## Abstract

In this paper the stimulated Raman scattering (SRS) effect in high-power fiber amplifiers seeded by the narrow-band filtered superfluorescent source (SFS) is firstly analyzed both theoretically and experimentally. Spectral models for the formation of the SFS and the spectral evolution in high-power fiber amplifiers seeded by filtered SFS are proposed. It is found that the SRS effect in high-power fiber amplifiers depends on the spectral width of the filtered SFS seed. The theoretical predictions are in qualitative agreements with the experimental results.

© 2016 Optical Society of America

## 1. Introduction

Stimulated Raman scattering (SRS) is one of the key limitations for power scaling in high-power fiber laser systems [1]. The traditional methods to analyze the SRS effect in the fiber amplifiers are based on the steady-state rate equations and the power coupling equations [2,3]. In previous theoretical works, the power characteristics of the SRS effect could be analyzed, while the instinct gain characteristics of the doped ions and the nonlinear propagation process were ignored. In high-power fiber amplifiers, the instinct gain characteristics of the doped ions will affect output spectra of the amplifiers, which partially determines the spectral characteristics of the Raman Stokes light. Meanwhile, the nonlinear propagation process is quite important for the formation of the Raman Stokes light [4]. In recent work, T. Schreiber et al. experimentally found that the threshold of SRS for kW fiber oscillators depends on the spectral width of the out coupling fiber Bragg grating (FBG) [5]. However, this phenomenon is not yet understood completely through the existed models.

Fiber amplifiers seeded by the narrow-band filtered superfluorescent sources (SFSs) provide an important way to obtain high-power narrow-band lasers [6–8]. In past several years, there are many investigations on scaling the output power from SFSs [9–12]. In 2011, O. Schmidt et al. [6] demonstrated a 697 W narrow-band SFS with 12 pm (3.5 GHz) bandwidth. In 2015, P. Ma et al. [7] obtained an 800 W polarization-maintained SFS with 0.2 ± 0.02 nm bandwidth. In 2015, J. Xu et al. [8] reported a 1.87 kW SFS with 1.7 nm bandwidth. In many practical applications where the narrow-band lasers are required, such as spectral beam combining [13,14] and optical frequency conversion [15,16], the spectral width of the laser partially determines the efficiency of the system. However, a few works focused on the spectral formation for the SFS, and the spectral evolution of the filtered SFS in high-power fiber amplifier has not been studied till now. In this work, spectral model for the SFS and the spectral evolution in high-power fiber amplifier seeded by filtered SFS are proposed for the first time. Meanwhile, the impacts of the spectral width of the filtered SFS on the SRS effect are investigated both theoretically and experimentally.

## 2. Theory for numerical modeling

For high-power narrow-band SFS, at a certain point SBS is not the main limiting nonlinear effect anymore but rather SRS [6]. However, no theoretical analysis was reported to study the SRS effect in high-power narrow-band SFS to the best of our knowledge. To analyze this phenomenon, both amplification process and the characteristics of filtered SFS seed should be incorporated into the theoretical model. Consequently, in the following theoretical analysis, both the spectral evolution during amplification and the spectral formation for the filtered SFS are investigated.

Along with the power scaling in high-power fiber amplifiers, there are mainly three typical phenomena in the spectral region, i.e. the spectral broadening [17], the asymmetrical spectra [18] and the amplified spontaneous emission spectrum [19]. The spectral broadening effect can be generally described by the nonlinear Schrödinger equation (NLSE) [20,21], while the existed models are mainly based on the pulsed seeds or master oscillators. For the asymmetrical spectra, it is intuitive to refer this phenomenon to the wavelength-dependent gain of the doped ions [22]. However, more specific descriptions of this phenomenon are still absence. To study the amplified spontaneous emission in the nonlinear propagation process, it is reasonable to analyze the spontaneous emission noise through its electric field characteristics. To conclude those spectral phenomena, we divide the spectral evolution during amplification into two main processes: the amplification of the optical field in active fiber and the energy transfer between different spectral components due to the nonlinear propagation. Accordingly, we use an amplitude equation to describe the nonlinear propagation by the NLSE with the electric field description of the initial spontaneous emission noise and the steady-state rate equations to calculate the wavelength-dependent gain in the active fiber.

#### 2.1 Spectral model for the amplification process

In the traditional analysis, the rate equations and NLSE are generally analyzed in the time domain [23]. When considering the wavelength-dependent characteristics of the active gain, it is convenient to transform the rate equations and NLSE into the frequency domain, and the set of unidirectional 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. $F\left\{\right\}$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.

Refer to the theory of spontaneous emission noise in open resonators [24], here we consider that the real and imaginary parts of spontaneous emission noise${f}_{SE}^{}(z,\omega )$ are Gaussian white noise. Accordingly, it can be analyzed as Gaussian stochastic process with zero mean value satisfies:

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.

#### 2.2 Spectral model for the SFS

The typical spectral model for the SFS is based on the steady-state bidirectional rate equations [25]. Through this model, both the power and spectral characteristics of the SFS can be obtained. Although the stimulated output power is quite close to the experimental results, the stimulated optical spectrum departures from the measured one. Meanwhile, the stimulated optical spectrum cannot be directly applied as the input signal of the Eqs. (1a) - (1c). To get the optical field of the SFS, the bidirectional NLSEs should be supplied [26]:

In Yb-doped fiber lasers (YDFLs), the steady-state output spectra can be calculated through the combined simulation of the rate equations with Eq. (3) [26]. However, this approach is not applicable for the SFS, as the effect of the spontaneous emission noise and the wavelength-dependent characteristics of the active gain are ignored in this model.

For the cavity-free SFS, the reflectivity of the end faces can be neglected and the laser origins from the amplified spontaneous emission in a single trip in the active fiber. When the signal power in the fiber is not so high, the interaction between the bidirectional optical fields is quite weak and the nonlinear effect (including the effect of the dispersion) can be omitted. Further ignoring the gain saturation effect induced by the bidirectional optical fields, the formation of the SFS may be simplified into two independent amplification processes in the opposite directions. Under those assumptions, the spectral formation for the bidirectional SFS degrades into the two unidirectional amplification processes with the independent pumps, and we define the effective pump power as the power which is only absorbed by one of the bidirectional signals.

According to the above discussions, the optical field of the SFS can be calculated through the similar model in section 2.1. When neglecting the nonlinear response and the dispersion, we get the simplified propagation equation for the SFS from Eq. (1a):

#### 2.3 Filtering process

The linewidth narrowing effect in the fiber laser system is conventionally performed by using a narrow-band FBG and a circulator [10,11]. When an optical field propagates into a FBG, the reflective optical field can be expressed as:

Here, ${A}_{in}\left(\omega \right)$ and ${A}_{out}\left(\omega \right)$are the input and output optical field, respectively; $R\left(\omega \right)$is the reflective spectrum of the FBG.

## 3. Theoretical analysis and experimental validation

#### 3.1 Simulation results for the SFS

Based on the theoretical model presented above, we first simulate the generation spectrum of the SFS. The major simulation parameters are corresponding to the commercial Yb-doped single-mode fiber with 6 μm core diameter and 125 μm inner cladding diameter, and the detail parameters are shown as follows: the length of the active fiber is 21m and the effective pump power is 15 W; *λ*_{p} = 976 nm, *N*_{0} = 3.05 × 10^{25} m^{−3}, *τ* = 840 µs, Г_{p} = 0.0064, Г_{s} = 0.85, *A* = 28.3 *µ*m^{2}, *β*_{1} = 4800 ps/m, *β*_{2} = 20.4 ps^{2}/km, *β*_{3} = 0.04 ps^{3}/km, *γ* = 3.47 km∙W^{−1}. For simplicity, we assume that the entire pump light is coupled into the active fiber here.

Figure 1 presents the simulated temporal evolution and corresponding normalized optical spectrum for the broadband SFS. As shown in Fig. 1(a), the SFS induces strong temporal fluctuations in the picosecond scale, which is very similar to that in the multi-longitudinal mode YDFL [27]. The average output power here is 12.8 W, while the peak power is over 150 W. The simulated spectrum in Fig. 1(b) is quite flat here due to the reabsorption of the active fiber as longer active fiber is used in the simulation. The probability density function (PDF) of the output power is exponential, which is shown in Fig. 1(c), thus it can be concluded that the temporal characteristics approximately obeys Gaussian distribution. A possible explanation for this phenomenon is that the SFS builds up from the spontaneous emission noise, which satisfies the Gaussian stochastic process.

#### 3.2 SRS effect in high-power fiber amplifiers

After generation of the SFS, we filter the SFS and simulate its nonlinear amplification process in an Yb-doped fiber amplifier. The detailed simulation parameters of the amplifier are shown as follows: the length of the commercial Yb-doped fiber and passive fiber length is 16 m and 3 m, respectively, with the 20 μm core diameter and 400 μm inner cladding diameter; the power of the seed is 12 W; *λ*_{p} = 976 nm, *N*_{0} = 7.8 × 10^{25} m^{−3}, *τ* = 840 µs, Г_{p} = 0.0025, Г_{s}(*ω*_{0}) = 0.85, *A* = 314 *µ*m^{2}, *β*_{1} = 4900 ps/m, *β*_{2} = 20.4 ps^{2}/km, *β*_{3} = 0.04 ps^{3}/km. Here, we assume that the entire pump light is coupled into the active fiber.

Figure 2 illustrates the simulated temporal evolution and corresponding normalized optical spectrum of the amplifier. The pump power is 1500 W and the full width at half maximum (FWHM) of the seed spectrum (Gaussian) is 0.2 nm. As shown in Fig. 2(a), the average output power here is about 1.3 kW, while the peak power is over 8 kW. Numerically simulated time dynamics of the amplifier denotes the total intensity also fluctuates strongly. In comparison between the spectra at different powers in Fig. 2(b), significant asymmetrical spectral broadening with the Raman stokes light is observed in the amplifier. The stimulated spectrum shows that this model is applicable to analyze the asymmetrical spectral broadening and the SRS effect in high-power fiber amplifiers seeded by filtered SFS seeds.

We further filter the SFS with several different spectral widths and calculate the corresponding spectra at 1.2 kW output power in the amplifiers. Meanwhile, the ratios of the Raman Stokes light are calculated at different output powers for the filtered SFS seeds. Figure 3(a) shows the normalized output spectra at 1.2 kW seeded by filtered SFS seeds with different FWHM spectral widths of 0.06 nm, 0.10 nm, 0.16 nm, 0.20 nm, 0.30 nm, 0.40 nm and 1.00 nm, respectively. As illustrated in Fig. 3(a), with the narrowing of the spectral widths, the amount of the Raman Stokes light increases quickly. Figure 3(b) shows the ratios of the Raman Stokes light at different output powers for the different seeds. Here, 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 called Raman ratio for short. With the narrowing of the spectral width or increasing of the output power, the Raman ratio increases quickly in Fig. 3(b).

#### 3.3 Experimental validations

To validate the theoretical models, high-power fiber amplifier system is established. The experimental setup of the filtered SFS seed is shown in Fig. 4, which mainly includes two parts. The first part is a low power half-opened cavity Yb-doped broadband SFS. In the experiment, a piece of 21 m active fiber with 6 *μm* core diameter and 125 *μm* cladding diameter is employed in the seed source, and we use longer active fiber here to shift the central wavelength of the SFS to about 1064nm. The second part is a linewidth narrowing module based on a FBG, a circulator and cascaded pre-amplifiers [13]. As shown in Fig. 5(a), the spectrum range of the broadband SFS is measured to be from 1030 nm to 1120 nm with FWHM of 26 nm [7]. The FWHM of the filtered SFS seeds are 0.55 nm, 0.23 nm and 0.15nm, which is shown in Figs. 5(b)-5(d).

The experimental configuration to amplify the filtered SFS seeds based on MOPA structure is shown in Fig. 6. Before launched into the main amplifier, the filtered SFS seeds are power scaled to be about 20 W through a pre-amplifier. In the main amplifier, six high-power laser diodes (LDs) with 976 nm central wavelength are incorporated into a (6 + 1) × 1 combiner to pump the active fiber. The double clad active fiber has a core diameter of 20 *μm* and an inner cladding diameter of 400 *μm*. The cladding absorption coefficient of the active fiber is about 1.7 dB/m at 976 nm, and 16 m active fiber is used in the main amplifier. Then, about 3 m double clad passive fiber with the same core and inner cladding diameters is spliced to the active fiber for power delivery and pump dump.

As shown in Fig. 7, the output powers grow linearly with the launched pump power for the three seeds. The maximum output powers are about 1.2 kW with slope efficiency of about 80%. From Fig. 7, it is shown that the spectral widths of the seeds have little influence on the power scaling characteristics of the amplifier.

Figure 8 shows the output spectra of the amplifiers at different power levels in logarithmic coordinate. The output spectra show significant asymmetrical spectral broadenings with an amount of Raman stokes light. This phenomenon is similar to the simulation results. The FWHM spectral width at 1.2 kW are measured to be about 4.2 nm, 1.48 nm and 0.95 nm with the corresponding spectral broadening factors of 7.9, 6.4 and 6.3, respectively in Fig. 8. The three spectral broadening factors are close to each other at the same amplified power.

By analysis, even that some distortions and discrepancies are existed, we find that the spectral distributions of the filtered SFS seeds shown in Fig. 5 are near Lorentzian distributions. The strictly mathematical expressions of the filtered spectra are difficult to achieve. For comparing the SRS ratios in Fig. 8 with the theoretically simulated results, we employ both the Gaussian and the Lorentzian functions to approximately describe the distributions of the three filtered spectra in the following simulations. The experimental and simulated Raman ratios at different output powers are described in Fig. 9. Here, the Raman ratio is also calculated through dividing the integrated spectrum from 1100 to 1150 nm by the integrated spectrum from 1050 to 1150 nm. Though the output powers are quite close to each other, the corresponding Raman ratios in the experiments are 0.05%, 0.37%, and 4% at 1.2 kW. As shown in Fig. 9, the experimental results also show that the Raman ratios in the amplifier increase quickly along with the narrowing of the spectral width of the seeds. The experimental results are in qualitative agreement with the theoretical simulations when the Raman ratio is not so small.

The deviations of the Raman ratios between the simulations and the experiments are mainly due to the inaccuracy of the simulated spectra for the filtered SFS seeds. The ideal spectral shapes of the FBGs are Gaussian, while the practical ones may be close to Lorentzian, hyperbolic secant or other irregular shape with small sidebands. Meanwhile, the pre-amplification processes may induce some spectral distortion. Therefore, the actual filtered SFS seeds are in irregular shapes and it’s difficult to get the relatively accurate simulated spectra for the seeds.

Figure 10 demonstrates the theoretical temporal characteristics of the three filtered seeds from the same SFS. As illustrated in Fig. 10, along with spectral narrowing of the filtered seed, the temporal fluctuations are stronger. This result may be the actual physical origin of the theoretical and experimental results shown in Fig. 9, which will be further studied in the future work.

## 4. Conclusions

In this work, the SRS effect for high-power fiber amplifiers seeded by the narrow-band filtered SFS seeds are investigated both theoretically and experimentally. Constructing seeds with different spectral widths through filtered SFS, and simulating their spectral evolution in the amplifiers, both the output spectra and the Raman ratios can be got theoretically for high-power fiber amplifiers. It is found that the spectral widths of the filtered SFS seeds have a significant effect on the SRS effect of the high-power fiber amplifiers. Both the simulation and experimental results show that the Raman ratios increases quickly in the amplifiers with the narrowing of the spectral widths of the filtered SFS seeds.

## Acknowledgments

The authors acknowledge support from the foundation for the author of National Excellent Doctoral Dissertation of China (Grant No.201329) and the National Natural Science Foundation of China (NSFC) (Grant No.11274386).

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