# Passive Optical Phase Stabilization on a Ring Fiber Network

Liang Hu, Xueyang Tian, Long Wang, Guiling Wu, and Jianping Chen Manuscript
received xxx xxx, xxx; revised xxx xxx, xxx. This work was supported in part
by the National Natural Science Foundation of China (NSFC) (61627871,
61535006, 61905143), and in part by science and technology project of State
Grid Corporation of China (No. SGSHJX00KXJS1901531). (Corresponding author:
Liang Hu and Guiling Wu)The authors are with the State Key Laboratory of
Advanced Optical Com- munication Systems and Networks, Department of
Electronic Engineering, Shanghai Jiao Tong University, Shanghai 200240, China,
with Shanghai Institute for Advanced Communication and Data Science, Shanghai
Jiao Tong University, Shanghai 200240, and also with Shanghai Key Laboratory
of Navigation and Location-Based Services, Shanghai 200240, China (e-mail:
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Optical frequency transfer provides the means for high-fidelity frequency
transfer across thousands of kilometers. A critical step in the further
developing optical frequency transfer is its capability to transfer a high
spectral purity feature from ultrastable lasers or optical clocks to any
remote locations and, at the same time, its adaptability to incorporate the
optical frequency transfer technique into any existing communication networks
with different topologies. Here we for the first time report a technique that
delivers optical-frequency signals to multiple independent remote hubs along a
ring optical-fiber network with passive phase stabilization. The technique
automatically corrects optical-fiber length fluctuations of arbitrary hubs
along the loop by mixing and shifting optical signals. Without the help of an
active phase tracker and a compensator, it could significantly mitigate some
technical problems such as the limited compensation speed and phase recovery
time, the phase jitter contamination caused by the servo bump in conventional
phase noise cancellation. Moreover, by transmitting optical signals along both
directions using the same optical source, it can improve the signal-to-noise
ratio at each hub. This technique maintains the same delay-limited phase noise
correction capability as in conventional techniques and, furthermore, improves
the phase jitter by a factor of 3, opening a way to a broad distribution of an
ultrastable frequency reference with high spectral purity and enabling a wide
range of applications beyond metrology over a ring fiber network with the
naturally impressive reliability and scalability.

###### Index Terms:

Optical clock, optical frequency transfer, passive phase stabilization, ring
fiber network, metrology.

##  I Introduction

Precision timekeeping is a prerequisite for so many applications, ranging from
navigation [1, 2], communication networks, radio astronomy [3, 4] to searching
for beyond-standard-model physics [5, 6]. Today’s most precise clocks are
optical clocks with trapped atoms or ions, which use the ultrastable lasers to
detect the optical frequency of an electron transitioning between two atomic
states as the timebase [7, 8, 9]. The outstanding performance makes the
optical clocks and ultrastable lasers become ideal tools for precision
measurements and fundamental physics tests, such as general relativity,
temporal variation of the fundamental constant [10], searching for dark
matter, chronometric geodesy [11], and gravitational waves [12, 13, 14].
However, these clocks and ultrastable lasers are cumbersome and expensive and
only available at national metrology institutes and several universities [7,
8, 15]. This causes a strong motivation to develop effective systems for
comparing and distributing these sources of ultraprecise frequency signals.
Among them, the fiber-optic frequency dissemination technique has been
recognized as an ideal solution for ultra-long haul dissemination because of
fiber-optic’s particular advantages of broad bandwidth, low loss, and high
immunity to environmental perturbations, etc [16].

Figure 1: A prospect hybrid ring and bus fiber topologies for the
dissemination of optical frequency to academic labs, industrial and scientific
applications used for navigation, communication networks, radio astronomy and
precise test of relativistic geodesy. A ring is created with some number of
optical frequency hubs. Each hub on the ring then acts as the center of the
star with multiple point-to-point links emanating and terminating at various
remote nodes.

Solutions based on fiber transmission have been aiming for suppressing the
fiber-induced phase noise to retrieve precise frequency information at remote
locations. To achieve this aim, active compensation schemes as first
demonstrated in 1994 by Ma et al. have been proposed to cancel the fiber-
induced phase drift and implement highly stable optical frequency distribution
[16, 17, 18, 19]. It generally utilizes the phase error from a round-trip
probe signal to achieve the feedback control of compensators. The compensators
mainly include variable delay lines [20] and phase-locked loops (PLL) [16].
Although this scheme can accomplish very high phase stability, the response
speed and phase recovery time are restricted by the compensators’ parameters
and optimization. Moreover, much attention has paid into the relative long-
term frequency instability and accuracy, while little into high spectral
purity of the transferred light. The possibility of transferring the spectral
purity of an ultrastable laser across different locations is beneficial to the
increasing requirement of high frequency stability lasers for optical atomic
clocks and high-resolution spectroscopy [21, 22]. Optical frequency transfer
with high spectral purity enables such performances to be copied to any laser
in any locations, with a simplification of the experimental setup. This is
especially relevant when several ultrastable lasers at different locations are
needed, but only one ultrastable cavity or clock exists.

In order to surmount the above mentioned barriers, passive phase noise
cancellation has drawn extensive attention for fiber-optic radio frequency
transfer [23, 24]. The passive phase noise cancellation technique can realize
rapid and endless phase fluctuation compensation, and also get rid of
complicated phase error detection and feedback circuits. However, the passive
phase noise cancellation technique used for RF frequency transfer is not
directly applicable for fiber-based optical frequency dissemination by
multiplying and dividing the frequency of the transferred optical carrier,
such as 1550 nm, itself. In our previous work, we have extended the passive
phase noise cancellation technique in optical frequency transfer by detecting
and compensating optical phase noise with different optical signals along the
single path [25]. The main drawback related to this technique is the different
frequencies between the detection and the compensation beam, leading to that
two different frequencies will be received at the remote site and, therefore,
a narrow bandpass optical filter has to be adopted to remove the undesired
signal, which may cause additional decoherence on the transferred light.

Over the last decade, extensions have been proposed that can provide
stabilized optical-frequency signals at intermediate sites along the length of
optical fiber [26, 27, 28, 29]. However, as phase stabilization at the
intermediate sites achieved by mixing signals received from the source and the
far end of the fiber, this approach is limited to fiber links with a bus
topology. Moreover, if the stabilization servo of the main link fails, then
transfer to all downstream remote sites will cease to be stabilized. To
overcome this main drawback, ultrastable optical frequency dissemination
schemes on a star topology optical fiber network have been proposed and
demonstrated [29, 30, 31]. Using this method, a highly synchronized optical
signal itself can be recovered at arbitrary remote locations by actively
compensating the phase noise of each fiber link at each user end [29, 30, 31].
However, the maximum node accommodation capability will be limited by the
radio frequency (RF) bandwidth of AOMs to distinguish the optical frequency
between the accommodated nodes and the bandwidth of the electrical bandpass
filters. Moreover, the existing schemes to support optical communication based
on bus and star topologies have limited scalability and reliability [32, 33,
34]. On the contrary, because of the self-healing characteristic of the ring
network, in particular, the dual-fiber ring, has a natural advantage in the
network reliability [35]. Although the number of fibers required in the dual-
fiber ring doubles that in the single-fiber ring, the dual-fiber ring network
has a protection mechanism and can carry out the protection of multiple
faults, resulting in shortening the recovery time and possessing higher
reliability [36, 37]. In addition, by deploying optical amplifiers in remote
nodes, the scale of the ring network can be increased dramatically [38]. With
the continuous extension of the optical frequency transfer network, the
reliability and scalability will become more important [33, 34, 30]. Owing to
the prominent advantages, the performance and compatibility of optical
frequency transfer on a fiber ring network have to be investigated
theoretically and experimentally.

Figure 2: Schematic diagram of our optical frequency transfer over a ring
fiber network with passive phase stabilization. We tapped bidirectional lights
with the assistance of a $2\times 2$ optical coupler at each hub. The optical
phase introduced by environment perturbations on the fiber links is passively
compensated at each hub. Electrical bandpass filters are not shown for
conciseness. AOM: acousto-optic modulator, FM: Faraday mirror, DDS: direct-
digital synthesizer, PD: photo-detector, FD: frequency divider, PC: power
combiner. The solid and dashed arrows represent the light propagation along
the clockwise and anticlockwise directions, respectively.

In this paper, a passive arbitrary-access stable optical phase delivery scheme
based on a ring fiber network is proposed and experimentally demonstrated. In
comparison with the previous schemes [16, 17, 18, 19], precise phase
correction is obtained by embedding the phase information into an RF signal
and shifting a copy of the optical frequency signal with the amount of phase
noise introduced by the fiber loop to avoid having to actively stabilize the
optical frequency signal. The scheme we proposed largely simplifies the setup
at the central station and the hubs simultaneously, and leaves the hubs to
independently control the fiber noise cancellation systems as performed in
[29, 30, 31]. Moreover, with the proposed configuration, one of the directions
will only provide one optical signal at each hub’s output instead of two
optical signals [25].

The proposed technique together with optical frequency transfer over a star
topology [29, 30, 31] provides a promising way to implement a robust optical
frequency transfer network as illustrated in Fig. 1. Depending on the size and
distance of the network, a ring can be created with some number of optical
frequency hubs which are all connected together to keep failure rate as low as
possible. At the same time, the various hubs on the ring then act as the
center of the star with multiple point-to-point links, emanating and
terminating at various remote nodes. These individual remote nodes may be
subject to failure, so they are generally located at non-critical positions
and can accept occasional outages. The ring, on the other hand, keeps the hubs
communicating constantly and makes the overwhelming majority of the network
fault-free [33, 34, 30]. This hybrid optical frequency transfer network could
be used in probes of fundamental physics and detection of submarine
earthquakes by means of deep-sea fiber cables [39], among other applications
[10, 11, 12, 13, 14]. At the same time, with the assistance of optical combs,
stable and accurate microwave signals can be obtained and can be used in a
variety of areas including communication, navigation, radar, radio astronomy,
and fundamental physics research as illustrated in Fig. 1.

The article is organized as follows. We illustrate the concept of coherent
optical phase dissemination with passive optical phase stabilization on a ring
fiber link in Sec. II and present in Sec. III the delay limited phase-noise
power spectral density (PSD). We discuss the experimental set-up and
experimental results in Sec. IV and illustrate representative features related
to the proposed scheme in V. Furthermore, we briefly present a discussion in
Sec. VI. Finally, we conclude in Sec. VII by summarizing our results.

##  II Concept of optical frequency transfer on a ring fiber network

A schematic diagram of the proposed technique is illustrated in Fig. 2. Here
we briefly describe the principle of our optical frequency transfer on a ring
fiber link. An optical-frequency signal $\nu$ is divided into 2. The two parts
are, respectively, sent from the signal source to the central site along the
clockwise and anticlockwise directions over a ring fiber link. The laser
frequency $\nu$ propagating clockwise is again split into 2. One part is
reflected by a Faraday mirror as a reference signal and the remaining one is
downshifted by an angular frequency $\omega_{s}$ with an acousto-optic
modulator (AOM) denoted as AOM$c$. The laser frequency propagating
anticlockwise is directly injected into the fiber loop, passes through the
fiber loop and then returns back in the AOM$c$ located at the central site.
The single-trip signal propagating along the anticlockwise direction is mixed
with the input ultrastable laser onto a photodetector 1 (PD1). The beat-note
frequency is $\omega_{s}$, exhibiting the single-trip fiber phase noise,
$-\phi_{p}$. After mixing with an another frequency of $\omega_{a}$
($\omega_{a}>\omega_{s}$) with the assistance of a frequency mixer, the lower
sideband signal is extracted and then applied to the RF port of the AOM$c$
together with $\omega_{s}$, resulting in a desirable clockwise optical signal
with the angular frequency of $\nu-\omega_{a}+\omega_{s}$.

Now we consider the extraction of the ultrastable signal along the fiber loop
with a $2\times 2$ optical coupler, enabling us to extract both the clockwise
and anticlockwise signals from the loop fiber link, at a distance $L_{a}$ from
the central site along the clockwise direction and $L_{b}$ from the central
site along the anticlockwise direction with the total fiber link length of $L$
($L=L_{a}+L_{b}$). The anticlockwise signal has a frequency $\nu$ and exhibits
the phase fluctuation of $\phi_{b}$, and the desirable clockwise signal with
the angular frequency $\nu-\omega_{a}+\omega_{s}$ at arbitrary hubs exhibits
the phase fluctuations $-\phi_{p}+\phi_{a}=-\phi_{b}$, where $\phi_{a}$ and
$\phi_{b}$ are the phase noise of the fiber sections $L_{a}$ and $L_{b}$,
respectively. To compensate the phase noise of the anticlockwise wave, we
detect the beat-note of the two extracted signals onto the PD2. The beat-note
frequency is thus $\omega_{a}-\omega_{s}$, exhibiting a phase fluctuation of
$2\phi_{b}$. The signal frequency is divided by 2, filtered, and drives an AOM
(AOM$a$, $-1$ order) to correct the phase fluctuations of the extracted
anticlockwise signal. The frequency of the extracted anticlockwise signal,
after passing through the AOM$a$, is thus downshifted to
$\nu-0.5(\omega_{a}-\omega_{s})$ and its phase fluctuation is cancelled. With
this configuration, the anticlockwise direction only includes one phase
stabilized optical signal. Compared to our previous passive phase noise
cancellation schemes [25], this represents another advantage, that is, no
optical filters are required to remove the unwanted optical signal. Similar
compensation can be obtained on the extracted clockwise signal with a positive
optical frequency shifter. However, in this case, the clockwise direction
signal includes two optical frequencies and needs an optical filter after the
AOM$a$ to select a stable optical frequency signal, which could introduce an
additional decoherence effect [25].

We can clearly see that the optical signal received at arbitrary hubs has the
same phase with the standard optical signal at the central station. Therefore,
the phase noise of the optical signal is effectively reduced by simply mixing
and shifting optical signals.

##  III Delay-limited phase noise PSD

In Sec. II, the description does not take the propagation delay of the fiber
sections into account. The capability of the phase noise rejection will be
limited by the propagation delay as first pointed out by Williams et al. [40].
By adopting the similar procedure performed in [28, 40], we find that the
residual phase noise power spectral density (PSD) at an arbitrary hub along
the fiber section $L_{b}$ in terms of the single-pass free-running phase noise
PSD, $S_{\text{fiber}}(\omega)$, and the propagation delay of the fiber loop,
$\tau_{0}$, can be calculated as,

$\begin{split}S_{E,b-}(\omega)&=F_{E,b-}\frac{(\omega\tau_{0})^{2}}{3}S_{\text{fiber}}(\omega)\\\
&=\frac{(\omega\tau_{0})^{2}}{3}\left(1-3\frac{L_{b}}{L}\frac{\tau_{b}}{\tau_{0}}+2\frac{L_{b}}{L}\left(\frac{\tau_{b}}{\tau_{0}}\right)^{2}\right)S_{\text{fiber}}(\omega).\end{split}$
(1)

where $\tau_{b}$ is the proragation delay along the fiber section $L_{b}$.
This coefficient factor $F_{E,b-}$ is zero for $L_{b}=L$, then increases up to
one at $L_{b}=0$. Following the same procedure, if we apply the phase
correction for the clockwise light, the residual phase noise PSD at arbitrary
hubs can have a form of,

$\begin{split}S_{E,a+}(\omega)&=F_{E,a+}\frac{(\omega\tau_{0})^{2}}{3}S_{\text{fiber}}(\omega)\\\
&=\frac{(\omega\tau_{0})^{2}}{3}\left(1-3\frac{L_{a}}{L}\frac{\tau_{a}}{\tau_{0}}+2\frac{L_{a}}{L}\left(\frac{\tau_{a}}{\tau_{0}}\right)^{2}\right)S_{\text{fiber}}(\omega).\end{split}$
(2)

where $\tau_{a}$ is the proragation delay along the fiber section $L_{a}$.

Figure 3: (a) Measured phase noise PSDs of the 100km free-running fiber link
(black curve) and the stabilized link with passive phase noise cancellation
for the 50/50 km hub(c, red curve) and the 30/70 km (b, blue curve) hub. Note
that strong servo bumps can be effectively eliminated in the passive phase
noise cancellation scheme. The green curve is the theoretical prediction based
on Eq. 1. (b) Measured fractional frequency instabilities of the 100 km free-
running fiber link (black circles) and the stabilized link for the 50/50 km
(red squares) hub and the 30/70 km (blue triangles) hub. The measurement is
derived from non-averaging ($\Pi$-type) frequency counters expressed as ADEV.
The measured noise floor of the interferometer is also shown (gray diamonds).

##  IV Experimental apparatus and results

###  IV-A Experimental apparatus

We have demonstrated this technique by using the simplest configuration as
shown in Fig. 2. The interferometer is built with fiber optics. The proposed
scheme was tested using a narrow-linewidth optical source (NKT X15) at a
frequency near 193 THz with a linewidth of 100 Hz. The signal was transmitted
along a 100 km fiber link loop. $2\times 2$ optical couplers were used to
extract both clockwise and anticlockwise light at the most symmetric position,
50/50 km ($L_{a}/L_{b}$), and a relative most asymmetric one, 30/70 km, over
the 100 km ring fiber link. Here we set $\omega_{s}=2\pi\times 45$ MHz and
$\omega_{a}=2\pi\times 80$ MHz. Before dividing the frequency of the beatnote
at the hub, we mix the beatnote with an assistant frequency of $115$ MHz, and
the lower sideband with a frequency of 80 MHz is extracted. All these RF
frequencies are provided by a direct-digital-synthesizer (DDS) generator,
phase locked to a 10 MHz rubidium clock. With this configuration, the AOMc is
simultaneously fed by $35$ MHz and $45$ MHz (downshifted mode), and the AOMa
is working at an angular frequency of 40 MHz (upshifted mode), resulting in an
out-of-loop beatnote of $40$ MHz for arbitrary hubs. To avoid the nonlinear
effect in the fiber, we keep the optical power into the ring fiber link below
5 dBm for each optical frequency: one for the anticlockwise direction ($\nu$)
and two for the clockwise direction ($\nu-2\pi\times 35$ MHz and
$\nu-2\pi\times 45$ MHz). However, in the conventional configuration [28], the
light transferred to the remote site will directly return back to the local
site, resulting in the power of the returning light of $-15$ dBm at the remote
site for the 100 km fiber link when the injection power is 5 dBm at the local
site and fiber loss is $0.2$ dB per kilometer. Consequently, we can obtain the
gain of the signal-to-noise ratio of approximately 20 dB without the
assistance of optical amplifiers in the proposed scheme.

To effectively measure the transfer stability at each hub, all hubs are co-
located at the same optical platform as the signal source. The out-of-loop
fiber connections were kept as short as practicable and were thermally and
acoustically isolated. We use non-averaging $\Pi$-type frequency counters,
which are referenced to the RF frequency source from the DDS at the central
site, to record the beating frequency between the fiber input light and the
output light. Additionally, to measure the phase noise of the optical carrier
frequencies at each hub, we perform the measurement by feeding the heterodyne
beat frequency together with a stable RF frequency reference produced by the
DDS to a phase detector. The voltage fluctuations at the phase detector output
are then measured with a fast Fourier transform (FFT) analyzer to obtain the
phase fluctuations.

###  IV-B Testing the phase noise rejection on hubs

To characterize optical transfer over the 100 km ring fiber loop, we measured
the phase noise PSDs of the 50/50 km hub and the 30/70 km hub for both the
stabilized and the unstabilized cases. Typically, the phase noise PSD is
usually parametrized as [41, 42],

$S_{\phi}(f)=\sum_{\alpha=-2}^{2}h_{\alpha}f^{\alpha-2},$ (3)

where $f^{\alpha}$ ($\alpha=-2,-1,0,1$ and 2), reflecting the various
contributions of noise in the system (i.e., random walk frequency noise,
flicker frequency noise, white frequency noise, flicker phase noise and white
phase noise).

The phase noise PSDs of the 50/50 km hub and the 30/70 km hub are plotted in
Fig. 3(a). Both hubs are very similar and typical for optical fiber links,
with noise of approximately 200 rad2/Hz at 1 Hz and $3\times 10^{-2}$ rad2/Hz
at 100 Hz with a $h_{0}f^{-2}$ dependency, indicating that the phase noise of
the free-running loop is mainly limited by the flicker phase noise. Both
compensated phase noise PSDs are below $10^{-3}$ rad2/Hz between 1 and 10 Hz
with a $h_{2}f^{0}$ dependency, illustrating that the loop is mainly
constrained by the white phase noise after the phase noise compensation. Noise
is corrected up to about 400 Hz, which is compatible with the theoretical
bandwidth of 500 Hz given by $1/(4\tau_{0})$ with $\tau_{0}$ being the
propagation delay of fiber loop $L=100$ km. This limit is the same for both
hubs and is mainly determined by the longest propagation delay $\tau_{0}$. We
checked that the noise floors of both outputs were below these PSDs. The noise
rejection of around $2\times 10^{5}$ at 1 Hz is also compatible with the
theoretical limit given by Eq. 1 as the green curve shown in Fig. 3(a). This
shows that the noise rejection is optimized.

###  IV-C Time-domain characterization

A time-domain characterization of the frequency stability in terms of
overlapping Allan deviation (ADEV) is shown in Fig. 3(b). In this plot, black
circle markers indicate the fractional frequency stability of optical carrier
frequency dissemination over the 100 km link when passive phase noise
cancellation is not activated. Curves with square and triangle markers
represent the stability of the signals with implementing passive phase noise
correction for the $50/50$ km hub and the $30/70$ km hub, respectively. With
the implementation of fiber noise cancellation at the $50/50$ km ($30/70$ km)
hub, optical frequency transfer achieves a fractional frequency stability of
$1.6\times 10^{-15}$ ($1.4\times 10^{-15}$) at the integration time of $1$ s,
decreases and reaches a floor of approximately $3.0\times 10^{-18}$
($3.0\times 10^{-18}$) at $1,000$ s.

We can clearly see that when the fiber noise cancellation setups are engaged,
frequency fluctuations can be effectively suppressed and no longer dominate
the instability of the optical signals at both hubs. In our experiment, we
observe that the stability of optical frequency dissemination is improved by
three orders of magnitude at the integration time of 10,000 s. Note that the
noise correction is very robust and that the set-up can operate several days
without any cycle slips. As a comparison, we measured the floor of optical
frequency dissemination by replacing each fiber spool with a 1 m fiber plus a
20-dB attenuator. We can observe that the floor of optical frequency
dissemination with a stability of $1.0\times 10^{-16}$ at 1 s and $1.1\times
10^{-18}$ at 10,000 s is obtained. Consequently, the stabilized link is mainly
limited by the noise floor. There are several reasons that lead to the floor
in the instability including the noise of the imperfect length adjustment and
thermal stabilization in the extraction optical set-up, and the
interferometric measurement set-up [43, 17, 18]. We estimate the path length
mismatch up to $10$ cm. For typical temperature perturbations due to our air
conditioning system, with the temperature fluctuation amplitude $1$ K and
cycle 3, 600 s, one expects a bump of the ADEV as high as $4\times 10^{-18}$
at approximately 1,800 s [44].

As calculated by Eq. 1, the ratio of the stability of the $50/50$ km and
$30/70$ km hubs should be $\mathcal{R}=0.64$. In our experiment, we obtain the
ratio of $\mathcal{R}=1.4\times 10^{-15}/1.6\times 10^{-15}=0.87$, which has a
large deviation from the theoretical one. We attribute this discrepancy to the
phase noise introduced by the hub itself such as the photo-detection process.
We have to note that the estimation in Eq. 1 acquired by the assumption that
the hub will introduce negligible phase noise. In our system, the phase noise
introduced by the hub itself dominates the total phase noise of the hub at the
short fiber section $L_{b}$ whereas the residual phase noise of the fiber link
becomes the domination when the fiber section $L_{b}$ is long enough, enabling
that the measured results are consistent with the theoretical one as increase
of the fiber section $L_{b}$.

###  IV-D Frequency transfer accuracy

Figure 4:  (a) Two-day frequency comparison between sent and transferred
frequencies after 50 km over the 100 km ring fiber. Data were taken with dead-
time free $\Pi$-type frequency counters with a 1 s gate time (green points,
left axis). We calculated unweighted mean ($\Pi$-type) values for all cycle-
slip free 100 s long segments, resulting in 1,803 data points (black dots,
right frequency axis, enlarged scale). Histograms (brown bars) and Gaussian
fits (red curves) for (b) frequency values as taken with $\Pi$-type frequency
counters with one second gate time and (c) 1,803 phase coherent 100-second
frequency averages with a mean of $2.3\times 10^{-18}$ and a standard
deviation of $1.2\times 10^{-17}$. Taking the long-term stability shown in
Fig. 3(b) into account, we determine the statistical uncertainty to be
$3\times 10^{-18}$.

We also performed an evaluation of the accuracy of frequency transfer at
arbitrary hubs. Figure 4 shows the frequency deviation of the beat-note’s data
for the 50/50 km hub, recorded with a 1 s gate time and $\Pi$-type counters,
over successive 180,300 s (green point, left axis) and the arithmetic mean of
all cycle-slip free 100 s intervals (black dots, right axis). Histograms
(brown bars) and Gaussian fits (red curves) of a frequency deviation for the
hub after 50 km are also illustrated in Fig. 4(b) and (c). According to the
Gaussian fit in Fig. 4(c), the calculated results demonstrate that the mean
frequency is shifted by 435 $\mu$Hz ($2.3\times 10^{-18}$). The standard
deviation of the 100 s data points is $2.3$ mHz ($1.2\times 10^{-17}$) which
is a factor of 100 smaller than the ADEV at 1 s as expected for this
$\Pi$-type evaluation. Considering the long-term stability of frequency
transfer as illustrated in Fig. 3(b) mainly limited by the flicker frequency
noise, we conservatively estimate the accuracy of the transmitted optical
signal as shown in the last data point of the ADEV, resulting in a relative
frequency accuracy of $3\times 10^{-18}$.

Figure 5: (a) Blue solid curve and red dashed curve show, respectively, the
residual phase noise PSDs with the active and passive phase noise cancellation
system for the phase noise PSD of the free-running fiber link $100/f^{2}$
rad2/Hz. To maintain a sufficient phase noise rejection capability, the gain
has to be tuned large enough, leading to the divergence of the gain amplitude
for frequencies equal to integer multiple of $f_{0}=1/(4\tau_{0})$. (b) The
phase jitter integrated from 1 Hz to 1 kHz for the active (blue solid curve)
and passive (red dashed curve) phase noise cancellation system, respectively.
(c) Measured phase noise PSD at the 50/50 km hub over the 100-km optical link
with passive (a, red dashed curve) and active (b, blue solid curve) phase
cancellation. Black lines represent the extrapolated noise components. Active
phase noise cancellation appears a strong servo bump compared to passive phase
cancellation. (d) The phase jitter integrated from 1 Hz to 1 kHz is $\sim 3.2$
rad and $\sim 1.0$ rad for the active (blue solid curve) and passive (red
dashed curve) phase noise cancellation system, respectively. As a comparison,
the phase jitter integrated from 1 Hz to 100 Hz for the active (d, blue dashed
dot curve) and passive (c, red long dashed curve) phase noise cancellation
systems, respectively is also shown.

Following the same procedure, the mean frequency offset for the 30/70 km hub
was calculated using the total 40,069 $\Pi$-type counter data to be -812
$\mu$Hz ($-4.2\times 10^{-18}$) and a standard deviation of the 100 s points
is $4.2$ mHz ($2.2\times 10^{-17}$). Considering the long-term ADEV at 10,000
s of the data set for the 30/70 km hub of $2.1\times 10^{-18}$, we
conservatively estimate that the mean frequency offset is $-4.2\times
10^{-19}$ with a statistical uncertainty of $2.1\times 10^{-18}$ for the 30/70
km hub. We can conclude that there is no systematic frequency shift arising in
the extraction setup at a level of a few $10^{-18}$.

##  V Representative features in the proposed technique

The above section is mainly devoted to characterizing the results of our
scheme from the perspective of conventional optical frequency transfer
parameters consisting of the fractional frequency stability, the phase noise
PSD and the accuracy as performed in most existing research work [16, 17, 18,
19]. In this section, we will theoretically study and experimentally
demonstrate the representative features of our proposed scheme, that is, a
ring fiber network with passive phase stabilization, including the lower phase
jitter and faster phase recovery capability.

###  V-A Lower phase noise and timing jitter

For active phase noise cancellation system similar with [40], the closed-loop
transfer function at arbitrary hubs along the anticlockwise direction in the
frequency domain can be expressed as,

$\begin{split}&H_{A}(\omega)=F_{E,b-}\int_{0}^{L}dz\exp({-i\omega(\tau_{0}+z/c_{n})})\\\
&\quad\times\bigg{[}\exp({-i\omega(z/c_{n})})-\frac{\cos(\omega\tau_{0}-\omega
z/c_{n})}{\cos(\omega\tau_{0})}\frac{G(\omega)}{1+G(\omega)}\bigg{]}\end{split}$
(4)

where $G(\omega)$ is the open-loop transfer function of the compensation
system, $L$ is the fiber link length and $c_{n}$ is the speed of light in the
fiber.

With the same procedure adopted in [40, 28], the transfer function in our
passive phase stabilization set-up at arbitrary hubs along the anticlockwise
direction can be calculated as,

$H_{P}(\omega)=F_{E,b-}\bigg{[}\frac{3}{2}-\cos(\omega\tau_{0})-\text{sinc}(\omega\tau_{0})+\frac{1}{2}\text{sinc}(2\omega\tau_{0})\bigg{]}$
(5)

Figure 5(a) shows the calculated phase noise PSDs for the stabilized link at
the 50/50 km hub by using active (blue solid curve) and passive (red dashed
curve) phase noise cancellation system with the phase noise PSD of the 100 km
free-running link of $10/f^{2}$ rad2/Hz. In typical servo controllers, the
gain has to be tuned large enough to maintain a sufficient phase noise
rejection capability. The infinite gain will lead to the divergence of the
gain for frequencies equal to integer multiple of $f_{0}=1/(4\tau_{0})=500$
Hz. Here the servo bandwidth is mainly limited by the total fiber length
instead of the fiber sections ($L_{a}$ and $L_{b}$). It is interesting to note
this issue is automatically disappeared in the passive phase stabilization
set-up with the optimized gain. To calculate the ratio of the phase jitter
between the active and passive phase noise cancellation technique, we
integrate the phase noise from 1 Hz to 1 kHz as shown in Fig. 5(b). We can see
that more than one order of magnitude of the reduction of the phase jitter can
be achieved for the proposed phase noise cancellation technique. Note that the
integration results for the active phase noise PSD are dependent on the
frequency resolution of the simulation. Here the frequency resolution is 1 Hz
and the phase jitter will increase more as improving the frequency resolution
due to the diverged bump effect.

To experimentally verify the calculated results, we used the set-up shown in
Fig. 2 as the passive phase noise system. The active phase noise system we
used is similar with our previous multiple-access optical frequency transfer
system [45]. Figure 5(c) shows the residual phase noise PSDs at the $50/50$ km
hub over the 100 km fiber link with passive (a, red dashed curve) and active
(b, blue solid curve) phase cancellation. In active phase noise cancellation,
the residual phase noise is essentially limited by the residual fiber noise in
the range from 1 Hz to $\sim 200$ Hz, with a strong bump appearing
significantly at 300 Hz. The shifted bump position from $f_{0}=500$ Hz could
be from the insufficient gain in the servo controller. On the contrary, the
spectral analysis does not report any strong noise contribution in the 300 Hz
range with passive phase noise cancellation, allowing that the bump does not
play a role in our passive optical phase noise cancellation concept. The total
integrated phase noise (1 Hz to 1 kHz) of the data in Fig. 5(d) for active
(blue solid curve) and passive (red dashed curve) phase noise cancellation are
3.2 rad and 1.0 rad, which corresponds to temporal jitters of $\sim$ 2.6 $f$s
and 825 as, respectively, enabling the reduction of the phase jitter by a
factor of about 3 by adopting passive phase stabilization. As a comparison,
the phase jitter integrated from 1 Hz to 100 Hz is almost identical for both
cases as shown in Fig. 5(d). The main bottleneck of our detection scheme is
the round-trip propagation delay, limiting the servo bandwidth. This can be
solved by dividing the fiber link into several sub-links which could serve to
further reduce the round-trip propagation delay, resulting in the improvement
of the signal-to-noise ratio in our scheme [46].

###  V-B Faster response speed and phase recovery time

To examine the characterization of the faster response speed and phase
recovery time, we compared two kinds of optical frequency transfer schemes
described above over a 20 km fiber link as performed in [25]. To simulate the
interruption, we inset one more AOM just after the laser source to switch the
light on/off. The RF port of the AOM is controlled by a TTL signal which has a
rising time of $\sim 20$ ns, which can be neglected. Both systems’ output was
analyzed based on the voltage generated by mixing down the out-of-loop beat to
the dc. Figure 6 illustrates the phase recovery time of 20 km optical path
length stabilization with active and passive phase correction. We observed
that the phase recovery time of optical path length stabilization with active
phase noise cancellation has a few strongly damped oscillations of the phase
lasting approximately $\sim 5$ ms, whereas this time is negligible for our
proposed passive phase noise cancellation. This feature is very beneficial for
the case in which the interruptions happen frequently on the long fiber links
[17, 18, 19].

Figure 6: Phase recovery behaviour of the 20 km optical path length
stabilization with active (red curve) and passive (blue curve) phase
correction, respectively. A delay is introduced between the TTL signal (black
curve) for switching the light on at 0 s and the activation of the phase
stabilization at $4\tau_{0}\simeq 400$ $\mu$s as indicated by the shaded green
area.

##  VI Discussion

The above analysis has ignored the effect of the backscattering noise on the
frequency transfer performance. Small-scale inhomogeneities of the refractive
index in the fiber cause Rayleigh scattering of the transferring waves. In our
case, the backscattered clockwise wave returns to the access hub and is
superimposed upon the extracted anticlockwise wave. Similarly, the
backscattered anticlockwise wave returns to the access hub and is superimposed
upon the extracted clockwise wave. Consequently, the Rayleigh scattering
effect can not be completely avoided in our application. According to the
results presented in [45], the Rayleigh backscattering induced fractional
frequency instability can be as low as a few $10^{-16}/\tau$ ($\tau$ being the
averaging time) over a 100 km fiber link. Thus, this effect can be neglected
at our precision.

Our dissemination loop can support multiple hubs simultaneously. Although
there is an insertion loss at every hub, proper optical amplifiers such as
erbium-doped-fiber-amplifiers (EDFA) and electrical amplifiers can be used to
amplify the desired optical signals and detected RF signals. Thus, it ensures
that multiple hubs can be mapped properly along the optical loop link. Though
$N$ copies of hardware for frequency recovery are needed if $N$ hubs are
required, all of these copies have the same configuration including fixed
optical and electronic components with no tunable parts. It should be noted
that several intermediate hubs along the fiber loop were tested besides the
50/50 km and 30/70 km hubs. Because of the similarity among the test results
of different hubs, we just show the test results of the two representative
hubs selected in the loop fiber link, the most symmetric one (50/50 km) and a
relative most asymmetric one (30/70 km).

##  VII Conclusion

In conclusion, we demonstrated a technique for dissemination of high-precision
optical-frequency signals to multiple independent hubs on a ring optical-fiber
network. The technique automatically corrects optical-fiber length
fluctuations of each hub along the loop. At the same time, using the same
optical source propagating along both directions can significantly improve the
signal-to-noise ratio. The results demonstrate relative frequency
instabilities, expressed as overlapping Allan deviation of $1.6\times
10^{-15}$ at 1 s averaging time, scaling down to $3.3\times 10^{-18}$ at 1,000
s with a $\tau^{-1}$ dependency at the intermediate hub over a 100 km fiber
ring. A similar performance is also demonstrated at another hub. We find no
systematic offset between the sent and transferred frequencies within the
statistical uncertainty of about $3\times 10^{-18}$.

This technique with passive phase compensation maintains the same phase noise
rejection capability as in conventional techniques and significantly shortens
the response speed and phase recovery time of optical frequency dissemination
and reduces the phase jitter by a factor of 3 compared to the conventional
technique, opening a way to a broad distribution of an ultrastable frequency
reference with high spectral purity and enabling a wide range of applications
beyond metrology over reliable and scalable ring fiber networks.

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Liang Hu received the B.S. degree from Hangzhou Dianzi University, China, in
2011, and the M.S. degree from Shanghai Jiao Tong University, China, in 2014.
He received the Ph.D. degree from University of Florence, Italy, in 2017
during which he was a Marie-Curie Early Stage Researcher at FACT project. He
is currently a Tenure-Track Assistant Professor in the State Key Laboratory of
Advanced Optical Communication Systems and Networks, Department of Electronic
Engineering, Shanghai Jiao Tong University, China. His current research
interests include photonic signal transmission and atom interferometry.
---
Xueyang Tian received the B.S. degree from Shanghai Dianji University, China,
in 2017. She is currently a graduate student in the State Key Laboratory of
Advanced Optical Communication Systems and Networks, Department of Electronic
Engineering, Shanghai Jiao Tong University, China. Her current research
interests include photonic signal transmission.
---
Long Wang  received the B.S. and M.S. degrees from Harbin Institute of
Technology, China, in 2017 and 2019, respectively. He has been admitted as a
doctoral student in the State Key Laboratory of Advanced Optical Communication
Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong
University, China. His current research interests include photonic signal
transmission.
---
Guiling Wu received the B.S. degree from Haer Bing Institute of Technology,
China, in 1995, and the M.S. and Ph.D. degrees from Huazhong University of
Science and Technology, China, in 1998 and 2001, respectively. He is currently
a Professor in the State Key Laboratory of Advanced Optical Communication
Systems and Networks, Department of Electronic Engineering, Shanghai Jiao Tong
University, China. His current research interests include photonic signal
processing and transmission.
---
Jianping Chen received the B.S. degree from Zhejiang University, China, in
1983, and the M.S. and Ph.D. degrees from Shanghai Jiao Tong University,
China, in 1986 and 1992, respectively. He is currently a Professor in the
State Key Laboratory of Advanced Optical Communication Systems and Networks,
Department of Electronic Engineering, Shanghai Jiao Tong University. His main
research interests include opto-electronic devices and integration, photonic
signal processing, and system applications. He is a Principal Scientist of
National Basic Research Program of China (also known as 973 Program).
---