Point-to-point stabilized optical frequency transfer with active optics

Timescale comparison between optical atomic clocks over ground-to-space and terrestrial free-space laser links will have enormous benefits for fundamental and applied sciences. However, atmospheric turbulence creates phase noise and beam wander that degrade the measurement precision. Here we report on phase-stabilized optical frequency transfer over a 265 m horizontal point-to-point free-space link between optical terminals with active tip-tilt mirrors to suppress beam wander, in a compact, human-portable set-up. A phase-stabilized 715 m underground optical fiber link between the two terminals is used to measure the performance of the free-space link. The active optical terminals enable continuous, cycle-slip free, coherent transmission over periods longer than an hour. In this work, we achieve residual instabilities of 2.7 × 10−6 rad2 Hz−1 at 1 Hz in phase, and 1.6 × 10−19 at 40 s of integration in fractional frequency; this performance surpasses the best optical atomic clocks, ensuring clock-limited frequency comparison over turbulent free-space links.

. Concurrent measurements of 265 m free-space link phase noise using an Ettus X300 (solid blue, tip-tilt off; solid orange, tip-tilt on) and a Microsemi 3120A (dashed cyan, tip-tilt off; dashed red, tip-tilt on). The Ettus and Microsemi are in very close agreement across the Microsemi's frequency range. The noise floor of the Ettus (gray) was obtained by measuring the phase noise of a low noise synthesizer at 1 MHz.

Note 2: Noise contributions in parallel compensated links
Here, we estimate the noise floor of the phase noise measurements presented in the Results section. The aim is to estimate the phase stabilization noise floor caused by atmospheric noise and laser frequency noise when compensating the two links.

Differential noise PSD
For a noise process x(t) that is stationary and characterized by a PSD S x ( f ), the PSD of the linear combination z(t) ≡ ∑ a i x(t − T i ) (where a i are arbitrary real constants) can be derived from first principles (see e.g. Duchayne 1 p.227) and is given by (1)

A single compensated link
Assume a compensated link (either of the two in Fig. 1 of main article) where a laser signal is frequency shifted by a transmission AOM before being injected into the link. The frequency of the laser signal is ν L (t) = ν L + ∆ν L (t) where ∆ν L (t) is the laser frequency noise. The transmission AOM shifts the optical frequency by a nominal frequency (ν tr ) which is varied by ∆ν tr (t) in order to suppress link noise. ∆ν tr (t) is ideally equal but opposite to the frequency shift that the signal will experience during propagation through the link (δ ν(t)). At the remote site, the optical signal is passed through a static anti-reflection AOM (ν ar ) before being passed out of the remote site. We will define t as the instant the signal is passed out of the remote site. The laser signal coming out at the remote site has frequency ν out (t).
where T is the propagation time through the link. The frequency shift δ ν(t) is compensated by measuring, at the local end, the beat-note between ν L (t) and a return signal (ν ret (t)) that went through the AOM twice (for simplicity we will ignore the remote site anti-reflection AOM, ν ar , and the nominal transmission AOM frequency, ν tr , applied by the transmission AOM): The phase-locked loop (PLL) 1 ensures that the beat-note signal (3) is zero, which means We linearize the expression by a Taylor expansion of the left side leading to a differential equation: where we neglect higher order terms in the Taylor expansion. The general solution of the differential equation is where C is an integration constant. As our loop is closed (the AOM frequency does not diverge) we have C = 0. Substituting that into (2) we finally get 1 We assume that the delay in the PLL is negligible with respect to T .

2/4
As one would expect, the expression reduces to ν out (t) = ν L + ∆ν L (t) when laser and link noise vary slowly with respect to T , i.e. the output signal is a copy of the input signal in spite of the presence of link noise.
Applying (1) to the laser noise contributions gives The second term quickly vanishes for f 1/T and we just have the input laser noise at the output. At approximately f ≥ 1/T the noise is increased by the compensation by a frequency dependent term that oscillates between 0 and 3S ∆ν ( f ).
For the link noise, we obtain also from (1) which, as expected, cancels completely when f 1/T . The total noise PSD of the output signal is the sum of (8) and (9).

Two parallel compensated links
We now consider two parallel compensated links A and B with delays T A and T B that are fed by the same input laser (see Fig. 1 of main article). We will be interested in the difference Dν out (t) ≡ ν outA (t) − ν outB (t), which is measured in the experiment. Each link is affected by the same laser noise ∆ν L (t), but by different link noises δ ν A (t) and δ ν B (t).
For the laser noise applying (7) we directly have When the two links have the same delay (T A = T B ) the laser noise cancels exactly. When they are different one can apply (1) to calculate the overall effect. We assume that the link noise is uncorrelated between the two links, and thus the link PSDs given by (9) simply add. The final result is then where ∆T ≡ T A − T B . As expected from (10), the laser noise contribution vanishes for T A = T B . In the limit when f 1/T the link noise contributions vanish, and (11) can be approximated as

Noise floor estimation
To estimate the noise floor in our compensated link we need to evaluate (11) using estimates of the free space link noise S δ νA ( f ), the fiber link noise S δ νB ( f ), and the laser noise S ∆ν L ( f ). We assume that the fiber link noise is negligible with respect to the atmospheric noise and use the "unstabilized, tip-tilt off" measurement (see Fig. 2 of main article) as our estimate of the latter. For the laser noise we use the "typical" noise curve given by the manufacturer 2 . The two delays were estimated from delay measurements in the fiber and from local maps as cT A ≈ 302 m (265 m free space + 25 m fibers with refractive index n = 1.45 between the telescopes and the beam splitters), and cT B = n × 715 m. The result is shown in supplementary Fig. 2.
We note that above about 200 Hz our measured results are well explained by the combined effects of atmospheric turbulence and laser noise, at least in terms of the orders of magnitude. The residual noise of the compensated links is dominated by atmospheric noise between 200 Hz and 2 kHz, and laser noise takes over at higher frequencies. The slight discrepancy in the atmospheric effect is probably due to the fact that the uncompensated measurement (solid red in supplementary was taken at a different time than the compensated one. The discrepancy at higher frequency is most likely due to the fact that the NKT X15 laser we used had more phase noise than the "typical" performance of that type of laser specified by the manufacturer in 2 . Nonetheless the similar shape and the fact that all measured curves on Fig. 2 of main article converge above 10 kHz comforts us in the hypotheses that laser noise is the common origin. The exception is the "system noise floor" shown in gray on Fig. 2 of main article for which two parallel fibers of equal length were used i.e. T A = T B in (11) and laser noise cancels, as can be observed.