Abstract

Gain dynamics study provides an attractive method to understand the intensity noise behavior in fiber amplifiers. Here, the gain dynamics of a medium power (5 W) clad-pumped Yb-fiber amplifier is experimentally evaluated by measuring the frequency domain transfer functions for the input seed and pump lasers from 10 Hz to 1 MHz. We study gain dynamic behavior of the fiber amplifier in the presence of significant residual pump power (compared to the seed power), showing that the seed transfer function is strongly saturated at low Fourier frequencies while the pump power modulation transfer function is nearly unaffected. The characterization of relative intensity noise (RIN) of the fiber amplifier is well explained by the gain dynamics analysis. Finally, a 600 kHz bandwidth feedback loop using an acoustic-optical modulator (AOM) controlling the seed intensity is successfully demonstrated to suppress the broadband laser intensity noise. A maximum noise reduction of about 30 dB is achieved leading to a RIN of −152 dBc/Hz (~1 kHz-10 MHz) at 2.5 W output power.

© 2017 Optical Society of America

1. Introduction

Low intensity noise laser sources are a fundamental prerequisite for demanding fundamental research such as spectroscopy, ultra-cold atom/molecular optical lattices, and high sensitive optical interferometers. In particular, laser sources used in interferometric gravitational-wave detector (GWD) simultaneously require high output power (>50 W) and extremely low intensity noise [1,2]. Besides, high power lasers with low intensity noise also attract great interest in the industry applications related to laser material processing, including nanolithography [3] and direct fiber Bragg grating writing [4]. Due to the intrinsic advantages of high conversion efficiency, outstanding thermo-optical properties, compact and robust configuration, and excellent mode output, fiber amplifiers are widely regarded as promising candidates for the next generation GWD [5]. The limiting factor in fiber power amplification comes from the stimulated scattering processes like stimulated Raman (SRS) or Brillouin (SBS) scattering. The development of large-mode-area double-clad micro-structured fiber has enabled single frequency Yb-fiber amplifiers to reach a record power level of 811 W [6]. Moreover, a kilowatt-class fiber amplifier operating at single frequency has also been demonstrated through coherent combing methods [7].

The output power of single frequency Yb-fiber amplifier has already exceeded the one of conventional Nd: YAG bulk crystal based high-power oscillators [2] or cascaded master oscillator power amplifier [8]. However, a significant amount of relative intensity noise (RIN) is introduced in the amplification process. In the low amplitude noise fiber amplifiers where the seed noise is optimized, the dominant additive intensity noise source originates from the pump laser noise converted through gain dynamics. As reported by Guiraud et al. [9], a 40 dB degradation of a seed laser RIN was observed during amplification up to 50 W. The suppression of this excess noise requires the understanding of noise dynamics. In addition, this knowledge is also important for the high energy pulsed amplifiers [10]. This behavior can be modeled by the rate equations, where the input seed and pump variations could be considered as perturbations. During amplification, the pump signal or seed fluctuations are not instantaneously transferred to the output, leading to significant frequency-dependent behavior. An analytical model for the gain dynamics proposed by Novak et al. [11] shows that the pump modulation transfer function is analog to a low-pass filter, while the seed modulation behaves like a damped high-pass filter. Later, the model was experimentally investigated for both low power Yb3+ and Er3+ fiber amplifiers [12, 13] as well as Er3+-Yb3+co-doped fibers amplifiers [14]. The gain dynamic behavior at higher output powers (>1 W) in clad-pumped amplifier has not been investigated yet. In particular, due to the relative low pump absorption in clad-pumped fiber amplifier, the residual pump power is significantly larger compared to core-pumping schemes.

In addition to the understanding of fiber amplifier noise dynamics, the precise knowledge of amplitude modulation transfer functions enables very effective intensity noise suppression by using an active feedback control loop approach. In order to suppress the laser intensity noise, various methods have been applied in the recent years. Direct pump current control is the simplest approach to suppress the intensity noise. However, the modulation of high current multi-mode pump diodes is not straightforward and achieving wide feedback control bandwidth beyond 200 kHz is challenging [15]. The Semiconductor optical amplifiers (SOAs) show a broad intensity noise suppression bandwidth of several GHz [16, 17]. Unfortunately, SOAs are only effective when operated under saturated regime. Recently, a broad bandwidth intensity noise suppression, in a single frequency Er3+-fiber laser, was demonstrated combining the SOA gain saturation effect with a slow electronic feedback control loop. Although a RIN of −150 dBc/Hz was achieved, the maximum output power remains limited to 17.5 mW [17]. For the laser sources used in GWD, the intensity noise could be suppressed down to ~-140 dBc/Hz (1 kHz-10 kHz) for an output power of 160 W [2]. However, this approach requires a quite complex setup. In this paper, a simple and robust scheme is developed through seed laser intensity feedback control by an acoustic-optical modulator (AOM). The achieved RIN is almost −152 dBc/Hz from below 1 kHz up to 10 MHz at an output power of 2.5 W.

2. Model of gain dynamics in frequency domain

The Yb3+-fiber amplifier system can be modeled with coupled rate equations, which describe the dynamic of pump and signal waves. By absorbing pump photons at 976 nm, the Yb3+-ions are excited from the ground state 2F7/2 to the excited state 2F5/2, creating population inversion. The signal photons at 1064 nm are then amplified by stimulated emission whereby the excited Yb3+-ions return to the ground state. The input pump or seed modulations could be considered as a perturbation of the steady-state operation. It is possible to define an effective lifetime of population inversion that describes the frequency response to the input power modulation. The key parameter of the gain dynamic is the corner frequency ωeff defined as the reciprocal of effective lifetime [11, 13]. The population inversion responds to the seed modulation at high frequency ω » ωeff or the pump modulation at low frequency ω « ωeff. The corner frequency impacts both pump and seed modulation. If non-linear effects such as Stimulated Raman and Brillouin Scattering (SRS, SBS) are neglected, the corner angular frequency is given by the relation [11, 13]:

ωeff=Ps0(L)Bs+Pp0(L)Bp+1τ
where Ps0 (L) and Pp0 (L) are the seed and pump average powers at the output of the gain fiber (z = L) in the units of number of photon per second. The coupling factors Bs = Гs (σsa + σse)/A and Bp = Гp σpa /A are given in terms of signal and pump overlap with the doped region Гs and Гp respectively, where σsa and σse are the absorption and emission cross sections at signal wavelength, σpa is the absorption cross section at pump wavelength, and A is the fiber mode area. The fluorescence lifetime τ of the doped Yb3+-ions (typically 0.7 ms) is related to spontaneous emission. In Yb3+-doped fiber amplifiers, the residual pump power Pp0 (L) is far less than the output signal power Ps0 (L), and the pump coupling factor Bp is also one order of magnitude less than signal coupling factor Bs [18]. Therefore, the impact of residual pump can be neglected. In addition, the contribution of the spontaneous emission is negligible, thus the corner frequency ωeff/2π is linearly dependent on the output power Ps0 (L).

Since the pump and seed modulation are perturbations for the amplified signal, the output intensity noise can be described by the means of transfer functions in the frequency domain. The modulated input seed and pump are defined as Ps,p0(0, t) = Ps,p0 (0)*(1 + ms,psinωt), where the input seed and pump powers are Ps0 (0) and Pp0 (0) in the units of number of photon per second. For the small seed modulation ms it leads to an amplitude modulation ms' of the amplified signal. The magnitude and the phase can be expressed as:

( ms'ms)2=(ω0ωeff)21+(ωω0)21+(ωωeff)2
θs'=arctanω(ω2+ωeffω0)/(ωeffω0)

This equation is analog to a damped high-pass filter, with a dominant zero angular frequency ω0 given by ω0 = Ps0 (0)Bs + Pp0 (L)Bp + 1/τ. The dominant zero angular frequency ω0 is constant at low output power (determined by the constant term related to the input seed power and the Yb3+-ions fluorescence lifetime). At higher output power where the residual pump and seed powers become comparable, ω0 deviates from the constant value and starts to increase linearly with the residual pump.

For high frequency modulation ω»ωeff, the seed modulation is amplified as the output power and hence ms' = ms. At low Fourier frequency ω<ω0, the seed modulation is suppressed and the Eq. (2) can be rewritten as ms'/ms = ω0eff. From the Eq. (3), we can see that phase shift vanishes for both low and high modulation frequencies. The maximum phase shift is located at the angular frequency ω = (ω0 ωeff)1/2. For low pump powers, the residual pump is negligible with respect to the input seed power. Therefore, the equation is approximately rewritten as ms'/ms = (Ps0 (0) + (τBs)−1)/Ps0 (L) and hence the seed modulation suppression ratio at low frequency is proportional to the output power. However, this approximation is no longer valid when the cladding pumped fiber amplifier is operated at medium or high output power regimes, where residual pump is not negligible anymore. Due to the contribution of residual pump, the seed modulation suppression ratio at low Fourier frequency (ω < ω0) can be expressed as ms'/ms = (Ps0 (0) + (τBs)−1)/Ps0 (L) + (BpPp0 (L)) / (BsPs0 (L)). This means that the low frequency seed modulation is also amplified if there is significant residual pump.

Since the residual pump increases with the amplified output signal, the suppression ratio of seed modulation at low Fourier frequency becomes dominated by the residual pump related term (BpPp0 (L))/ (BsPs0 (L)). For a given seed power and gain fiber length, the residual pump power is proportional to the amplified output signal power [19]. Therefore, the suppression ratio of seed modulation at low Fourier frequency no longer increases with the amplified signal output power.

In an equivalent way, we can define the output signal modulation index mp' is a function of the pump power modulation mp. Then, the output signal modulation index magnitude and phase shift are,

(mp'mp)2=[Bs(Pp0(0)-Pp0(L))/ωeff]21+(ωωeff)2
θp'=arctan(ω/ωeff)
The frequency response to pump power modulation is equivalent to a low pass filter with a corner frequency ωeff/2π. The expression Bs (Pp0 (0) -Pp0 (L)) ≈ωeff is a good approximation even in the presence of residual pump. According to the energy conservation law, the absorbed pump photons are equal to the amplified signal photons Pp0 (0) -Pp0 (L) = Ps0 (L) -Ps0 (0), if we neglect the spontaneous emission.

3. Experimental measurement of transfer functions for pump and seed power modulation

In order to measure the frequency response, pump and seed modulation experiments were carried out separately. Compared to the previous core-pumping scheme [12–14], the effect of residual pump due to the lower pump absorption can be fully explored and potentially scaled-up to higher output powers. The frequency measurement range was also extended to 1 MHz (10 MHz for pump modulation), as the corner frequency increases with the higher output power. The experimental setup for pump and seed modulation is schematically shown in Fig. 1. A narrow linewidth (<50 kHz) external cavity diode laser emitting at 1064 nm with an optical power of 50 mW was used as seed source. The seed power intensity was modulated by a fiber pigtailed AOM with 3dB insertion losses. The pump laser was a 9 W grating stabilized multi-mode diode laser emitting at 976 nm. The pump and seed were coupled together via a multimode pump combiner. The gain media was a 3 m polarization-maintained large-mode-area Yb-doped fiber (PLMA-YDF-10/125, Nufern) which had a core diameter of 10 µm and a cladding diameter of 125 µm respectively. The achieved maximum output power of the fiber laser was 4.5 W for a pump power of 7 W. Fiber pigtailed isolators were inserted behind each stage to prevent detrimental backward beam propagating and the output face end was angle cleaved at 8 degrees to suppress Fresnel reflection. The output beam intensity was controlled by a combination of half-wave plate (λ/2) and a polarization beam splitter (PBS). Then, it was further attenuated by a 20 dB neutral density filter (NF) and finally detected by a homemade low noise photo detector (PD). The generated electronic signal from PD was analyzed by the vector signal analyzer (VSA, HP89410A). The internal VSA chirped sine wave generator drive both the modulation input of the AOM driver and the VSA reference channel.

 

Fig. 1 Schematic setup for seed and pump power modulation transfer function measurement. AOM: acoustic-optical modulator; PLMA-YDF: polarization-maintained large-mode-area Yb-doped fiber; λ/2: half-wave plate; PBS: pol. beam splitter; NF: neutral density filter; PD: photo detector; VSA: vector signal analyzer (S: signal source; 1: input port 1; 2: input port 2).

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3.1 Pump modulation

In order to properly modulate the pump signal, we avoided current modulation of the multimode laser diode. We took advantage of the second port of the multimode pump combiner to inject an auxiliary 976 nm single mode laser diode with 100 mW average output power. This additional pump signal was amplitude-modulated by an AOM. In order to avoid the nonlinear effects, the modulation index ms,p were set to 2%. These plots have been obtained by careful calibration of the frequency response of both the AOM modulation system and the photo-detection system. The amplitude and phase transfer functions of PD, AOM and AOM driver have been subtracted to access to the gain medium response only. This is critical for accurate phase measurements at high frequency (>100 kHz). It is mentioned that all the measured magnitude transfer functions are normalized.

The measured transfer function for different output optical powers is shown in Fig. 2. As expected, the pump modulation transfer function behaves as a low-pass filter. One can observe that pump fluctuations are integrally transferred to the signal at low Fourier frequencies. Since the loss of pump photons induced by residual pump is negligible, the pump modulation is totally converted into amplified signal modulation. As predicted by the model, the corner frequency ωeff/2π shifts to higher frequency by increasing the output power and the coupling coefficient starts to decrease as the pump modulation frequency exceeds ωeff/2π. It is worth nothing that beyond 100 kHz the low signal-to-noise ratio of the detected signal prevent from a perfect phase determination.

 

Fig. 2 Output power modulation index relative to pump modulation index in magnitude (a) and phase (b) transfer function for different output powers.

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3.2 Seed modulation

Figure 3 shows the measured transfer function of the output power modulation relative to the seed modulation. As expected the frequency response of the seed modulation behaves as a damped high-pass filter. It can be seen that the seed modulation suppression ratio at low Fourier frequencies increases with the output power. For higher output powers (≥ 2.7 W), the suppression ratio starts to saturate and reaches a maximum value of ~25 dB.

 

Fig. 3 Measured normalized amplitude (a) and phase (b) transfer function of output power relative to seed modulation at varied output power.

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The transfer functions behavior with respect to the seed power was also investigated. Two situations were considered at the seed power below and above seed saturation. The calculated saturation power for our fiber amplifier was 40 mW. Previously, we operated the amplifier slightly below saturation (injected seed power 27 mW). An auxiliary fiber amplifier was used between the seed output isolator and fiber coupled AOM to increase the seed power to 130 mW, well above the seed saturation power. From the measured seed modulation transfer function, we can determine both the corner frequency ωeff/2π and the dominant zero frequency ω0/2π (3dB values) [20].

In Fig. 4(a) both corner frequency ωeff/2π and dominant zero frequency ω0/2π are plotted with respect to the output power for the 27 mW and 130 mW of injected power respectively. The ωeff/2π and ω0/2π behaviors with respect to the output power are quite different. The corner frequency ωeff/2π is proportional to the output power, consistently with the Eq. (1) regardless of seed power. This confirms the model accuracy at moderate output power. On the other hand, the dominant zero frequency ω0/2π is nearly constant at low output power, but then starts to linearly increase from 1.5 W of output power. Again, this is consistent with the model predictions. Because the residual pump is reduced by increasing seed power, thus, increasing the injected seed power shifts this threshold to about 3.5 W. As shown in Fig. 4(b), the saturated seed modulation suppression ratio at low Fourier frequencies (equivalent to ωeff0) slightly increases. From Fig. 4(b), it can be seen that the saturation ratios are about 20 and 26 for 27 mW and 130 mW seed power respectively. From these observations, it is then possible to determine the signal emission cross section σse with great accuracy. Indeed, since σse is about two order larger than the signal absorption cross section σsa, its expression can be simplified as σse = ABs/Гs. From a linear fit of the corner frequency as a function of output power Ps0 (L), we obtain a value of (3.3 ± 0.3) × 10−15 for Bs. The calculated signal/doped region overlap Гs is 0.8 [9], leading to a value σse = (3.2 ± 0.4) × 10−25 m2 for our gain fiber. The signal cross-section emission value depends on the dopant concentration and the composition of the host glass. For instance, this value range from ~1.5 × 10−25 m2 to ~4 × 10−25 m2 depending on the ratio between Aluminum (Al3+) and Phosphorus (P3+) in silica preforms [21].

 

Fig. 4 (a) Corner frequency ωeff/2π (squares) and dominant zero frequency ω0/2π (circles) versus output power for 27 mW (solid symbols) and 130 mW (cross symbols) seed power. (b) The ratio ωeff/ω0 for 27 mW and 130 mW of seed power respectively vs. output power.

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In order to check the validity of the amplifier model, we fit a damped high-pass filter to the measured seed modulation transfer functions (Fig. 5). We select three different output powers levels. Both the fitted magnitude and phase transfer functions well agree with the experimental ones within the error bars. The maximum magnitude offset is less than 1.5 dB (< 10%) and the maximum phase difference is about 5 degrees (< 10%) for all modulated Fourier frequencies. This again confirms the validity of the model for intermediate power (up to 5 W) fiber amplifiers.

 

Fig. 5 Magnitude (a) and relative phase (b) comparison of experimental measured (solid curve) and numerical modelled (dash curve) transfer function of output power relative to seed modulation at 0.4 W (black), 1.8 W (red) and 4.5 W (blue).

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4. Intensity noise characterization and analysis

The intensity noise of this type of fiber amplifier has already been reported by Guiraud et al. [9] at the low output power level (<1 W). Here we extend the characterization for the fiber amplifier, as well as that of pump diode, seed source and noise floor. From Fig. 6 we can notice the fiber amplifier measured RIN is well explained by the gain dynamics analysis. Below 200 kHz the amplifier measured RIN is well approximated by filtering the pump diode RIN with the pump transfer function (low frequency seed RIN is negligible in our configuration). The small discrepancy at low frequency (<1 kHz) is likely to be due to the non-stationary behavior of the pump diode noise. It should be noted that all the measured traces are obtained with 6 mW of optical power coupled into the PD. The fiber amplifier high frequency RIN (beyond 200 kHz) is dominated by the seed source. The relatively smooth fiber amplifier RIN simplifies the electronic requirements for the feedback loop. The parasitic lines around 100 kHz are related to electronic-magnetic coupling (EMC) issues which can be confirmed by measuring the PD noise floor.

 

Fig. 6 Measured RIN of the fiber amplifier at 2.5 W (black plot), seed source (blue plot), pump diode (green plot), photodiode noise floor (dark yellow plot) and the pump induced noise (red dash plot).

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5. Broad bandwidth intensity noise suppression with the AOM feedback

In order to suppress the amplifier pump induced excess noise we implemented a feedback control loop. To avoid direct modulation of the seed current, which can impact the seed frequency noise, an AOM is used as actuator on the amplifier input. The seed power modulation analysis, shows that the low frequency transfer function is attenuated up to 25dB for high output power (~4.5 W). However, this attenuation can be in principle compensated by shaping the low frequency gain of the electronic controller. The scheme with broadband intensity noise suppression is depicted in Fig. 7. The setup contains a polarization beam splitter (PBS) together with a half-wave plate (λ/2) and a 4 degrees plano-wedged mirror (Wedge) to generate two separated beams with nearly identical optical power. Both beams are detected by a pair of homemade low noise PDs for in-loop and out-of-loop measurements. The in-loop signal is AC coupled (high-pass filter with a bandwidth ~30 Hz).

 

Fig. 7 Schematic of the experimental setup for intensity noise suppression. P-offset AMP: proportional-offset electronic amplifier; LPF: low-pass filter.

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This error signal was amplified by a homemade controller with a maximum gain up to 80 dB and with very flat amplitude and phase response. We design the circuit to achieve a bandwidth of about 10 MHz with a total delay below 20 ns (negligible with respect to the 220 ns time delay of the AOM). The electronic circuit transfer function is an ideal low pass filter starting at 100 Hz (red dashed plot in Fig. 8(b)). The correction signal is summed with a constant offset (AOM bias) and coupled to the AOM driver. The out-of-loop RIN measurement is performed with the VSA. The RIN measurements with and without feedback loop are shown in Fig. 8(a). The fiber amplifier is operated at 2.5 W output power. The free-running RIN (black curve) is measured with the AOM driven by a low RF power (attenuation of about 3%). Compared to the measurement at the same output power (black line in Fig. 6), the AOM intensity noise degradation is negligible except at high frequency (>1 MHz) where is increased by about 2 dB. When we turn on the servo loop, broadband intensity noise is effectively suppressed down to-152 dBc/Hz (marked by blue dashed line). The maximum noise suppression ratio exceeds 30 dB. It should be pointed out that the bump around 600 kHz is mainly due to the AOM delay. The spikes around 100 kHz are caused by the EMC related issues.

 

Fig. 8 Measured RIN at 2.5 W output power for free-running (black curve) and for the AOM feedback loop active (green curve) (a). Figure (b) shows the measured open loop servo transfer function of the seed modulation (blue dash), servo box (red dot) and total gain (green line).

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The RIN suppression results can be predicted by the feedback loop frequency response. Since the feedback signal acts on the seed source, both the seed modulation and servo box frequency response are considered. We scale the open loop transfer function by setting the unity gain around 300 kHz. The resulted total gain is illustrated in Fig. 8(b) which is the sum of magnitude transfer function of seed modulation and the gain of servo box. The open-loop maximum noise suppression ratio is 35 dB (1 kHz-10 MHz) which indicates our feedback loop is operating near its full capacity.

6. Conclusions

In this paper we have extensively investigated the gain dynamics of a medium power clad-pumped Yb-fiber amplifier. We have experimentally verified that the previously developed models are valid in clad pumped amplifiers even when residual pump power is no longer negligible. In addition, we have shown that using the measured value of corner frequency ωeff/2π, an accurate value of the signal emission cross-section can be retrieved. Furthermore, we have shown that the seed modulation transfer function is significantly affected by the residual pump, resulting in strongly saturated seed modulation suppression ratio at low Fourier frequency. The saturated behavior has been fully investigated with different seed power levels. Moreover, the fiber amplifier RIN is well predicted by the gain dynamic analysis. Finally, by the means of feedback controlling seed intensity via an AOM, a wide feedback bandwidth (~600 kHz) and a maximum RIN reduction more than 30 dB is demonstrated resulting in a broadband noise floor of −152 dBc/Hz (~1 kHz-10 MHz) at 2.5 W output power. The noise suppression performance agrees well with the open-loop frequency response. The extension of this technique to higher powers (>50 W) and implementation of RIN suppression of a 50 W single frequency fiber amplifier is in progress.

Funding

Agence Nationale de la Recherche (ANR) (ANR14 LAB05 0002 01) and Conseil Regional d’Aquitaine (2014-IR60309-00003281); Post-doctor scholarship grant from the La Fondation Franco-Chinoise pour la Science et ses Applications (FFCSA).

References and links

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2. P. Kwee, C. Bogan, K. Danzmann, M. Frede, H. Kim, P. King, J. Pöld, O. Puncken, R. L. Savage, F. Seifert, P. Wessels, L. Winkelmann, and B. Willke, “Stabilized high-power laser system for the gravitational wave detector advanced LIGO,” Opt. Express 20(10), 10617–10634 (2012). [CrossRef]   [PubMed]  

3. G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992). [CrossRef]   [PubMed]  

4. D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007). [CrossRef]   [PubMed]  

5. N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011). [CrossRef]  

6. C. Robin, I. Dajani, and B. Pulford, “Modal instability-suppressing, single-frequency photonic crystal fiber amplifier with 811 W output power,” Opt. Lett. 39(3), 666–669 (2014). [CrossRef]   [PubMed]  

7. N. A. Naderi, A. Flores, B. M. Anderson, and I. Dajani, “Beam combinable, kilowatt, all-fiber amplifier based on phase-modulated laser gain competition,” Opt. Lett. 41(17), 3964–3967 (2016). [CrossRef]   [PubMed]  

8. C. Basu, P. Wessels, J. Neumann, and D. Kracht, “High power single frequency solid state master oscillator power amplifier for gravitational wave detection,” Opt. Lett. 37(14), 2862–2864 (2012). [CrossRef]   [PubMed]  

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10. M. Müller, C. Jauregui-Misas, M. Kienel, F. Emaury, C. J. Saraceno, U. Keller, J. Limpert, and A. Tünnermann, “Amplitude Noise Reduction in Yb-doped Fiber Amplifiers,” Lasers Congress2016(ASSL, LSC, LAC), paper JTh2A.33. [CrossRef]  

11. S. Novak and A. Moesle, “Analytic model for gain modulation in EDFAs,” J. Lightwave Technol. 20(6), 975–985 (2002). [CrossRef]  

12. M. Tröbs, P. Weßels, and C. Fallnich, “Power- and frequency-noise characteristics of an Yb-doped fiber amplifier and actuators for stabilization,” Opt. Express 13(6), 2224–2235 (2005). [CrossRef]   [PubMed]  

13. H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “Gain dynamics and refractive index changes in fiber amplifiers: a frequency domain approach,” Opt. Express 20(12), 13539–13550 (2012). [CrossRef]   [PubMed]  

14. M. Steinke, J. Neumann, D. Kracht, and P. Wessels, “Gain dynamics in Er3+:Yb3+ co-doped fiber amplifiers,” Opt. Express 23(11), 14946–14959 (2015). [CrossRef]   [PubMed]  

15. N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007). [CrossRef]  

16. G. Danion, F. Bondu, G. Loas, and M. Alouini, “GHz bandwidth noise eater hybrid optical amplifier: design guidelines,” Opt. Lett. 39(14), 4239–4242 (2014). [CrossRef]   [PubMed]  

17. Q. Zhao, S. Xu, K. Zhou, C. Yang, C. Li, Z. Feng, M. Peng, H. Deng, and Z. Yang, “Broad-bandwidth near-shot-noise-limited intensity noise suppression of a single-frequency fiber laser,” Opt. Lett. 41(7), 1333–1335 (2016). [CrossRef]   [PubMed]  

18. L. Zhang, S. Cui, C. Liu, J. Zhou, and Y. Feng, “170 W, single-frequency, single-mode, linearly-polarized, Yb-doped all-fiber amplifier,” Opt. Express 21(5), 5456–5462 (2013). [CrossRef]   [PubMed]  

19. Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009). [CrossRef]  

20. H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “On the Effective Ion Lifetime in Fiber Amplifiers,” in CLEO: Science and Innovations, OSA Technical Digest (Optical Society of America, 2013), paper CTu1K.1.

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References

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  1. H. Grote, “Overview and Status of Advanced Interferometers for Gravitational Wave Detection,” J. Phys. Conf. Ser. 718, 022009 (2016).
    [Crossref]
  2. P. Kwee, C. Bogan, K. Danzmann, M. Frede, H. Kim, P. King, J. Pöld, O. Puncken, R. L. Savage, F. Seifert, P. Wessels, L. Winkelmann, and B. Willke, “Stabilized high-power laser system for the gravitational wave detector advanced LIGO,” Opt. Express 20(10), 10617–10634 (2012).
    [Crossref] [PubMed]
  3. G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
    [Crossref] [PubMed]
  4. D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007).
    [Crossref] [PubMed]
  5. N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
    [Crossref]
  6. C. Robin, I. Dajani, and B. Pulford, “Modal instability-suppressing, single-frequency photonic crystal fiber amplifier with 811 W output power,” Opt. Lett. 39(3), 666–669 (2014).
    [Crossref] [PubMed]
  7. N. A. Naderi, A. Flores, B. M. Anderson, and I. Dajani, “Beam combinable, kilowatt, all-fiber amplifier based on phase-modulated laser gain competition,” Opt. Lett. 41(17), 3964–3967 (2016).
    [Crossref] [PubMed]
  8. C. Basu, P. Wessels, J. Neumann, and D. Kracht, “High power single frequency solid state master oscillator power amplifier for gravitational wave detection,” Opt. Lett. 37(14), 2862–2864 (2012).
    [Crossref] [PubMed]
  9. G. Guiraud, N. Traynor, and G. Santarelli, “High-power and low-intensity noise laser at 1064 nm,” Opt. Lett. 41(17), 4040–4043 (2016).
    [Crossref] [PubMed]
  10. M. Müller, C. Jauregui-Misas, M. Kienel, F. Emaury, C. J. Saraceno, U. Keller, J. Limpert, and A. Tünnermann, “Amplitude Noise Reduction in Yb-doped Fiber Amplifiers,” Lasers Congress2016(ASSL, LSC, LAC), paper JTh2A.33.
    [Crossref]
  11. S. Novak and A. Moesle, “Analytic model for gain modulation in EDFAs,” J. Lightwave Technol. 20(6), 975–985 (2002).
    [Crossref]
  12. M. Tröbs, P. Weßels, and C. Fallnich, “Power- and frequency-noise characteristics of an Yb-doped fiber amplifier and actuators for stabilization,” Opt. Express 13(6), 2224–2235 (2005).
    [Crossref] [PubMed]
  13. H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “Gain dynamics and refractive index changes in fiber amplifiers: a frequency domain approach,” Opt. Express 20(12), 13539–13550 (2012).
    [Crossref] [PubMed]
  14. M. Steinke, J. Neumann, D. Kracht, and P. Wessels, “Gain dynamics in Er3+:Yb3+ co-doped fiber amplifiers,” Opt. Express 23(11), 14946–14959 (2015).
    [Crossref] [PubMed]
  15. N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007).
    [Crossref]
  16. G. Danion, F. Bondu, G. Loas, and M. Alouini, “GHz bandwidth noise eater hybrid optical amplifier: design guidelines,” Opt. Lett. 39(14), 4239–4242 (2014).
    [Crossref] [PubMed]
  17. Q. Zhao, S. Xu, K. Zhou, C. Yang, C. Li, Z. Feng, M. Peng, H. Deng, and Z. Yang, “Broad-bandwidth near-shot-noise-limited intensity noise suppression of a single-frequency fiber laser,” Opt. Lett. 41(7), 1333–1335 (2016).
    [Crossref] [PubMed]
  18. L. Zhang, S. Cui, C. Liu, J. Zhou, and Y. Feng, “170 W, single-frequency, single-mode, linearly-polarized, Yb-doped all-fiber amplifier,” Opt. Express 21(5), 5456–5462 (2013).
    [Crossref] [PubMed]
  19. Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
    [Crossref]
  20. H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “On the Effective Ion Lifetime in Fiber Amplifiers,” in CLEO: Science and Innovations, OSA Technical Digest (Optical Society of America, 2013), paper CTu1K.1.
  21. M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
    [Crossref]

2016 (4)

2015 (1)

2014 (3)

2013 (1)

2012 (3)

2011 (1)

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

2009 (1)

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

2007 (2)

N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007).
[Crossref]

D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007).
[Crossref] [PubMed]

2005 (1)

2002 (1)

1992 (1)

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Alouini, M.

Anderson, B. M.

Basu, C.

Behringer, R. E.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Berggren, K. K.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Bogan, C.

Bondu, F.

Byer, R. L.

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

Codemard, C. A.

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

Cui, S.

Cunningham, J. E.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Dajani, I.

Danion, G.

Danzmann, K.

Deng, H.

Digonnet, M. J.

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

Fallnich, C.

Feng, Y.

Feng, Z.

Flores, A.

Frede, M.

Grote, H.

H. Grote, “Overview and Status of Advanced Interferometers for Gravitational Wave Detection,” J. Phys. Conf. Ser. 718, 022009 (2016).
[Crossref]

Guiraud, G.

Jiang, S.

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

Kim, D. I.

D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007).
[Crossref] [PubMed]

Kim, H.

King, P.

Kracht, D.

Kwee, P.

Lee, Y. W.

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007).
[Crossref] [PubMed]

Li, C.

Liu, C.

Loas, G.

Machida, K.

N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007).
[Crossref]

Mavalvala, N.

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

McClelland, D. E.

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

Mio, N.

N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007).
[Crossref]

Moesle, A.

Moriwaki, S.

N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007).
[Crossref]

Mueller, G.

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

Naderi, N. A.

Neumann, J.

Novak, S.

Ozeki, T.

N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007).
[Crossref]

Peng, M.

Pöld, J.

Prentiss, M.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Pulford, B.

Puncken, O.

Reitze, D. H.

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

Rhee, H. G.

D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007).
[Crossref] [PubMed]

Robin, C.

Santarelli, G.

Savage, R. L.

Schnabel, R.

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

Seifert, F.

Sinha, S.

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

Song, J. B.

D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007).
[Crossref] [PubMed]

Steinke, M.

Tennant, D. M.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Timp, G.

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Traynor, N.

Tröbs, M.

Tünnermann, H.

Urbanek, K. E.

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

Wessels, P.

Weßels, P.

Wessels, P.

Weßels, P.

Willke, B.

P. Kwee, C. Bogan, K. Danzmann, M. Frede, H. Kim, P. King, J. Pöld, O. Puncken, R. L. Savage, F. Seifert, P. Wessels, L. Winkelmann, and B. Willke, “Stabilized high-power laser system for the gravitational wave detector advanced LIGO,” Opt. Express 20(10), 10617–10634 (2012).
[Crossref] [PubMed]

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

Winkelmann, L.

Xu, S.

Yang, C.

Yang, Z.

Zervas, M. N.

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

Zhang, L.

Zhao, Q.

Zhou, J.

Zhou, K.

Gen. Relativ. Gravit. (1)

N. Mavalvala, D. E. McClelland, G. Mueller, D. H. Reitze, R. Schnabel, and B. Willke, “Lasers and optics: looking towards third generation gravitational wave detectors,” Gen. Relativ. Gravit. 43(2), 569–592 (2011).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

Y. W. Lee, M. J. Digonnet, S. Sinha, K. E. Urbanek, R. L. Byer, and S. Jiang, “High-power-doped phosphate fiber amplifier,” IEEE J. Sel. Top. Quantum Electron. 15(1), 93–102 (2009).
[Crossref]

M. N. Zervas and C. A. Codemard, “High Power Fiber Lasers: A Review,” IEEE J. Sel. Top. Quantum Electron. 20(5), 219–241 (2014).
[Crossref]

J. Lightwave Technol. (1)

J. Phys. Conf. Ser. (1)

H. Grote, “Overview and Status of Advanced Interferometers for Gravitational Wave Detection,” J. Phys. Conf. Ser. 718, 022009 (2016).
[Crossref]

Jpn. J. Appl. Phys. (1)

N. Mio, T. Ozeki, K. Machida, and S. Moriwaki, “Laser Intensity Stabilization System Using Laser-Diode-Pumped Nd:YAG Rod-Laser Amplifier,” Jpn. J. Appl. Phys. 46(8A), 5338–5341 (2007).
[Crossref]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

G. Timp, R. E. Behringer, D. M. Tennant, J. E. Cunningham, M. Prentiss, and K. K. Berggren, “Using light as a lens for submicron, neutral-atom lithography,” Phys. Rev. Lett. 69(11), 1636–1639 (1992).
[Crossref] [PubMed]

Rev. Sci. Instrum. (1)

D. I. Kim, H. G. Rhee, J. B. Song, and Y. W. Lee, “Laser output power stabilization for direct laser writing system by using an acousto-optic modulator,” Rev. Sci. Instrum. 78(10), 103110 (2007).
[Crossref] [PubMed]

Other (2)

M. Müller, C. Jauregui-Misas, M. Kienel, F. Emaury, C. J. Saraceno, U. Keller, J. Limpert, and A. Tünnermann, “Amplitude Noise Reduction in Yb-doped Fiber Amplifiers,” Lasers Congress2016(ASSL, LSC, LAC), paper JTh2A.33.
[Crossref]

H. Tünnermann, J. Neumann, D. Kracht, and P. Weßels, “On the Effective Ion Lifetime in Fiber Amplifiers,” in CLEO: Science and Innovations, OSA Technical Digest (Optical Society of America, 2013), paper CTu1K.1.

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Figures (8)

Fig. 1
Fig. 1 Schematic setup for seed and pump power modulation transfer function measurement. AOM: acoustic-optical modulator; PLMA-YDF: polarization-maintained large-mode-area Yb-doped fiber; λ/2: half-wave plate; PBS: pol. beam splitter; NF: neutral density filter; PD: photo detector; VSA: vector signal analyzer (S: signal source; 1: input port 1; 2: input port 2).
Fig. 2
Fig. 2 Output power modulation index relative to pump modulation index in magnitude (a) and phase (b) transfer function for different output powers.
Fig. 3
Fig. 3 Measured normalized amplitude (a) and phase (b) transfer function of output power relative to seed modulation at varied output power.
Fig. 4
Fig. 4 (a) Corner frequency ωeff/2π (squares) and dominant zero frequency ω0/2π (circles) versus output power for 27 mW (solid symbols) and 130 mW (cross symbols) seed power. (b) The ratio ωeff/ω0 for 27 mW and 130 mW of seed power respectively vs. output power.
Fig. 5
Fig. 5 Magnitude (a) and relative phase (b) comparison of experimental measured (solid curve) and numerical modelled (dash curve) transfer function of output power relative to seed modulation at 0.4 W (black), 1.8 W (red) and 4.5 W (blue).
Fig. 6
Fig. 6 Measured RIN of the fiber amplifier at 2.5 W (black plot), seed source (blue plot), pump diode (green plot), photodiode noise floor (dark yellow plot) and the pump induced noise (red dash plot).
Fig. 7
Fig. 7 Schematic of the experimental setup for intensity noise suppression. P-offset AMP: proportional-offset electronic amplifier; LPF: low-pass filter.
Fig. 8
Fig. 8 Measured RIN at 2.5 W output power for free-running (black curve) and for the AOM feedback loop active (green curve) (a). Figure (b) shows the measured open loop servo transfer function of the seed modulation (blue dash), servo box (red dot) and total gain (green line).

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

ω eff = P s 0 ( L ) B s + P p 0 ( L ) B p + 1 τ
(   m s ' m s ) 2 = ( ω 0 ω eff ) 2 1+ ( ω ω 0 ) 2 1+ ( ω ω eff ) 2
θ s ' =arctan ω ( ω 2 + ω eff ω 0 )/( ω eff ω 0 )
( m p ' m p ) 2 = [ B s ( P p 0 ( 0 )- P p 0 ( L ) )/ ω eff ] 2 1+ ( ω ω eff ) 2
θ p ' =arctan( ω/ ω eff )

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