Abstract

In this paper, the spectral evolution properties of different transverse modes with the stimulated Raman scattering (SRS) effect are analyzed in large-mode-area (LMA) fiber amplifiers for the first time. Both the ratios of laser power in Raman Stokes light and high order modes (HOMs) can be calculated through the comprehensive analysis of transverse mode competition and nonlinear transverse mode coupling processes. The theoretical study reveals that SRS-induced inter-modal wave mixing (IM-WM) effect would transfer power from signal light in LP01 mode to Raman Stokes light in LP11 mode and lead to the onset of the mode distortion phenomenon in high-power LMA fiber amplifiers. Different from the traditional thermal-induced mode instability (MI) phenomenon, the SRS-induced mode distortion could occur by just with the contribution of quantum noise.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Fiber lasers are promising candidates for industrial applications due to their essential properties such as high conversion efficiency, excellent beam quality, convenient heat management and compact configuration [1–3]. For the high intensity and relative long length, nonlinear effects have a significant impact on brightness scaling properties of fiber lasers, one of which is the stimulated Raman scattering (SRS) [4, 5]. The SRS effect results in power transfer to longer wavelength and restricts power scaling of fiber laser systems. In order to suppress SRS effect, the large-mode area (LMA) fibers are commonly applied in high-power fiber laser systems [6]. The LMA fibers support a few transverse modes, thus the output beam profile may exhibit modal instability (MI) phenomenon after a certain threshold has been reached in high-power LMA fiber amplifiers [7]. The MI phenomenon degrades the beam quality and sets the practical upper brightness limit for fiber laser systems [8]. In the past decade, there are a lot of theoretical and experimental reports on the mechanism and mitigation strategies of MI phenomenon [9–18]. In theory, it is proposed that the thermal-induced refractive index change could lead to strong transverse mode coupling between fundamental mode (FM) and higher order modes (HOMs) (dominantly LP11 mode) in the situation that a portion of HOMs is injected into the amplification [9–12]. In practical experiment studies, the threshold of MI phenomenon could be increased based on the strategies such as weaken the mode coupling between the FM and HOMs [13–15], increase the relative loss of HOMs [16, 17], decrease the quantum defeat and optimizing the seed power and pump manners [3, 18].

In recent experimental report, K. Hejaz et al. has found a new type of MI phenomenon in which rapid output beam profile deterioration occurs after the onset of SRS effect in high-power fiber amplifiers [19]. Nevertheless, no further theoretical explanation is given and the physical origin of this phenomenon is unclear. In the previous studies of SRS effect in high-power fiber lasers, the impact of transverse mode properties are ignored [20–23], since the SRS effect and mode coupling process seem to be irrelevant to each other. Thus, it is worthwhile to investigate on the mechanism and properties of SRS-induced mode distortion phenomenon in high-power LMA fiber amplifiers, which helps to establish comprehensive mitigation strategies for the two effects and provide design guidelines for further brightness scaling in high-power LMA fiber amplifiers.

The purpose of this paper is to establish a model for describing the spectral evolution properties of different transverse modes and illustrate the mechanism behind SRS-induced mode distortion phenomenon. The model is based on the comprehensive analysis of the rate equations and nonlinear propagation equations in LMA fiber amplifiers. Based on the spectral evolution model, the mechanism and properties of the SRS-induced mode coupling process are demonstrated.

2. Spectral evolution model for a LMA fiber amplifier

Apart from the thermal-induced refractive index change, there are four possible factors which may lead to the generation of HOMs in high-power LMA fiber amplifiers. The first one is the intrinsic spontaneous emission noise, which provides the initial production of HOMs in fiber amplifiers. The next two factors are the direct amplification of HOMs pumped by the signal light in FM mode through SRS effect, and the inter-modal four-wave-mixing (IM-FWM) process among different transverse modes. Those two factors could be analyzed through the coupled amplitude equations or the nonlinear propagation equation [24]. The last factor is the active gain of HOMs from the doped ions. The transverse mode competition properties in LMA fiber amplifiers could be described through the rate equations [25].

The four factors are all wavelength-dependent processes in high-power LMA fiber amplifiers, thus it is appropriate to study the influence of the four factors on the mode coupling processes in the frequency domain through a spectral model. The spectral model with SRS effect has been established by our group for a single-transverse mode (SM) fiber amplifier [26], in which the spectral evolution process could be theoretically described through the combined simulation of the rate equations and the nonlinear propagation equations. When it comes to multi-transverse mode case, the nonlinear propagation process is generally described through multimode generalized nonlinear Schrödinger equation (MM-GNLSE) [27, 28]. As for the individual analysis of the four factors, the theoretical description is separately established, while the interaction among these four aspects has not been conducted and it is complex to encompass the four aspects in a unified investigation.

The set of unidirectional spectral-spatial equation describing the power amplification and spectral evolution processes of a transverse mode is given by the combination of four terms:

Au(z,ω)z=Du(z,ω)+Nu(z,ω)+Gu(z,ω)+fu(z,ω),
whereDu(z,ω)and Nu(z,ω)refer to the dispersion and nonlinear terms in MM-GNLSE, Gu(z,ω) andfu(z,ω) stand for the net gain term and spontaneous emission noise in the fiber amplifier, respectively.

When mode distortion phenomenon occurs in a fiber amplifier, the LP11 mode is the dominant component in HOMs. Without loss of generality, we focus the spectral evolution of two scalar-modes, LP01 and LP11 modes and the interactions between different degenerate vector-modes are ignored in the following analysis. Then, the dispersion and nonlinear terms could be simplified from the MM-GNLSE:

Du(z,ω)=i[(β0(u)β0(01))+(β1(u)β1(01))ω+n2βn(u)n!ωn]Au(z,ω)
Nu(z,t)=iγ(1+ωω0)F{QuuAu(z,t)R(t)|Au(z,t)|2+QuvAu(z,t)R(t)|Av(z,t)|2+QuvAv(z,t)R(t)[Av(z,t)Au(z,t)]},
where, the scripts (u) and (v) stand for the different modes among LP01 and LP11 modes, and superscript (01) denotes for the LP01 mode;βnstands for the nth order derivative of the propagation constant with respect to the angular frequency;γis the nonlinear Kerr coefficient; ωis the reference angular frequency of the spectrum andω0is the carrier angular frequency of the signal; the nonlinear response functionR(t)=(1fR)δ(t)+fRh(t), fRis the fractional contribution of the Raman response to the total nonlinearity and h(t) is the delayed Raman response function; A(z,ω)(A(z,t))is the complex amplitude of signal in the frequency (time) domain; F{}denotes the Fourier transform and ‘⨂’ denotes the convolution operation; Qis the normalized overlap integral, and we assume thatQvvQuu=1.

The three terms in the brace of Eq. (3) correspond to the contribution of the mode u, mode v and the interaction between the two modes to the nonlinearity. Each term in the brace of Eq. (3) contains the instantaneous intensity-dependent phase modulation and the delayed Raman response. Specifically, the first term corresponds to the self-phase modulation (SPM) effect and Raman amplification pumped by the signal light in the same mode. The second term corresponds to the cross-phase modulation (XPM) effect and Raman amplification pumped by the signal light in the other mode. The third term corresponds to the XPM effect and SRS-induced inter-modal wave mixing (IM-WM) effect.

To obtain the net gain term, the gain competition between different transverse modes should be considered. Generally, the transverse mode competition process could be described in detail through incorporating the transverse distribution of the light-intensity in rate equations [29]. In LMA fiber amplifiers, a significant issue is the brightness scaling properties of the high power fiber system. Thus, the influence of HOMs is mainly concentrated on the characteristics of LP11 mode near threshold value. In this case, the traditional overlap factors of individual modes could be approximately used to analyze the superposed intensity profile. A point should be noted is that inter-phase-related coherent superstation of spatial modes should be carefully considered when multiple HOMs are involved. In most cases, the doping area in LMA active fiber is uniform and symmetric. Thus, it is simpler and more efficient to analyze this process through different power overlap factors between the transverse modes and doped area in this two-mode case. Then, the net gain term can be calculated through the following rate equations:

Gu(z,ω)=12[Γu(ω)(σas(ω)+σes(ω))N2(z)σa(ω)N0αu]Au(z,ω)
dPp(z)dz=Γp{σa(ωp)N0(σa(ωp)+σe(ωp))N2}Pp(z)αpPp(z)
N2N0=ΓpωpAσa(ωp)Pp+12πTmAσa(ω˜)ω˜(Γu|Au(z,ω)|2+Γv|Av(z,ω)|2)dωΓpωpA(σa(ωp)+σe(ωp))Pp+1τ+12πTmAσa(ω˜)+σe(ω˜)ω˜(Γu|Au(z,ω)|2+Γv|Av(z,ω)|2)dω,
where, indexpstands for pump wave; Γis the power overlap factor between the transverse modes and doped area; σaand σeare the corresponding absorption and emission cross sections at different angular frequency; N0is the ytterbium dopant concentration and N2is the total number of Yb-ions in excited state; αis the loss coefficient. τis the life of the excited state population. Tmis the time window during the calculation; is the Planck’s constant; A is the doped cross-section area; ω˜is the actual angular frequency of the signal.

Based on classical electromagnetic theory, the intensity of spontaneous emission noise satisfies the Gaussian stochastic process with zero mean value, and its variance is proportional to the gain coefficient [30]. As the initial spontaneous emission noise mainly originates from the doped ions, the corresponding gain coefficient for spontaneous emission noise is G(z,ω) and the spontaneous emission noise term could be expressed as:

{fu(z,ω)fu(z,ω)=2DFF(z,ω)δ(zz)δ(ωω)fu(z,ω)=0DFF(z,ω)=(ω+ω0)3πc2G(z,ω)nsp,
here, nsp=1/(exp((ω+ω0)/kBT)1)represents the average mode occupation number in equilibrium; kBis the Boltzmann constant; T is the environmental temperature.

3. Properties of SRS-induced mode coupling

To simulate the spectral evolution properties of different transverse modes through the above model, we need to obtain the spectral property of the seed. The spectral property of laser in LP01 mode in the seed could be calculated through a SM spectral model, which has been established by our group through the combined simulation of the rate equations and nonlinear propagation equations with the boundary conditions [21].

Figure 1 illustrates a typical simulated spectrum for a SM fiber oscillator. The output power is about 50 W and the 3 dB spectral width is about 0.3 nm. As shown in Fig. 1, the intensity of laser in Raman Stokes light is below −90 dB compared to the signal light, thus the power of Raman Stokes light is negligible in the seed. The signal light and the Raman Stokes light correspond to the laser around 1070 nm and 1120 nm here and in the following analysis, respectively. In most cases, the inserted seed source is operated in LP01 mode state to ensure high beam quality in the following amplification. To compare with traditional thermal-induced MI and emphasize the impact of the SRS effect, we assume that there is no injected laser in LP11 mode, and the initial production of laser in LP11 mode just originates from the spontaneous emission noise in the fiber amplifiers.

 figure: Fig. 1

Fig. 1 The spectrum of the seed source.

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Based on the model in the above section, we simulate the spectral evolution properties of different transverse modes in a typical high-power fiber amplifier with the commercial double-clad Yb-doped fiber. The core diameter is 20 μm (NA = 0.065) and the inner cladding diameter is 400 μm. This fiber supports two scalar-modes, LP01 and LP11 modes. The calculated dispersion parameters and overlap factors for the two modes in this fiber are show in Table 1, and the other major simulation parameters are shown in Table 2. The power overlap factor for the LP11 mode is normalized by the LP01 mode in Table 1 and the iterative solution of the spectral-spatial equation is fulfilled by using the fourth-order Runge-Kutta method in our simulation. For simplicity, we assume that 95% of the pump power is coupled into the active fiber in all the simulations.

Tables Icon

Table 1. Dispersion parameters and power overlap factors for LP01 and LP11 modes

Tables Icon

Table 2. Major simulation parameters for the fiber amplifier

Figure 2 illustrates the power distribution along the fiber amplifier when the pump power is increased to be 2 kW. The four legends denote pump power, total laser power and the percentages of laser power in Raman Stokes light and LP11 mode, respectively. Here, the percentage of Raman Stokes light is calculated through dividing integrated spectrum from 1100 to 1150 nm by integrated spectrum from 1050 to 1150 nm. As shown in Fig. 2, the corresponding output percentages of Raman Stokes light and LP11 are about 4% (68 W) and 1.2% (21 W) at the fiber end, respectively. Thus, significant SRS effect and mode coupling phenomenon occur here at the pump power of 2 kW. For the curves of pump power and total laser power, after the fiber length of 10 m, most of the pump power is absorbed, and the total laser power nearly keeps unchanged, about 1.7 kW at the fiber end. For the curves of the two percentages, the percentage of Raman Stokes light exceeds 0.1% and increases quickly after the fiber length of 13.5 m. At the fiber length of 17 m, the percentage of LP11 mode begins to exceed 0.1% when the percentage of Raman Stokes light is about 1.3%. Thus, the occurrence of the mode coupling process lags behind the SRS effect in the fiber amplifier.

 figure: Fig. 2

Fig. 2 The power distribution along the fiber amplifier.

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Figures 3(a)-3(c) illustrate the normalized output spectra and corresponding temporal properties of lasers in the two modes. As shown in Fig. 3(a), the simulated spectra of laser in LP01 mode and LP11 mode are quite different from each other. The spectrum of laser in LP01 mode is asymmetrical with a mount of the Raman Stokes light, and the corresponding percentage of the Raman Stokes light in LP01 mode is about 2.8% (48 W). Nevertheless, the dominant spectral component of laser in LP11 mode is the Raman Stokes light, and the element around 1070 nm is quite small. Besides, the corresponding percentage of Raman Stokes light in LP11 mode is about 1.2% (20.8 W). Based on the simulation results, we could infer that SRS effect leads to the mode coupling from the signal light in LP01 mode to the Raman Stokes light in LP11 mode.

 figure: Fig. 3

Fig. 3 The normalized output spectra and corresponding temporal properties of lasers in the two modes: (a) the optical spectra for lasers in the two modes; (b) the temporal evolution for laser in LP11 mode; (c) the ACFs for lasers in the two modes.

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As shown in Fig. 3(b), due to the randomness of spontaneous emission noise, the temporal property of the generated LP11 mode is intense pulsed light in nanosecond scale with varied pulse separation and peak power. The generated laser power in LP11 mode would fluctuate quickly in nanosecond scale and the peak power is over 400 times than the average power. Accordingly, the mode coupling process from the signal light in LP01 mode to the Raman Stokes light in LP11 mode is dynamically changing. To compare the temporal properties of lasers in the two modes, we also simulate the normalized intensity auto-correlation function (ACF) for the temporal evolution of lasers in the two modes. As shown in Fig. 3(c), the ACFs are quite different from each other. Specifically, the background level of the ACF for laser in LP01 mode tends to be about 0.68 as it should be for the partially coherent quasi-continuous wave radiation. The background level of the ACF for laser in LP11 tends to be about zero as the laser emission in LP11 mode is intense pulsed light.

To further demonstrate properties of this mode coupling process, we analyze the spectral evolution properties of laser in the two modes. Figures 4(a) and 4(b) illustrate the normalized spectral intensity for laser in the two modes along the fiber amplifier. As shown in Fig. 4(a), the normalized spectra of laser in LP01 mode is asymmetrical along the fiber amplifier, which could be explained by the wavelength-dependent gain of the doped ions. After the fiber length of 5.3 m, the component of Raman Stokes light begins to arise (in the region A). As shown in Fig. 4(b), the component of laser in LP11 begins to arise after the fiber length of 7.8 m (in the region B) and the dominant component of laser in LP11 mode is always the Raman Stokes light. Comparing Fig. 4(b) with Fig. 4(a), it is clear that the occurrence of the mode coupling process lags behind the SRS effect and the SRS-induced mode coupling leads to the generation and amplification of Raman Stokes light in LP11 mode.

 figure: Fig. 4

Fig. 4 The spectral evolution of laser in the two modes along the fiber amplifier: (a) spectra of laser in LP01 mode; (b) spectra of laser in LP11 mode.

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We might also notice that the intensity of laser in LP11 mode is below −100 dB (in the region C), while the maximum output intensity is over −25 dB at the fiber end. Thus, the SRS-induced mode distortion is free from the initial injection of laser in LP11 mode and could occur by just with the contribution of quantum noise. As for thermal-induced MI, in a practical system, due to that the level of quantum noise and thermal Rayleigh scattering noise is much lower than the intensity noise, so the origin of HOM is mainly attributed to the intensity noise [31, 32]. Besides, technical insufficient (for example forward fused point in the main amplifier) is also a significant source to incorporate HOM. In typically theoretical analysis of the TMI phenomenon, normally the portion of the HOM mode is generally set to originate from the inserted initial weak HOM seed [9–11]. Thus, the origin of the SRS-induced mode distortion is different from that of the traditional thermal-induced MI.

In the above analysis, the laser power in LP11 mode originates from the weak spontaneous emission noise and gets amplified along the fiber amplifier. There are several terms in Eq. (1) which might lead to the amplification of the Raman Stokes light in LP11 mode, i.e. the active gain termG11, Raman amplification term pumped by signal light in LP01 modeA11h(τ)|A01|2dτ and the SRS-induced IM-WM termA01h(τ)A11A01*Adτ. To achieve the dominant term on the mode coupling process, we make contrast simulations by neglecting one of the three terms in Eq. (1), respectively.

Figures 5(a)-5(c) illustrate the power distributions along the fiber amplifier when neglecting one of the three terms, respectively. As shown in Figs. 5(a)-5(c), the power distributions of the pump power and total laser power keep nearly unchanged in all the three cases compared to the original simulation (shown in Fig. 2). Besides, the percentages of Raman Stokes light are 3.5%, 4.1% and 2.9% at the fiber end, respectively.

 figure: Fig. 5

Fig. 5 The power distributions along the fiber amplifier for three cases: (a) neglecting active gain term; (b) neglecting Raman amplification term; (c) neglecting SRS-induced IM-WM term.

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As shown in Fig. 5(a), when the active gain is neglected in the simulation, the percentage of laser power in LP11 mode at the fiber end is about 0.6%, which is only half of that in original case in Fig. 2, thus the active gain term would contribute the amplification of Raman Stokes light in LP11 mode. As shown in Fig. 5(b), when the Raman amplification pumped by signal light in LP01 mode is neglected in the simulation, the percentage of laser power in LP11 mode at the fiber end is about 1.2%, which is near identical to original case in Fig. 2, thus this term would not impact the amplification of Raman Stokes light in LP11 mode. As shown in Fig. 5(c), when the SRS-induced IM-WM effect is neglected in the simulation, the percentage of laser power in LP11 mode at the fiber end is less than 0.01%, thus this term also contributes the amplification of Raman Stokes light in LP11 mode. In summary, both the active gain and the SRS-induced IM-WM effect contribute to amplification of Raman Stokes light in LP11 mode. Specifically, it is the SRS-induced IM-WM term which dominates in the mode coupling process from the signal light in LP01 mode to Raman Stokes light in LP11 mode.

As shown in Fig. 4(b), the signal power in LP11 mode is negligible. In addition, even when the Raman stokes light in LP01 mode is eliminated in the simulation, the mode coupling process still happens. Thus, the signal power in LP11 mode and the Raman stokes light in LP01 mode are nearly not participated in the SRS-induced IM-WM process. And direct mode transformation occurs here from the signal light in LP01 mode to the Raman Stokes light in LP11 mode. Accordingly, the SRS-induced IM-WM phenomenon is different from the typical IM-FWM or IM-SRS cases [24, 33, 34]. In the previous studies of the IM-FWM or IM-SRS effects, four waves in different frequencies will be participated and phase-matching condition is required. In fact, this SRS-induced IM-WM process could effectively occur as a result of the participation of the active gain [35, 36].

Another important feature for the SRS-induced mode coupling is the mode distortion threshold of the fiber amplifier. To describe the mode distortion threshold in this case, we simulate the output power properties at different pump powers. Figure 6(a) illustrates the output signal power in LP01 mode verse pump power, and the linear dash-dotted line is the trend line of the ideal output signal power. As shown in Fig. 6(a), a decrease in the slope efficiency takes place beyond the pump power of 1.8 kW, and this point could be corresponded to the threshold of SRS effect. Figure 6(b) illustrates the ratios of laser power in LP11 mode and Raman Stokes light verse the pump power on logarithmic coordinates. As shown in Fig. 6(b), both the ratios of laser power in LP11 mode and Raman Stokes light increase along with the pump power. The difference between them decreases from about −9 dB to −6 dB beyond the pump power of 1.8 kW, i.e. the laser power in LP11 mode keeps about 25% of that in Raman Stokes light at high-power cases. By defining the SRS threshold as the output signal power when the ratio of laser power in Raman stokes light is over 2% (−17 dB), the SRS threshold is about 1.5 kW at the pump power of 1.8 kW. Applying similar definition for the mode distortion threshold, the mode distortion threshold is about 1.8 kW at the pump power of 2.3 kW. The threshold discrepancy of the two effects above is compatible with the results shown in Fig. 2 and Fig. 4.

 figure: Fig. 6

Fig. 6 The properties of the output powers verse the pump power: (a) the power slope of signal light in LP01 mode; (b) the ratios of laser power in LP11 mode and Raman Stokes light.

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4. Conclusions

In this work, the spectral evolution properties for different transverse modes are established through including the transverse mode competition and nonlinear transverse mode coupling processes in high-power fiber amplifiers. The theoretical analysis shows that there exists the mode coupling process from LP01 mode to LP11 mode after the onset of SRS effect, and the occurrence of the mode distortion phenomenon lags behind the SRS effect. Different from the traditional thermal-induced MI phenomenon, the SRS-induced mode coupling process could occur by just with the contribution of quantum noise. The contrast analysis reveals that the physical mechanism behind the mode distortion phenomenon is the SRS-induced IM-WM effect.

Funding

Foundation for the Author of National Excellent Doctoral Dissertation of the People’s Republic of China (201329); Huoying Dong Education Foundation of China; National Natural Science Foundation of China (61705264).

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33. S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24(26), 30338–30349 (2016). [CrossRef]   [PubMed]  

34. M. Ziemienczuk, A. M. Walser, A. Abdolvand, and P. St. J. Russell, “Intermodal stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” J. Opt. Soc. Am. B 29(7), 1563–1568 (2012). [CrossRef]  

35. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15(2), 577–582 (2007). [CrossRef]   [PubMed]  

36. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012).

References

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  2. M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0904123S (2014).
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  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–A36 (2017).
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  4. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
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  5. R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by Stimulated Raman and Brillouin scattering,” Appl. Opt. 11(11), 2489–2494 (1972).
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  6. Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004).
    [Crossref] [PubMed]
  7. C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
    [Crossref]
  8. M. N. Zervas, “Power scaling limits in high power fiber amplifiers due to transverse mode instability, thermal lensing, and fiber mechanical reliability,” Proc. SPIE 10512, 1051205 (2018).
  9. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
    [Crossref] [PubMed]
  10. 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] [PubMed]
  11. C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012).
    [Crossref] [PubMed]
  12. K. R. Hansen and J. Lægsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22(9), 11267–11278 (2014).
    [Crossref] [PubMed]
  13. T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
    [Crossref]
  14. L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
    [Crossref]
  15. C. Robin, I. Dajani, and B. Pulford, “Modal instability-suppressing, single-frequency photonic crystal fiber amplifier with 811 W output power,” Opt. Lett. 39(3), 666–669 (2014).
    [Crossref] [PubMed]
  16. 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] [PubMed]
  17. F. Beier, C. Hupel, S. Kuhn, S. Hein, J. Nold, F. Proske, B. Sattler, A. Liem, C. Jauregui, J. Limpert, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Single mode 4.3 kW output power from a diode-pumped Yb-doped fiber amplifier,” Opt. Express 25(13), 14892–14899 (2017).
    [Crossref] [PubMed]
  18. R. Tao, X. Wang, and P. Zhou, “Comprehensive theoretical study of mode instability in high power fiber lasers by employing a universal model and its implications,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0903319 (2018).
    [Crossref]
  19. 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] [PubMed]
  20. C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
    [Crossref] [PubMed]
  21. W. Liu, P. Ma, H. Lv, J. Xu, P. Zhou, and Z. Jiang, “General analysis of SRS-limited high-power fiber lasers and design strategy,” Opt. Express 24(23), 26715–26721 (2016).
    [Crossref] [PubMed]
  22. V. Bock, A. Liem, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Explanation of stimulated Raman scattering in high power fiber systems,” Proc. SPIE 10512, 105121F (2018).
  23. H. Xu, M. Jiang, C. Shi, P. Zhou, G. Zhao, and X. Gu, “Spectral shaping for suppressing stimulated-Raman-scattering in a fiber laser,” Appl. Opt. 56(12), 3538–3542 (2017).
    [Crossref] [PubMed]
  24. S. J. Garth and R. A. Sammut, “Theory of stimulated Raman scattering in two-mode optical fibers,” J. Opt. Soc. Am. B 10(11), 2040–2047 (1993).
    [Crossref]
  25. Z. Jiang and J. R. Marciante, “Impact of transverse spatial-hole burning on beam quality in large-mode-area Yb-doped fibers,” J. Opt. Soc. Am. B 25(2), 247–254 (2008).
    [Crossref]
  26. W. Liu, P. Ma, H. Lv, J. Xu, P. Zhou, and Z. Jiang, “Investigation of stimulated Raman scattering effect in high-power fiber amplifiers seeded by narrow-band filtered superfluorescent source,” Opt. Express 24(8), 8708–8717 (2016).
    [Crossref] [PubMed]
  27. 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]
  28. Y. Xiao, R. J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers,” Opt. Express 22(26), 32039–32059 (2014).
    [Crossref] [PubMed]
  29. 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] [PubMed]
  30. C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4(3), 288–297 (1986).
    [Crossref]
  31. A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013).
    [Crossref]
  32. 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), 3000112 (2014).
    [Crossref]
  33. S. M. M. Friis, I. Begleris, Y. Jung, K. Rottwitt, P. Petropoulos, D. J. Richardson, P. Horak, and F. Parmigiani, “Inter-modal four-wave mixing study in a two-mode fiber,” Opt. Express 24(26), 30338–30349 (2016).
    [Crossref] [PubMed]
  34. M. Ziemienczuk, A. M. Walser, A. Abdolvand, and P. St. J. Russell, “Intermodal stimulated Raman scattering in hydrogen-filled hollow-core photonic crystal fiber,” J. Opt. Soc. Am. B 29(7), 1563–1568 (2012).
    [Crossref]
  35. J.-P. Fève, “Phase-matching and mitigation of four-wave mixing in fibers with positive gain,” Opt. Express 15(2), 577–582 (2007).
    [Crossref] [PubMed]
  36. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012).

2018 (3)

M. N. Zervas, “Power scaling limits in high power fiber amplifiers due to transverse mode instability, thermal lensing, and fiber mechanical reliability,” Proc. SPIE 10512, 1051205 (2018).

R. Tao, X. Wang, and P. Zhou, “Comprehensive theoretical study of mode instability in high power fiber lasers by employing a universal model and its implications,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0903319 (2018).
[Crossref]

V. Bock, A. Liem, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Explanation of stimulated Raman scattering in high power fiber systems,” Proc. SPIE 10512, 105121F (2018).

2017 (4)

2016 (4)

2014 (5)

2013 (3)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013).
[Crossref]

2012 (4)

2011 (1)

2010 (1)

2009 (1)

2008 (3)

2007 (2)

2004 (1)

1993 (1)

1986 (1)

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4(3), 288–297 (1986).
[Crossref]

1972 (1)

Abdolvand, A.

Abedinajafi, A.

Agrawal, G. P.

Azizi, S.

Barty, C. P. J.

Beach, R. J.

Begleris, I.

Beier, F.

Bock, V.

V. Bock, A. Liem, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Explanation of stimulated Raman scattering in high power fiber systems,” Proc. SPIE 10512, 105121F (2018).

Chen, Z.

Clarkson, W. A.

Codemard, C. A.

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0904123S (2014).
[Crossref]

Dajani, I.

Dawson, J. W.

Desgroseilliers, M.

Dong, L.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Eberhardt, R.

Eidam, T.

Essiambre, R. J.

Fève, J.-P.

Foy, P.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Friis, S. M. M.

Galvanauskas, A.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Garth, S. J.

Gong, M.

Gu, G.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Gu, X.

Haarlammert, N.

Hansen, K. R.

Hawkins, T.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Heebner, J. E.

Hein, S.

Hejaz, K.

Henry, C. H.

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4(3), 288–297 (1986).
[Crossref]

Horak, P.

Hu, I.-N.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Hupel, C.

Jansen, F.

Jauregui, C.

Jeong, Y.

Jiang, M.

Jiang, Z.

Jung, Y.

Kong, F.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Koponen, J. J.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Kuhn, S.

Kuznetsov, A.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Lægsgaard, J.

Leng, J.

Li, C.

Liao, S.

Liem, A.

Limpert, J.

Liu, W.

Liu, Z.

Lv, H.

Ma, P.

Ma, X.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Marciante, J. R.

Maynard, R.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Mcclane, D.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
[Crossref]

McComb, T. S.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Messerly, M. J.

Mumtaz, S.

Nilsson, J.

Nold, J.

Otto, H. J.

Parmigiani, F.

Pax, P. H.

Payne, D.

Petropoulos, P.

Poletti, F.

Proske, F.

Pulford, B.

Rezaei-Nasirabad, R.

Richardson, D. J.

Robin, C.

Roohforouz, A.

Rottwitt, K.

Russell, P. St. J.

Ryf, R.

Sahu, J.

Saitoh, K.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers,” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Sammut, R. A.

Sattler, B.

Schreiber, T.

Shayganmanesh, M.

Shi, C.

Shverdin, M. Y.

Siders, C. W.

Smith, A. V.

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), 3000112 (2014).
[Crossref]

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

Smith, J. J.

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), 3000112 (2014).
[Crossref]

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

Smith, R. G.

Sosnowski, T.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Sridharan, A. K.

Stappaerts, E. A.

Stutzki, F.

Su, R.

Tao, R.

R. Tao, X. Wang, and P. Zhou, “Comprehensive theoretical study of mode instability in high power fiber lasers by employing a universal model and its implications,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0903319 (2018).
[Crossref]

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] [PubMed]

Tulino, A. M.

Tünnermann, A.

Vatani, V.

Walser, A. M.

Wang, X.

R. Tao, X. Wang, and P. Zhou, “Comprehensive theoretical study of mode instability in high power fiber lasers by employing a universal model and its implications,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0903319 (2018).
[Crossref]

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] [PubMed]

Ward, B.

Xiao, H.

Xiao, Y.

Xu, H.

Xu, J.

Yan, P.

Yuan, Y.

Zervas, M. N.

M. N. Zervas, “Power scaling limits in high power fiber amplifiers due to transverse mode instability, thermal lensing, and fiber mechanical reliability,” Proc. SPIE 10512, 1051205 (2018).

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0904123S (2014).
[Crossref]

Zhang, H.

Zhao, G.

Zhou, P.

Zhu, C.

T. Sosnowski, A. Kuznetsov, R. Maynard, X. Ma, C. Zhu, I.-N. Hu, A. Galvanauskas, J. J. Koponen, and T. S. McComb, “3C Yb-doped Fiber Based High Energy and Power Pulsed Fiber Lasers,” Proc. SPIE 8601, 86011M (2013).
[Crossref]

Ziemienczuk, M.

Appl. Opt. (2)

IEEE J. Sel. Top. Quantum Electron. (3)

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), 3000112 (2014).
[Crossref]

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 0904123S (2014).
[Crossref]

R. Tao, X. Wang, and P. Zhou, “Comprehensive theoretical study of mode instability in high power fiber lasers by employing a universal model and its implications,” IEEE J. Sel. Top. Quantum Electron. 24(3), 0903319 (2018).
[Crossref]

IEEE Photonics J. (1)

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013).
[Crossref]

J. Lightwave Technol. (1)

C. H. Henry, “Theory of spontaneous emission noise in open resonators and its application to lasers and optical amplifiers,” J. Lightwave Technol. 4(3), 288–297 (1986).
[Crossref]

J. Opt. Soc. Am. B (6)

Nat. Photonics (1)

C. Jauregui, J. Limpert, and A. Tünnermann, “High-power fibre lasers,” Nat. Photonics 7(11), 861–867 (2013).
[Crossref]

Opt. Express (15)

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] [PubMed]

F. Beier, C. Hupel, S. Kuhn, S. Hein, J. Nold, F. Proske, B. Sattler, A. Liem, C. Jauregui, J. Limpert, N. Haarlammert, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Single mode 4.3 kW output power from a diode-pumped Yb-doped fiber amplifier,” Opt. Express 25(13), 14892–14899 (2017).
[Crossref] [PubMed]

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

Y. Jeong, J. Sahu, D. Payne, and J. Nilsson, “Ytterbium-doped large-core fiber laser with 1.36 kW continuous-wave output power,” Opt. Express 12(25), 6088–6092 (2004).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

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] [PubMed]

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012).
[Crossref] [PubMed]

K. R. Hansen and J. Lægsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22(9), 11267–11278 (2014).
[Crossref] [PubMed]

Y. Xiao, R. J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers,” Opt. Express 22(26), 32039–32059 (2014).
[Crossref] [PubMed]

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] [PubMed]

C. Jauregui, J. Limpert, and A. Tünnermann, “Derivation of Raman treshold formulas for CW double-clad fiber amplifiers,” Opt. Express 17(10), 8476–8490 (2009).
[Crossref] [PubMed]

W. Liu, P. Ma, H. Lv, J. Xu, P. Zhou, and Z. Jiang, “General analysis of SRS-limited high-power fiber lasers and design strategy,” Opt. Express 24(23), 26715–26721 (2016).
[Crossref] [PubMed]

W. Liu, P. Ma, H. Lv, J. Xu, P. Zhou, and Z. Jiang, “Investigation of stimulated Raman scattering effect in high-power fiber amplifiers seeded by narrow-band filtered superfluorescent source,” Opt. Express 24(8), 8708–8717 (2016).
[Crossref] [PubMed]

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[Crossref] [PubMed]

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[Crossref] [PubMed]

Opt. Lett. (2)

Proc. SPIE (4)

M. N. Zervas, “Power scaling limits in high power fiber amplifiers due to transverse mode instability, thermal lensing, and fiber mechanical reliability,” Proc. SPIE 10512, 1051205 (2018).

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[Crossref]

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[Crossref]

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Other (1)

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2012).

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Figures (6)

Fig. 1
Fig. 1 The spectrum of the seed source.
Fig. 2
Fig. 2 The power distribution along the fiber amplifier.
Fig. 3
Fig. 3 The normalized output spectra and corresponding temporal properties of lasers in the two modes: (a) the optical spectra for lasers in the two modes; (b) the temporal evolution for laser in LP11 mode; (c) the ACFs for lasers in the two modes.
Fig. 4
Fig. 4 The spectral evolution of laser in the two modes along the fiber amplifier: (a) spectra of laser in LP01 mode; (b) spectra of laser in LP11 mode.
Fig. 5
Fig. 5 The power distributions along the fiber amplifier for three cases: (a) neglecting active gain term; (b) neglecting Raman amplification term; (c) neglecting SRS-induced IM-WM term.
Fig. 6
Fig. 6 The properties of the output powers verse the pump power: (a) the power slope of signal light in LP01 mode; (b) the ratios of laser power in LP11 mode and Raman Stokes light.

Tables (2)

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Table 1 Dispersion parameters and power overlap factors for LP01 and LP11 modes

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Table 2 Major simulation parameters for the fiber amplifier

Equations (7)

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A u ( z,ω ) z = D u ( z,ω )+ N u ( z,ω )+ G u ( z,ω )+ f u ( z,ω ),
D u ( z,ω )=i[ ( β 0 ( u ) β 0 ( 01 ) )+( β 1 ( u ) β 1 ( 01 ) )ω+ n2 β n ( u ) n! ω n ] A u ( z,ω )
N u ( z,t )=iγ( 1+ ω ω 0 )F{ Q uu A u (z,t)R(t) | A u (z,t) | 2 + Q uv A u (z,t)R(t) | A v (z,t) | 2 + Q uv A v (z,t)R(t)[ A v (z,t) A u (z,t) ] },
G u ( z,ω )= 1 2 [ Γ u ( ω )( σ a s ( ω )+ σ e s ( ω ) ) N 2 ( z ) σ a ( ω ) N 0 α u ] A u ( z,ω )
d P p (z) dz = Γ p { σ a ( ω p ) N 0 ( σ a ( ω p )+ σ e ( ω p ) ) N 2 } P p (z) α p P p (z)
N 2 N 0 = Γ p ω p A σ a ( ω p ) P p + 1 2π T m A σ a ( ω ˜ ) ω ˜ ( Γ u | A u (z,ω) | 2 + Γ v | A v (z,ω) | 2 )dω Γ p ω p A ( σ a ( ω p )+ σ e ( ω p ) ) P p + 1 τ + 1 2π T m A σ a ( ω ˜ )+ σ e ( ω ˜ ) ω ˜ ( Γ u | A u (z,ω) | 2 + Γ v | A v (z,ω) | 2 )dω ,
{ f u ( z,ω ) f u ( z , ω ) =2 D FF ( z,ω )δ( z z )δ( ω ω ) f u (z,ω) =0 D FF (z,ω)= ( ω+ ω 0 ) 3 π c 2 G(z,ω) n sp ,

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