Theoretical study of narrow-linewidth hybrid rare-earth-Raman fiber amplifiers

Wei Liu; Yu Miao; Pengfei Ma; Pu Zhou; Zongfu Jiang

doi:10.1364/OE.27.014523

1. Introduction

Raman fiber lasers (RFLs) have drawn extensive interests due to their wavelength agility, high efficiency, and power scalability [1,2]. It could operate at arbitrary wavelength in the transmission window of optical fibers, when the wavelengths of the pump and the signal lasers satisfy the Raman-Stokes shift [3]. One of the key components in RFLs is the high-power wavelength division multiplexer (WDM), which combines or splits the pump and signal lasers. Constrained by the power handling ability of commercially available WDM, the reported output powers of RFLs are limited to a few hundred watts [4–6]. To avoid the restriction of the WDM and further scale RFLs to over kilowatt, hybrid rare-earth–Raman fiber amplifiers (H-RFAs) are proposed, which are capable of achieving effective Raman amplification based on standard Yb-doped fiber amplifiers (YDFAs) [7–12]. To achieve Raman amplification in a high-power YDFA, multiple seed lasers are inserted, and the wavelength separations of the seed lasers are close to the Raman-Stokes shift. After the proof-of principle experiment in 2014 [7], L. Zhang et al. reported the first demonstration of a kilowatt level RFL operating at 1120 nm later in the same year [9]. Q. Xiao et al. reported a 3.89 kW RFL at 1123 nm in 2016 [11], and up to 6 kW output has been predicted recently [12], which proves that the hybrid rare-earth–Raman architecture has a good perspective for the power scaling of RFLs.

Apart from the output power, the linewidth is another important issue in the wavelength-sensitive applications of RFLs, especially for the nonlinear frequency conversion processes [13,14]. However, most of the reported H-RFAs have broadband linewidth, and H-RFAs seem to be unsuitable to achieve narrow-linewidth operation. It is natural to speculate that the spectral broadening phenomenon is inevitable in H-RFAs according to the previous studies [15–18]. On the one hand, for fiber amplifiers based on the multi-longitudinal mode fiber seeds, the linewidth of the signal laser would get broadened due to the self-phase modulation (SPM) and the four-wave-mixing (FWM) effects, which have been well explained in YDFAs or RFLs [15–17]. On the other hand, even when applying the single-frequency seed, the linewidth of the signal laser also got broadened in co-pumped single-frequency RFAs [18].

In this work, we aim to clarify the spectral broadening phenomenon and propose possible techniques to achieve narrow-linewidth operation in H-RFAs theoretically. Through numerical analysis of the spectral evolution properties and gain dynamics in H-RFAs, the mechanisms for the spectral broadening phenomena are clarified, and the design strategies to maintain narrow-linewidth operation are given.

2. Theory for numerical modeling

The rare-earth doped fiber in most reported H-RFAs is ytterbium-doped fiber (YDF). Thus, we construct the numerical model based on the YDF here, and the related theoretical analysis could also be promoted for other types of rare-earth doped fibers. The spectral evolution in fiber amplifiers could be mainly divided into the energy absorption from pump laser to signal laser and the energy transformation among different spectral components. Consequently, the basic approach to model the spectral evolution in fiber amplifiers is the joint analysis of the two energy conversion processes. As for YDFAs, the two energy conversion processes could be described through rate equation and nonlinear propagation equation, respectively [16,17,19]. Comprehensive numerical analysis of the two equations has shown that the spectral broadening property in YDFAs is mainly determined by the temporal properties of seed laser. Specifically, the temporal stable seed laser can maintain the linewidth while the temporal unstable seed laser would induce spectral broadening during amplification [20]. As for RFAs, the two energy conversion processes could be included through the coupled amplitude equations. Numerical study based on the coupled amplitude equations has shown that the intensity fluctuations in the pump laser would transfer into the signal laser and lead to the spectral broadening in co-pumped single-frequency RFAs [21]. Therefore, both the temporal stabilities of the pump and the seed lasers might impact the spectral evolution properties in fiber amplifiers.

Figure 1 shows the typical structure diagram of an H-RFA. The Raman seed (signal, λR) laser and pump laser of Raman gain (λs) are combined through a WDM, and further coupled into the main amplifier through a pump combiner together with the pump laser of rare-earth gain (λp). The pump lasers of rare-earth gain and Raman gain are defined by initial pump laser and Raman-pumped laser for short, respectively. In the first half of the YDF, initial pump laser is mainly converted into Raman-pumped laser through rare-earth gain. In the second half, initial pump power and Raman-pumped laser would mainly convert into Raman signal laser through rare-earth gain and Raman gain, respectively [7,8]. Thus, both the rare-earth gain and Raman gain contribute to the amplification of Raman signal laser in H-RFAs.

Fig. 1 Structure diagram of an H-RFA. WDM: wavelength division multiplexer; ISO: isolator; YDF: ytterbium-doped fiber; CPS: cladding pump stripper.

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To combine the contributions of rare-earth gain and Raman gain, and include the temporal properties of pump and seed lasers, the numerical model for H-RFAs is established by incorporating time-dependent rate equations and coupled amplitude equations. The set of unidirectional temporal-spatial equations could be expressed as

\frac{\partial N_{2}}{\partial t} = - \frac{N_{2}}{τ} + \frac{Γ_{p} λ_{p}}{h c A_{e f f}} (σ_{p}^{a} N_{1} - σ_{p}^{e} N_{2}) P_{p} + \frac{Γ_{s} λ_{s}}{h c A_{e f f}} (σ_{s}^{a} N_{1} - σ_{s}^{e} N_{2}) P_{s} + \frac{Γ_{R} λ_{R}}{h c A_{e f f}} (σ_{R}^{a} N_{1} - σ_{R}^{e} N_{2}) P_{R}

\frac{\partial P_{p}}{\partial z} + \frac{1}{v_{p}} \frac{\partial P_{p}}{\partial t} = Γ_{p} (σ_{p}^{e} N_{2} - σ_{p}^{a} N_{1}) P_{p} - α_{p} P_{p}

\begin{matrix} \frac{\partial A_{s}}{\partial z} + \frac{1}{v_{s}} \frac{\partial A_{s}}{\partial t} + \frac{i β_{2 s}}{2} \frac{\partial^{2} A_{s}}{\partial t^{2}} = i γ_{s} [{| A_{s} |}^{2} + (2 - f_{R}) {| A_{R} |}^{2}] A_{s} \\ - \frac{g_{R}}{2} {| A_{R} |}^{2} A_{s} - \frac{α_{s}}{2} A_{s} + \frac{1}{2} Γ_{s} (σ_{s}^{e} N_{2} - σ_{s}^{a} N_{1}) A_{s} \end{matrix}

\begin{matrix} \frac{\partial A_{R}}{\partial z} + \frac{1}{v_{R}} \frac{\partial A_{R}}{\partial t} + \frac{i β_{2 R}}{2} \frac{\partial^{2} A_{R}}{\partial t^{2}} = i γ_{R} [{| A_{R} |}^{2} + (2 - f_{R}) {| A_{s} |}^{2}] A_{R} \\ + \frac{g_{R}}{2} \frac{λ_{s}}{λ_{R}} {| A_{s} |}^{2} A_{R} - \frac{α_{R}}{2} A_{R} + \frac{1}{2} Γ_{R} (σ_{R}^{e} N_{2} - σ_{R}^{a} N_{1}) A_{R} \end{matrix}

where, the subscripts p, s, and R stand for initial pump laser, Raman-pumped laser, and Raman seed or signal laser, respectively; σ^a and σ^e are the corresponding absorption and emission cross section; v is the group velocity; N₁ and N₂ are the ion densities in the ground state or excited state, N₁ + N₂ = N₀, N₀ is the dopant density in the fiber core; Г is the power overlap factor; τ is the lifetime of the excited state; λ is the wavelength, h is the Planck constant, c is the laser velocity and A_eff is the effective mode area of the fiber; β₂ is the second order dispersion coefficient; γ is the nonlinear coefficient; g is the Raman gain coefficient and α is the loss coefficient; f_R represents the fractional contribution of the delayed Raman response to nonlinear polarization. An iterative solution of Eqs. (1)-(4) is obtained numerically using the finite-different time-domain (FDTD) method with the initial conditions.

To construct the initial inserted lasers which have different temporal stabilities, two representative cases are considered. One is temporal stable laser, such as the multi-tone laser by phase-modulation of single-frequency laser for spectral broadening (defined by phase-modulated single-frequency laser for short here). The other is temporal unstable laser which induces strong intensity fluctuations, such as multi-longitudinal mode fiber oscillator [22,23]. Here, we apply the polarized thermal radiation model to characterize lasers which induces strong intensity fluctuations [24]. The corresponding optical field could be expressed as

{\tilde{A}}_{p} (ω) \propto \exp (- 2 I n (2) \frac{ω^{2}}{Ω_{L}^{2}}) \exp (i φ (ω))

where Ω_L is the full width at half maximum (FWHM) linewidth and the random spectral phase φ(ω) obeys the uniform probability distribution between –π to π.

Figures 2(a) and 2(b) illustrate the typical spectral and temporal properties of phase-modulated single-frequency laser. As shown in Fig. 2(a), there exists many discrete single-frequency peaks in the spectrum. The root-mean-square (RMS) linewidth is about 0.5 GHz, which could ensure about ten times enhancement of the stimulated Brillouin scattering (SBS) threshold compared to the single-frequency laser [25]. As shown in Fig. 2(b), the temporal property of the phase-modulated single-frequency laser is stable.

Fig. 2 The temporal and spectral properties of the phase-modulated single-frequency laser: (a) spectral intensity; (b) temporal evolution.

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Figures 3(a) and 3(b) illustrate the typical spectral and temporal properties for the laser which induces strong intensity fluctuations. Here, the FWHM linewidth is set to be 0.1 nm in Eq. (5) and the average power is 1 W. As shown in Fig. 3(a), the spectral shape of the laser is Gaussian. As shown in Fig. 3(b), there exists strong intensity fluctuations in the laser, and the maximum power is over ten times of the average power. The normalized standard deviation (NSD) of the intensity fluctuations is about 1.0.

Fig. 3 The temporal and spectral properties of the laser which induces strong intensity fluctuations: (a) spectral intensity; (b) temporal evolution.

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In the previous experimental report, more than 800 W single-frequency fiber laser has been obtained [26]. Considering an SBS threshold enhancement of ten times, it is possible to achieve 2 kW narrow-linewidth fiber laser based on the phase-modulated single-frequency seed laser shown in Fig. 2. In the following simulations, the temporal stable Raman seed laser or Raman-pumped laser is set to be the phase-modulated single-frequency laser shown in Fig. 2, and the initial pump laser or Raman-pumped laser which induces strong intensity fluctuations is set to be the polarized thermal radiation shown in Fig. 3.

3. Spectral evolution in H-RFAs

Based on the above model, the spectral evolution in H-RFAs could be analyzed. The major simulation parameters of the H-RFA are shown in Table 1. The core and inner cladding diameters of the YDF are 20 μm and 400 μm, respectively. The absorption coefficient of the initial pump laser is set to be 1.5 dB/m, and the lengths of the active fiber and passive fiber are 15 m and 1 m, respectively. For simplicity, the loss coefficient is set to be the same at different wavelength, and the insertion loss of the pump laser is omitted. Due to that our main purpose is to investigate the spectral evolution of the H-RFAs while not power scaling limitations, we ignore the SBS effect and focus on the case well below the SBS threshold.

Table 1. Major simulation parameters for the H-RFAs

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3.1 Influence of temporal stability of the Raman-pumped laser

To analyze the influence of temporal stability of the Raman-pumped laser on the H-RFA, two types of fiber lasers which have different temporal stabilities are considered, i.e. the multi-longitudinal mode fiber oscillator and the phase-modulated single-frequency laser. The initial pump laser is set to be temporal stable laser diode operating at 976 nm, and the Raman seed laser is set to be temporal stable phase-modulated single-frequency laser in this section.

Figures 4(a)-4(c) illustrate the power distribution, the temporal and spectral properties of the Raman signal light in the H-RFAs, respectively, when applying the multi-longitudinal mode fiber oscillator as the Raman-pumped laser. As shown in Fig. 4(a), after the fiber length of about 4.5 m, the power of the Raman-pumped laser begins to decrease and the power of the Raman signal laser increases quickly. The powers of the unabsorbed initial pump laser, the Raman-pumped laser, and the Raman signal laser at the output port are about 14 W, 100 W, and 1.64 kW, respectively. Thus, most of the initial pump laser is converted into the Raman signal laser, and the corresponding power conversion efficiency is about 82%. As shown in Fig. 4(b), despite that the Raman seed laser is stable, there exists strong intensity fluctuations in the output Raman signal laser. The average output power is about 1.64 kW, while the maximum power is over 15 kW. In addition, the NSD of the intensity fluctuations is about 1.18, which is bigger than that of the Raman-pumped laser. As shown in Fig. 4(c), the narrow-linewidth part in Raman signal laser is almost covered by the background noise.

Fig. 4 The power distribution, the temporal and spectral properties of the Raman signal light when applying the multi-longitudinal mode fiber oscillator as the Raman-pumped laser: (a) power distribution; (b) temporal evolution; (c) corresponding optical spectrum.

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To further demonstrate the spectral evolution in the H-RFA, we obtain the output spectra and powers of the Raman signal laser at different pump powers. Figure 5(a) illustrates the output spectra of the Raman signal laser when the pump powers are 0.2 kW, 0.4 kW, 0.6 kW, 0.8 kW, and 1.0 kW, respectively. Here, each optical spectrum is shifted by 2 GHz at different pump powers to avoid the overlap of the spectra in the figure. As shown in Fig. 5(a), the narrow-linewidth part of the Raman signal laser would decrease and only the background noise in the sideband is amplified along with the increase of the pump power. Based on the output spectra and the total output power, the ratio of the narrow-linewidth part in the Raman signal laser could be calculated at different pump powers. Here, the ratio of the narrow-linewidth part is calculated through dividing the integrated spectral intensity ranging from −1 GHz to 1 GHz near the center frequency by the total spectral intensity. As shown in Fig. 5(b), the total output power would increase along with the pump power. However, when the pump power is over 0.3 kW, the ratio of the narrow-linewidth part in the Raman signal laser would always be below 10%. Therefore, when applying the multi-longitudinal mode fiber oscillator as the Raman-pumped laser, the Raman seed laser could get effective amplification, while the narrow-linewidth operation could not be maintained in the H-RFA.

Fig. 5 The output spectra and powers of the Raman signal laser at different pump powers: (a) output spectra; (b) output powers and ratios of the narrow-linewidth part.

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Figures 6(a)-6(c) illustrate the power distribution, the temporal and spectral properties of the Raman signal light in the H-RFA, when applying the phase-modulated single-frequency laser as the Raman-pumped laser. As shown in Fig. 6(a), the powers of the unabsorbed pump laser, the Raman-pumped laser, and the Raman signal laser at the output port are about 14 W, 32 W, and 1.71 kW, respectively. The power conversion efficiency here is a little higher than that in Fig. 4(a). As shown in Fig. 6(b), the temporal property of the Raman signal laser is stable. As shown in Fig. 6(c), the spectral shape of the output Raman signal laser is identical to the Raman seed laser (shown in Fig. 2(a)).

Fig. 6 The power distribution, the temporal and spectral properties of the Raman signal light when applying the phase-modulated single-frequency laser as the Raman-pumped laser: (a) power distribution; (b) temporal evolution; (c) corresponding optical spectrum.

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To further demonstrate the spectral evolution in the H-RFA, we obtain the output spectra and powers of the Raman signal laser at different pump powers. Figure 7(a) illustrates the output spectra of the Raman signal laser when the pump powers are 0.2 kW, 0.4 kW, 0.6 kW, 0.8 kW, and 1.0 kW respectively. Here, each optical spectrum is also shifted by 2 GHz at different pump powers to avoid the overlap of the spectra in the figure. As shown in Fig. 7(a), the narrow-linewidth part of the Raman signal laser gets effectively amplified. Figure 7(b) illustrates the total power of the Raman signal laser and the ratio of the narrow-linewidth part at different pump powers. As shown in Fig. 7(b), the total output power would increase along with the pump power, and the ratio of the narrow-linewidth part in the Raman signal laser is always over 99%. Therefore, when applying the temporal stable laser as the Raman-pumped laser, the narrow-linewidth amplification could be achieved in the H-RFA.

Fig. 7 The output spectra and powers of the H-RFA at different pump powers: (a) output spectra; (b) output powers and ratios of the narrow-linewidth part.

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3.2 Influence of the temporal stability of the initial pump laser

To analyze the influence of temporal stability of the initial pump laser on the H-RFA, contrast simulation is conducted by assuming that there exists strong intensity fluctuations in the initial pump laser operating at 976 nm. The Raman-pumped laser and the Raman seed laser are both set to be temporal stable phase-modulated single-frequency lasers in the section.

Figures 8(a)-8(c) illustrate the power distribution, the temporal and spectral properties of the Raman signal light in the H-RFA, when there exists strong intensity fluctuations in the initial pump laser operating at 976 nm. As shown in Fig. 8(a), the power distribution is similar to that shown in Fig. 4(a), and the output power of the Raman signal laser is about 1.71 kW. As shown in Fig. 8(b), despite that there exists strong intensity fluctuations in the initial pump laser, the output Raman signal laser could still keep relatively stable over time. In addition, the NSD of the intensity fluctuations in the output laser is about 0.015, which is significant smaller than that in the initial pump laser. As shown in Fig. 8(c), the spectral shape of the output Raman signal laser is also identical to the Raman seed laser (shown in Fig. 2(a)). Therefore, even when there exists strong intensity fluctuations in the initial pump laser operating at 976 nm, narrow-linewidth amplification could be achieved in the H-RFA.

Fig. 8 The power distribution, the temporal and spectral properties of the Raman signal light when there exists strong intensity fluctuations in the initial pump laser operating at 976 nm: (a) power distribution; (b) temporal evolution; (c) corresponding optical spectrum.

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Apart from diode-pumped scheme, tandem-pumped scheme provides another effective way to achieve high-power fiber lasers [27–29], which offers another route to generate high-power Raman-pumped laser. In typical tandem-pumped fiber amplifiers, the laser diode is replaced by multi-longitudinal mode fiber oscillator operating at 1018 nm [29]. In this case, the Raman gain would also contribute to the energy conversion from the initial pump laser to the Raman-pumped laser. In commercial YDF, the absorption coefficient of the pump light at 1018 nm is about 1/20 of that at 976 nm, thus much longer YDF is required in the typical tandem-pumped fiber amplifiers. For better comparisons with the diode-pumped scheme and avoiding the change of the fiber length, the absorption coefficient of the active fiber which is suitable for tandem-pumped scheme is assumed to be 1.5 dB/m. In practical, this absorption coefficient could be achieved by increasing the dopant density or decreasing the inner cladding diameter in the YDF. In fact, the pump absorption of YDF has been reached 18 dB/m at 976 nm and employed in high-power pulsed amplification [30].

Figures 9(a)-9(c) illustrate the power distribution, the temporal and spectral properties of the Raman signal light in the H-RFA, when there exists strong intensity fluctuations in the initial pump laser operating at 1018 nm. As shown in Fig. 9(a), the power distribution is similar to that shown in Fig. 4(a), and the output power of the Raman signal laser is about 1.67 kW. As shown in Fig. 9(b), there exists strong intensity fluctuations in the output Raman signal laser. The average output power is about 1.67 kW, while the maximum output power is over 2 kW. In addition, the NSD of the intensity fluctuations in the output laser is about 0.5, which is smaller than that in the initial pump laser. As shown in Fig. 9(c), the narrow-linewidth part of the Raman signal laser is partially covered by the background noise. The ratio of the narrow-linewidth part in the Raman signal laser is only about 4%. Therefore, when applying the multi-longitudinal fiber oscillator operating at 1018 nm as the initial pump laser, the narrow-linewidth operation could not be maintained in the H-RFA.

Fig. 9 The power distribution, the temporal and spectral properties of the Raman signal light when there exists strong intensity fluctuations in the initial pump laser operating at 1018 nm: (a) power distribution; (b) temporal evolution; (c) corresponding optical spectrum.

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3.3 Influence of the temporal stability of the Raman seed laser

To analyze the influence of temporal stability of the Raman seed laser on the H-RFA, contrast simulation is conducted by assuming that there exists strong intensity fluctuations in the Raman seed laser. The temporal property of the Raman seed laser is similar to that shown in Fig. 3(b), and the NSD of the intensity fluctuations in the Raman seed laser is about 1.0. The initial pump laser is set to be temporal stable laser diode operating at 976 nm, and the Raman-pumped laser is set to be temporal stable phase-modulated single-frequency laser in this section.

Figures 10(a)-10(c) illustrate the power distribution, the temporal and spectral properties of the Raman signal light in the H-RFA, when there exists strong intensity fluctuations in the Raman seed laser. As shown in Fig. 10(a), the power distribution is similar to that shown in Fig. 4(a), and the output power of the Raman signal laser is about 1.62 kW. As shown in Fig. 10(b), despite that there exists strong intensity fluctuations in the Raman seed laser, the output Raman signal laser could still keep relatively stable over time. In addition, the NSD of the intensity fluctuations in the output laser is about 0.2, which indicates that part of the intensity noise in the Raman seed laser had been filtered during amplification. As shown in Fig. 10(c), the spectrum of output Raman signal laser gets broadened compared to the spectrum of the Raman seed laser, which is similar to the spectral broadening phenomenon in multi-longitudinal mode YDFAs. The corresponding RMS linewidth of the Raman signal laser broadens from 0.5 GHz to 1.12 GHz during amplification. Therefore, the output Raman signal laser would get broadened when there exists strong intensity fluctuations in the Raman seed laser. Consequently, temporal stable Raman seed laser is required to maintain narrow-linewidth operation in H-RFAs.

Fig. 10 The power distribution, the temporal and spectral properties of the Raman signal light when there exists strong intensity fluctuations in the Raman seed laser: (a) power distribution; (b) temporal evolution; (c) corresponding optical spectrum.

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The major simulation results for the five representative cases are summarized in Table. 2. The output powers of the Raman signal laser are all close to 1.7 kW, thus the temporal stabilities of the Raman seed, the initial pump, and Raman-pumped lasers would not impact the output power of the Raman signal laser in H-RFAs significantly. However, their temporal stabilities would change the spectral broadening properties in H-RFAs. Specifically, applying the temporal stable Raman seed laser is one of the basic conditions to maintain narrow-linewidth operation in H-RFAs. For the diode-pumped scheme to obtain the Raman-pumped laser, narrow-linewidth operation could be maintained in H-RFAs when applying the temporal stable Raman-pumped laser. For the tandem-pumped scheme to obtain the Raman-pumped laser, both temporal stable initial pump and Raman-pumped lasers are required to maintain narrow-linewidth operation in H-RFAs.

Table 2. Major results for the five contrast simulations

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3.4 Gain dynamics in H-RFAs

To clarify the physical mechanism behind the different spectral evolution properties, the gain dynamics in H-RFAs is investigated. Gain dynamics analysis provides an effective way to describe the noise transfer properties from the pump or the seed lasers into the signal laser in fiber lasers [31,32]. To quantify the noise transfer properties, it is useful to apply the modulation transfer function in the frequency domain [31]. To obtain the pump modulation transfer function, a small cosine modulation is imposed into the pump power:

P_{p} (t) = P_{p}^{0} [1 + δ_{0} \cos (2 π f t)]

Where, P⁰ is the initial pump power, δ₀ is the modulation depth, and f is the modulation frequency. Then, the output power of the Raman signal laser and the corresponding amplitude of the pump modulation transfer function could be expressed as:

P_{R} (t) = P_{R}^{0} * [1 + δ_{p} (f) \cos (2 π f t + ϕ)]

T_{p} (f) = δ_{p} (f) / δ_{0}

Where, P_R⁰ is the steady state output power of the Raman signal laser and ϕ is the phase delay between the Raman signal laser and the pump laser.

Incorporating Eqs. (6)-(8) into Eqs. (1)-(4), both the pump modulation transfer functions for the initial pump and Raman-pumped lasers could be obtained. The major parameter which impacts the noise transfer property is the amplitude of the modulation transfer function, thus only the amplitude of the transfer function is given and compared in the following analysis.

Figure 11 illustrates the simulated magnitudes of the pump modulation transfer functions for the initial pump and Raman-pumped lasers. As for initial pump power operating at 976 nm, the magnitude decreases along with the increase of the modulation frequency. Thus, overall curve of the magnitude satisfies the properties of a low-pass filter approximately. As for initial pump power operating at 1018 nm, the magnitude decreases along with the increase of the modulation frequency when the modulation frequency is below 50 MHz, and keeps about −7.4 dB when the modulation frequency is over 50 MHz. Thus, overall curve of the magnitude satisfies the properties of an all-pass filter with an attenuation of about 7.4 dB approximately. As for Raman-pumped laser, the magnitude is about 1.4 dB for all the modulation frequency. Thus, overall curve of the magnitude satisfies the properties of an all-pass filter with a gain of about 1.4 dB approximately. Therefore, the intensity fluctuations in the Raman-pumped laser and the initial pump laser operating at 1018 nm would transfer into the Raman signal laser, while most of the intensity fluctuations in the initial pump laser operating at 976 nm would be filtered in the H-RFAs.

Fig. 11 The magnitudes of the modulation transfer functions for the initial pump and Raman-pumped lasers.

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Further combining the results in Table. 2 and in Fig. 11, it could be concluded that narrow-linewidth operation could be maintained in H-RFAs, on the condition that there are no strong intensity fluctuations in the initial pump and Raman-pumped signal lasers, or the strong intensity fluctuations in the initial pump and Raman-pumped signal lasers could be filtered during amplification. It should be noted that the diversity of the pump modulation transfer functions is induced by the different gain characteristics. For a co-pumped fiber amplifier based on only rare-earth gain, the pump modulation transfer function would behave as a low-pass filter due to time required for the population inversion process. However, for a co-pumped fiber amplifier based on only Raman gain, the pump modulation transfer function would behave as an all-pass filter because the Raman response are instantaneous in optical fibers.

4. Conclusions

In this work, we establish a spectral evolution model for narrow-linewidth H-RFAs. Five representative cases are simulated to compare the spectral evolution properties and the gain dynamics in H-RFAs based on two typical pump schemes. The possible techniques to achieve narrow-linewidth operation in H-RFAs are proposed based on the simulation results. Specifically, the temporal stable Raman-pumped laser is required for narrow-linewidth H-RFAs when applying the diode-pumped scheme to obtain the Raman-pumped laser, and both the temporal stable initial pump laser and Raman-pumped laser are required for narrow-linewidth H-RFAs when applying the tandem-pumped scheme to obtain the Raman-pumped laser. Our work could provide a well reference to obtain high-power narrow-linewidth H-RFAs.

Funding

National Natural Science Foundation of China (61705264); HuoYingDong Education Foundation of China.

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Parameter	Value	Parameter	Value
λ_p	976 nm	λ_s	1070 nm
λ_R	1120 nm	α	0.002 dB/m
Γ_s	0.85	Γ_R	0.85
N	7.85 × 10²⁵ /m³	τ	0.85 ms
σ_p^a	1.77 × 10⁻²⁴ m²	σ_p^e	1.71 × 10⁻²⁴ m²
σ_s^a	4.5 × 10⁻²⁷ m²	σ_s^e	3.6 × 10⁻²⁵ m²
σ_R^a	7.56 × 10⁻²⁹ m²	σ_R^e	4.7 × 10⁻²⁶ m²
g_R	1.08 × 10⁻¹³ m/W	P_p	2000 W
P_s	20 W	P_R	10 W

	Raman seed laser	Initial pump laser	Raman-pumped laser	Output signal power (kW)	Narrow-linewidth operation
laser Stability	Instable	Stable	Stable	1.62	×
	Stable	Stable	Instable	1.65	×
	Stable	Stable	Stable	1.71	√
	Stable	Instable	Stable	1.71	√
	Stable	Instable (1018nm)	Stable	1.73	×

Parameter	Value	Parameter	Value
λ_p	976 nm	λ_s	1070 nm
λ_R	1120 nm	α	0.002 dB/m
Γ_s	0.85	Γ_R	0.85
N	7.85 × 10²⁵ /m³	τ	0.85 ms
σ_p^a	1.77 × 10⁻²⁴ m²	σ_p^e	1.71 × 10⁻²⁴ m²
σ_s^a	4.5 × 10⁻²⁷ m²	σ_s^e	3.6 × 10⁻²⁵ m²
σ_R^a	7.56 × 10⁻²⁹ m²	σ_R^e	4.7 × 10⁻²⁶ m²
g_R	1.08 × 10⁻¹³ m/W	P_p	2000 W
P_s	20 W	P_R	10 W

	Raman seed laser	Initial pump laser	Raman-pumped laser	Output signal power (kW)	Narrow-linewidth operation
laser Stability	Instable	Stable	Stable	1.62	×
	Stable	Stable	Instable	1.65	×
	Stable	Stable	Stable	1.71	√
	Stable	Instable	Stable	1.71	√
	Stable	Instable (1018nm)	Stable	1.73	×

Theoretical study of narrow-linewidth hybrid rare-earth-Raman fiber amplifiers

Abstract

1. Introduction

2. Theory for numerical modeling

3. Spectral evolution in H-RFAs

3.1 Influence of temporal stability of the Raman-pumped laser

3.2 Influence of the temporal stability of the initial pump laser

3.3 Influence of the temporal stability of the Raman seed laser

3.4 Gain dynamics in H-RFAs

4. Conclusions

Funding

References

Cited By

Figures (11)

Tables (2)

Equations (8)

Optics Express