## Abstract

In this paper, a spectral model by incorporating SRS effect is proposed and established, which is feasible for analyzing the SRS effect both in high-power fiber oscillator and master oscillator power amplifier (MOPA) system. The theoretical results show that the SRS effect is tightly related to the bandwidths of the fiber Bragg gratings (FBGs) and it can be efficiently suppressed by optimizing the bandwidth of the FBGs. Besides, the established theoretical model is also feasible for analyzing the influence of seed power on the SRS effect. The theoretical predictions agree well with the previous experimental results.

© 2016 Optical Society of America

## 1. Introduction

The stimulated Raman scattering (SRS) effect is one of the dominant nonlinear processes in the continuous wave (CW) fiber based high-power laser systems, which sets the upper limit to the power scaling capabilities of systems [1–3]. Though the SRS effect in high-power fiber lasers has steadily been investigated over the last three decades [4–6], most of the studies are based on the steady-state rate equations together with the power coupling equations. In this classic analysis, the SRS threshold in fiber amplifier is mainly dependent on the fiber characteristics of the amplifier stage, such as the effective mode area, fiber length and new fiber design [7], while the properties of the seeds are not considered. In recent work, A. E. Bednyakova et al. theoretically and experimentally demonstrated that quasi-CW Yb-doped fiber oscillator exhibits intensity fluctuations in the nanosecond scale [8]. As the SRS effect is associated with the peak power of the laser, those temporal fluctuations should be included in the SRS analysis. In 2014, T. Schreiber et al. experimentally reported the threshold of SRS for kW fiber oscillators depends on the spectral width of the out coupling fiber Bragg grating (FBG), while the experimental result was not completely explained by the theoretical analysis [9]. In 2016, J. Wang et al. experimentally found that the SRS threshold was inversely proportional to the seed power in high-power fiber amplifiers [10], while the theoretical explanation was not given. In order to study the power scaling property, we propose and establish a spectral model by focusing on SRS effect. Based on the established model, the relationship between SRS effect and the bandwidth of FBGs are analyzed for the first time. In addition, some design guidelines are proposed based on the theoretical results, which is significant for further power scaling in SRS-limited high-power fiber laser systems.

## 2. Theory for numerical modeling

As for a practical high-power fiber laser system, two approaches are conventionally employed, one is power scaling directly based on an oscillator, and the other is based on a MOPA structure. In addition, in a MOPA structure, several types of seeds can be used in the following amplification processes. Despite that, different fiber sources are formed by different structures (cavity/cavity-free) with different gain characteristics (active gain/Raman gain), the spectral formation process with the SRS effect can be mainly explained by two energy conversion processes, one is the energy absorption from the pump light to the signal light through the doped ions and the other is the energy transformation between different spectral components during the nonlinear propagation. Those two processes could be included through the rate equations and the nonlinear propagation equations, respectively. In the following analysis, we mainly focus on the fiber oscillators, which can be directly applied for the MOPA structure by just considering one direction.

In the numerical model, the lasering process in a fiber oscillator is regarded as the amplification of the spontaneous emission noise with the cavity effect. Thus, this process can be divided into the reciprocating amplification of the signal light along the active fiber with FBGs acted as the filters. Here, we directly improve the spectral model with SRS effect for the amplification process in fiber amplifier [11] into the fiber oscillator, and the set of bidirectional spectral-spatial equations describing the optical field during amplification are given by

Here, the active gain$g\left(\omega \right)={\Gamma}_{s}\left(\omega \right)\left[{\sigma}_{a}\left(\omega \right)+{\sigma}_{e}\left(\omega \right)\right]{N}_{2}-{\sigma}_{a}\left(\omega \right){N}_{0}$and the nonlinear response function$R\left(t\right)=\left(1-{f}_{R}\right)\delta \left(t\right)+{f}_{R}h\left(t\right)$; ${\beta}_{n}$is the *n*-order derivative of the propagation constant with respect to the angular frequency and ${\omega}_{0}$is the carrier frequency of the signal; $\gamma $is the nonlinear Kerr coefficient; $\tilde{A}(z,\omega )\left(A(z,t)\right)$is the complex amplitude of signal in the frequency (time) domain; ${P}_{p}$is the pump power. $F\{\ast \}$denotes the Fourier transform and ‘⨂’ denotes the convolution operation. Index$p$stands for pump wave; $\Gamma $is the overlap factor; ${\sigma}_{a}$and ${\sigma}_{e}$are the corresponding absorption and emission cross sections at different angular frequency; ${N}_{0}$is the ytterbium dopant concentration and ${N}_{2}$is the total number of Yb-ions in excited state; $\alpha $is the loss coefficient. $\tau $is the life of the excited state population. ${T}_{m}$is the time window during the calculation; $\hslash $is the Planck’s constant; *A* is the doped cross-section area. The spontaneous emission noise is analyzed as Gaussian stochastic process with zero mean value satisfies [12]

Here, ${n}_{sp}=1/\left(\mathrm{exp}\left(\hslash \left(\omega +{\omega}_{0}\right)/{k}_{B}T\right)-1\right)$represents the average mode occupation number in equilibrium; ${k}_{B}$is the Boltzmann constant; *T* is the environmental temperature.

The dissipation of the signal is mainly induced by FBGs and the boundary conditions that related to the reflective spectrum of FBGs can be expressed as

Here, ${R}_{1}\left(\omega \right)$and ${R}_{2}\left(\omega \right)$ are the reflective spectra of the high reflectivity (HR) FBG and the output coupling (OC) FBG, respectively.

The output optical field and the corresponding optical spectrum of the fiber oscillator can be expressed as

## 3. The SRS effect in high-power fiber oscillators

Based on the above model, we analyze the SRS effect in a typical high-power fiber oscillator with the commercial double-clad Yb-doped fiber with the 20 μm core diameter and 400 μm inner cladding diameter. The simulation parameters of the oscillator are shown in Table 1. For simplicity, we assume that the reflective spectra of the HR and OC FBGs satisfy the Gaussian distributions, and the effective pump power coupled into the active fiber can be boosted to 1500 W in the forward pumping configuration.

For proposing some instructions for a practical high-power fiber system, we analyze the cavities formed by OC FBGs with different bandwidths. Figures 1(a) and 1(b) present the comparisons between the OC FBGs with 0.3 nm, 0.5 nm, 1.0 nm and 2.0 nm bandwidths, respectively. As shown in Fig. 1(a), the high-power fiber oscillators induce strong temporal fluctuations in the picosecond scale. Though, the average output powers are both close to 1.2 kW when the bandwidths of the OC FBGs are 0.5 nm or 2.0 nm, the fiber oscillator induces stronger temporal fluctuations when using the narrower OC FBG.

The simulated spectra (shown in Fig. 1(b)) are asymmetrical with a mount of the Raman stokes light, which can be explained by the wavelength-dependent gain of the doped ions. There also exists an obvious decrease of the ratios of the Raman Stokes light along with the broadening of the bandwidths of the OC FBGs, and the corresponding ratios of the Raman stokes light in the signal light are −25 dB, −37dB, −54 dB and −60 dB, respectively. The simulation results are compatible with the experimental results shown in Re [9]. The ratio of the Raman Stokes light is calculated through dividing the integrated spectrum from 1100 to 1150 nm by the integrated spectrum from 1050 to 1150 nm, which is defined by Raman ratio in the following analysis.

## 4. SRS effect in high-power MOPA structure

In MOPA structures, the parameters of the seeds will have a significant effect on the SRS effect. In this section, we will give the detailed analysis on the properties of the fiber seeds with different structural parameters.

The major simulation parameters of the medium-power seed are shown in Tab. 2. For simplicity, we also assume that the reflective spectra of the FBGs satisfy the Gaussian distributions, and the effective pump power coupled into the active fiber can be boosted to 60 W in the forward pumping configuration. And the same simulation parameters are used for the amplifier stage as in Tab. 1.

#### 4.1 Influence of bandwidth of each individual FBG

First, we consider the impact of the bandwidth of the OC FBG in the fiber seed. When the bandwidth of the OC FBG vary from 0.2 nm to 2.0 nm, the output spectrum broadens from about 0.15 nm to about 0.6 nm (shown in Fig. 2(a)). The output powers are all close to 50 W (shown in Fig. 2(b)) and the variance of the output powers are less than 3% when using the OC FBGs with different bandwidths.

Although the seed powers are close to each other, the SRS effect in the amplifier stage is different. Figures 3(a) and 3(b) present the simulated optical spectra for the amplifier stage and the Raman ratios verse the bandwidth of the OC FBG in the logarithmic coordinates, respectively. As illustrated in Fig. 3(b), there is a sharp transformation in the SRS properties when the bandwidth of the OC FBG changes from the 0.8 nm to 1.0 nm. When the bandwidth of OC FBG is over 1.0 nm, the ratio of the Raman stokes light decreases by about 15 dB, and the Raman ratio is changed a little along with the increase of the bandwidth of the OC FBG. This phenomenon may be attributed to the fact that the bandwidth of the HR FBG has played a dominant function in the SRS process. This point can be inferred in the following section by changing the bandwidth of the HR FBG.

Similar results are obtained when changing the bandwidth of the HR FBG alone. Figures 4(a) and 4(b) present the simulated optical spectra for the amplifier stage and the Raman ratios verse the bandwidth of the HR FBG in the logarithmic coordinates, respectively. As illustrated in Fig. 4(b), there is also a sharp transformation in the SRS properties when the bandwidth of the HR FBG changes from the 0.4 nm to 0.6 nm. When the bandwidth of HR FBG is over 0.6 nm, the ratio of the Raman stokes light decreases by about 20 dB, and the Raman ratio is changed a little along with the increase of the bandwidth of the HR FBG. This phenomenon may be attributed to the fact that the bandwidth of the OC FBG has played a dominant function in the SRS process.

#### 4.2 Influence of the bandwidths of the FBG pairs

In the following section, we will discuss the SRS effect by simultaneously changing the bandwidths of the HR and OC FBGs. In each case, we ensure that the bandwidths of the HR and OC FBGs are identical. Figures 5(a) and 5(b) present the simulated optical spectra for the amplifier stage and the Raman ratios verse the bandwidths of the two FBGs in the logarithmic coordinates. When changing the bandwidths of the OC and HR FBGs simultaneously, the ratios of the Raman stokes light almost decrease linearly along with the increasing bandwidth of the FBG (shown in Fig. 5(b)).

As for a practical high-power fiber laser system, the scaling ability of the SRS threshold is highlighted. Based on this consideration, we compare the SRS threshold scaling ability when the bandwidths of the FBGs are 0.4 nm and 2.0 nm, respectively. Based on our theoretical model, the SRS threshold is calculated to be about 1.34 kW and 2.5 kW in the two cases above, respectively. In the aforecited analysis, the SRS threshold is defined as −20 dB. Accordingly, theoretical results show that the Raman threshold for the high-power fiber amplifiers can be scaled up to 2 times through optimization of the FBGs of the oscillator.

#### 4.3 Influence of seed power

As pointed above, in MOPA configuration, the SRS threshold is also related to the seed power in the experiments [10]. In this section, we analyze the SRS effect by changing the seed powers, and the parameters in the simulations are shown in Tab. 2. Figures 6(a) and 6(b) present the simulated optical spectra and the Raman ratios verse the seed power in the logarithmic coordinates. As shown in Fig. 6(b), the ratios of the Raman stokes light almost increase linearly along with the increasing seed power. The simulation results are compatible with the experimental results shown in Re [10]. Thus, it is concluded that the theoretical model can be also employed to analyze the influence of the seed power on the SRS effect.

## 5. Conclusions

It is to be noted that recently there are several independent studies on high-power fiber amplifiers seeded by fiber sources without standard cavity configuration (i.e. fiber-based ASE source [13, 14] or random distributed feedback fiber source [15]), and kilowatt output power had been demonstrated. Those fiber sources have different temporal properties with oscillators [16], the power scaling property and SRS effect in high-power amplifiers seeded by those fiber sources would be investigated in future endeavors.

The SRS effect in high-power fiber oscillator and MOPA structure is analyzed based on a SRS-related spectral model for the first time. The analysis results show that the bandwidths of FBGs play an important role in high-power scaling by suppressing the SRS effect. From the theoretical results, the SRS threshold can be effectively scaled by either increasing the bandwidths of the FBGs or decreasing the seed power appropriately. Our theoretical model gives a useful reference to design the high-power fiber laser system in the future.

## Funds

Foundation for the Author of National Excellent Doctoral Dissertation of China (201329); National Natural Science Foundation of China (NSFC) (11274386).

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