High-accuracy coherent optical frequency transfer over a doubled 642-km fiber link

Abstract

To significantly improve the frequency references used in radio-astronomy and the precision measurements in atomic physics, we provide frequency dissemination through a 642-km coherent optical fiber link. On the frequency transfer, we obtained a frequency instability of $3\times 10^{-19}$ at 1,000 s in terms of Allan deviation on a 5-mHz measurement bandwidth, and an accuracy of $5\times 10^{-19}$. The ultimate link performance has been evaluated by doubling the link to 1,284 km, demonstrating a new characterization technique based on the double round trip on a single fiber. This method is an alternative to previously demonstrated techniques for link characterization. In particular, the use of a single fiber may be beneficial to long hauls realizations in view of a continental fiber network for frequency and time metrology, as it avoids the doubling of the amplifiers, with a subsequent reduction in costs and maintenance. A detailed analysis of the results is presented, regarding the phase noise, the cycle-slips detection and removal and the instability evaluation. The observed noise power spectrum is seldom found in the literature; hence, the expression of the Allan deviation is theoretically derived and the results confirm the expectations.

Appendix

This appendix demonstrates the formula for the delay-unsuppressed noise on a link with both ends in the same laboratory, and where the same fiber is used to loop the link, instead of two independent fibers. An approach similar to the one used in [13] has been followed.

Let us write the expression for the round-trip optical phase $\varphi _{\mathrm{rt}}(t)$ and for the forward signal optical phase $\varphi _{\mathrm{fw}}(t)$, as a function of the fiber noise $\delta \varphi (z,t)\hbox {d}z$ of an infinitesimal fiber segment dz at time t and position z along the fiber and as a function of the phase correction needed for the link stabilization $\varphi _{\mathrm{c}}(t)$.

$$\begin{aligned} \varphi _{\mathrm{rt}}(t)&= \varphi _{\mathrm{c}}(t- 2\tau ) +\varphi _{\mathrm{c}}(t)\nonumber \\&\quad + \int \nolimits _0^{L/2} \left( \delta \varphi \left( z,t-2 \tau + n\frac{z}{c}\right) + \delta \varphi \left( z,t- \tau - n\frac{z}{c}\right) \right. \nonumber \\&\left. \quad+\, \delta \varphi \left( z,t- \tau + n \frac{z}{c}\right) + \delta \varphi \left( z,t- n \frac{z}{c}\right) \right) \,\hbox {d}z \nonumber \\ \varphi _{\mathrm{fw}}(t)&= \varphi _{\mathrm{c}}(t- \tau )\nonumber \\&\quad +\int \nolimits _0^{L/2} \left( \delta \varphi (z,t- \tau + n \frac{z}{c})+ \delta \varphi \left( z,t- n\frac{z}{c}\right) \right) \,\hbox {d}z \end{aligned}$$

(3)

where $L$ is the loop length, in our case L = 1,284 km, and $\tau =nL/c$ is the link delay. For the sake of clarity, we integrate the length only between 0 and L/2, as the two fiber halves are indeed the same fiber travelled in opposite directions.

Let us now assume that the fiber perturbations evolve linearly with time; this is justified for perturbations which act on timescales much longer than $\tau$, as in the case of interest in this context. Within this approximation, Eq. 3 is simplified into

$$\begin{aligned} \varphi _{\mathrm{rt}}(t)&= 2\varphi _{\mathrm{c}}(t-\tau ) + \int \nolimits _0^{L/2} 4\delta \varphi (z,t-\tau )\,\hbox {d}z\nonumber \\ \varphi _{\mathrm{fw}}(t)&= \varphi _{\mathrm{c}}(t-\tau ) +\int \nolimits _0^{L/2} 2 \delta \varphi \left( z,t- \frac{\tau }{2}\right) \,\hbox {d}z \end{aligned}$$

(4)

Now, considering that in the closed feedback loop configuration $\varphi _{\mathrm{rt}}(t,z)=0$, Eq. 4 is rewritten as

$$\begin{aligned} \varphi _{\mathrm{fw}}(t)&= 2\int \nolimits _0^{L/2}\left( \delta \varphi \left( z, t-\frac{\tau }{2}\right) -\delta \varphi (z,t-\tau )\right) \,\hbox {d}z\nonumber \\&= 2\int \nolimits _0^{L/2}\frac{\tau }{2}{\frac{\hbox {d}}{{\hbox {d}t}}}\delta \varphi (z,t-\tau )\,\hbox {d}z \end{aligned}$$

(5)

In the last equation, the evolution of the fiber noise is expressed as a function of its time derivative. For the fundamental theorem of the signal analysis [39], the noise power spectrum of the output of a linear and time-invariant system can be written in terms of the noise of the input; in our case, this theorem can be applied to each fiber segment separately, i.e.:

$$\begin{aligned} S_\varphi (z,f) = \vert H(z,f) \vert ^2 S_{\mathrm{fiber}}(z,f), \end{aligned}$$

(6)

where $S_\varphi (z,f)$ is the contribution of a fiber segment with length dz to the compensated forward signal phase noise, $H(z,f)=\mathcal {F}(\tau {\frac{\hbox {d}}{{\hbox {d}t}}})=2\pi i f\tau$ and $S_{\mathrm{fiber}}(z,f)$ is the phase noise power spectrum of each fiber segment. Assuming that the contributions of each fiber segment are independent, we can perform the integration and end up with the stated result that

$$\begin{aligned} S_\varphi (f) =\frac{1}{4}(2 \pi f \tau )^2 S_{\mathrm{fiber}}(f), \end{aligned}$$

(7)

where $S_{\mathrm{fiber}}(f)$ is the phase noise of the 1,284-km-long link, and it has been used in the relation

$$\begin{aligned} S_{\mathrm{fiber}}(f)=4 \int \nolimits _0^{L/2} \vert \delta \varphi (z,t) \vert ^2\,\hbox {d}z. \end{aligned}$$