1. Introduction

State-of-the-art optical clocks such as optical lattice clocks [1–3] and single-ion clocks [4,5] have demonstrated fractional frequency uncertainties at the $10^{-18}$ level and are expected to soon attain the $10^{-19}$ level [6,7]. Such clocks enable us to measure height differences at the Earth’s surface with centimeter to millimeter uncertainty [8,9], which serves as a quantum benchmark for geopotential [10–12], and networking them will provide a wide range of applications [13]. In particular, optical lattice clocks [14] employing thousands of atoms to improve the quantum projection noise (QPN) limit [15] can achieve remarkably low frequency instabilities. Cryogenic optical lattice clocks [3,16] have been demonstrated with instability of $2\times 10^{-16} (\tau /\textrm {s})^{-1/2}$ for an averaging time $\tau$. Lower instability of $3.5\times 10^{-17} (\tau /\textrm {s})^{-1/2}$ has been demonstrated [17] by using an ultralow-noise interrogation laser. These highly stable clocks are capable of detecting dynamic variations in geopotential due to tidal effects and crustal activities [10–12]. In addition, continental-scale optical clock networks have been constructed in Europe [13]. Such clock networks are particularly beneficial in countries with latent risks of earthquakes and active volcanoes.

The use of optical fiber links is a proven approach for networking distant optical clocks with high stability and accuracy [10,18–21]. Frequency noise induced on a transfer light through an optical fiber can be approximated by white frequency noise [22], and it follows a power-law dependence of the Fourier frequency $f$ of $S_{\nu }^\textrm {fiber}(f) = h_\textrm {L} L f^{0}$, where $h_\textrm {L}$ is the frequency noise coefficient of fiber and $L$ is fiber length. In a fiber noise compensation (FNC) technique [23], the transfer light is sent back through the same fiber to detect a beat signal with the round-trip fiber noise. The noise is suppressed with a phase-locked loop (PLL). For long fiber links, the one-way propagation delay time limits the feedback bandwidth and imposes a fundamental limit due to the difference in fiber noise between the forward and return paths. The delay-uncompensated fiber noise is approximately white phase noise, and the link instability scales as $h_\textrm {L}^{1/2} L^{3/2} \tau ^{-3/2}$ [22,24] in the modified Allan deviation, until reaching the system noise floor. For quiet fibers with a $h_\textrm {L}\approx 0.5$ Hz$^2$/Hz/km level as reported in Germany [25,26], link instabilities lower than $1\times 10^{-16} (\tau /\textrm {s})^{-3/2}$ can be achieved over 100-km-long fiber. On the other hand, fiber noises observed in Japan [10,27–29] are as large as $h_\textrm {L}\approx 70$ Hz$^2$/Hz/km, which increases the link instability by more than a factor of ten. For such noisy fibers, a 30-km-long link has been demonstrated with the instability of $4\times 10^{-17} (\tau /\textrm {s})^{-3/2}$ [29]. In these conditions, the link instabilities are lower than the laser instabilities of the $10^{-15}$ level at the averaging time of several hundred milliseconds, and the master-slave clock configuration [10] is applicable. By using a master laser shared via the fiber link to simultaneously operate distant slave clocks, the stability of the clock comparison is improved owing to common-mode rejection of laser noise [10,30]. In practice, as residual fiber noise at high frequencies and servo peaks are present in the delivered light, a slave laser stabilized with cavities is used as a tracking oscillator to eliminate those noise [10]. For longer fiber links [31–34], the short-term link instabilities become worse than the laser instabilities. In this situation, independent clocks operated with respective lasers are required. With the development of ultralow-noise lasers with instabilities of the $10^{-17}$ level [35], highly stable fiber links become increasingly important to make full use of the laser stability.

A key technique for improving the length-scalability of such noisy fibers is the cascaded link [36,37], which is achieved by dividing a long fiber into a number of short spans, connecting them with laser repeater stations (LRSs), and applying FNC for each span. Assuming a uniform spatial distribution of fiber noise, a cascaded link consisting of $N$ spans with equal length decreases the total link instability by a factor of $N$ [27]. However, multiple cascades may limit the link instability due to accumulated system noise of LRSs, such as out-of-loop path noise of optical interferometers [22,38] and electric noise from phase-locking circuits.

In this paper, we demonstrate a cascaded optical fiber link connecting three laboratories—one at RIKEN, another at the University of Tokyo (UTokyo), and the other at NTT—within 100 km. The link employs a transfer light at 1397 nm, a subharmonic of the Sr clock frequency, which has the advantage of being able to connect distant Sr clocks without optical frequency combs [29]. To reduce accumulated system noise, we developed ultralow-noise LRSs equipped with optical interferometers fabricated on planar lightwave circuits (PLCs) [39]. The interferometers exhibit an instability of $1\times 10^{-18}$ at 1 s and $1\times 10^{-21}$ at 5,000 s, which does not limit current clock stabilities even when tens of them are cascaded. The LRSs are installed in the three laboratories and two telecommunications offices (TCOs). We send a transfer light from RIKEN to NTT via a 150-km-long fiber with three spans, and the light received at NTT is sent back to UTokyo via a 120-km-long fiber with two spans. The 240-km-long loop link UTokyo–NTT–UTokyo exhibits the instability of $3\times 10^{-16} (\tau /\rm {s})^{-3/2}$ for $\tau\;<\;10$ s and reaches $1\times 10^{-18}$ at 2,600 s. The over-100-km link employing a subharmonic of the Sr clock frequency will allow construction of nationwide clock networks based on the master-slave configuration.

2. LRS with laser distributor

The LRS is schematically shown in Fig. 1. It consists of a repeater laser, PLC-based optical interferometers (blue shaded area), an optical phase-locking system (OPLS) and a polarization controller (PC) (green shaded area), and FNC (red shaded area). The four optical interferometers (OI1, 2, 3, and 4) were fabricated on a PLC chip $44\times 42$ mm$^2$ in size. Black lines in the chip represent optical waveguides. The repeater laser, acousto-optic modulators (AOMs), and differential photodetectors (PDs) are connected to the PLC chip by optical fibers depicted as green lines. The repeater laser is distributed to the OIs with an arbitrary power ratio using three variable couplers. The lights are output from fiber ports (FP1, 2, 3, and 4) as transfer lights. Received lights input from the FPs are heterodyned with the repeater laser, and optical beat signals are detected by the PDs. OI1 and OI2 are designed to detect beat signals between the transfer and received lights with linear polarization orthogonal to each other at the fiber ports, and they are used for long fiber links via telecom single-mode fibers (SMFs). The received lights are split from paths of the transfer lights by polarizing beamsplitters (PBSs). Reference lights tapped from the transfer lights are polarization-rotated by half-wave plates (HWPs) and coupled with the received lights for beat detection. FP1 is a receive port and acts like a Faraday mirror which receives a light with arbitrary elliptic polarization and sends back a light with an orthogonal polarization to compensate for the polarization rotation in the return path [40]. FP2 is a transfer port with functions for round-trip beat detection and FNC. OI3 and OI4 are designed to detect beat signals between the transfer and received lights with the same linear polarization at the fiber ports, and FP3 and FP4 are used as distribution ports for intra-office links via polarization-maintaining fibers (PMFs) with FNC.

 

Fig. 1. Schematic of the laser repeater station (LRS). A repeater laser is phase-locked to a received light in FP1, which is sent from a previous station via a long SMF. The repeater laser light is transferred to the next station from FP2 with FNC. FP3 and FP4 are used for intra-office distribution of the light via PMFs. PLC: planar lightwave circuit. OPLS: optical phase-locking system. PC: polarization controller. FNC: fiber noise compensation. OI: optical interferometer. FP: fiber port. HWP: half-wave plate. PBS: polarizing beamsplitter. PD: differential photodetector. DPFD: digital phase-frequency discriminator. DBM: double-balanced mixer. AOM: acousto-optic modulator.

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In each interferometer, optical path lengths for a reference light, a transfer light, and a received light are designed to minimize out-of-loop path noise as common-mode noise. Moreover, all paths from the repeater laser to each interferometer are designed to be of equal length. The interferometer noise floor was evaluated to be $1\times 10^{-18}$ at 1 s and $1\times 10^{-21}$ at 5,000 s by a back-to-back measurement, which is better than that of free-space [22,31] and fiber-based [38] interferometers. Notably, this instability was obtained without a special antivibration technique or temperature control. As the optical path lengths of the interferometer arms are balanced on the order of the optical wavelength, the temperature sensitivity is estimated to be $10^{-5}$ cycles/K level, assuming uniform temperature distribution in the chip. The detailed design and stability evaluation of the PLC-based interferometers will be described in a future publication.

The repeater laser at 1397 nm is an external cavity diode laser (ECDL) with an interference filter for wavelength selection [41]. Typical output power after fiber coupling is 30 mW at a laser current of 300 mA. The frequency instability of the free-running ECDL is less than 30 kHz at averaging times in the range of 10 $\mu$s to 100 ms. The laser frequency is controlled by the laser current and a piezo actuator mounted on an output coupler of the external cavity. A light sent from a previous station is heterodyned with the repeater laser in OI1 after it has been frequency-shifted by an AOM. A phase error signal is obtained by using a digital phase-frequency discriminator (DPFD) to compare the beat signal with a radio-frequency oscillator (RF-osc) signal. The error signal is proportional to the phase difference $\Delta \phi$ between the two signals in a range of $|\Delta \phi |\;<\;\pi$ rad, and it saturates for $|\Delta \phi | \geq \pi$ rad to provide a frequency discrimination function. A loop filter circuit generates feedback signals for the laser current and piezo voltage with 1.5 MHz and 100 Hz bandwidths, respectively.

The beat signal is also used for stabilizing the polarization of the received light. The PC consists of two pairs of piezo actuators which squeeze an optical fiber in orthogonal directions. A polarization error signal is obtained from the amplitude of the beat signal modulated by the polarization with a deviation of 0.2 rad at 0.1 Hz. The polarization is controlled to maximize the beat signal with a feedback bandwidth of 0.1 rad/min (2 mrad/s) against a typical deviation speed of $<\;0.02$ rad/min for a 100-km-long free-running SMF in our link. Because of such a low modulation frequency and feedback bandwidth, additional phase noise from the PC is sufficiently suppressed by the FNC.

The transfer light from FP2 is sent to the next station, where the light is regenerated and sent back to the present station. The returned light is heterodyned with the repeater laser in OI2 to detect a beat signal containing round-trip fiber noise. A phase error signal is obtained by using a 6-bit DPFD with a phase detection range of $|\Delta \phi |\;<\;64\pi$ rad to compare the beat signal with an RF signal. The 6-bit DPFD can store uncompensated phase bursts arising in the acoustic frequency region, for instance, due to subway operation [10]. A feedback signal is generated by a loop filter circuit, which controls the driving frequency of an AOM inserted in FP2 to compensate for fiber noise. The feedback bandwidth is set to $1/(4\tau _\textrm {d}) \approx 50~\textrm {kHz} \times (L/\textrm {km})^{-1}$, where $\tau _\textrm {d}=L/c_n$ is the one-way propagation delay time to the next station with the speed of light $c_n\approx 2\times 10^8$ m/s in the fiber. Distribution ports FP3 and FP4 have similar FNC. For intra-office links via PMFs with lengths of tens of meters, a double-balanced mixer (DBM) is used as a phase detector whose output is proportional to $\sin (\Delta \phi )$, and the feedback bandwidth of the FNC is about 10 kHz. The whole LRS is remotely operated.

3. Cascaded link

Figure 2(a) shows a schematic of the cascaded link connecting the three laboratories. RIKEN and UTokyo are 15 km apart and are connected by a 30-km-long dedicated fiber. The previous link setup of this span is described in Ref. [10,29], where optical interferometers were configured in free space. UTokyo and NTT are 50 km apart and are connected by two 120-km-long dedicated fibers. We implement cascaded links from NTT to UTokyo and back using two RLSs installed in telecommunications offices (TCOs). The 150-km-long link from RIKEN to NTT consists of spans from RIKEN to UTokyo (30 km), UTokyo to TCO1 (49 km), and TCO1 to NTT (69 km). The backward link from NTT to UTokyo consists of spans from NTT to TCO2 (96 km) and TCO2 to UTokyo (22 km). By connecting the two links, we implemented a 240-km-long UTokyo–NTT–UTokyo loop for stability measurement. Transmission losses shown in Fig. 2(a) include the attenuation of the fiber, $\sim 0.5$ dB/km at 1397 nm, and fiber connection losses. TCO3 was temporarily used to divide the TOC1–NTT span for fiber noise measurement.

 

Fig. 2. (a) Schematic diagram of the optical fiber network connecting the three laboratories and two telecommunications offices (TCOs). LL and RL denote a local laser and repeater laser at 1397 nm. White squares represent fiber connectors which are temporarily used for fiber noise measurement between the sites. Orange arrows indicate directions of frequency transfer. (b) Schematic of the experimental setup. LLs are stabilized to cavity-stabilized clock lasers (CLs) at 698 nm via second-harmonic generation (SHG). RLs are stabilized to received lights sent from previous stations. RIKEN–UTokyo (RL1–CL3) and RIKEN–NTT (RL3–LL2) beat signals are used for remote clock laser comparisons. A UTokyo–NTT–UTokyo (RL1–RL5) beat signal is used to evaluate the link stability of the 240-km-long fiber loop. $\nu _0 \approx 214.6$ THz is a subharmonic of the Sr clock frequency. PD: photodetector. BS: beam splitter. DM: dichroic mirror.

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The experimental setup is depicted in Fig. 2(b). Cavity-stabilized clock lasers (CLs) at 698 nm are operated in the three laboratories. A 40-cm-long reference cavity at RIKEN provides CL1 with the thermal noise limited instability of $3\times 10^{-16}$ [10], whereas a 12-cm-long cavity at NTT and a 7.5-cm-long cavity at UTokyo provide CL2 and CL3 with instabilities of $1\times 10^{-15}$ and $2\times 10^{-15}$ respectively. Local lasers LL1 and LL2 at 1397 nm at RIKEN and NTT are phase-locked to CL1 and CL2, respectively, via second-harmonic generation (SHG). The setup of the first input station at RIKEN is similar to the LRS. A distribution port (FP3 or 4) of a PLC chip is connected to an SHG module incorporating a periodically poled lithium niobate (PPLN) waveguide. Frequency-doubled LL1 is heterodyned with CL1 for phase-locking, while an unconverted light at 1397 nm is retro-reflected for FNC of a short fiber. LL2 at NTT is directly connected to an SHG module and used as a light source at 1397 nm and 698 nm. A beat measurement between a repeater laser (RL3) and LL2 corresponds to a remote frequency comparison between CL1 and CL2 (RIKEN–NTT beat signal). RL1 at UTokyo is also frequency-doubled and heterodyned with CL3, which corresponds to a remote frequency comparison between CL1 and CL3 (RIKEN–UTokyo beat signal). Remote optical lattice clock comparisons can be implemented with this setup by stabilizing the CLs to the clock transition of Sr atoms. The RIKEN–NTT beat signal is also used to stabilize CL2 to RL3 with a feedback bandwidth of 10 Hz, where CL2 acts as a narrow-linewidth optical tracking oscillator [10] (see Sec. 4). The stability and accuracy of the 240-km-long loop link were evaluated by measuring the beat signal between RL1 and RL5 (UTokyo–NTT–UTokyo beat signal).

In all the spans of the cascaded link, transfer lights are frequency-shifted by $-80$ MHz with AOMs for FNC, and received lights are frequency-shifted by $+80$ MHz with AOMs driven by RF oscillators. LL1 and LL2 are tuned to $\nu _0 -5$ MHz, where $\nu _0 \approx 214.6$ THz is the subharmonic of the Sr clock frequency. RL1, RL3, and RL4 are tuned to $\nu _0 +5$ MHz by setting offset frequencies of phase-locking to $+10$ MHz, whereas RL2 and RL5 are tuned to $\nu _0 -5$ MHz by setting them to $-10$ MHz. The round-trip beat frequency for FNC is 10 MHz for all spans. Frequencies of the RIKEN–NTT, RIKEN–UTokyo, and UTokyo–NTT–UTokyo beat signals are also 10 MHz. In this configuration, frequency fluctuations of RF oscillators are canceled out with FNC by using a technique similar to that described in Ref. [37]. Note that in our system, the transfer frequency $\nu _0$ can be distributed without its being affected by the RF oscillators.

4. Results and discussion

The power spectral density (PSD) of frequency noise induced on a light propagating through an optical fiber approximately follows a power-law dependence [22] of $S_{\nu }^\textrm {fiber}(f) = h_\textrm {L} L f^{0}$ for the Fourier frequency $f$ below a cutoff $f_\textrm {c} \approx 100$ Hz [29] and decreases as $f^{-2}$ for $f\;>\;f_\textrm {c}$. The round-trip fiber noise is not simply twice the one-way noise, since the noise in the forward path and return path in the same fiber are correlated at low frequencies. Assuming the uniform spatial distribution of fiber noise, the round-trip fiber noise is given by $S_{\nu }^\textrm {fiber,RT}(f) = 2 \left [ 1+\textrm {sinc}(4\pi f \tau _\textrm {d}) \right ] S_{\nu }^\textrm {fiber}(f)$ [22], which can be obtained from the round-trip beat signal without FNC. In measurements of fiber noise, we use a temporary LRS in TCO3 that divides the TCO1–NTT span. Figure 3(a) summarizes the frequency noise PSD of the fibers. Dotted curves depict the one-way fiber noise $S_{\nu }^\textrm {fiber}(f)$ for the UTokyo–TCO2 (22 km, violet), TCO2–TCO1 (27 km, light blue), TCO1–TCO3 (45 km, light green), and TCO3–NTT (24 km, orange) spans. The corresponding noise coefficients are estimated to be $h_\textrm {L}\approx 1$, 30, 10, and 40 Hz$^2$/Hz/km for the respective spans. A red solid curve represents noise of twice the whole UTokyo–NTT span, corresponding to the 240-km-long fiber loop with the assumption that the noise induced by the two fibers in this span are at the same level but uncorrelated. This indicates that $S_{\nu }^\textrm {fiber}(f) \approx 5\times 10^{3} f^{0}$ Hz$^2$/Hz for $f\;<\;f_\textrm {c}=100$ Hz, and from this curve, an average noise coefficient of $h_\textrm {L}\approx 20$ Hz$^2$/Hz/km is derived. In addition to limiting the feedback bandwidth, the one-way propagation delay time $\tau _\textrm {d}$ imposes imperfect noise compensation due to the difference in fiber noise between the forward and return paths [24]. The delay-uncompensated fiber noise with FNC is given by $S_{\nu }^\textrm {stab}(f) = a(2\pi f\tau _\textrm {d})^2 S_{\nu }^\textrm {fiber}(f) = (2\pi /c_n)^2 a h_\textrm {L} L^3 f^2$ for $f\;<\;1/(4\tau _\textrm {d})$ [22], where $a=1/3$ for the uniform spatial distribution of fiber noise and $a \approx 1$ for non-uniform noise. Assuming a single-span 240-km-long link and $a=1$, $S_{\nu }^\textrm {stab}(f)$ is calculated as a green solid curve in Fig. 3(a) with a feedback bandwidth of $1/(4\tau _\textrm {d}) \approx 210$ Hz for FNC. For the cascaded configuration shown in Fig. 2(a), the total $S_{\nu }^\textrm {stab}(f)$ of the four spans is calculated as a blue solid curve, which is improved by a factor of ten from that for a single-span 240-km-long link. It is mainly limited by the feedback bandwidth of 530 Hz for the 96-km span from NTT to TCO2. The calculated noise for the 120-km-long cascaded link from UTokyo to NTT is shown as a black solid curve.

 

Fig. 3. Power spectral densities of frequency noise for fibers and lasers. (a) Free-running fiber noises are depicted by dotted curves for the UTokyo–TCO2 (violet), TCO2–TCO1 (light blue), TCO1–TCO3 (light green), and TCO3–NTT (orange) spans and by a red solid curve for the 240-km-long UTokyo–NTT–UTokyo loop. The delay-uncompensated fiber noises are calculated as the other solid curves for a single-span 240-km-long link (green), a cascaded 240-km-long link (blue), and a cascaded 120-km-long link from UTokyo to NTT (black). Dashed lines represent the laser noise, assuming the thermal noise limit of the reference cavities. (b) Frequency noise of the RIKEN-NTT beat signal in the un-steered (blue) and steered (red) conditions of CL2 to RL2 with a 10-Hz bandwidth. The gray curve depicts the delay-uncompensated fiber noise calculated for a cascaded 150-km-long link from RIKEN to NTT.

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Dashed lines indicate laser noise of CL1, CL2, and CL3, assuming the thermal noise limits [42] of the reference cavities, which are $S_{\nu }^\textrm {laser}(f) = 0.012 f^{-1}$ (black), $0.13 f^{-1}$ (blue), and $0.34 f^{-1}$ (red), respectively. As the noise of CL1 is an order of magnitude lower than that of CL2, the stability of CL2 can be improved by tracking RL3. In the 240-km-long link measurement, CL2 is stabilized to RL3 using the RIKEN–NTT beat signal with a feedback bandwidth of 10 Hz [see Fig. 2(b)], which effectively eliminates the residual fiber noise and system noise for higher frequencies [10]. Figure 3(b) shows frequency noise of the RIKEN-NTT beat signal when CL2 is unstabilized (blue curve) and stabilized (red cureve) to RL2. The gray curve is calculated from the free-running noise and represents the fundamental limit in the cascaded 150-km-long link from RIKEN to NTT. In the measured noise PSD, some residual acoustic peaks are present. In particular, servo peaks caused by FNC in the three spans of 30 km, 49 km, and 69 km are present at 1.7 kHz, 1.0 kHz, and 0.7 kHz, respectively. The residual fiber noise is estimated to be larger than the laser noise of CL2 for $f\;>\;3$ Hz. In the stabilized condition of CL2 to RL3 with the feedback bandwidth of 10 Hz, the laser noise at $f\;<\;5$ Hz is suppressed in the beat signal. Servo bumps are not measured at 10 Hz, due to low feedback gain. In the current setup, the short-term frequency instability of 30 kHz of the free-running RLs results in $-30$-dBc level (1-kHz resolution bandwidth) servo peaks at 1.5 MHz. The servo peaks accumulate as LRSs increase. In the round-trip beat detection, beat notes between the servo peaks are detected at the same frequency as those between carriers, which degrades the FNC. The accumulated servo peaks are also eliminated from LL3 by tracking CL2 to RL3. We expect that using more stable RLs such as Whispering-Gallery-Mode stabilized lasers [43], which have a short-term instability of less than 1 kHz, will improve the servo peaks and enable us to implement more cascaded links with higher bandwidth for FNC.

Figure 4 shows fractional frequency instabilities of the optical beat signals via the fiber links, evaluated by the modified Allan deviation (MDEV). The RIKEN–UTokyo (red squares) and RIKEN–NTT (blue circles) beat signals contain both the residual fiber noise and laser noise and exhibit $\tau ^{-3/2}$ slopes of the link instabilities until the laser noise limits are reached. These results indicate the short-term link instabilities of $6\times 10^{-17} (\tau /\rm {s})^{-3/2}$ for the 30-km-long link from RIKEN to UTokyo [29] and $2\times 10^{-16} (\tau /\rm {s})^{-3/2}$ for the 150-km-long cascaded link from RIKEN to NTT, as depicted by red and blue dotted lines, respectively. The laser instabilities are indicated by dashed lines, assuming the thermal noise limit of the reference cavities, which are $3\times 10^{-16}$ (black), $1\times 10^{-15}$ (blue), and $2\times 10^{-15}$ (red) for CL1, CL2, and CL3 respectively. An orange dashed line shows the system noise floors of the LRS due to PLC interferometer and PLL circuit, which are $1\times 10^{-18}$ at 1 s and $1\times 10^{-21}$ at 5,000 s, evaluated by a back-to-back measurement. Green triangles shows the instability of the UTokyo–NTT–UTokyo beat signal. This measurement cancels out the laser noise as common-mode noise and indicates link instability below the laser noise limit. The MDEV for $\tau\;<\;10$ s is calculated from frequency data recorded for 100 s with zero dead time, which improves with $3\times 10^{-16} (\tau /\rm {s})^{-3/2}$. A gray dotted line represents an expected fundamental limit when a single-span 240-km-long link is implemented. The measured instability in the 240-km-long cascaded link is lower than this limit by factor of 3. The longer data set used to calculate the MDEV for $\tau \geq 10$ s includes dead time because of removed cycle slips with a slip rate of $\sim 4 \times 10^{-3}$ s$^{-1}$, which reduces the slope to $\sim \tau ^{-1/2}$. The cycle slips are mainly caused by the phase-locking of RL4 and FNC for the 96-km span with 55-dB loss, as the slip rate depends on the signal-to-noise ratio of the beat signal [37]. As a single cycle slip causes fractional frequency deviation of $5\times 10^{-15} \tau ^{-1}$, it can be detected by monitoring the loop link. For spectroscopy application, clock frequency data including the slips need to be corrected or removed. We expect that the cycle slip rate can be improved by using more stable repeater lasers or cavities as clean-up filters in UTokyo and TCOs. We also presume that the remaining free-space optical interferometers for detecting the RIKEN–NTT and UTokyo–NTT–UTokyo beat signals [see Fig. 2(b)] may limit the long-term stability. The measured link instability reaches $1\times 10^{-18}$ at $\tau =2,600$ s. This stability is better than that of cryogenic optical lattice clocks [3,16].

 

Fig. 4. Fractional frequency instability of the RIKEN–UTokyo (red squares), RIKEN–NTT (blue circles), and UTokyo–NTT–UTokyo (green triangles) beat signals. A gray dotted line represent $1\times 10^{-15} \tau ^{-3/2}$, an expected instability when a single-span 240-km-long link is implemented. The other dotted lines are eye guides showing $\tau ^{-3/2}$ dependence. An orange dashed line represent the system noise floor of a LRS due to PLC interferometer and PLL circuit. The other dashed lines represent the laser instabilities, assuming the thermal noise limit of the reference cavities. The instabilities of the beat signals between distant lasers are limited by the laser instability, whereas that of the 240-km-long loop link eliminates the laser noise limit and reaches $1\times 10^{-18}$ at 2,600 s.

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Figure 5 summarizes link instabilities at 1 s with respect to the fiber length for fiber links over 100 km reported by groups in the USA (violet square) [24], Germany (orange circles) [26,34], France (green triangles) [36,37], and Italy (blue reversed triangle) [33], and in this work (red diamond). The fiber noise coefficients in these links are estimated to be $h_\textrm {L}\approx 4$ (USA), 0.5 (Germany), 1 (France), 20 (Italy), and 20 Hz$^2$/Hz/km (this work). The delay-time-limited instabilities calculated with these fiber noise coefficients and assuming $a=1$ are depicted by solid lines in corresponding colors. While most of the reported link instabilities are a few times larger than the calculated ones, the 1480-km-long link in France [37] and the 240-km-long link in this work indicate instabilities below the lines because they are cascaded by four spans. Instabilities of cascaded links consisting of four spans with equal length for $h_\textrm {L}\approx 1$ and 20 Hz$^2$/Hz/km are calculated as dashed green and red lines, respectively. These results indicate that the cascaded link in this work successfully improves the instability in an environment where fiber noise is more than ten times larger. Whereas the other over-100 km links are operated with 1.55-$\mu$m lights, our cascaded link is the longest fiber link employing a transfer light at 1.4 $\mu$m, a subharmonic of the Sr clock frequency.

 

Fig. 5. Link instabilities at an averaging time of 1 s with respect to fiber length, reported by groups in the USA (violet square) [24], Germany (orange circles) [26,34], France (green triangles) [36,37], Italy (blue reversed triangle) [33], and this work (red diamond). Solid lines represent the delay-time limited instabilities for $h_\textrm {L}$ estimated in these links, indicated by corresponding colors, assuming $a=1$. Dashed lines represent expected instabilities of cascaded links consisting of four spans with equal length for $h_\textrm {L}\approx 1$ (green) and 20 Hz$^2$/Hz/km (red).

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5. Conclusion

We have demonstrated a cascaded optical fiber link connecting three laboratories located within a 100-km region with a transfer light at 1397 nm. Three remote clock lasers operating at 698 nm have been compared via the fiber link and second-harmonic generation, which provide a simple network of Sr clocks without applying optical frequency combs. Using noisy fibers with $h_\textrm {L}\approx 20$ Hz$^2$/Hz/km, we implemented a 240-km-long cascaded loop link with a fractional frequency instabilities of $3\times 10^{-16}$ at an averaging time of 1 s and $1\times 10^{-18}$ at 2,600 s. In the link, laser repeater stations incorporating arm-balanced interferometers fabricated on planar lightwave circuits provide a compact, robust, and highly stable optical system by minimizing interferometer noise [39]. Our system will improve the length-scalability of accurate optical frequency transfer through noisy environments.