GRIN fiber. Figure 2(a) illustrates the generated Stokes light power and residual pump power from the end cap as a function of the launched pump power. In the amplifier stage, the Raman conversion starts with a relatively slow pace when pump laser is injected. When 3.34 kW pump power is launched, the total output power of 3.26 kW is obtained at maximum, in which the signal power at 1130 nm is 2.087 kW. The residual pump power increases at the start as a result of the inadequate conversion to 1130 nm light. After that, more pump light shifts to signal light with the increase of pump power and the 1080 nm laser power decreases to 1.167 kW gradually. The fractional pump leakage and conversion efficiency versus launched pump power and absorbed pump power are shown in Fig. 2(b) and 2(c), respectively. The power ratio of residual pump power is higher than the depleted pump power in the beginning, and keeps decreasing along with the Raman conversion. The o-o conversion efficiency at maximum power is 90.1% and 59.33% according to the signal power versus absorbed pump power and launched pump power, respectively. This result indicates that the shortening of fiber length brings about the low loss and more residual pump laser. The slope efficiency is obtained by linear fitting, which is 64.6% and 89.7% corresponding to the slope of signal laser power versus the injected pump power and the absorbed pump power, respectively. One should note that although we use the word “absorption”, it is to be appreciated that SRS is quasi-instantaneous and the pump power is not absorbed in the way it is by a RE-doped fiber.
Fig. 2. Output characteristics of the RFA, including (a) the output power with varying launched pump power, (b) fractional pump leakage and conversion efficiency versus launched pump power, and (c) fractional pump leakage and conversion efficiency versus absorbed pump power.
The optical spectra are measured by optical spectrum analyzer (OSA) YOKOGAWA AQ6370D, and the spectrum evolution with various signal output power is shown in Fig. 3. One can see that the unabsorbed laser at 1080 nm together with the pump laser transfers into the signal laser at 1130 nm progressively through Raman conversion, and higher order Raman light centered at 1185 nm arises when signal power is 1079.2 W. With more launched pump laser, the higher order Raman laser increases. Nevertheless, the growth rate is still slow, and the intensity difference between the first and second stokes light is over 36.5 dB at maximum output. If more pump laser is injected, the signal laser will obtain power scaling with the reduction of residual pump laser and relatively low power level of 2nd order Raman laser. In our amplifier, the power scaling is limited by the available pump laser at this time, and with more injected pump laser the slope efficiency will be higher without doubt. The full width at half maximum (FWHM) of signal spectra increases from 1.54 nm of seed laser to 2.12 nm of maximum laser, which is overall higher than those in Ref. [23,32]. The spectrum of the two pump laser at 1080 nm is monitored before the amplification, and the FWHM of single pump laser with 1.8-kW power is ∼2.88 nm. In comparison, the pump lasers in Ref. [23,32] are homemade Ytterbium-doped fiber lasers at 1018 nm and each one has smaller FWHM of ∼ 0.5 nm at power of 180 W. Moreover, the bandwidth of seed laser with output power of 131.5 W is also higher than that of the several-tens-watts seed laser in Ref. [23,32]. In addition, the broadening of the spectrum owing to Four wave mixing (FWM) is also shown in the output spectra [43].
The beam quality of signal light is measured by the beam quality analyzer M2-200s based on 4-sigma method. The beam quality parameter M2 of signal laser, launched pump laser and residual pump laser as a function of output power at 1130 nm are illustrated in Fig. 4. The M2 of seed laser is 15.84, and the beam quality ameliorates with rising signal power which improves to 8.9 at 1551.6 W. The beam quality of injected pump laser at 1080 nm after the TFB remains stable at around 11 during power scaling, while that of residual pump laser decreases in the beam cleanup process. There is an obvious improvement of brightness of signal laser compared with the brightness of the seed laser. According to the previous study, the beam quality should be upgradable with higher output power. However, this beam cleanup process comes to slow down. Moreover, distortion of laser modes is observed at and above this power level accompanied by the continuous deformation of beam shape, as can be seen in Visualization 1. When more pump laser is injected, this deformation is more severe and the energy center together with beam shape keeps changing. At this time it is incapable of continuing the M2 measurement, thus we use high-speed camera to catch and record the beam spot at high power level, of which the sampling frequency and the image resolution are 250 Hz and 1280×1024 pixels, respectively. As is illustrated in Fig. 5, the appearance of deformation is noticeable at maximum output signal power. This instability also happens when a certain power threshold is reached, which is similar to mode instability (MI) in high power fiber laser systems [44–50]. Nevertheless, the lasing power does not show hysteresis at high power level since that all laser, regardless of the modes, is confined and transferring in the core of GRIN fiber rather than dumped from cladding.
Fig. 4. The beam quality parameter M2 of the signal laser at 1130 nm, launched pump laser and the residual pump laser as a function of the generated stokes power.
Fig. 5. The beam shape measured by high-speed camera at maximum output power with the time interval of 1.6 s.
In order to characterize the beam fluctuation dynamics more adequately at diverse frequency magnitudes, we raise the recording rate of high-speed camera to 10 kHz with frame size of 1280×240 pixels. The pixel size of the high-speed camera is 10 µm. The recording time is limited by the memory of the high-speed camera which is 16 G, and the maximum acquization time is ∼3.7 s corresponding to frequency of 10 kHz. In the 100-µm fiber core the thousands of modes result in irregular beam geometry, besides, at high power level the shape and position of beam spot are changing continuously. To better cover the moving range of beam spot, apertures with varying sizes (i.e. 10×10 pixels, 20×20 pixels, etc.) and positions are chosen in the picture, and within individual sampling aperture the total power intensity in each frame is integrated, meanwhile the related normalized standard deviation (Sn) in a 3.7 s acquisition time is calculated. Since that the variation tendency of the Sn is similar which is only different in amplitudes, here a nine-box-aperture synthetic sampling is applied for demonstration around the spot center with a square of 10×10 pixels, as is depicted in Fig. 6. The average Sn of pump and signal laser at different levels of output power is shown in Fig. 7. Temporal stabilization of the pump laser is obvious with small average Sn (0.01-0.02), and no fluctuation of beam spot is observed as can be seen in the corresponding frame. By contrast, the average Sn of seed laser and signal laser under 1020 W is below 0.2. However, the average Sn surges up to 0.43 at a power threshold of 1397 W and beam fluctuation is preliminarily observed. With increasing pump laser, the average Sn has a slightly downtrend and finally rises back to 0.48. The beam cleanup process is inferred through comparing the insets of seed and amplified laser in Fig. 7, since that the laser is increasingly concentrated. The changing tendency of the average Sn curve is similar to that in Ref. [45,46,50] which are employed to characterize MI, and this indicates the incremental instability of generated laser.
Fig. 6. Single frame of beam spot of maximum pump laser. The red square represents the size and position of nine-box sampling aperture.
Fig. 7. The average normalized standard deviation of signal laser at 1130 nm and corresponding launched pump power of nine sampling positions in 3.7 s acquisition time, and the insets are beam profile captured at various power level.
The Fourier spectra of the beam trace of signal laser is calculated under various power levels. Figure 8 and Fig. 9 depict the Fourier spectra of seed, pump and generated signal laser at varying power. As is seen in Fig. 8, by comparing the Fourier spectrum of various signal laser power, several peaks gradually emerge and rise at kHz level, more specifically, 1 kHz at 1.552 kW and 4 kHz at 2.09 kW. The rising tendency of 1-kHz peak is obvious and occurs earlier, and it corresponds to the beginning of deformation of beam spot. It should be mentioned that at several-hundred-Hz level there are obvious spectrum peaks, and we make comparison of this low-frequency signal which is shown in Fig. 9. The Fourier spectrum reveals that this signal exists in seed, pump and generated Raman laser simultaneously at different power levels, and it can be inferred that this part of signal is the common noise signal caused by vibration of water-cooling and other devices rather than the Raman laser. In the nine-box sampling apertures the changing tendencies of Fourier spectrum are consistent, which have same position of spectrum peaks but relatively different amplitudes.
Fig. 9. Fourier spectra within 1.2 kHz of beam spot dynamics at various output power levels, including seed laser, amplified signal laser at 226 W and 1551.6 W, and the corresponding required launched pump laser.
4. Discussions
In the deformation process of beam spot, the interconversion of modes is unobvious since the mode number is far more than that in previous studies with low order modes. Moreover, the fluctuation of beam spot observed by naked eye is similar to that caused by thermal distortion of lens which has relatively slow change. In order to remove the influence of thermal effects of possible devices, several implements are employed. On one hand, the length-width ratio of end cap is specially designed to adapt to the NA of GRIN fiber, in this way all laser transmits from the end surface of end cap. This end cap spliced with a small piece of GRIN fiber (40 cm) is employed to measure the high power pump laser (up to 3.5 kW) individually for comparison. On the other hand, the temperatures of end cap as well as lens employed in the beam splitting system are monitored. By utilizing the cooling action, the temperatures of GRIN fiber, end cap and the collimating mirror keep below 40 ℃ in the amplifying process. Moreover, the beam splitting system is installed when the pure pump laser is measured at high power level. In the above contrast experiments, no fluctuation of beam spot is observed, therefore the thermal influence of end cap and beam splitting system is excluded.
To eliminate the influence of the specificity of this GRIN fiber such as manufacturing flaws, we also establish RFAs based on additional GRIN fibers with various core/cladding diameters from other manufacturers, which reach around 2 kW. The details will not be described here, however in all these experiments the distortion of beam shape is observed, and kHz-frequency peaks in the Fourier spectrum of beam spot dynamics are obtained. Meanwhile, the threshold of fluctuation exists in all experiments, yet it seems that the core size has little influence on it. The possible reason is that the core size of GRIN fibers are far larger than that in previous amplifier with MI, which all have rather large mode numbers and similar sophisticated mode intensity distribution, hence the fluctuation will not be sensitive to this change. From the experimental results it can be summarized that in several-kilowatt high power CW RFL based on GRIN fiber, the distortion of beam spot occurs with characteristic frequency at kHz-level, which is related to the possible nonlinear effects in the GRIN fiber, i.e. thermal lens effect, MI or the thermal-induced gain competition and coupling of modes [51–55]. It is a sort of instability of modes in RFA, however, there are different features compared with the common MI in RE-doped fiber laser systems, which still deserves more theoretical and experimental study.
5. Conclusion
In summary, we report the power scaling of high power CW RFA. The gain scheme is based upon SRS in MM GRIN fiber, which is totally irrelevant with RE gain regime, while offers power boosting and improvement of beam quality simultaneously. When 3.34 kW pump laser is injected, the output power is 2.087 kW at 1130 nm with conversion efficiency of 90.1% (the output signal power versus the depleted pump power). To the best of our knowledge, this is the highest power in the fields of Raman fiber lasers based merely on Stokes radiation. The beam quality parameter M2 improves from 15 to 8.9 during the power boosting process, then the spot distortion at beam waist with fluctuation of modes appears at high power level. The characteristic parameter of the distortion dynamics is analyzed which shows obvious properties, including clear threshold property, characteristic frequency in Fourier spectrum of fluctuation and rising trend of normalized standard deviation similar to that in previous amplifier with MI. This is the first observation of erratic spot dynamics in 2 kW-level CW RFA based on GRIN fiber. Additional experiments and numerical analysis should be carried on in future research to fully understand and inhibit this effect.
Funding
Huo Yingdong Education Foundation (151062); Hunan Provincial Innovation Foundation for Postgraduate (2019RS3017); National Natural Science Foundation of China (11704409, 61605246, 61911530134).
Acknowledgement
Yizhu Chen and Tianfu Yao contributed equally to this work.
Disclosures
The authors declare no conflicts of interest.
References
1. D. J. Richardson, J. Nilsson, and W. A. Clarkson, “High power fiber lasers: current status and future perspectives [Invited],” J. Opt. Soc. Am. B 27(11), B63–B92 (2010). [CrossRef]
2. H. Otto, C. Jauregui, J. Limpert, and A. Tünnermann, “Average power limit of fiber-laser systems with nearly diffraction-limited beam quality,” Proc. SPIE 9728, 97280E (2016). [CrossRef]
3. P. Zhou, H. Xiao, J. Leng, J. Xu, Z. Chen, H. Zhang, and Z. Liu, “High-power fiber lasers based on tandem pumping,” J. Opt. Soc. Am. B 34(3), A29 (2017). [CrossRef]
4. W. Shi, Q. Fang, X. Zhu, R. A. Norwood, and N. Peyghambarian, “Fiber lasers and their applications [Invited],” Appl. Opt. 53(28), 6554–6568 (2014). [CrossRef]
5. V. R. Supradeepa, Y. Feng, and J. Nicholson, “Raman fiber lasers,” J. Opt. 19(2), 023001 (2017). [CrossRef]
6. Y. Feng, Raman Fiber Lasers (Springer, 2017).
7. G. P. Agrawal, Nonlinear Fiber Optics (4 Edition) ((Springer, 2006).
8. Y. Glick, Y. Shamir, Y. Sintov, S. Goldring, and S. Pearl, “Brightness enhancement with Raman fiber lasers and amplifiers using multi-mode or multi-clad fibers,” Opt. Fiber Technol. 52, 101955 (2019). [CrossRef]
9. S. A. Babin, “High-brightness all-fiber Raman lasers directly pumped by multimode laser diodes,” High Power Laser Sci. Eng. 7(1), e15 (2019). [CrossRef]
10. Z. Wang, H. Wu, M. Fan, Y. Rao, X. Jia, and W. Zhang, “Third-order random lasing via Raman gain and Rayleigh feedback within a half-open cavity,” Opt. Express 21(17), 20090–20095 (2013). [CrossRef]
11. M. Bernier, V. Fortin, M. El-Amraoui, Y. Messaddeq, and R. Vallée, “3.77 µm fiber laser based on cascaded Raman gain in a chalcogenide glass fiber,” Opt. Lett. 39(7), 2052–2055 (2014). [CrossRef]
12. J. Xu, Z. Lou, J. Ye, J. Wu, J. Leng, H. Xiao, H. Zhang, and P. Zhou, “Incoherently pumped high-power linearlypolarized single-mode random fiber laser: Experimental investigations and theoretical prospects,” Opt. Express 25(5), 5609–5617 (2017). [CrossRef]
13. Z. Wang, H. Wu, M. Fan, L. Zhang, Y. Rao, W. Zhang, and X. Jia, “High Power Random Fiber Laser With Short Cavity Length: Theoretical and Experimental Investigations,” IEEE J. Sel. Top. Quantum Electron. 21(1), 10–15 (2015). [CrossRef]
14. L. Zhang, J. Dong, and Y. Feng, “High-Power and High-Order Random Raman Fiber Lasers,” IEEE J. Sel. Top. Quantum Electron. 24(3), 1–6 (2018). [CrossRef]
15. V. Balaswamy, S. Aparanji, S. Arun, S. Ramachandran, and V. R. Supradeepa, “High-power, widely wavelength tunable, grating-free Raman fiber laser based on filtered feedback,” Opt. Lett. 44(2), 279–282 (2019). [CrossRef]
16. T. Yao and J. Nilsson, “835 nm fiber Raman laser pulse pumped by a multimode laser diode at 806 nm,” J. Opt. Soc. Am. B 31(4), 882–888 (2014). [CrossRef]
17. H. J. Hoffman, J. J. Koponen, M. J. Söderlund, and S. K. T. Tammela, “Measuring photodarkening from single-mode ytterbium doped silica fibers,” Opt. Express 14(24), 11539 (2006). [CrossRef]
18. A. S. Kurkov, “Oscillation spectral range of Yb-doped fiber lasers,” Laser Phys. Lett. 4(2), 93–102 (2007). [CrossRef]
19. Y. Feng, L. R. Taylor, and D. B. Calia, “150 W highly-efficient Raman fiber laser,” Opt. Express 17(26), 23678–23683 (2009). [CrossRef]
20. C. A. Codemard, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580(6), 75801N (2010). [CrossRef]
21. V. R. Supradeepa and J. W. Nicholson, “Power scaling of high-efficiency 1.5 µm cascaded Raman fiber lasers,” Opt. Lett. 38(14), 2538–2541 (2013). [CrossRef]
22. Y. Shamir, Y. Glick, M. Aviel, A. Attias, and S. Pearl, “250 W clad pumped Raman all-fiber laser with brightness enhancement,” Opt. Lett. 43(4), 711 (2018). [CrossRef]
23. Y. Chen, J. Leng, H. Xiao, T. Yao, and P. Zhou, “High-efficiency all-fiber Raman fiber amplifier with record output power,” Laser Phys. Lett. 15(8), 085104 (2018). [CrossRef]
24. Y. Glick, Y. Shamir, A. A. Wolf, A. V. Dostovalov, S. A. Babin, and S. Pearl, “Highly efficient all-fiber continuous-wave Raman graded-index fiber laser pumped by a fiber laser,” Opt. Lett. 43(5), 1027–1030 (2018). [CrossRef]
25. Y. Glick, Y. Shamir, M. Aviel, Y. Sintov, S. Goldring, N. Shafir, and S. Pearl, “1.2 kW clad pumped Raman all-passive-fiber laser with brightness enhancement,” Opt. Lett. 43(19), 4755–4758 (2018). [CrossRef]
26. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the Conversion to the First Stokes in Cladding-Pumped Fiber Raman Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009). [CrossRef]
27. A. Polley and S. E. Ralph, “Raman Amplification in Multimode Fiber,” IEEE Photonics Technol. Lett. 19(4), 218–220 (2007). [CrossRef]
28. N. B. Terry, T. G. Alley, and T. H. Russell, “An explanation of SRS beam cleanup in graded-index fibers and the absence of SRS beam cleanup in step-index fibers,” Opt. Express 15(26), 17509–17519 (2007). [CrossRef]
29. S. I. Kablukov, E. A. Zlobina, M. I. Skvortsov, I. N. Nemov, A. A. Wolf, A. V. Dostovalov, and S. A. Babin, “Mode selection in a directly diode-pumped Raman fibre laser using FBGs in a graded-index multimode fibre,” Quantum Electron. 46(12), 1106–1109 (2016). [CrossRef]
30. Y. Glick, V. Fromzel, J. Zhang, N. Ter-Gabrielyan, and M. Dubinskii, “High-efficiency, 154 W CW, diode-pumped Raman fiber laser with brightness enhancement,” Appl. Opt. 56(3), B97 (2017). [CrossRef]
31. E. Zlobina, S. Kablukov, I. Nemov, A. Wolf, A. V. Dostovalov, V. Tyrtyshnyy, D. Myasnikov, and S. Babin, “High-efficiency LD-pumped all-fiber Raman laser based on a 100 µm core graded-index fiber,” Laser Phys. Lett. 15(9), 095101 (2018). [CrossRef]
32. Y. Chen, J. Leng, H. Xiao, T. Yao, and P. Zhou, “Pure Passive Fiber Enabled Highly Efficient Raman Fiber Amplifier With Record Kilowatt Power,” IEEE Access 7, 28334–28339 (2019). [CrossRef]
33. Y. Chen, T. Yao, H. Xiao, J. Leng, and P. Zhou, “1.6 kW Raman fiber amplifier based on multimode graded index fiber,” in ICOCN’2019 (Huangshan, China, 2019), paper W1D.2
34. S. H. Baek and W. B. Roh, “Single-mode Raman fiber laser based on a multimode fiber,” Opt. Lett. 29(2), 153–155 (2004). [CrossRef]
35. S. Longhi, “Modulational instability and space time dynamics in nonlinear parabolic-index optical fibers,” Opt. Lett. 28(23), 2363–2365 (2003). [CrossRef]
36. F. Poletti and P. Horak, “Description of ultrashort pulse propagation in multimode optical fibers,” J. Opt. Soc. Am. B 25(10), 1645–1654 (2008). [CrossRef]
37. W. H. Renninger and F. W. Wise, “Optical solitons in graded-index multimode fibres,” Nat. Commun. 4(1), 1719 (2013). [CrossRef]
38. Z. Liu, L. G. Wright, D. N. Christodoulides, and F. W. Wise, “Kerr self-cleaning of femtosecond-pulsed beams in graded-index multimode fiber,” Opt. Lett. 41(16), 3675–3678 (2016). [CrossRef]
39. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012). [CrossRef]
40. A. V. Smith and J. J. Smith, “Overview of a Steady-Periodic Model of Modal Instability in Fiber Amplifiers,” IEEE J. Sel. Top. Quantum Electron. 20(5), 472–483 (2014). [CrossRef]
41. A. S. Ahsan and G. P. Agrawal, “Graded-index solitons in multimode fibers,” Opt. Lett. 43(14), 3345–3348 (2018). [CrossRef]
42. H. Zhou, Z. Chen, X. Zhou, J. Hou, and J. Chen, “All-fiber 7 × 1 signal combiner with high beam quality for high-power fiber lasers,” Chin. Opt. Lett. 13(6), 061406 (2015). [CrossRef]
43. S. A. Babin, D. V. Churkin, A. E. Ismagulov, S. I. Kablukov, and E. V. Podivilov, “Four-wave-mixing-induced turbulent spectral broadening in a long Raman fiber laser,” J. Opt. Soc. Am. B 24(8), 1729–1738 (2007). [CrossRef]
44. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental Observations of the Threshold-Like Onset of Mode Instabilities in High Power Fiber Amplifiers,” Opt. Express 19(14), 13218–13224 (2011). [CrossRef]
45. H. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20(14), 15710–15722 (2012). [CrossRef]
46. L. Huang, T. Yao, J. Leng, S. Guo, R. Tao, P. Zhou, and X. A. Cheng, “Mode instability dynamics in high-power low-numerical-aperture step-index fiber amplifier,” Appl. Opt. 56(19), 5412–5417 (2017). [CrossRef]
47. M. N. Zervas, “Transverse mode instability analysis in fiber amplifiers,” Proc. SPIE 10083, 100830M (2017). [CrossRef]
48. K. Hejaz, M. Shayganmanesh, R. Rezaei-Nasirabad, A. Roohforouz, S. Azizi, A. Abedinajafi, and V. Vatani, “Modal instability induced by stimulated Raman scattering in high-power Yb-doped fiber amplifiers,” Opt. Lett. 42(24), 5274–5277 (2017). [CrossRef]
49. Z. Li, Z. Huang, X. Xiang, X. Liang, H. Lin, S. Xu, Z. Yang, J. Wang, and F. Jing, “Experimental demonstration of transverse mode instability enhancement by a counter-pumped scheme in a 2 kW all-fiberized laser,” Photonics Res. 5(2), 77–81 (2017). [CrossRef]
50. R. Tao, H. Xiao, H. Zhang, J. Leng, X. Wang, P. Zhou, and X. Xu, “Dynamic characteristics of stimulated Raman scattering in high power fiber amplifiers in the presence of mode instabilities,” Opt. Express 26(19), 25098–25110 (2018). [CrossRef]
51. D. Brown and H. Hoffman, “Thermal, stress, and thermo-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37(2), 207–217 (2001). [CrossRef]
52. M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007). [CrossRef]
53. A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19(12), 11318–11329 (2011). [CrossRef]
54. W. Liu, J. Cao, and J. Chen, “Study on the adiabaticity criterion of the thermally-guided very-large-mode-area fiber,” Opt. Express 26(7), 7852–7865 (2018). [CrossRef]
55. W. Liu, J. Cao, and J. Chen, “Study on thermal-lens induced mode coupling in step-index large mode area fiber lasers,” Opt. Express 27(6), 9164–9177 (2019). [CrossRef]
