1. Introduction

Narrow-linewidth fiber amplifiers around 1 µm have a wide range of applications in the field of directed energy, scientific research, material processing, nonlinear frequency conversion, etc. Especially, several power scaling techniques such as spectral beam combining and coherent beam combining call for fiber amplifiers with narrow linewidth (e.g., below 20 GHz) and high spectral brightness. However, the power scaling of the high spectral brightness for a monolithic fiber amplifier is limited by nonlinearities especially stimulated Brillouin scattering (SBS). Up to now, several SBS suppressing methods have been proposed, including using fibers with a larger effective mode area, controlling the temperature or stress gradient to change the Brillouin gain spectrum, reducing the effective length of the fiber, and external phase modulation. Among them, external phase modulation is considered as a promising method, such as white noise modulation, optimized waveform modulation and PRBS phase modulation. In 2019, Z. Chang et al. [1] used white noise phase modulation to broaden the spectrum of a single frequency seed source, and when the FWHM linewidth is 13 GHz, the output power is 1.5 kW. In 2015, A. V. Harish et al. [2] investigated the modulation signals that produce top-hat-shaped optical spectra of discrete lines with the highest total power within a limited bandwidth and peak spectral power density through algorithm optimization. In 2021, Y. Panbiharwala et al. [3] used optimized waveform modulation, which provided 1.4 times higher enhancement in the SBS threshold than noise modulation did. In the same year, Y. Wang et al. [4] demonstrated the superiority of a nearly top-hat-shaped spectrum to SBS suppression in experiments and achieved 3.25 kW linearly polarized laser output while maintaining 20 GHz linewidth. In 2023, M. Shi et al. [5] proposed the spectral broadening and shape control of seed sources based on high-order phase modulation of binarized multi-tone signals. A maximum power of 2234 W is obtained with a rectangular spectrum with 10 GHz bandwidth. Among them, the scheme of using pseudo-random binary sequence (PRBS) external phase modulation has the characteristics of simple and controllable operation, which has become the dominant technique for SBS suppressing in a high-power narrow linewidth fiber amplifier [6–10].

The PRBS is a bit sequence with a pattern length of $N = {2^n} - 1$ and a period of $NT$, where n is the bits number of the Linear Feedback Shift Register (LFSR), and $T$ is the bit period. The “0” and “1” bits appear with equal and random probability, and its spectrum distribution is a frequency comb with $\textrm{Sin}{\textrm{c}^\textrm{2}}\textrm{}$ as the envelope, because of the periodic clock rate, which produces a discrete and controllable spectral interval, whose frequency spacing is given by $\varDelta f = {{{f_{cr}}} / N}$, where ${f_{cr}} = {1 / T}$ is the clock rate, and whose FWHM linewidth and first zero values are both equal to the clock rate. In this case, the optical phase can change faster than the accumulation time of the SBS, thus preventing the acoustic wave from accumulating to large amplitude and thus reducing the SBS gain [2,11].

In 2012, C. Zeringue et al. [12] theoretically derived the SBS suppression through PRBS phase modulation. In the following year, C. Robin et al. [8] verified the effectiveness of PRBS phase modulated signals in suppressing SBS effects in high-power lasers by experimental and theoretical investigations and obtained a maximum output power of 1 kW at 6 GHz clock rate with PRBS6. In 2014, A. Flores et al. [13] used a (non-PM) 25/400 ${\mathrm{\mu} \mathrm{m}}$ Yb-doped fiber (YDF) and achieved an output power of 1.17 kW near diffraction limited at a PRBS clock rate of 3 GHz. The further increase in power is mainly limited by the SBS effect. In 2016, I. Dajani et al. [14] at a PRBS clock rate of 5 GHz with PRBS6 achieved 1.47 kW output power in a (non-PM) 25/400 µm YDF and 1.48 kW output power in a (non-PM) 20/400 µm YDF. In 2018, M. Kanskar et al. [15] at a PRBS clock rate of 20 GHz with PRBS7 achieved 2.2 kW output power in a Yb-doped 20/400/0.064 DC LMA fiber and 2.6 kW output power in a Yb-doped 21.9/400/0.059 DC chirally-coupled core (3C) fiber. In 2017, B. M. Anderson et al. [16] showed that filtered PRBS can improve the SBS threshold of high-power fiber amplifiers for optimal patterns (e.g. 2⁹-1) over conventional unfiltered PRBS. In 2020, M. Liu et al. [9] achieved 1.2 kW output power at a distributed Bragg reflector (DBR) laser seed and 1.27 kW output power at a distributed feedback (DFB) laser seed by optimizing the filtered PRBS spectral line spacing using a 2.2 GHz low-pass filter. In both cases, the beam quality is close to the diffraction limit (${\textrm{M}^\textrm{2}}$ < 1.2). At the highest output power, a more pronounced SBS effect was generated, and further power enhancement was mainly limited by the SBS effect. In 2021, we investigate that the enhancement of the SBS threshold of the filtered PRBS phase modulated lightwave in the passive fiber system is evaluated by numerically simulating the coupled three-wave SBS interaction equations. We found that the SBS suppression depends non-monotonically on the parameters of the filtered and amplified PRBS waveform and the simulations indicate that the normalized SBS threshold reaches a maximum for an RMS modulation depth of 0.56π and a ratio of filter cutoff frequency to clock rate of 0.54 and that PRBS9 is superior to other investigated patterns [17]. However, no experimental verification has been performed. For the practical high-power fiber laser system, SBS occurs in the main amplifier stage and also needs to be considered. Modeling considering SBS in amplifiers has also been reported in the literature [3,18–24], But few reports have been seen that consider the time dynamic model of SBS when it occurs both in the active fiber and passive fiber.

In this work, we report a theoretical and experimental study on SBS suppression in a monolithic fiber amplifier with filtered and amplified PRBS phase modulation. Theoretically, we use a time-dependent three-wave coupled nonlinear system in both active fiber and passive fiber to solve the laser, acoustic phonon, and Stokes characteristics of the fiber amplifier. Through the active fiber spliced passive fiber model by introducing a YDF structure, the SBS suppressing capability of a filtered and amplified PRBS phase modulating high-power narrow-linewidth fiber amplifiers is described more accurately compared with our previous work [17,25]. Experimentally, we established a multi-stage fiber amplifier system to verify our theory. The configuration of the fiber amplifier system includes a filtered PRBS phase-modulated single-frequency fiber laser, a three-stage pre-amplification stage, and a counter-pumping main stage. 2.5 kW output power has been achieved with an FWHM linewidth of 9.63 GHz. The experimental results fully demonstrate the proposed model in this paper. The present work provides a practical and reliable solution for the realization of localized high-power, high-beam-quality narrow-linewidth fiber lasers based on PRBS phase modulation.

This paper is structured as follows. Section 2 describes the SBS dynamics based on filtered and amplified PRBS phase modulation in a fiber system with both active fiber and passive fibers. Section 3 simulations predict the SBS suppressing capability for filtered PRBS phase modulation in an active fiber spliced passive fiber model under the optimal modulation parameters. In Section 4, a narrow linewidth YDF amplifier based on filtered PRBS phase modulation is experimentally illustrated. In Section 5, the simulation results obtained from the active fiber and passive fiber model and the previously proposed passive fiber model are compared with the experimental results. Section 6 concludes the paper.

2. Theoretical framework

The PRBS phase modulation and SBS suppression scheme is shown in Fig. 1. A single-frequency seed laser with a wavelength of 1067.9 nm is externally modulated by an electro-optic phase modulator (EOPM). A PRBS waveform is filtered by a low-pass filter and amplified by an RF wideband amplifier and then drives the EOPM. The modulated lightwave is then amplified in an active fiber with length L_YDF, the output of which is spliced to a passive fiber with length L_GDF. Normally, the active fiber refers to the ytterbium (Yb)-doped double-clad fiber, and the passive fiber refers to the large-mode-area (LMA) germanium-doped fiber (GDF). The spectral broadening is controlled by the pattern length and the clock rate of the PRBS, the cutoff frequency of the filter, and the RF power, related to the modulation voltage and depth. In order to accurately describe the effect of this modulation scheme on high-power fiber systems, the amplifier gain is required to be modeled accurately since it alters the distribution of the laser power along the fiber length, contributing to the Brillouin Stokes power and leading to an effective length that is even shorter than the physical fiber length [3]. We first consider the SBS dynamic in this active fiber, and then in the passive fiber. The SBS occurring in the pre-amplifier can be ignored.Our model captures the propagation of four waves in a spliced fiber of total length: L= L_YDF+ L_GDF. We use pure counter pumping to improve the SBS suppression [19], so the $P_p^ - $ is the backward pumping power applied to the cladding of the active fiber. The optical waves propagating in the active fiber and passive fiber core can be represented by the electric field $A_L^ + $ of the forward propagating laser and the electric field $A_S^ - $ of the backward propagating Brillouin Stokes wave [12]. The acoustic wave that couples the laser and Brillouin Stokes wave is propagating in the forward direction [3]. In the main stage of a fiber amplifier system, the laser from the initial end is amplified in the active fiber and then delivered through a passive fiber before transmitting from the endcap. Throughout this process, SBS occurs during both the laser amplifying and transmitting process. In this scenario, the SBS dynamic theoretical model should be considered by the rate equation associated with the three-wave system. According to the 0.8 ms [19,26] upper-state lifetime and commonly 10-ns-level SBS build-up time [17], we consider the laser establish process to be steady state. Thus, the Stokes evolution in the amplifier can be calculated in a similar time-scale as in passive fiber [3].

 

Fig. 1. Schematic diagram of the PRBS phase modulation and fiber system. z = 0 and z = L denote the input and output end of both active and passive fiber sections. EOPM, electro-optic phase modulator.

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For the steady-state equations, the laser light output is considered to be constant according to the number of particles ${N_2}$ in the upper energy level of the gain fiber reaches a steady state. The transformation relationship between optical power and electric field in an optical fiber is given by: ${P_i}(z,t) = \frac{1}{2}{\varepsilon _0}c{n_L}{A_{eff}}{|{{A_i}(z,t)} |^2}$, $i = S,L$. We transform the electric field into the corresponding power. $P_L^ + $ is the laser power [27]. The rate equations in this system are represented by Eqs. (1 )-(3) as follows:

Equation (3) describes the particle number concentration of the upper energy level of the gain medium Yb according to the fiber length. The effect of Stokes on ${N_2}$ [28,29] is neglected because the calculated Stokes optical power is much less than the laser power [3]. ${\Gamma _p}$ and ${\Gamma _L}$ are the overlap factors of the YDF for the pump and laser, respectively. ${\sigma _{ap}}$ and ${\sigma _{ep}}$ are the absorption and emission cross sections of the backward pumping light, while ${\sigma _{aL}}$ and ${\sigma _{eL}}$ are the absorption and emission cross sections of the laser, respectively. N is Yb³⁺ doping concentration in the fiber core, $\tau $ is the average lifetime of the upper energy level of Yb³⁺ particles, h is Planck's constant, A is the cross-sectional area of the core that is equal to the effective mode area ${A_{eff}}$ in this simulation. ${v_p}$ and ${v_L}$ are the frequencies of the pumped light and the laser, respectively. To calculate the laser output power when ${N_2}$ is in a steady state, we iterated 20 times using the Euler method to amplify the laser power obtained in the gain fiber. Among them, the power of the amplified laser is constant along the passive fiber in a steady state. The losses of the laser and the pump light in the fiber are neglected.

Here we further consider the SBS dynamics of the phase-modulated lightwave occurring in the amplifier. We extend the time-dependent three-wave coupled nonlinear system used in [17] to include the laser gain induced by the Yb-ions. We also neglect the effect on the ${N_2}$gain term of the laser field in the YDF caused by the Stokes. To simplify the discussion, it is assumed here that the fiber is polarization-preserving, the incident laser and Stokes light are linearly polarized, and the polarization direction coincides with the fast axis of the fiber [3]. The acoustic wave is assumed to be a purely longitudinal pressure wave represented by a variation in the density of the medium (ρ). Our model employs the slowly varying envelope approximation, ignoring group velocity dispersion and background propagation losses. The temporal and spatial evolution of the laser field, Stokes field, and acoustic field propagating within the core of the gain fiber are represented by the following coupled system:

In the discrete case, the quantity $f(z,t) = \sqrt {{{{n_\textrm{L}}Q} / {{{(\varDelta t)}^2}c}}} S(z,t)$ in RHS of Eq. (6) represents the initiation of the SBS process from Langevinian noise [30]. Where $S(z,t)$ is a standard complex Gaussian random distribution obeying a mean of 0 and a variance of 1, $\varDelta t$ is the time step, and ${n_L}$ is the refractive index of the medium. Q is a parameter describing the intensity of the fluctuations and is generally given by the thermodynamic process [12]: $Q = 2{k_B}{T_c}{\rho _0}{\Gamma _B}/v_A^2{A_{eff}}$. For a CW input laser, the Brillouin Stokes power still fluctuates in time. Where ${\Omega _B} = 2{n_L}{v_A}\omega /c$ denotes (resonant acoustic angular frequency) the intrinsic resonant frequency of the medium, the phonon decay rate is [12]${\Gamma _B} = 2\pi \varDelta {v_B} = 2\pi /17.5n{s^{ - 1}} = 2\pi \times 57.1 \times {10^6}{s^{ - 1}}$, and for Brillouin scattering, the phonon decay rate can be expressed as the reciprocal of the phonon lifetime: ${\Gamma _B} = 1/{\tau _B}$. ${k_B}$ is Boltzmann’s constant, ${T_c}$ is the temperature in the fiber core, and ${A_{eff}}$ is the fiber effective mode area, respectively. Furthermore, $\alpha = {\Gamma _B}/2{\Omega _B}$, $\chi = {\varepsilon _0}{\gamma _e}{q^2}/2{\Omega _B}$, respectively. The laser angular frequency $\omega = 2\pi c/\lambda $ and the acoustic wave number $q = 4\pi /\lambda $. The definition of some other parameters is like this: ${\rho _0}$ is the mean density of the fiber medium, ${\gamma _e}$ is the electrostrictive constant, ${\varepsilon _0}$ is the dielectric constant, ${n_L}$ is the core refractive index, c is the velocity of light in vacuum, and ${v_A}$ is the speed of the acoustic wave, respectively.

For the passive fiber, the time-dependent coupled three-wave SBS interaction equations are the same as our previous work [17]. Note that backward Stokes is generated and accumulates throughout the entire fiber length L. The initial and boundary conditions are ${A_S}(L,t) = 0$ and ${A_S}(z,0) = 0$. The boundary condition and the initial conditions for the laser electric field at the input end of the active fiber and passive fiber (at location $\textrm{z = 0}$): ${A_L}(0,t) = A_L^0{e^{i\varphi (t)}}$, ${A_L}(z,0) = \sqrt {{{2P_L^ + (z)} / {{A_{eff}}{\varepsilon _0}c{n_L}}}} $, where the input laser amplitude $A_L^0 = \sqrt {{{2P_L^0} / {{A_{eff}}{\varepsilon _0}c{n_L}}}} $, where $P_L^0$ is the input seed power identical to that of our experiment. PRBS phase modulation can be introduced by the optical phase $\varphi (t)$ encoded by the PRBS waveform. Finally, the initial and boundary conditions for the phonon field are $\rho (z,0) = \sqrt {{{{n_L}Q} / {c{\Gamma _B}}}} S(z,0)$ and $\rho (0,t) = \sqrt {{{{n_L}Q} / {c{\Gamma _B}}}} S(0,t)$.

3. Simulation

For a fiber system consisting of the active and passive fiber, we solve the time-dependent coupled three-wave SBS interaction equations starting with steady-state assumed for fiber system. First of all, by solving Eqs. (1 )-(3), the ${N_2}$ of the amplifier at steady state is obtained. Then, the coupled three-wave SBS interaction equations Eqs. (4 )-(6) and the equations for the passive fiber shown in [17] are solved numerically using the Euler method along the characteristic lines $dz/dt ={\pm} c/{n_L}$ [12]. The parameters and the values we use are shown in Table 1.

Table 1. Simulation parameters

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In our simulation, the total fiber length L is 15.17 m, and the single fiber transit time is ${n_L}L/c$ = 73.3 ns. The Stokes power at location z=0 versus time is evaluated by the time-averaged reflectivity, which is calculated for at least five transit times, and discarding the initial four transits to obtain a relatively stable value [3]. The time-averaged reflectivity is defined as $R = {{\left\langle {|{{A_S}(0,t)} |} \right. {^2} \rangle } / {\left\langle {|{{A_L}(L,t)} |} \right. {^2} \rangle }}$, where the brackets indicate time-averaging at least one transit time. Increasing the incident light power, the Stokes reflectivity will gradually increase exponentially. When the reflectivity reaches a certain threshold value ${R_{th}}$, the incident light power is considered to be the SBS threshold power under the current system. In this work, ${R_{th}}$ is defined as 0.1‰.

We use the normalized SBS threshold to quantify the SBS suppression (or the SBS threshold enhancement factor). This is the ratio of the SBS threshold with phase modulation to the unmodulated SBS threshold. In our previous work [17], it is known that the SBS suppression non-monotonically depends on parameters of the filtered and amplified PRBS, such as pattern, modulation depth, and the ratio of the cutoff frequency of the low-pass filter to the modulation rate. Our previous simulation results [17] (SBS considered in passive fiber system) show that when the RMS modulation depth ${k_{RMS}}$ is 0.56π and the ratio of filter cutoff frequency to clock rate ${r_c} = {{{f_{co}}} / {{f_{cr}}}}$= 0.54 the normalized SBS threshold reaches its maximum value with PRBS9. Here, in the active fiber spliced passive fiber model with introducing the YDF structure, we further investigate the dependence of normalized SBS threshold on the clock rate at ${{{f_{co}}} / {{f_{cr}}}}$= 0.54 and ${k_{RMS}}$= 0.56π for different bit patterns, as shown in Fig. 2. It is clear from this figure that the n = 9 pattern is superior to other patterns within ${f_{cr}}$∈ (0, 11.32) GHz for ${k_{RMS}}$= 0.56π and ${r_c} = {{{f_{co}}} / {{f_{cr}}}}$= 0.54.

 

Fig. 2. Normalized SBS threshold with low-pass filtered (${\textrm{r}_\textrm{c}}\textrm{ = 0}\textrm{.54}$) and ${k_{RMS}}$ = 0.56 π vs. clock rate for n = 5, 7, 9, and 10 for clock rates from 0 GHz to 11.32 GHz.

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Based on the fiber system introduced in this work, we calculate the normalized SBS threshold (for PRBS9) as a function of ${k_{RMS}}$ and ${r_c}$, which is shown in Fig. 3(a). The value of ${f_{cr}}$ is chosen to be 5 GHz. The local maxima for normalized SBS threshold follow a significant structure. In order to examine the dependence of optical linewidth to the modulation parameters, we calculate the RMS linewidth of the filtered and amplified PRBS phase modulated optical spectra versus ${k_{RMS}}$ and ${r_c}$, shown in Fig. 3(b). Normally, a larger linewidth does not necessarily correspond to a higher SBS threshold. Therefore, we calculate the ratio of the normalized SBS threshold to $\varDelta {v_{RMS}}$ versus ${k_{RMS}}$ and ${r_c}$, shown in Fig. 3(c). Only one global maximum can be found at (0.54, 0.56 π) (marked in purple circles), which indicated a maximized SBS suppression in unit optical linewidth. The optimum modulation parameters in the active and passive fiber system are obtained and found to coincide with those in our previous work (i.e., only passive fibers are considered) [17].

 

Fig. 3. The normalized SBS suppression vs. ${r_c}$ and ${k_{RMS}}$ for PRBS9. (b) The RMS linewidth vs. ${r_c}$ and ${k_{RMS}}$. (c) The ratio of normalized SBS threshold to RMS linewidth vs. ${r_c}$ and ${k_{RMS}}$. The local maxima is marked in purple circle.

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According to the above simulation results, at the optimal modulation ratio, our simulations use two different sets of modulation parameters (Group I: clock rate 7.037 GHz, low-pass filter 3.8 GHz, ${k_{RMS}}$= 0.56π, PRBS9; Group II: clock rate 11.32 GHz, low-pass filter 6 GHz, ${k_{RMS}}$= 0.56π, PRBS9). The time-averaged distribution of the three waves along the fiber length for at least one transit time when reaching the SBS threshold is obtained, as shown in Fig. 4. The purple line shows the distribution of the applied counter pump power in the active fiber along the fiber length. The orange line represents the laser power in the active fiber spliced passive fiber. The blue line shows the Stokes power accumulated in the whole spliced fiber. It is shown that in counter pumping, the laser power grows slowly at the seed input side, increases dramatically in the last short section before reaching the end of the active fiber, and finally passes through the passive fiber to maintain a high-power output. In contrast, the backward propagating Brillouin Stokes power starts from noise at the output of the passive fiber end and is amplified when it reaches the initial end of the active fiber. The results show that under the first set of modulation parameters, the SBS threshold is exactly reached when the applied counter pump power is 1970 W. At this time, the output power of the laser at z=L is ∼1701 W, and the Stokes power generated at z=0 is ∼ 0.176 W. Under the second set of modulation parameters, the SBS threshold is exactly reached when the applied counter pump power is 3101 W. At this point, the laser output power at z=L is ∼2659 W, and the Stokes power generated at z=0 is ∼ 0.296 W.

 

Fig. 4. Counter-pumped power (purple line), laser power (orange line), and Stokes power (blue line) distributions along the fiber length for two sets of modulation parameters (a) Group I and (b) Group II.

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4. Experimental setup

The configuration of the experiment system is illustrated in Fig. 5. We use NKT Photonics BASIK Y10 series as the seed laser (FWHM linewidth is of kHz-level), operating at 1067.9 nm. The two sets of experiments use different modulation parameters (Group I and Group II), the same as our simulations. For Group II, the phase modulation is PRBS9 with a clock rate of 11.32 GHz. The RF signal generated by PRBS passes through a 6 GHz low-pass filter, and then is amplified by a 20 GHz bandwidth RF amplifier. And the amplified RF signal is applied to the $\textrm{LiNb}{\textrm{O}_3}$ electro-optic phase modulator (EOPM). The FWHM optical linewidth of the seed laser is broadened to 9.63 GHz. The ratio of the clock rate to the FWHM of the low-pass filter is ${r_c}$= 0.54 while the modulation depth is ${k_{RMS}}$= 0.56π to achieve the optimal SBS suppression effect. For Group I, we perform the same experimental operations. The unfiltered (black line) and filtered (red line) RF power spectral density (PSD) of PRBS signal for both Groups are measured by spectrum analyzer (Rohde & Schwarz, FSW43), as shown in Fig. 6.

 

Fig. 5. Schematic of the PRBS phase modulated, low pass filtered, four-stage monolithic fiber amplifier. (MFA: mode-field adapter; LD: laser diode; YDF: ytterbium-doped fiber; CPS: cladding power stripper; PM: power meter; OSA: optical spectrum analyzer; EOPM: electro-optic phase modulator.)

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Fig. 6. PSD of PRBS9 RF signal with a clock rate of 7.037 GHz (black line) and filtered by a 3.8 GHz low-pass filter (red line). (b) PSD of PRBS9 RF signal with a clock rate of 11.32 GHz (black line) and filtered by a 6 GHz low-pass filter (red line).

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A group of circulators are placed after each stage to protect against back-reflected power. In the first three stages, we use the Nufern LMA-YDF -10/125-9 M gain fiber with an absorption coefficient of 5.7 dB/m@976 nm and a length of ∼3 m. The first and second stages are pumped by a 3 W laser diode at 976 nm via a (2+1) ×1 signal-pump combiner. The third stage is pumped by two wavelengths stabilized 27 W laser diode at 976 nm via a (2+1) ×1 signal-pump combiner. The backward light power is detected at the port 3 of circulator 4. The output power from the cladding power stripper (CPS1) is controlled to be 26.8 W to provide sufficient laser power for the main amplifier to prevent amplified spontaneous emission (ASE). The pre-amplified laser is then injected into an ∼11.8 m length homemade YDF with the core/cladding diameter of 20.2/400 µm through a mode-field adapter (MFA) and CPS1. The core/cladding diameters of the input and output fibers of the MFA are 10/125 µm and 20/400 µm, respectively. The main amplifier is counter-pumped by using thirty-five 976 nm pumps in a cascaded 7×1 combiner and a homemade (8+1) ×1 backward combiner scheme to improve gain saturation and increase the unmodulated SBS threshold. The amplified laser then passes through the backward combiner and the CPS2, and then outputs from the endcap. The majority of the laser beam is measured by a power meter (PM2). The temporal dynamics of the output laser are monitored by detecting light scattered from the PM2 using a photodetector (Thorlabs, PDA05CF2). The length of the 20/400 delivery fiber from the output of the seed laser through the circulator 4 to the input of the gain fiber is ∼ 2.7 m, and the length of the delivery fiber from the output of the gain fiber to the endcap is ∼ 3.4 m. The core/cladding diameters of the CPS1 are 20/400 µm and match the gain fiber.

5. Results and discussion

In the above two sets of experiments using different modulation parameters (Group I and Group II), the performances of the narrow-linewidth fiber amplifier with the counter-pumping scheme are investigated and the validation of the above theoretical model is completed.

For Group I, with the increase of pump power, the output power increases almost linearly with slope efficiencies of ∼86.1%, as shown in Fig. 7(a). The output power increases from ∼ 32 W to 1690 W when the pump power is increased to 1963 W. At the same time, the monitored backward scattering power steadily increases and finally reaches 0.115 W, as shown in Fig. 7(b). At maximum power, the measured reflectivity is 0.068‰, which is not up to the SBS threshold, and is very close to the theoretically calculated SBS threshold result of 1701 W at a pump power of 1970 W, as shown in Fig. 4(a). In order to ensure the safety of the system, the pump power is not further increased. The FWHM linewidth of the seed spectrum measured by an optical spectrum analyzer (OSA1, HyperFine, HN-8995-2, Lightmachinery) with a resolution of 0.8 pm after filtered PRBS modulation (red line) is 5.08 GHz, compared with the unmodulated seed spectrum (blue line), as shown in Fig. 8(a).

 

Fig. 7. Fiber amplifier output character. (a) Group I: output power as a function of pump power; (b) Group I: backward scattering power as a function of output power. (c) Group II: output power as a function of pump power; (d) Group II: backward scattering power as a function of output power.

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Fig. 8. (a) Group I: Optical spectra of seed laser with (red line) and without (blue line) filtered PRBS modulation. (b) Group II: Optical spectra of seed laser with (red line) and without (blue line) filtered PRBS modulation. (c) Backward light corresponding to different output powers. (d) Optical spectrum of three-stage pre-amplifier showing ASE suppression of better than 40 dB.

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For Group II, with the increase of pump power, the output power increases almost linearly with slope efficiencies of ∼ 86.0%, as shown in Fig. 7(c). When the pump power is increased to 2886 W, the output power increases from ∼ 32 W to 2500 W. At the same time, the monitored backward scattering power increases steadily and finally reaches 0.122 W, as shown in Fig. 7(d). At the maximum power, the measured reflectivity is 0.049‰, well below the SBS threshold. There is potential room for further power scaling, as shown in Fig. 4(b). The FWHM linewidth of the seed spectrum measured after the EOPM by the OSA1 after filtered PRBS modulation (red line) is 9.63 GHz and the RMS linewidth is 9.66 GHz, compared with the unmodulated seed spectrum (blue line), as shown in Fig. 8(b). And it is very close to the theoretically calculated RMS linewidth of 9.64 GHz. The power amplification process did not affect the FWHM linewidth. The backward output spectra of the seed laser and the amplifier at different output powers are measured at the third port of circulator 4 using the OSA2 (YOKOGAW AQ6370D) with a resolution of 0.02 nm, as shown in Fig. 8(c). The forward spectral content for fiber amplifier configurations is measured as shown in Fig. 8(d), with the three-stage pre-amplifier providing over 40 dB of ASE suppression at full power output.

The single frequency threshold of ∼15.5 W is measured using the same SBS threshold criterion, and the enhancement factor of ∼109 is attained at the maximum output power of 1.69 kW for Group I, and the SBS threshold is not yet reached, based on the close proximity to the simulation that yielded an enhancement factor of ∼110.5 when the SBS threshold is reached as shown in Fig. 9(b). For Group II, the enhancement factor of ∼161 is obtained at the maximum output power of 2.5 kW, with potential room for further power enhancement. The enhancement factor of ∼172.9 is obtained from the simulation when the SBS threshold is reached, as shown in Fig. 9(b).

 

Fig. 9. Simulations of (a) SBS threshold, and (b) RMS linewidth (orange line) of the optical spectra and normalized SBS threshold vs. clock rate, PRBS9, ${r_c}$=0.54 and ${k_{RMS}}$=0.56π in case of active fiber and passive fiber (red dots) and only passive fiber (blue dots), respectively. Results measured in the experiment (black dots), note that the SBS threshold is not reached at this point.

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The trends of the maximum normalized SBS thresholds obtained from the PRBS9 with ${{{f_{co}}} / {{f_{cr}}}}$= 0.54 and ${k_{RMS}}$= 0.56π for the active fiber spliced passive fiber model (red dots) with a seed power of 26.8 W and the passive fiber model (blue dots) with the clock rate (straight lines are the corresponding linear fitting curves) are plotted against the experimental results. We also calculate the RMS linewidth of this optimized filtered and amplified PRBS phase modulation with ${f_{cr}}$∈ (0, 12) GHz, as shown in Fig. 9(b) (orange line). It is shown that both the RMS linewidth of the optical spectra and the maximum normalized SBS threshold with the optimized parameters increase linearly with ${f_{cr}}$. Among them, the variation curves of SBS threshold with clock rate for both models are shown in Fig. 9(a). It is very consistent with the experimental results. In the figure, the pump power is not further increased to ensure the absolute safety of the system when the experimental results are closer to the threshold.

6. Conclusion

In summary, the previously proposed SBS three-wave coupling generation in the passive fiber model can be optimized into an active fiber spliced passive fiber model by introducing a YDF structure, using the optimal parameters of PRBS phase modulation obtained by the numerical method, which can accurately predict the high-power narrow linewidth fiber under PRBS modulation in a system's power scaling capability when limited by the SBS effect. A narrow-linewidth high-power counter-pumping amplifier based on filtered PRBS phase modulation is experimentally designed. A homemade YDF with a core/cladding diameter of 20.2/400 µm and a homemade (8+1) × 1 backward combiner are used to achieve a power output of 2.5 kW at an FWHM linewidth of 9.63 GHz. The reflectivity is 0.049‰ when reaches the maximum output power, which indicates SBS free at this power level. The comparison between experimental results and simulation shows that the maximum normalized SBS threshold increases linearly with the PRBS clock rate under the selection of optimal parameters. The enhancement factor of active fiber spliced passive fiber is obviously more realistic than that of the enhancement factor considering only the energy-transmitting fiber to fit a curve with a higher linear slope, which is a better fit to the experimentally obtained results. The SBS threshold results obtained from the simulation for active fiber spliced passive fiber are significantly better than that of the case where only the passive fiber is considered. It is possible to accurately predict the reverse pumping optical power that needs to be applied to the complete fiber system, the resulting Stokes power, and thus the expected power output. Interpreting the power amplification capability of the laser system is highly instructive in experiments. This model is also applicable to the prediction of SBS generation and the build-up process in active fiber spliced passive fiber structures with different modulation schemes.

Appendix

In this appendix, the recent progress of high-power narrow linewidth fiber lasers based on PRBS phase modulation is shown in Table 2.

Table 2. Progress of high-power narrow-linewidth fiber lasers based on PRBS modulation

View Table | View all tables in this article

Funding

Youth Innovation Promotion Association of the Chinese Academy of Sciences (2020252); Transformation Program of Scientific and Technological Achievements of Jiangsu Province (BA2020004).

Acknowledgments

We would like to thank Prof. Chunlei Yu's research team for providing the active fiber.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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