1. Introduction

Laser frequency stabilization techniques such as optical frequency combs [1, 2], have led to unprecedented levels of precision of optical frequency standards, to an accuracy of approximately 5 parts in 10¹⁷ [3]. With such high stability, it is believed that atomic clocks based on optical standards will replace those based on microwave standards.

The current stabilization techniques, however, are complex, expensive and not very portable, and therefore methods for distributing highly coherent stable frequencies over long distances has become vital in furthering the field. In the past satellite based methods [4] have been used with some success, for transferring microwave frequency standards. However, for transferring optical frequency standards, a fiber-optic based transmission is generally the dissemination technique adopted, due to its relatively easy installation, flexibility and the fact that it can yield high fractional frequency stabilities [5, 6].

Remote transfer of an optical frequency standard over an optical fiber network is also desirable for large area science projects that require synchronization of various systems over the facility. Some examples of such facilities include radio telescope arrays and particle accelerators. A more specific application exists in the lock acquisition of interferometric gravitational wave detectors such as the Advanced Laser Interferometric Gravitational Wave Observatory (AdvLIGO) [7]. The lock acquisition scheme currently being designed for AdvLIGO involves pre-stabilizing the 4 km long Fabry-Perot cavities in the arms of the instrument to an auxiliary laser [8]. A strict requirement on the auxiliary lasers, which are located at the ends of the 4 km long interferometer, is that the relative frequency noise between the main science and auxiliary lasers is less than 70 mHz/√Hz within the signal bandwidth for locking the arm cavities (around 400 Hz). To achieve this, the stable frequency reference of the main science laser will be transferred via a 4 km fiber to the auxiliary laser.

The biggest issue with fiber-optic transfer is that optical path length fluctuations arising from mechanical and temperature variations in the fiber degrades the phase noise and decreases the stability of the transmitted optical frequency. Several techniques exist to suppress this fiber-induced phase noise and ultra-stable standards have been successfully transferred over fiber links [9, 10]. These methods implement a common scheme to stabilize these fiber fluctuations, first suggested by Ma et al [11], where the standard is first passed through an acousto-optic modulator (AOM), before being sent through the fiber to some remote station. Some of the fiber destabilized standard is reflected back through the fiber and AOM, and combined onto a photodetector with some of the original stabilized frequency standard light. A correction signal is derived and fed back to the AOM to suppress the fiber phase noise being added.

In this paper we report on a different technique for the stable dissemination of an optical frequency standard over an optical fiber link. The technique, which is similar to the LISA (Laser Interferometric Space Antenna) back-link measurement [12], utilizes sensitive phasemeters to phase-lock an auxiliary laser located in a remote station to the science laser producing the frequency standard in the local station. The phasemeters are highly immune to amplitude fluctuations and they allow for direct feedback to the laser actuator eliminating the need for AOMs.

We also report on the demonstration of this technique in a bench top experiment and characterize its performance. This experiment was specifically developed for the remote transfer of the stable frequency reference for the AdvLIGO lock acquisition scheme, however, the technique can also be applied to larger scale optical frequency standard transfer.

2. Technique

The diagram in Fig. 1 shows the basic layout of the technique. The science laser, which is located in the local station, produces the frequency standard. Light from this science laser is sent via the optical fiber link to the remote station, which houses the auxiliary laser. The light that exits the fiber at the remote station has extra phase noise that has been introduced by the fiber path length fluctuations. This light is combined with light from the auxiliary laser. The resulting heterodyne signal, which is at a frequency equal to the relative frequency between the two beams, is detected with a photodetector (PD1) and read out by a phasemeter.

 

Fig. 1. Basic layout of the technique where the sum of the phasemeter readouts are fed back to the auxiliary laser to cancel out fiber induced phase noise. BS = Beam Splitter, SM = Steering Mirror, PD = Photodetector.

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The phase measurement is used to adjust the frequency of the auxiliary laser to phase-lock it to the light exiting the fiber. In this way the phase of the auxiliary laser follows that of the science laser plus the fiber induced phase fluctuations.

To eliminate these fiber phase fluctuations, the auxiliary laser light is sent back down the fiber to the local station, where it is combined on another photodetector (PD2) with the science laser light. Another phasemeter reads out the phase of this heterodyne signal. This phase measurement is then sent via a digital link to the remote station. The local station signal is added to the remote station signal and then fed back to the auxiliary laser.

Counter propagating the two laser beams through the fiber is the key to eliminating the fiber noise. The observed phase fluctuations of the local, ϕ_loc, and remote, ϕ_rem, station phasemeters in the frequency domain are given by:

where ω = 2πf is the Fourier frequency (radians/second) and ϕ_sci, ϕ_aux and ϕ_fib, are the phase fluctuations of the science laser, auxiliary laser and fiber optic link respectively. The delay, τ, is the travel time of the light through the fiber and is equal to the fiber length divided by the speed of light in the fiber (approx. 2 × 10⁸ m/s). The local station signal takes some finite time, ε to travel to the remote station. A consequence of this being that the remote signal is added to the past local signal. This is represented by the product of the local signal ϕ_loc and an exponential term e ^-iωε being added to the remote signal ϕ_rem, as can be seen in Equation 3.

Bode plots of the transfer functions from the phase fluctuations of the science laser, the auxiliary laser and the fiber to the sum of the local and remote signals, can be seen in Fig. 2. The length of the fiber was set at 4.6 km, giving a τ of approximately 23 μs, and the electronic delay ε was approximated to be slightly longer than τ at 26 μs.

 

Fig. 2. Transfer functions from the science laser (a), auxiliary laser (b) and fiber (c) phase to the sum of the local and remote signals.

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Plot (a) in Fig. 2 shows that response of the summed signal to the science laser phase is a factor of 2 (6 dB) and flat up to about 50 kHz. It then rolls off into a null at a frequency given by half the inverse of the difference between the local signal travel time and the fiber delay (i.e. 1/ [2(τ − ε)] ≈ 167 kHz). The phase rolls off much earlier than this, with a phase delay of 180° at 20 kHz.

Plot (b) in Fig. 2 shows that the response to the auxiliary laser phase is also a factor of 2 and flat up to about 2 kHz before rolling off to a null at approximately 10 kHz. The first null here is given by half of the inverse of the sum of the local signal travel time and fiber delay (i.e. 1/[2(τ + ε)] ≈ 10.2 kHz). Each null is accompanied by 180° phase jump. This discontinuity is not due to phase wrapping, it is instead a result of the response passing through zero.

Plot (c) in Fig. 2 shows that the response of the summed signal to the fiber phase is less than unity for Fourier frequencies below 10 kHz, and within this frequency range the magnitude of the response is proportional to the Fourier frequency.

Feeding this summed signal back to the auxiliary laser with appropriate amplification and filtering, will result in the auxiliary phase fluctuations tracking those of the science laser below 1 kHz, while effectively not tracking the fiber phase fluctuations.

3. Experiment

The technique was tested in a bench-top experiment. The science laser was an Innolight Prometheus Nd:YAG laser, with an optical wavelength of 1064 nm, while a Lightwave Nd:YAG laser was used as the auxiliary laser.

The optical fiber link used in the experiment was a 4.6 km long fiber that was single mode for 1064 nm. The fiber link had the potential to introduce other non-linear forms of noise to our experiment, such as stimulated brillouin scattering (SBS) and Rayleigh back scattering. We minimised these effects by reducing the optical power of the light going into the fiber down to 50 μW. The fiber is also non-polarization maintaining and the polarization of the light passing through the fiber would drift, causing the amplitude of the heterodyne signal to also drift. However the phasemeters that we used had an amplitude to phase coupling of about 0.06 μradians/% and therefore the deviation of the amplitude due to polarization drift was not a factor.

The photodetectors that we used as our local and remote photodetectors were built in-house and have a flat frequency response up to approximately 600 MHz [13]. The RF (Radio Frequency) output from each photodiode was digitized by sending it through an anti-aliasing filter and then a high-speed 40 MHz Analog-Digital Converter (ADC). Each digitized signal was sent to its own phasemeter, which was a digital phase-locked loop implemented on a Field Programmable Gate Array (FPGA) - National Instruments PXI-7833R. The phasemeters were similar to the LISA phasemeter [14] and can measure either frequency or phase of the beatnote, referenced to a local 10 MHz oscillator. The phasemeter also included a digital controller and analog outputs to phase-lock one laser to another with a controllable frequency offset.

The remote phasemeter measured the phase of the heterodyne signal on the remote photode-tector and contained information shown in Equation 2. At the same time the local phasemeter measurement (represented in Equation 1) was sent to the remote phasemeter via a digital link. For this benchtop demonstration, both phasemeters were co-located. To simulate the time-delay present in a distributed system we added a 26 μs delay to the local phasemeter signal before combining it with the remote phasemeter signal. The two phasemeter signals were summed together so that the fiber-induced fluctuations would be cancelled as shown in Equation 4. This ϕ ₊ signal was then sent through the phasemeter’s digital controller and fed back to the auxiliary laser by the digital-to-analog converter (DAC) outputs. For fast frequency control one DAC output was fed back to a piezo-electric transducer (PZT) that exerted pressure on the laser crystal. A second DAC output was fed back to the temperature of the laser crystal to provide larger range feedback. The DAC channels had an independent update rate of 1 MHz.

To characterize the performance of the technique, light from each laser was tapped off before entering the fiber and combined on another photodetector. The RF signal from this photodetector was provided to a third phasemeter to directly measure the phase fluctuations between the two lasers.

4. Results

Initially the system was locked up with remote signal feedback (Equation 2) only to observe frequency stability degradation caused by fiber phase noise. A ten second trace of the phase difference was recorded with a 100 kHz sampling rate and is plotted as trace (a) in Fig. 3. As can be seen, the phase difference between the two lasers varies over a range of about ±7 radians.

The local station signal was then added to the remote station signal and fed back to suppress fiber fluctuations. Another ten second trace was taken and is plotted as trace (b) in Fig. 3. There is a significant suppression of the phase fluctuations between the two lasers when the local signal is added. The insert in Fig. 3 shows a two second segment of the suppressed phase noise exhibiting a variation of about ±0.1 radians, indicating a suppression factor of approximately 70.

 

Fig. 3. The time series of the relative phase fluctuations between the two lasers when (a) the remote signal was fed back to the tracking laser and (b) the sum of the local and remote signals was fed back to the tracking laser eliminating the fiber phase fluctuations.

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A root power spectral density of the two time series in Fig. 3 (a) and (b) was taken and converted from phase noise to the equivalent frequency noise, and then plotted in Fig. 4 (a) and (b) respectively. To characterize our fiber noise cancellation technique, the fiber link was bypassed by placing a steering mirror in front of each fiber input coupler and the two beams were directed towards each other. The relative frequency noise for this setup was also measured and plotted as (c) in Fig. 4. This plot is the measurement noise floor for our experiment and is due to the relative motion of the sensing optics at both ends. The gain settings were kept the same for all three cases. The AdvLIGO lock acquisition requirement has also been included as (d) in Fig. 4. The fraction of the frequency noise over the optical frequency has been included on the right axis of Fig. 4.

 

Fig. 4. The relative frequency noise between the two lasers and the fractional frequency when (a) the remote signal was fed back to the tracking laser, (b) the sum of the local and remote signals was fed back to the tracking laser, (c) the fiber was bypassed and (d) the AdvLIGO requirement

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In plot (a) of Fig. 4 (remote signal feedback only) the relative frequency is considerably noisier than the measurement noise floor. As the only difference between these two setups is the inclusion of the 4.6 km fiber, it can be deduced that fiber phase noise is the limiting noise source in (a). It is also worth noting that the fiber did not receive any isolation from the surrounding environment and was therefore subject to acoustic, seismic and thermal noise.

The addition of the local signal feedback in plot (b) of Fig. 4 (remote + local signal feedback), shows a considerable suppression of the relative frequency noise. For Fourier frequencies below 100 Hz the suppression from (a) to (b) is more than two orders of magnitude. A comparison of (b) with (c), shows that the fiber induced noise has been suppressed down to the measurement noise floor. Between 5 Hz and 20 Hz the level of stability is approximately 0.5 mHz/√Hz, equivalent to a fractional frequency stability of 1.7 × 10^-18/√Hz.

In plot (b) of Fig. 4, from about 100 Hz to 1 kHz, the relative frequency noise rolls up as approximately f ². This part of the spectra is gain limited in that the relative frequency noise between the two lasers rolls down as f ^-1 when not locked, and the digital controller response has a f ^-3 slope in this frequency range. The peak just below 10 kHz can be explained by the lack of phase margin, which has been reduced by the fiber and digital delays. It was found that increasing the gain of the controller resulted in the loop becoming unstable. However when the fiber was bypassed the gain could be increased substantially before losing stability. This is further evidence that the fiber and digital delays limited the phase margin.

The relative frequency noise in (b) meets the AdvLIGO requirement with more than 10 times margin at frequencies below 100 Hz. The noise below 20 Hz doesn’t match that of the controller response and can be explained by the measurement noise floor resulting from the relative motion of the sensing optics. Another possible noise source that is not included in Equation 3 is the frequency noise between the local and remote phasemeters’ 10 MHz reference oscillators. We estimate that for standard off-the-shelf signal generators this noise source is below our measurement noise floor.

5. Conclusion

We have reported a technique for transferring a stable optical frequency standard to a remote location via an optical fiber link. We have tested this technique in the laboratory using a 4.6 km fiber, with simulated digital signal delay and highly sensitive phasemeters to phase-lock two lasers. The technique has been shown to meet the requirements for the AdvLIGO lock acquisition scheme for Fourier frequencies below 100 Hz. A relative frequency noise of 0.5 mHz√Hz was reached for Fourier frequencies between 5 Hz and 20 Hz which is equivalent to a fractional frequency stability of 1.7 × 10^-18/√Hz.