1. Introduction

Several designs for space based gravitational wave detectors have been proposed [1–3]. Of these the Laser Interferometer Space Antenna (LISA), a proposed joint mission between NASA and ESA, is the most studied. Even though the future status of LISA is uncertain, its design specifications will be used here as the interferometry of the other proposed designs are similar to that of LISA.

LISA is designed to observe gravitational radiation at frequencies around 1 mHz. The LISA mission uses heterodyne laser interferometry to measure fluctuations in the path length between three spacecraft (SC) caused by gravitational waves. In order to reach the precision required to observe gravitational waves, the laser frequency noise must be reduced using frequency stabilization and time delay interferometry (TDI). TDI is a technique in which the phase measurements on each SC are combined with time delayed versions of the same signals in a way that cancels the laser frequency noise [4]. For LISA to be successful the SC must be able to communicate with each other in order to transfer clock signals, measure the light travel time between the SC, and to share recorded data. All of these auxiliary functions will be accomplished using the laser links between the SC.

Each SC measures the phase of the heterodyne signals relative to its own clock. Since TDI requires measurements from each SC, the clock noise enters the TDI combinations [5] and is non-negligible. Each phase measurement is made in comparison to an oscillator of the same frequency of the beat note being measured. This oscillator is derived from the local clock and has a phase noise of $\frac{\nu }{{\nu }_{\mathit{clk}}}{\mathrm{\Phi }}_{\mathit{clk}}(t)$ where ν is the frequency of the merasured laser beat note and ν_clk and Φ_clk(t) are the frequency and phase noise of the clock. To cancel this noise, the clock signals on each SC will be frequency up-converted to a GHz signal using a frequency synthesizer, modulated onto the laser signals using an Electro-Optic Modulator (EOM), and sent to the other SC. There the sidebands will beat against similar sidebands generated by the clock of the local SC. These beat signals will be recorded and used in post-processing to measure and then cancel the clock noise [6]. Since the sidebands will only contain ten percent of the laser power their signal to noise ratio will be lower than that of the main carrier beat note. Up-converting multiplies the phase noise of the clock signals by the up-conversion factor reducing the precision required to measure the clock phase noise. The clock noise is measured by subtracting the upper sideband-sideband measurement from the lower sideband-sideband measurement. This cancels the common laser phase noise and doubles the clock noise signal which is contained in both upper and lower sideband beat notes. This results in a measured clock noise signal of S = 2αΦ_clk(t), where α is the frequency up-conversion factor.

The LISA science team has set a maximum noise requirement of

The clock transfer requires frequency synthesizers, high frequency electronic cables, and EOMs. All of these components must be investigated to ensure that they do not add phase noise beyond the requirement in Eq. (3). Other groups investigated EOMs at 8 GHz [7], EOMs at 2 GHz [8], and high frequency cables at 2 GHz [8]. The 8 GHz experiment measured the differential phase noise added between two EOMs using a single laser source. The 2 GHz experiment measured the phase noise added by a single EOM in a set-up with two lasers utilizing a low frequency LISA Technology Package (LTP) type phasemeter to measure the phase fluctuations. We present results on a differential measurement of the phase noise added by two frequency synthesizers, a measurement of the phase noise added by a single EOM at a modulation of 2 GHz using a LISA like phasemeter, a differential measurement of the phase noise added by the combination of two frequency synthesizers and two EOMs, and an electronic test of the clock noise transfer concept using the frequency synthesizers.

2. Experimental results

The experiments in this report make use of the University of Florida LISA Interferometry Simulator (UFLIS) phasemeter [9]. The UFLIS phasemeter is an IQ tracking phasemeter programmed onto the FPGA of a Pentek digital signal processing board. The UFLIS phasemeter is capable of measuring the phase of a 20 MHz signal to an accuracy approximately $6\times {10}^{-6}\mathit{cycles}/\sqrt{Hz}$ at 1 mHz. This limitation is due to timing jitter and improves as the signal frequency is lowered. All of the experiments in this report were performed with the phasemeter clocked at 50 MHz and the phase data down sampled to 11.9 Hz.

2.1. Frequency synthesizers

Figure 1 is a diagram of the experimental set up to measure the differential noise between two frequency synthesizers. The tested synthesizers were manufactured by Rupptronik (www.rupptronik.de) and are capable of converting a 50 MHz signal to a frequency between 1.980 and 2.020 GHz in steps of 100 kHz. A 50 MHz signal was split and sent to two frequency synthesizers to produce two 2.001 GHz signals. Each of these signals was mixed with a common 2.000 GHz signal. The outputs of those mixers were low pass filtered and the resulting 1 MHz signals were measured with the UFLIS phasemeter. The difference of the two measurements is plotted in Fig. 2. Also plotted for comparison is the result of the same experiment using the output of two Stanford Research Systems CG635 clock generators phase locked to a common 10 MHz signal. The frequency synthesizers meet the requirement for all frequencies while the clock generators do not. The differential phase noise generated by the two synthesizers declines with $1/\sqrt{f}$ dependence towards higher frequencies. It has ample margin at all frequencies except around 3 mHz where LISA and most other space based gravitational wave proposals reach their maximum strain sensitivity.

 

Fig. 1 Set up of the experiment to measure the differential noise added to the clock transfer by two frequency synthesizers. A common MHz signal is frequency up-converted to the GHz range by two separate frequency synthesizers. These two signals are heterodyned down to 1 MHz signals by electronic mixing with a common GHz signal.

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Fig. 2 Linear spectral density of the differential phase noise added by two Rupptronik frequency synthesizers. Also plotted is the linear spectral density of the same measurement using Stanford Clock Generators as frequency synthesizers and the phase noise requirement on the clock noise transfer. The Rupptronik frequency synthesizers meet the requirement for all frequencies while the clock generators do not.

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2.2. Electro-Optic modulators

Figure 3 is a diagram of the experiment to determine the noise added to the clock noise transfer by the EOM when it is modulated at 2 GHz. The input and output of the tested EOM (Jenoptik, model PM1060HF, www.jenoptik.com) are coupled to polarization maintaining fibers. This EOM was also tested by a separate group of researchers [8]. A 2 GHz signal is modulated onto Laser 2 which is offset phase locked to a reference laser at 1.999 GHz. The reference laser also passes through an EOM because the same set up was used for other experiments in which both lasers were modulated. In this experiment the reference laser only serves to create a beat note with laser 2 and any phase noise added by the EOM will be canceled along with the intrinsic phase noise of the reference laser. The bandwidth of the phase-locked loop is 10 kHz, which is much smaller than the GHz sideband signals. Thus the sidebands are safely outside of the servo bandwidth and will not be suppressed by the phase-locked loop. The modulation results in three signals at the photodetector; the lower sideband at 1 MHz (aliased from −1 MHz), the carrier at 1.999 GHz, and the upper sideband at 3.999 GHz (which is well outside of the bandwidth of the photodetector). The phase of the lower sideband and the carrier are

 

Fig. 3 The experimental set up for the test of the EOM’s phase stability at 2 GHz. Laser 2 is phase locked to the reference laser at an offset frequency of 1.999 GHz. Laser 2 is also modulated with a 2.000 GHz signal creating a lower sideband at 1 MHz. The lower sidebands is filtered and measured at channel 1 of the phasemeter while the carrier signal is electronically mixed with the modulation signal. The output of the mixer is filtered leaving a 1 MHz signal which is measured at channel 2.

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Fig. 4 Linear spectral density of the results of the noise in the clock noise transfer using the EOMs at 2 GHz. The EOM meets the requirement at all frequencies.

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2.3. Frequency synthesizer and EOM combination

In space based gravitational wave detectors the frequency synthesizers and EOMs will be used together to transmit the clock noise signals. An experiment was performed to test both devices at the same time in a single experiment. Figure 5 is a differential measurement of the combined phase noise of the frequency synthesizers and the EOMs. A 50 MHz signal was frequency up-converted by two frequency synthesizers, one to 2.000 GHz and the other to 2.009 GHz. This same 50 MHz signal was used to clock the phasemeter. Each signal was modulated onto separate lasers using EOMs. The beat note between the lasers was formed as in section 2.2, and Laser 2 was offset phase locked to the reference laser at 10 MHz. Two signals from the photodetector were measured with the phasemeter, the 10 MHz carrier-carrier signal, and the 1 MHz lower sideband-sideband signal. If ν_clk and Φ_clk(t) are the frequency and phase noise of the 50 MHz signal, α₁, and α₂ are the up-conversion factors of the frequency synthesizers, and β₁, and β₂ are the combined phase noises of the frequency synthesizer and EOM combinations, then the phase noise of the carrier-carrier beat note is measured as

 

Fig. 5 The experimental set up of the combined test of both the frequency synthesizers and the EOMs. Not shown is that the 50 MHz signal that is frequency up-converted is also used to clock the phasemeter.

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Fig. 6 Linear spectral density of the results of the differential phase noise added by the frequency synthesizer and EOM combination. The frequency synthesizers and EOMs together meet the requirement at all frequencies.

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2.4. Electronic test of clock noise transfer

The previous experiments used a single clock to measure the phase noise added by the frequency synthesizers and the EOMs. LISA and other proposed space based gravitational wave detectors will measure the different signals with independent clocks. The differential clock noise and clock rate must be measured and included in the TDI combinations. Figure 7 is a diagram of an electronic test of the clock noise transfer using frequency synthesizers. A 20 MHz signal was split and measured with two separate phasemeters, each clocked by independent 50 MHz oscillators. Our digital signal processing hardware allows for two separate pairs of analog to digital converters (ADCs) and FPGAs to be externally clocked by different sources. Channels 1 and 2 were clocked by clock 1, while channels 3 and 4 were clocked by clock 2. Both clocks were run independently of each other and set to 50 MHz. The clock signals were both frequency up-converted with the frequency synthesizers; clock 1 to 2.00 GHz, and clock 2 to 2.001 GHz. The up-converted signals were mixed, low pass filtered, and measured by the phasemeter on channel 2. Let f and ϕ(t) be the common 20 MHz oscillator’s frequency and phase noise, F₁, Φ₁(t) and F₂, Φ₂(t) be the frequency and phase noise of clocks 1 and 2 respectively, and α₁ and α₂ be the up-conversion factor of the frequency synthesizers. The signals measured by the phasemeter are

 

Fig. 7 The set-up for the electronic test of the clock noise transfer concept using frequency synthesizers. A common 20 MHz signal is split and measured on channels 1 and 3 of the phasemeter. Channels 1 and 3 are each clocked by independent 50 MHz clocks. Each clock signal is up-converted, one to 2.000 GHz and the other to 2.001 GHz. The up-converted clock signals are electronically mixed, filtered, and sent to channel 2.

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Figure 8 shows the results of the clock noise transfer. The initial 20 MHz signal is plotted in red, while the contribution due to the clock noise is found from S₁ − S₃ and is plotted in blue. The combination in Eq. (13) is plotted in cyan. Since the measurements of S₁ and S₃ are made relative to independent clocks, there is a delay that is smaller than the sampling rate (a fractional delay) between the signals which amplifies the noise. A traditional fractional delay filter cannot be used to remove this delay as the clocks run at slightly different frequencies causing the fractional delay between the signals to slowly grow larger. Instead a time varying fractional delay filter was implemented by recalculating the filter coefficients for every data point. The change in the fractional delay was approximated to be linear. The initial fractional delay was found by applying a traditional fractional delay filter to the first ten minutes of data and searching for the fractional delay value that minimized Eq. (13). The difference in the frequency between the clocks was measured by subtracting the means of the measured frequencies of S₁ and S₃. This value was used to calculate the rate of change of the fractional delay. The results of Eq. (13) with the time varying fractional delay filter are plotted in magenta. The measurement was limited by the expected phasemeter system noise for a 20 MHz signal. This limitation, at its worst, is slightly less than a factor of six above the requirement.

 

Fig. 8 Linear spectral density of the results of the electronic test of the clock noise transfer concept using frequency synthesizers. The red curve is the 20 MHz common signal, the blue curve is the mixed signal from the frequency synthesizers, the cyan and magenta curves are the result of Eq. (13) before and after the application of a time varying fractional delay filter.

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3. Conclusions

We have performed several experiments testing the phase fidelity of both frequency synthesizers and EOMs. These components are necessary for the clock noise transfer in space based gravitational wave detectors including LISA. We found that the differential noise between two frequency synthesizers was less than the requirement set by the LISA science team. The differential measurement suffers from two drawbacks, that there may be some common noise between the two synthesizers that is canceled in the differential measurement, and that the frequency synthesizers were both operated at the same frequency up-conversion factor. In LISA the up-conversion factor must be different in order to produce sideband-sideband beat notes at frequencies separate from the main carrier-carrier beat note. The experimental test of the EOMs did not suffer from these drawbacks as they were tested individually. They were found to meet the requirement at a modulation frequency of 2 GHz.

The frequency synthesizers and the EOMs were also tested in a differential measurement of their combined phase noise. The differential phase noise of the frequency synthesizer and EOM combination was found to be lower than the requirement. The frequency synthesizers were also used in an electronic test of the clock noise transfer. This experiment tested the ability of two synthesizers, operating at different up-conversion factors, to cancel the clock noise incurred by measuring the phase of a 20 MHz signal with two independently running clocks. The clock noise was measured and removed to the performance limitation of our phasemeter system by employing a time varying fractional delay filter. This limitation is at worst a factor of six higher than the requirement for clock noise cancelation.