## 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 LP_{01} mode to Raman Stokes light in LP_{11} 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 LP_{11} 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:

When mode distortion phenomenon occurs in a fiber amplifier, the LP_{11} mode is the dominant component in HOMs. Without loss of generality, we focus the spectral evolution of two scalar-modes, LP_{01} and LP_{11} 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:

*u*) and (

*v*) stand for the different modes among LP

_{01}and LP

_{11}modes, and superscript (01) denotes for the LP

_{01}mode;${\beta}_{n}$stands for the

*n*th order derivative of the propagation constant with respect to the angular frequency;$\gamma $is the nonlinear Kerr coefficient; $\omega $is the reference angular frequency of the spectrum and${\omega}_{0}$is the carrier angular frequency of the signal; the nonlinear response function$R\left(t\right)=\left(1-{f}_{R}\right)\delta \left(t\right)+{f}_{R}h\left(t\right)$, ${f}_{R}$is the fractional contribution of the Raman response to the total nonlinearity and $h\left(t\right)$ is the delayed Raman response function; $A(z,\omega )\left(A(z,t)\right)$is the complex amplitude of signal in the frequency (time) domain; $F\left\{\right\}$denotes the Fourier transform and ‘⨂’ denotes the convolution operation; $Q$is the normalized overlap integral, and we assume that${Q}_{vv}\approx {Q}_{uu}=1$.

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 LP_{11} 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:

*A*is the doped cross-section area; $\tilde{\omega}$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 $G\left(z,\omega \right)$ and the spontaneous emission noise term could be expressed as:

*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 LP_{01} 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 LP_{01} 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 LP_{11} mode, and the initial production of laser in LP_{11} mode just originates from the spontaneous emission noise in the fiber amplifiers.

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, LP_{01} and LP_{11} 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 LP_{11} mode is normalized by the LP_{01} 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.

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 LP_{11} 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 LP_{11} 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 LP_{11} 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.

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 LP_{01} mode and LP_{11} mode are quite different from each other. The spectrum of laser in LP_{01} mode is asymmetrical with a mount of the Raman Stokes light, and the corresponding percentage of the Raman Stokes light in LP_{01} mode is about 2.8% (48 W). Nevertheless, the dominant spectral component of laser in LP_{11} mode is the Raman Stokes light, and the element around 1070 nm is quite small. Besides, the corresponding percentage of Raman Stokes light in LP_{11} 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 LP_{01} mode to the Raman Stokes light in LP_{11} mode.

As shown in Fig. 3(b), due to the randomness of spontaneous emission noise, the temporal property of the generated LP_{11} mode is intense pulsed light in nanosecond scale with varied pulse separation and peak power. The generated laser power in LP_{11} 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 LP_{01} mode to the Raman Stokes light in LP_{11} 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 LP_{01} 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 LP_{11} tends to be about zero as the laser emission in LP_{11} 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.

We might also notice that the intensity of laser in LP_{11} 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 LP_{11} 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 LP_{11} 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 LP_{11} mode, i.e. the active gain term${G}_{11}$, Raman amplification term pumped by signal light in LP_{01} mode${A}_{11}{\displaystyle \int h\left(\tau \right){\left|{A}_{01}\right|}^{2}d\tau}$ and the SRS-induced IM-WM term${A}_{01}{\displaystyle \int h\left(\tau \right){A}_{11}{A}_{01}^{\text{*}}Ad\tau}$. 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.

As shown in Fig. 5(a), when the active gain is neglected in the simulation, the percentage of laser power in LP_{11} 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 LP_{11} mode. As shown in Fig. 5(b), when the Raman amplification pumped by signal light in LP_{01} mode is neglected in the simulation, the percentage of laser power in LP_{11} 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 LP_{11} 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 LP_{11} mode at the fiber end is less than 0.01%, thus this term also contributes the amplification of Raman Stokes light in LP_{11} mode. In summary, both the active gain and the SRS-induced IM-WM effect contribute to amplification of Raman Stokes light in LP_{11} mode. Specifically, it is the SRS-induced IM-WM term which dominates in the mode coupling process from the signal light in LP_{01} mode to Raman Stokes light in LP_{11} mode.

As shown in Fig. 4(b), the signal power in LP_{11} mode is negligible. In addition, even when the Raman stokes light in LP_{01} mode is eliminated in the simulation, the mode coupling process still happens. Thus, the signal power in LP_{11} mode and the Raman stokes light in LP_{01} mode are nearly not participated in the SRS-induced IM-WM process. And direct mode transformation occurs here from the signal light in LP_{01} mode to the Raman Stokes light in LP_{11} 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 LP_{01} 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 LP_{11} 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 LP_{11} 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 LP_{11} 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.

## 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 LP_{01} mode to LP_{11} 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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