The interferometric gravitational-wave detector LISA requires laser sources with 1W of output power and low frequency and power noise as well as actuators for further power and frequency stabilization. We report on power- and frequency noise measurements of an Yb-doped fiber amplifier seeded by a nonplanar ring oscillator and identify actuators for both power and frequency stabilization of such a system.
©2005 Optical Society of America
The spaceborne interferometric gravitational-wave detector LISA (Laser Interferometer Space Antenna) requires single-frequency radiation with 1W of output power in a diffraction-limited beam . The most promising candidates for the generation of such output characteristics are a stand-alone high-power nonplanar ring oscillator [2, 3] (NPRO) operating at 1064 nm or a fiber amplifier seeded by a low-power NPRO . Advantages of the latter system are the high efficiency of the fiber amplifier and the power scalability of the overall laser system.
LISA is in principle a Michelson interferometer with 5 million km armlength. It comprises three satellites in heliocentric orbits. Each satellite contains two test masses, one for each interferometer arm. The test masses, cubes of the approximate dimensions 4×4×4 cm3 made of an Au/Pt alloy of low magnetic susceptibility, reflect a portion of the light arriving from the remote spacecraft and define the reference mirror of the interferometer arm. The laser power reflected from the test masses produces a force on them. On their back face, an interferometer using a few 100 µW of light measures the position of the test masses with respect to the satellite. Fluctuations in the laser power will thus produce fluctuating forces on the test masses which could limit the sensitivity of LISA. Hence, a relative power stability of better than at a Fourier frequency of 1 mHz  is required. Although nonplanar ring oscillators achieve shot-noise limited operation at MHz frequencies , technical noise such as pump power noise is dominating at low frequencies . Accordingly, an active power stabilization is required for LISA.
To date, frequency-resolved power noise of fiber amplifiers has only been measured in the kHz to MHz range and significant excess noise has been reported for frequencies up to 100 kHz in Yb-doped fiber amplifiers [7, 8], in a Nd:glass fiber amplifier  as well as in injection locked Nd:YAG lasers [10, 11]. LISA requires relative power noise measurements down to 0.1 mHz which will be presented in this paper.
Celestial mechanics of the LISA orbits will cause relative armlength variations in the order of 10-2 (=50000 km) and these would produce spurious signals from laser phase fluctuations . From LISA’s strain sensitivity goal of 10-23 one can calculate an upper limit of allowed laser frequency fluctuations in the µHz/Hz0.5 range which is very difficult to achieve and may even be impossible to implement in hardware. Hence, a sophisticated algorithm, which is described elsewhere , will be required to cancel out laser frequency noise and obtain the measurement signal. Therefore, laser phase noise is a crucial quantity for the sensitivity of LISA. When using a master-oscillator power-amplifier scheme, the use of a pre-stabilized master-oscillator is attractive, but only possible if the frequency noise produced by the amplifier is negligible compared to the noise of the seed laser.
To date, linewidth measurements have been performed to investigate the phase noise properties of a solid-state amplifier  and of fiber amplifiers [14, 15, 16]. However, these measurements do not provide sufficient information to evaluate the suitability of such systems for LISA. Spectrally resolved measurements in the frequency range from 0.1 mHz to 1Hz are required. Recently, spectrally resolved phase noise measurements of a fiber amplifier for frequencies between 1Hz and 1MHz have been reported .
In Section 2 the fiber amplifier is described followed by power noise measurements of the seed laser, the amplifier output, and the amplifier pump diode in Section 3. In Section 4 phase noise measurements are presented, and in Section 5 appropriate actuators for power and frequency stabilization of the master-oscillator fiber-amplifier system are discussed.
2. Amplifier setup
Our fiber amplifier used 3m of Yb-doped single-mode, double-clad fiber (6500 mol ppm Yb2O3) with 5.2µm core diameter, 120 µm pump cladding diameter and numerical apertures of 0.15 and 0.38 for core and pump cladding, respectively. The amplifier has been pumped at the wavelength 976 nm counterdirectionally to the seed signal by a fiber-coupled pump diode. To monitor the optical seed power, a fiber coupler has been incorporated into the input port of the amplifier. All fiber ends were angle-polished by 8° to avoid optical back reflections. According to the LISA specifications , 1W of optical power has been extracted from the fiber amplifier. As seed power level 10mW have been chosen since lower seed power levels would not be sufficient for saturated operation of the amplifier as is discussed in detail in Section 5.
3. Power noise measurements
The power noise of seed laser and amplifier output has been measured simultaneously. A photo detector was placed behind the second output port of the fiber coupler, and a glass wedge was used to direct a portion of the amplifier output to a second photo detector. In order to be insensitive to polarization changes the glass wedge was used under a small angle of incidence. For the low-frequency measurements up to 1Hz Fourier frequency, temperature-stabilized In-GaAs PIN photodiodes were used, their signals recorded and a modified version of the popular WOSA method  that will be described elsewhere, has been used to compute relative power noise spectral densities. For the higher frequency part, InGaAs PIN photodiodes with a flat frequency response up to 100 kHz were used and their signals were directly fed into a commercial spectrum analyzer (SR785 by Stanford Research Systems). For the five frequency decades above 1 Hz, up to five measurements were appended for each curve.
Figure 1 shows the results of the power noise measurements. The curve labelled “Amplifier” denotes the relative power fluctuations of the amplifier output, “Pump” the fluctuations of the pump diode, and “NPRO” the fluctuations of the seed laser. The “bump” between 10 kHz and 20 kHz in the NPRO trace was caused by the NPRO drive electronics. The noise floor above 10 Hz is likely to be dominated by NPRO pump power noise as has been shown for the NPROs discussed in reference . The curve shows peaks at 50 Hz and multiples that originate from the electrical power grid. The remaining peaks in the frequency range between 50 Hz and 3 kHz are likely to be caused by acoustic noise and vibrations of mechanical components within the optical setup. Pump power fluctuations are not more than a factor of two below the amplifier fluctuations, which indicates that pump power fluctuations are the dominant noise source for amplifier power fluctuations. Seed laser power fluctuations at low frequencies are suppressed by the amplifier gain as will be discussed in Section 5. The cross-over of amplifier power fluctuations and pump power fluctuations at 10 kHz is no contradiction. For frequencies above the inverse effective life-time of the upper laser level in the amplifier, pump power noise is low-pass filtered as will be discussed in Section 5. The pump power noise measurements showed non-stationary behavior, and for frequencies above 1Hz variations up to a factor of three have been observed when the pump power was changed by a fraction of a percent. One should note that all three traces measure power fluctuations of fiber-coupled signals. Hence, the possible effect of varying coupling efficiency is included.
4. Frequency noise measurements
In principle the frequency (and hence the phase) of the amplifier output in an oscillator amplifier system is determined by the frequency of the oscillator. Frequency fluctuations of the oscillator directly cause frequency fluctuations of the amplifier. Fluctuations of the amplifier pump power, temperature fluctuations or acoustic noise represent other, significantly smaller sources for frequency noise at the amplifier output. The setup presented in the following section allows the measurement of this excess noise.
Heterodyne Mach-Zehnder interferometers as shown in Fig. 2 have been used for the phase measurements. The setup in Fig. 2a has been used for the low-frequency measurements from 10-4 Hz up to 0.5 Hz and the one in Fig. 2b for measurements from 0.1 Hz up to 100 kHz. In both setups light of a 1064 nm NPRO (Mephisto 800 by Innolight) was shielded from back-reflections by an optical isolator. The subsequent half-wave plate (HWP) in combination with the polarizing beam splitter (PBS) was used to adjust the power distribution in the two interferometer arms. One arm contained the fiber amplifier, the other arm contained an acousto-optic modulator driven by a 40MHz local oscillator (LO). The light of both arms was brought to interference at a beam splitter (BS) and detected by an InGaAs photodiode (PD). In order to obtain interferometer arms with similar length, which reduces the influence of seed laser frequency fluctuations on the interferometer output phase, a 5m long angle-polished single-mode fiber was inserted after the AOM.
For the low-frequency measurements, we have used a phase counter (53132A by Agilent) to measure the phase between the 40MHz signal from the LO and the 40MHz signal from the PD with the advantage that the interferometer does not need to be locked. Locking the interferometer over many hours would require an actuator with a large dynamic range which was not available. Since the phase counter permitted one phase reading per second, spectral estimates up to the Nyquist frequency of 0.5 Hz could be obtained. The possible effect of aliasing has been estimated to less than using the high-frequency measurements described below and found to be negligible.
For the high-frequency measurements we have locked the interferometer to constant output phase with the advantage that phase information can be obtained at high frequencies. The drawback is that the dynamic range of our PZT-mirror limited the measurement time to a few minutes. As shown in Fig 2b, mirror M1 has been replaced by a mirror with a longitudinal PZT actuator. The 40MHz photodiode signal has been mixed down with the LO signal in a double-balanced mixer with subsequent 100 kHz low-pass filter. A preamplifier (SRS560 by Stanford Research Systems) and an integrator as servo have been used to lock the interferometer output phase. By changing the gain of the preamplifier, the unity gain frequency of the interferometer control loop could be varied between 47 Hz and 3 kHz.
When the interferometer was locked with a bandwidth of 47 Hz the error signal was used as measure for the fiber amplifier phase noise for Fourier frequencies above 400 Hz. Phase noise measurements between 0.02 Hz and 400 Hz have been performed by locking the interferometer with 3 kHz unity gain frequency and measuring the PZT actuator signal. The respective signal was fed to a commercial spectrum analyzer (SR785 by Stanford Research Systems).
Figure 3 shows the measurement results obtained with setup a. The solid curve shows the excess phase noise produced by the pumped fiber amplifier while the curve “unpumped” shows the measurement sensitivity of the setup measured with the unpumped amplifier. In comparison to preliminary experiments (not shown) with the unpumped amplifier we were able to reduce the noise floor to the level presented here by placing the setup in an isolation housing thereby suppressing air convection. Towards 0.1 mHz and 0.5 Hz, the solid curve approaches the curve “unpumped”, which means that the measurement result is limited by the sensitivity of the setup. Towards 0.5 Hz the measurement sensitivity was limited by the phase counter: the curve “func. gen.” has been obtained by feeding two identical 40MHz signals from a function generator to the phase counter. Towards 0.1 mHz the measurement was limited by frequency noise of the seed laser because the interferometer was not fully balanced: we have measured the sensitivity of the interferometer phase to frequency changes to 46.8 rad/GHz which corresponds to an optical interferometer armlength difference of 2.2 m. The laser frequency noise density as function of Fourier frequency f follows a 1/f-slope and was estimated as 10 kHz/Hz0.5 at 1Hz . The curve “NPRO induced” shows the resulting phase noise induced by the seed laser frequency noise.
Figure 4 shows the results of our phase noise measurements obtained with setup b. The solid curve represents the measured phase noise of the pumped fiber amplifier, the dashed curve represents the phase noise of the unpumped fiber amplifier, which is a measure for the sensitivity of the setup. The curve of the pumped amplifier is significantly above the curve of the unpumped amplifier, i. e. the interferometer is sensitive enough to measure excess phase noise of the fiber amplifier. The roll-off at 7 kHz of the curve labelled “pumped” can be explained by filtering effects due to the effective lifetime of the upper laser level in the fiber amplifier as discussed below. We have measured the transfer function from pump power to phase at the interferometer output. The resulting transfer function has the shape of a low-pass filter with a corner frequency of 7 kHz. The transfer function from pump power to amplifier output power also shows low-pass filter characteristics with the same corner frequency which indicates that the same physical mechanism might be present in both cases: pump power variations slower than the corner frequency are translated to output power and phase variations of the amplifier.
For faster frequencies than 7 kHz the inversion cannot follow the pump power variations and the effective lifetime of the Yb3+-ions acts as a low-pass filter. Multiple mechanisms are possible for phase changes due to pump power changes: heat generating processes from the upper laser level, direct modulation of the refractive index due to modulation of the inversion, and the nonlinear refractive index in combination with pump induced amplifier power fluctuations. The magnitude of latter effect has been estimated and found to be negligible. From the nonlinear refractive index n 2(glass)=3·10-20 m2/W  we estimated the transfer function from pump power changes to phase changes and derived a transfer function a factor 170 below the measured transfer function.
The trace “pump induced” shows the influence of pump power noise on phase noise of the pumped amplifier. It was obtained by multiplying the measured power noise of the pump radiation (see Section 3) by the transfer function from pump power noise to phase noise (see Section 5). The trace “pump induced” indicates that pump power noise is a relevant contributor to the phase noise of the pumped amplifier.
Since the optical frequency ν is the derivative of the optical phase ϕ with respect to time, the linear phase noise spectral density Sϕ(f) in units of can be converted to linear frequency noise spectral density Sν(f) in units of Hz/√Hz, both as function of Fourier frequency f, by Eq. (1)
that has been used to convert the data from Figs. 3 and 4 and plot the results in Fig. 5. Since the phase noise measurement in Fig. 3 was limited by the sensitivity of the interferometer (we demanded a signal-to-noise ratio of 3) for frequencies towards 0.5 Hz, the data between 0.1 Hz and 0.5 Hz have not been shown. Despite having been measured with two different setups, the two sets of measurements fit well. The highest excess frequency noise of the fiber amplifier has been measured as 0.4 Hz/Hz0.5 at 6 kHz.
The excess frequency noise of our fiber amplifier in the kHz frequency range is mainly dominated by pump power noise as has been analyzed in the high-frequency phase noise measurements shown in Fig. 4. In the mHz range pump power noise does not seem to be the dominating noise source. Instead, comparisons of ambient air temperature fluctuations above the amplifier for unpumped and pumped operation indicate higher temperature noise levels for pumped operation, which could be the dominating noise source for phase noise at mHz frequencies. Measurements showing lower phase-noise of the pumped amplifier might be obtained by further improved reduction of air convection or even by operation in vacuum.
5. Power and frequency actuators for stabilization
The frequency and power stabilities presented in the preceding sections need to be actively improved for LISA to reach its design sensitivity. The simultaneous control of frequency and power requires separate actuators for both parameters. The frequency can be controlled by controlling the seed laser frequency. A PZT crystal glued onto the laser crystal of the NPRO is a commonly used actuator for its frequency. As will be shown in the following, the control of the output power requires two actuators: modulation of seed laser pump current and modulation of amplifier pump current are possible actuators.
The investigations discussed in Section 4 have shown that amplifier pump power variations also induce amplifier phase variations. In nonplanar ring oscillators it is known that pump power variations change the laser frequency and a parasitic effect of signals applied to the NPRO PZT on the laser output power has also been measured . A detailed discussion of these correlations is however out of the scope of this article. Experiments on pump power induced laser frequency control have been reported in refs.  and .
The design of optimized control loops requires the knowledge of the dynamic behavior of the laser system. Therefore, we present transfer functions from the seed laser PZT to laser frequency and from seed laser pump power and amplifier pump power to output power.
Figure 6 shows the measured transfer function from relative seed laser power variations to relative amplifier output power variations. It has been obtained by modulating the seed laser pump current, measuring the seed laser output power and the amplifier output power. The transfer function has the shape of a high-pass filter with DC bypass.
In a saturated amplifier, slow seed laser power variations of 1mW translate to output power variations of 1mW, since an increase in seed power reduces the inversion and hence the gain in the amplifier. The transfer function for the relative power variations is hence 1/G. For modulation frequencies above the inverse lifetime of the upper laser level, the inversion cannot follow the seed power changes, and a seed power modulation of ΔP s translates to
of output power modulation where
is the gain of the amplifier. Hence, the amplifier acts as a linear amplifier for high modulation frequencies. Then, using Eqs. (2) and (3) the transfer function from relative seed power variations to relative output power variations equals 1:
Hence, for a fully saturated amplifier the amplifier gain G=g sat can be read from the difference between the low-frequency and the high-frequency amplifier response as discussed below.
The magnitude shape shown in Fig. 6 can mathematically be modelled by the product of a real zero at frequency f 2 and a real pole at frequency f 3 with a scaling factor H 2 as shown in Eq. (5). This and the following mathematical models can be found in e.g. .
Equation 5 has been fitted to the measured complex transfer function and f 2=313 Hz and f 3=12.6 kHz have been obtained.
From the high-frequency limit
and the fact that the relative transfer function approaches 1, it follows that the scaling factor is given by H 2=f 2/f 3.
The fit of Eq. 5 to the data of Fig. 6 and all following fits have been carried out with the complex logarithm of the data to account for the large variations in magnitude as explained below. From the measured magnitude r and phase ϕ, the complex transfer function z=r·exp(iϕ) has been obtained, where r, ϕ, and z depend on frequency. When performing a least square fit of H 2(f) to the complex transfer function z obtained from Fig. 6, a faithful representation of the measured data by the fit function is complicated by the large variations in magnitude, i.e. a fit to the data between 10 Hz and 100 Hz is likely to be inaccurate due to the low magnitude in this frequency range. More accurate fits are obtained by fitting the logarithm log(z)=log(r)+iϕ of the measured data to the logarithm of the fit function.
With the fitted values for f 2 and f 3 in Eq. (5) a saturated gain of g sat=40 results. The amplifier on the other hand produced 1W of output power from 10mW of seed power which corresponds to an overall gain of G=100 (20 dB). This means the amplifier was not fully saturated. As model for the not fully saturated amplifier we use a purely linear amplifier with gain g lin and a fully saturated amplifier with gain g sat in series. The overall gain G=g lin·g sat then is the product of linear gain g lin=2.5 and saturated gain g sat=40. This is in good agreement with the theoretical saturation power of 15.5mW.
For frequencies below f 2, seed laser power modulation is not a suitable actuator because relative seed laser power modulations are suppressed by the saturated gain of the amplifier and large absolute seed laser power variations would be required. Above f 3 power modulation of the seed laser is a good power actuator of the overall system.
Figure 7 shows the transfer function from relative amplifier pump power variations to relative amplifier output power variations. The pump current of the fiber amplifier pump diode has been modulated and the transfer function to amplifier output power has been measured. The transfer function has the shape of a low-pass filter, where the corner frequency f 4 is given by the inverse effective lifetime of the upper laser level in the amplifier.
Equation 7 has been fitted to the measured complex transfer function and f 4=12.5 kHz has been obtained.
Since amplifier pump current variations directly modulate the amplifier inversion and hence the amplifier gain, the transfer function of relative pump power variations to relative output power variations approaches 0 dB towards DC (H 3=1). The higher corner frequency of 12.5 kHz in comparison to 7 kHz, as measured in Section 4 is due to a higher pump power in the transfer function measurements. Due to the higher pump power the effective lifetime of the upper laser level is reduced and hence the corner frequency increased .
Pump current is a good actuator for the amplifier output power for frequencies below f 4. For higher frequencies, high pump current changes would only lead to small amplifier output power variations.
Seed laser pump current modulation and amplifier pump current modulation are each not sufficient as actuators but the sum of both is a suitable actuator. To prove this, the sum of the fit-functions (Eqs. (5), (7)) has been plotted in Fig. 8. The magnitude and phase of the sum have been plotted as trace “combined” in Fig. 8. The magnitude of the combined transfer function is flat and the phase is zero with only a tiny wiggle at the cross-over frequency around 12.5 kHz. One should note that the complex functions have been added.
Figure 9 shows the measured transfer function from signals applied to the seed laser PZT to laser frequency behind the amplifier. Two measurements were performed to obtain Fig. 9: the shape of the graphs has been determined with the setup shown in Fig. 2 b), where the interferometer output phase was locked with a bandwidth of 47 Hz and the transfer function from laser PZT to interferometer phase error signal has been measured. Due to the unequal interferometer arms, laser frequency changes directly translate to phase changes at the interferometer output. The overall gain of the transfer function (the frequency tuning coefficient at DC) was obtained using a scanning Fabry-Perot interferometer.
The measured transfer function was flat and the DC-frequency tuning coefficient was measured as 1.67 MHz/V. Typically, the first mechanical resonance of the PZT can be found around or above 100 kHz and a PZT resonance of an NPRO can be fitted by a complex pole. However, since the transfer function is flat up to 100 kHz and data at higher frequency were not available, no faithful fit was possible.
In conclusion we have presented sensitive power- and frequency-noise measurements of an Yb-doped fiber amplifier and investigated actuators for power and frequency stabilization. We have found that the master-oscillator power-amplifier system is suitable for the gravitational wave detector LISA concerning its phase noise properties. We have found that the excess frequency noise of the fiber amplifier is negligible compared to the frequency noise of free-running non-planar ring oscillators but above the allowed level of frequency fluctuations for LISA. Depending on the measurement procedure and the laser stabilization scheme that will be implemented in LISA, the measured phase noise of the fiber amplifier has to be considered. Excess power noise of the amplifier due to the pump light has been measured. It needs to be suppressed by a factor of at least 200 to meet the LISA requirement within the measurement band of 0.1 mHz to 1 Hz. Power stabilization of an NPRO that shows similar power fluctutations as the fiber amplifier at 1 mHz has been successfully demonstrated in independent measurements and will be reported on elsewhere. We have identified suitable actuators for the system: a PZT crystal on the seed laser, the seed laser pump current, and the amplifier pump current. An implementation of power stabilization and frequency stabilization of the master-oscillator power-amplifier system is in progress.
We thank Gerhard Heinzel and Dietmar Kracht for useful discussions.
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