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Experimental demonstration of weak-light inter-spacecraft clock jitter readout for TianQin

Hanyu Zeng, Hao Yan, Siyuan Xie, Sicheng Jiang, Yingzi Li, Yuhang Pan, Diaomin He, Yuanbo Du, and Hsien-chi Yeh

Open Access Open Access

Abstract

The space-based gravitational wave detection mission, TianQin, requires high-level synchronization between independent clocks of all spacecrafts to extract the gravitational wave signals. It is necessary to measure the inter-spacecraft relative clock jitter based on laser phase-sideband clock transfer. The main challenge is the tracking and locking of clock sideband beatnote signals with low signal-to-noise ratio and frequency variation. In this paper, a systematic scheme of inter-spacecraft clock jitter readout is reported. The requirement of the clock transfer link for TianQin based on the time-delay interferometry algorithm is derived. A bi-directional laser interferometer system with a transmission optical power below 1 nW and a time delay of ∼50 µs is built up to demonstrate the weak-light clock transfer. In this scheme, frequency modulation is performed on the laser to simulate the inter-spacecraft Doppler frequency shift and its variation. Based on electrical and optical clock transfer comparison experiments, it is demonstrated that the GHz frequency synthesizer is the main noise source below the 50 mHz frequency range. The residual clock jitter noise introduced by the optical transfer link is below 40 fs/Hz1/2 above the 6 mHz frequency range, and the fractional frequency instability is less than 6.7 × 10−17 at 1000 s, which meets the requirement of the TianQin mission. Ultimately, The carrier phase measurement accuracy reaches 1 × 10−4 cycles/Hz1/2 above 6 mHz after differential clock noise correction using measured clock jitter.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

The space-based gravitational wave detection missions include LISA [1], TaiJi [2], and TianQin [3] etc., which aim to detect gravitational waves in the millihertz range. TianQin is a space-based gravitational wave detection mission on geocentric orbits. It constructs a large-scale laser interferometer with an arm length of ∼1.7 × 105 km, aiming to measure the inter-satellite displacement change with a limitation resolution of 1 pm/Hz1/2. The incident laser beats with the local laser at each spacecraft. The phase of the beatnote signal contains the information of the inter-spacecraft displacement change that is indirectly obtained by measuring the phase of the beatnote signal in the phasemeter. The phasemeter and analog-to-digital converter (ADC) are driven by the onboard ultra-stable oscillator (USO). In the process of ADC sampling the beatnote signal, the clock jitter noise δt is directly coupled with the beatnote frequency fh, inducing the phase measurement noise fh·δt [4,5]. In TianQin mission, the budget of displacement measurement noise δx is 0.5 pm/Hz1/2 at 6 mHz, and the required phase measurement sensitivity δφ is approximately 5 × 10−7 cycles/Hz1/2 at 6 mHz for laser with a wavelength of 1064 nm. For a heterodyne beatnote signal of 10 MHz, the required clock jitter of the USO should be less than δt=δφ/fh≈50 fs/Hz1/2 at 6 mHz. However, there is no such stable aerospace-qualified USO that could meet the requirement up to now [6,7]. Fortunately, a data post-processing algorithm, named time-delay interferometry (TDI), was proposed to eliminate the contribution of clock noise for LISA [5,8–14]. Nevertheless, in this TDI algorithm, the inter-spacecraft differential clock jitter noise needs to be measured based on clock transfer.

On the ground, high-performance transfer of clock signals has been realized through both the optical fiber and free space. A microwave signal was transferred through an 86-km urban optical link with a fractional frequency instability of less than 1.3 × 10−15 at 1 s and below 10−18 at one day [15]. However, a round-trip link is necessary for phase noise compensation of the optical link, which will render an optical power attenuation of more than 1016 in the space-based gravitational wave detection. Moreover, this scheme will also increase the complexity of the optical system in gravitational wave detection missions. An ultra-stable laser was disseminated with an instability of 4 × 10−19 at 10,000 s through a free-space link of 113 km [16]. The link involves a high-power optical frequency comb, which is still immature for onboard applications. A scheme of inter-spacecraft clock transfer was proposed to measure the inter-spacecraft clock jitter noise and verified for the first time in an experiment for LISA [4,5]. In this scheme, the clock signal is modulated onto the phase of the laser and transferred to the distant spacecraft, where the incoming laser beats with the local laser which carries the local clock signal in a photodetector (PD). Both the carrier and the sideband beatnote signals are obtained, and the inter-spacecraft differential clock jitter need to be measured by tracking the sideband beatnote signals in a phasemeter. An electronic test was performed to demonstrate the practicability of the phasemeter that tracks the clock sideband beatnote signal with low carrier-to-noise (CNR) characteristics, and the noise floor of the phasemeter meets the requirement of clock transfer in LISA [17]. On a hexagonal optical bench, Kohei Yamamoto et al. verified the clock synchronization performance reaches LISA levels based on clock transfer [18].

However, the current work still has several shortcomings: Firstly, the calculation model for the requirements of the clock transfer chain is relatively simple and does not take into account the data processing process of the three satellites for LISA [5,19]. Secondly, several actual on-orbit conditions, including the low signal-to-noise ratio (SNR) of the beatnote signal induced by the received weak light (∼1 nW), beatnote signal frequency variation due to the inter-satellite Doppler motion, and so on, are still not considered in the previous experiments of inter-spacecraft clock jitter readout. These extreme conditions may result in a low SNR output of the clock sideband digital phase-locked loop (DPLL), even to the point of out-of-lock. Besides, the experimental verification of the inter-spacecraft clock jitter readout on an optical path with a large time delay, which could be simulated the inter-spacecraft clock transfer performance over a long distance, has not been carried out up to now.

In this work, we report the systematical scheme of inter-spacecraft clock jitter readout based on clock transfer. Initially, we strictly derive the requirements of inter-satellite clock transfer link for TianQin based on TDI algorithm, and give the requirement curves according to the parameters of TianQin. Moreover, the experimental setup of clock transfer is established to simulate the on-orbit conditions of received weak light, the beatnote frequency variation, and long-baseline laser propagation. Based on the experimental setup, the relative clock jitter and frequency difference of the two free-running USOs are measured, and the coupling characteristics of the clock noise with the internal devices and the ambient environment are verified, which is consistent with the coupling model of signal and noise established in this paper. Through the comparison experiment of electrical and optical clock transfer, it is found that the GHz frequency synthesizer is the main noise source in this scheme. The residual noise of the optical chain is below 40 fs/Hz1/2 above 6 mHz, and the fractional frequency instability is less than 6.7 × 10−17 at the average time of 1000 s. Ultimately, the measured differential clock jitter noise is used to eliminate the clock noise in the carrier phase measurement.

The paper is organized as follows: Section 2 outlines the inter-spacecraft clock transfer scheme based on an interferometric arm for TianQin. The signal model and the requirement for clock transfer are derived and the design of clock sideband DPLL is implemented. Section 3 demonstrates the experimental setup of weak-light clock transfer and differential clock noise measurement. Section 4 shows the measurement results, including the noise floor of weak-light clock transfer chain, the frequency difference between the two free-running USOs, and the differential clock jitter for the under the simulated on-orbit conditions. Section 5 is conclusion.

2. Inter-spacecraft clock transfer scheme

The schematic TianQin configuration is shown in Fig. 1, which contains three satellites, three independent USOs and six laser links, etc., forming an inter-satellite laser interferometer. The USO information is transmitted in the laser link, which shares the same optical link with the inter-satellite scientific interferometer.

 figure: Fig. 1.

Fig. 1. Schematic TianQin configuration. Each of the three spacecrafts (SC) is equipped with the same optical bench, which is labeled by 1, 2, 3 ($\mathrm{1^{\prime}}$, $\mathrm{2^{\prime}}$, $3^{\prime}$). The three spacecrafts are connected through laser, which forms six laser links. The armlength is denoted by Li, and i is primed or unprimed depending on the direction of the light propagation. Each satellite carries a free-running USO for frequency reference, and its clock noise is q.

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The preliminary scheme of inter-spacecraft clock transfer for clock jitter readout is given in Fig. 2 (a), which only shows one interferometric arm in Fig. 1. In spacecraft 1 (SC1), the 100 MHz output signal of the USO is scaled up to GHz by frequency synthesizer to drive an EOM to modulate the phase of the laser source (an ultra-stable laser). Then the laser carrying the clock message transmits a distance of ∼170,000 km and arrives at the spacecraft 2 (SC2). The incoming laser beats with the local laser in a PD, where the phase of the local laser is locked to the incoming laser with a heterodyne frequency fh of 10 MHz using a DPLL. The local laser is also modulated by the local USO signal, and thus the beatnote signal includes both the heterodyne carrier signal (beatnote between the incoming laser from SC1 and the local laser in SC2) and the clock sideband signals (beatnote between the clock sidebands of the two lasers). Two channels of clock sideband DPLLs are introduced to track the phase of the two clock sideband signals independently to realize the differential clock jitter readout. The same process is performed in SC1 except the heterodyne phase locking. The optical spectrum structure of the two lasers before beating in SC1 is as shown in Fig. 2 (b). In order not to reduce the signal-to-noise (SNR) of the carrier beatnote signal severely, the optical power modulated for one single clock sideband is usually below 10% of the total optical power, which results in a low SNR for clock sideband beatnote signals. The design of the phasemeter needs to consider the case of input signal with low power and frequency variation.

 figure: Fig. 2.

Fig. 2. (a) Scheme diagram of inter-spacecraft clock transfer for an interferometric arm in TianQin. The USO provides the frequency reference for phase measurements on each satellite. The USO frequency is scaled up to GHz and drives the EOM to modulate the phase of the laser source. The beatnote signals are measured in a phasemeter whose frequency reference is a USO. (b) The spectrum structure of the two lasers after phase modulation.

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2.1 Clock sideband signal model

In order to analyze the signal structure in clock transfer and facilitate the design of the loop parameters for the clock sideband DPLL, the signal model in the inter-satellite clock transfer scheme is established here. For appropriate simplification, the polarization of the laser is ignored and the scalar electric field is used to describe the propagation process of the laser. After phase modulation by the EOM, the electric field of the local laser at SC1 in time domain can be described as

$$\begin{array}{c} {E_1}(t) = {E_1}{e^{i[{2\pi f_1^ct + \varphi_1^c(t )} ]+ im\sin [{2\pi f_1^{Mod}t\textrm{ + }{N_1} \cdot \varphi_1^{USO}(t )+ \varphi_1^{FS}(t )+ \varphi_1^{EOM}(t )} ]}}\\ \approx {E_1}{J_0}(m ){e^{i[{2\pi f_1^ct + \varphi_1^c(t )} ]}}\textrm{ + }{E_1}{J_1}(m ){e^{i[{2\pi ({f_1^c + f_1^{Mod}} )t\textrm{ + }{N_1} \cdot \varphi_1^{USO}(t )+ \varphi_1^{FS}(t )+ \varphi_1^{EOM + }(t )+ \varphi_1^c(t )} ]}}\\ - {E_1}{J_1}(m ){e^{i[{2\pi ({f_1^c - f_1^{Mod}} )t - {N_1} \cdot \varphi_1^{USO}(t )- \varphi_1^{FS}(t )- \varphi_1^{EOM - }(t )+ \varphi_1^c(t )} ]}}, \end{array}$$
where E1 is amplitude of the electric field, f1c is the carrier frequency of the laser, φ1c is the phase noise of the laser, m is the phase modulation factor, f1Mod is the modulation frequency of the USO1, N1 is the scale factor f1Mod/f1USO, φ1USO is the phase noise of the USO1, φ1EOM is the phase modulation noise of the EOM sideband, φ1FS is the additional noise of the frequency synthesizer, φiEOM +  and φiEOM- are the phase noises of the upper sideband and the lower sideband induced by the EOM, respectively. The additional phase noise of EOM and frequency synthesizer have been measured in Ref. [19–22]. When the modulation factor m is set small (<0.6 rad), the optical power percentage of the Bessel modulation terms with the order higher than the 1st order in the laser is less than 0.4%, and thus only the carrier and the 1st order of the sidebands are considered here. After the heterodyne phase locking process in SC2, the carrier phase of Laser 2 is written as
$$\phi _2^c(t) = 2\pi f_1^c \cdot [{t - {\tau_{12}}(t )} ]+ 2\pi {f_h}t + {\varphi _h}(t) + \varphi _1^c[{t - {\tau_{12}}(t )} ]+ {\varphi _{OPLL}}(t )+ 2\pi \int_0^{t - {\tau _{12}}(t)} {{f_D}(t )dt}. $$

Here, fD is the Doppler shift induced by inter-satellite relative motion, τ is the light flight time between satellites, and φh is the residual phase noise in the weak-light heterodyne phase locking which is subject to the phase noise of the USO2. Thus, the electric field of laser 2 after phase modulation by the EOM is written as

$$\begin{array}{c} {E_2}(t )\approx {E_2}{J_0}(m ){e^{i\phi _2^c(t )}}\textrm{ + }{E_2}{J_1}(m ){e^{i[{2\pi f_2^{Mod}t\textrm{ + }{N_2} \cdot \varphi_2^{USO}(t )+ \varphi_2^{FS}(t) + \varphi_2^{EOM + }(t )+ \phi_2^c(t )} ]}}\\ - {E_2}{J_1}(m ){e^{i[{ - 2\pi f_2^{Mod}t - {N_2} \cdot \varphi_2^{USO}(t )- \varphi_2^{FS}(t )- \varphi_2^{EOM - }(t )+ \phi_2^c(t )} ]}}. \end{array}$$

The laser 2 arrives at SC1 after the propagation time τ21 and heterodynes with the local laser at SC1. One PD with a built-in transimpedance is used to convert the optical beatnote signal into a voltage signal, and the expression of the AC term of the PD1 output signal is given by

$$\scalebox{0.9}{$\begin{aligned} &V_{PD1}^{AC} = 2{R_{pd}}{G_{TIA}}\sqrt {{\eta _{het}}{P_{Local}}{P_{rec}}} J_0^2(m )\times \\ &\left\{ \begin{array}{@{}c@{}} \cos ({2\pi f_{beat1}^ct + \varphi_{beat1}^c} )\\ + \dfrac{{J_1^2(m )}}{{J_0^2(m )}}\cos [{2\pi ({f_{beat1}^c + f_2^{Mod} - f_1^{Mod}} )t + {N_2} \cdot \varphi_2^{USO}[{t - {\tau_{21}}(t )} ]- {N_1} \cdot \varphi_1^{USO}(t )+ \varphi_{SB\_beat1}^{noise} + \varphi_{beat1}^c} ]\\ + \dfrac{{J_1^2(m )}}{{J_0^2(m )}}\cos [{2\pi ({f_{beat1}^c - f_2^{Mod} + f_1^{Mod}} )t - {N_2} \cdot \varphi_2^{USO}[{t - {\tau_{21}}(t )} ]+ {N_1} \cdot \varphi_1^{USO}(t )- \varphi_{SB\_beat1}^{noise} + \varphi_{beat1}^c} ]\end{array} \right\} + {N_0}, \end{aligned}$}$$
where Rpd is the photodiode responsivity, GTIA is the gain of the transimpedance amplifier, ηhet is the heterodyne efficiency, Plocal and Prec are the local optical power and the received optical power, respectively. N0 is readout noise, which mainly includes shot noise, relative intensity noise and electrical noise. Here ${f}_{{beat1}}^{{c}}$ is the carrier frequency of the beatnote signal, ${\varphi }_{{beat1}}^{c}$ is the phase noise of the carrier beatnote signal, and ${\varphi }_{{SB\_beat1}}^{{noise}}$ is the additional noise of the clock transfer chain. Their specific expressions are described as
$$\left\{ \begin{array}{l} f_{beat1}^c(t) = 2{f_D}(t) + {f_h},\\ \varphi_{beat1}^c = \varphi_1^c[{t - {\tau_{12}}(t) - {\tau_{21}}(t)} ]- \varphi_1^c(t) - 2\pi {f_h}{\tau_{21}}(t) - 2\pi (f_1^c + 2{f_D}) \cdot [{{\tau_{12}}(t) + {\tau_{21}}(t)} ]\\ + {\varphi_h}[{t - {\tau_{21}}(t)} ]+ {\varphi_{OPLL}}[{t - {\tau_{21}}(t)} ],\\ \varphi_{SB\_beat1}^{noise} ={-} 2\pi f_2^{Mod}{\tau_{21}}(t) + \varphi_2^{FS}[{t - {\tau_{21}}(t)} ]- \varphi_1^{FS}(t) + \varphi_2^{EOM + }[{t - {\tau_{21}}(t)} ]- \varphi_1^{EOM + }(t). \end{array} \right.$$

The output of PD consists of three frequency signals: carrier beatnote, lower sideband beatnote, and the upper sideband beatnote. Each of them is subject to the same Doppler frequency shift of ∼MHz. In addition, due to weak-light interference and readout noise contribution, the SNR of the sideband signal is very low. The multiple beatnote signals are fed into the phasemeter, in which the carrier DPLL is used to measure the phase of the carrier beatnote signal and two clock sideband DPLLs are used to track and demodulate the phase of the clock sideband beatnote signals. Based on the equations above, the expression of the inter-spacecraft differential clock jitter noise can be written as

$$\scalebox{0.86}{$\begin{aligned} \delta {t_{21}} &= \frac{{{\phi _{lower\_SB}} - {\phi _{upper\_SB}}}}{{4\pi {f^{Mod}}}}\\ &= \frac{{{N_1} \cdot \varphi _1^{USO}(t) - {N_2} \cdot \varphi _2^{USO}(t - {\tau _{21}}(t)) + [\varphi _1^{FS}(t) + \varphi _1^{EOM - }(t)] - [\varphi _2^{FS}(t - {\tau _{21}}(t)) + \varphi _2^{EOM + }(t - {\tau _{21}}(t))]}}{{\textrm{ }2\pi {f^{Mod}}}}\\ &\quad + \frac{{2\pi f_2^{Mod}{\tau _{21}}(t) + {N_0}}}{{\textrm{ }2\pi {f^{Mod}}}}, \end{aligned}$}$$
where ϕupper_SB and ϕlower_SB are the phase of the upper sideband beatnote and the lower sideband beatnote, respectively. In addition to clock noise, it also includes additional noise introduced by frequency synthesizers, EOMs, and optical links in Eq. (6). In addition, the readout noise of the PD also contributes to the phase measurement process of clock sideband signal.

2.2 Requirement of the inter-spacecraft clock transfer for TianQin

The additional noise requirements of clock transfer link need to be derived according to the clock sideband signal model described in Section 2.1. The measurement data of each satellite is sent to the ground after down-sampling and processed by the TDI algorithm to eliminate the laser frequency noise and clock noise. However, the additional noise in the inter-satellite clock transfer chain cannot be suppressed, which affects the extraction of gravitational wave signals. Therefore, it is necessary to strictly derive the requirements of the inter-satellite clock transfer based on TDI algorithm. The first generation Michelson TDI combination X without laser frequency noise and clock noise is given by the following formula [9,11,13,23]:

$$\begin{aligned} {X_1} &= X_1^q - \frac{{{b_{1^{\prime}}}}}{2}[(I - {D_3}{D_{3^{\prime}}})({r_{1^{\prime}}} + {D_{2^{\prime}}}{r_3}) + (I - {D_{2^{\prime}}}{D_2})({r_1} + {D_3}{r_{2^{\prime}}})] + {a_1}[{r_{1^{\prime}}} + {D_{2^{\prime}}}{r_3}]\\ &\quad - {a_{1^{\prime}}}[{r_1} + {D_3}{r_{2^{\prime}}}] + {a_{2^{\prime}}}[{r_{1^{\prime}}} - (I - {D_{2^{\prime}}}{D_2}){r_1} + {D_{2^{\prime}}}{r_3}] - {a_3}[{r_1} - (I - {D_3}{D_{3^{\prime}}}){r_{1^{\prime}}} + {D_3}{r_{2^{\prime}}}]. \end{aligned}$$

The notation in the above formula is shown in Fig. 1. Here, ${X}_{1}^{{q}}$ is the TDI algorithm without laser frequency noise as given by

$$\begin{aligned} X_1^q &= {b_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}})(I - {D_{2^{\prime}}}{D_2}){q_1} + {a_1}(I - {D_{2^{\prime}}}{D_2}){q_1} - {a_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}}){q_1}\\ &\quad + {a_{2^{\prime}}}{D_3}(I - {D_{2^{\prime}}}{D_2}){q_2} - {a_3}{D_{2^{\prime}}}(I - {D_3}{D_{3^{\prime}}}){q_3}, \end{aligned}$$
and the remaining term in Eq. (7) is the calibration expression for clock noise. I represents the identity operator, Di is the delay operator, and Di satisfies this relationship Di f(t)=f(t-Li), where the speed of light c = 1 and Li represents the inter-spacecraft arm length as shown in Fig. 1. The qi is the clock jitter noise of spacecraft i, ${{b}_{{i^{\prime}}}}$, ${{a}_{{i^{\prime}}}}$ and ${{a}_{i}}$ are beatnote frequency and are given by
$${b_{1^{\prime}}} = {v_1} - {v_{1^{\prime}}}\textrm{ , }{a_1} = {v_{2^{\prime}}} - {v_1}\textrm{ , }{a_{1^{\prime}}} = {v_3} - {v_{1^{\prime}}}\textrm{ , }{a_{2^{\prime}}}\textrm{ = }{v_1} - {v_{2^{\prime}}}\textrm{ , }{a_3} = {v_{1^{\prime}}} - {v_3}, $$
where vi is the center frequency of laser i. The ri and ${\textrm{r}_{\mathrm{i^{\prime}}}}$ represent the measured inter-spacecraft differential clock jitter noise on the left and right optical benches of spacecraft i in Eq. (7), respectively. Here, taking r1 as an example, the specific expression for r1 is given by Eq. (6):
$${r_1} = {q_1} - {D_3}{q_2}\textrm{ + }N_1^c - {D_3}N_2^c, $$
where D3 represents the time delay from SC2 to SC1, denoted as τ21, The ${N}_{i}^{c}$ is the total additional noise of clock transfer link introduced by the GHz frequency synthesizer, EOM. The q1, q2, ${N}_{1}^{c}$ and ${N}_{2}^{c}$ are described as
$${q_1} = \frac{{{N_1} \cdot \varphi _1^{USO}(t)}}{{2\pi {f^{Mod}}}},\textrm{ }{q_2} = \frac{{{N_2} \cdot \varphi _2^{USO}(t)}}{{2\pi {f^{Mod}}}},\textrm{ }N_1^c = \frac{{\varphi _1^{FS}(t) + \varphi _1^{EOM - }(t)}}{{2\pi {f^{Mod}}}},\textrm{ }N_2^c = \frac{{\varphi _2^{FS}(t) + \varphi _2^{EOM + }(t)}}{{2\pi {f^{Mod}}}}. $$

Similarly, the ${{r}_{{1^{\prime}}}}$, ${{r}_{{2^{\prime}}}}$ and r3 are given by

$${r_{1^{\prime}}} = {q_1} - {D_{2^{\prime}}}{q_3} + N_1^c - {D_{2^{\prime}}}N_3^c\textrm{, }{r_{2^{\prime}}} = {q_2} - {D_{3^{\prime}}}{q_1} + N_2^c - {D_{3^{\prime}}}N_1^c\textrm{, }{r_3} = {q_3} - {D_2}{q_1} + N_3^c - {D_2}N_1^c. $$

Substituting the Eq. (10) and Eq. (12) into Eq. (7), the remaining noise is written as

$$\begin{array}{c} X_1^{residual} ={-} [{b_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}})(I - {D_2}{D_{2^{\prime}}}) + {a_1}(I - {D_2}{D_{2^{\prime}}}) - {a_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}})]N_1^c\\ + {a_{2^{\prime}}}{D_3}(I - {D_2}{D_{2^{\prime}}})N_2^c - {a_3}{D_{2^{\prime}}}(I - {D_3}{D_{3^{\prime}}})N_3^c. \end{array}$$

Here, it is assumed that ${{L}_{2}} = {{L}_{{2^{\prime}}}} = {{L}_{3}} = {{L}_{{3^{\prime}}}} = {L}$, and the power spectral density (PSD) of the additional noise in the clock transfer chain of the three satellites satisfies the relationship ${{S}_{{N}_{1}^{{c}}}}{(f)}$=${}{{S}_{{N}_{2}^{{c}}}}{(f)}$=${}{{S}_{{N}_{3}^{{c}}}}{(f)}$=${}{{S}_{{{N}^{{c}}}}}{(f)}$. The PSD of the Eq. (13) is derived and can be written as

$$S_{{X_1}}^{residual}(f) = 4{\sin ^2}(2\pi fL)[({b_{1^{\prime}}} + {a_1} - {a_{1^{\prime}}})4{b_{1^{\prime}}}{\sin ^2}(2\pi fL) + {({a_1} - {a_{1^{\prime}}})^2} + {a_{2^{\prime}}}^2 + {a_3}^2]{S_{{N^c}}}(f). $$

The secondary noise in space gravitational wave detection are optical path noise and acceleration noise, and their expressions are given by [24]

$$S_X^{secondary}(f) = [8{\sin ^2}(4\pi fL) + 32{\sin ^2}(2\pi fL)]{S_{acc}}(f) + 16{\sin ^2}(2\pi fL){S_{opt}}(f). $$

The requirement of the optical path noise and acceleration noise are 1 pm/Hz1/2 and 1 × 10−15 m/s2/Hz1/2 for TianQin [3], and their PSD is

$${S_{acc}}(f) = {[\frac{1}{\lambda } \cdot \frac{{{s_a}}}{{{{(2\pi f)}^2}}}]^2} = \frac{{5.7 \times {{10}^{ - 22}}}}{{{f^4}}}\textrm{ }\frac{{\textrm{cycle}{\textrm{s}^2}}}{{\textrm{Hz}}}\textrm{ , }{S_{opt}}(f) = {[\frac{1}{\lambda } \cdot {s_x}]^2} = 8.8 \times {10^{ - \textrm{ }13}}\textrm{ }\frac{{\textrm{cycle}{\textrm{s}^2}}}{{\textrm{Hz}}}, $$
where λ=1064 nm, is the wavelength of the laser. The remaining clock transfer chain noise ${S\; }_{{X1}}^{{residual}}$ must be less than the secondary noise, so the requirement for additional clock jitter noise in the clock transfer chain can be described as
$$\sqrt {{S_{{N^c}}}(f)} \le \sqrt {\frac{{[8{{\sin }^2}(4\pi fL) + 32{{\sin }^2}(2\pi fL)]{S_{acc}}(f) + 16{{\sin }^2}(2\pi fL){S_{opt}}(f)}}{{4{{\sin }^2}(2\pi fL)[({b_{1^{\prime}}} + {a_1} - {a_{1^{\prime}}})4{b_{1^{\prime}}}{{\sin }^2}(2\pi fL) + {{({a_1} - {a_{1^{\prime}}})}^2} + {a_{2^{\prime}}}^2 + {a_3}^2]}}}. $$

Here, the laser beatnote frequency is assumed to be ±10 MHz, and the requirement curve of the clock transfer chain is shown in Fig. 3. The red curve represents the derived requirement of the clock transfer, and the blue curve represents 77 fs/Hz1/2·(1 + (6 mHz/f)4)1/2. The two curves are consistent within the frequency range of 0.1 mHz to 1 Hz. This requirement is then used to evaluate the performance of experimental setup for clock jitter readout.

 figure: Fig. 3.

Fig. 3. The requirement curve in the clock transfer chain. The red curve is derived based on TDI combination X. The blue curve is described as 77 fs/Hz1/2·NSF. The NSF represents the noise shape function, which is (1 + (6 mHz/f)4)1/2.

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2.3 Clock sideband DPLL design

The USO information in the clock sideband beatnotes, as shown in Eq. (4), is demodulated by DPLL in this scheme. The DPLL is widely used to measure the phase and frequency of beatnote signals in space-based gravitational wave detection missions, and its basic principle has been elaborated in Ref. [25–29]. Unlike the conventional phasemeter, in the scheme of clock transfer, a phasemeter that simultaneously measures the phases of the carrier beatnote and the two sideband beatnote signals is needed. In this work, the carrier DPLL is improved based on the prototype given in Ref. [30,31].

The block diagram of the DPLL is shown in Fig. 4. The phase output of the DPLL follows the change of the input signal within the designed bandwidth. When the DPLL operates in the locked state, the phase difference between the input signal and the output signal of NCO is small, and then the characteristics of the DPLL can be described by its transfer function. The sampling clock frequency of DPLL is set as 100 MHz, which is much higher than the frequency range of the beatnote signal. In this condition, the DPLL can be simplified to a continuous system model and described in Laplace domain. According to Ref. [32], the open-loop transfer function and closed-loop transfer function of the DPLL are described as

$$G(s) = \frac{{{\varphi _{nco}}(s)}}{{{\varphi _e}(s)}} = \frac{{{G_{PD}} \cdot {G_{PA}} \cdot {K_P} \cdot T \cdot s + {G_{PD}} \cdot {G_{PA}} \cdot {K_i}}}{{{s^2}{T^2}}},\textrm{ }H(s) = \frac{{{\varphi _{nco}}(s)}}{{{\varphi _{in}}(s)}} = \frac{{2\zeta {w_n}s + w_n^2}}{{{s^2} + 2\xi {w_n}s + w_n^2}}, $$
where φnco is the phase of the NCO, φe is the phase difference between the input signal and the NCO output signal, GPD is the scale factor of the phase detector, whose value is related to the bit widths of both the ADC and the NCO output signal after quantization. GPA is the phase control gain of the NCO, KP and Ki are proportional and integral parameters, and T is the sampling interval, respectively. wn and ζ jointly determine the performance of DPLL, and their specific expressions are given by ${{w}_{n}}{ = }\sqrt {{{G}_{{PD}}} \cdot {{G}_{{PA}}} \cdot {{K}_{i}}} /{T}$ and $\zeta$ = ${}{{K}_{P}}/{2} \cdot \sqrt {{{G}_{{PD}}} \cdot {{G}_{{PA}}}/{{K}_{i}}} $.

 figure: Fig. 4.

Fig. 4. The block diagram of the DPLL. It mainly contains a phase discriminator (PD), a proportional-integral regulator (PIR), a numerically controlled oscillator (NCO), and a decimation filter.

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In practice, the SNR of the clock sideband beatnote signals output by PD is extremely low, as shown in Fig. 5 (b). The clock sideband DPLL must track and lock the input signal of this weak power. The current research indicates that the SNR of loop output greater than 6 dB is required to ensure the DPLL loop can properly lock with the input signal. The relationship between the output SNR and the input SNR of DPLL can be described as

$${\left( {\frac{S}{N}} \right)_o} = {\left( {\frac{S}{N}} \right)_{in}} \cdot \frac{{{B_i}}}{{{B_L}}} = {\left( {\frac{S}{{{N_0}}}} \right)_{in}} \cdot \frac{1}{{{B_L}}}, $$
where Bi is the input signal bandwidth, BL is the loop noise bandwidth, which is approximately equal to wn(1 + 4ζ 2)/8ζ. The N0 (=N/Bi) is the noise power spectral density, and S/N0 represents the carrier to noise ratio (CNR). In this scheme, assuming the CNR of the clock sideband beatnote signals is 40 dB-Hz, so the loop bandwidth BL meets BL < 2.5 kHz. Then, all loop parameters of the clock sideband DPLL can be determined [33].

 figure: Fig. 5.

Fig. 5. (a) The designed scheme of the carrier DPLL and the clock sideband DPLLs. The PIR of the carrier DPLL is ±1 MHz offset and fed back to the two sideband DPLLs. The clock sideband DPLLs are used to track the clock sideband beatnote signals. (b) The frequency spectrum of the PD output signal. The sideband beatnote signal has the characteristics of low SNR and its center frequency varies in the ∼MHz range.

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According to the above analysis, KP and Ki parameters can be optimized to reduce the loop bandwidth and improve the SNR of loop. However, since the frequency of the clock sideband beatnote signal varies in the range of ∼MHz, the reduction of loop bandwidth will result in the clock sideband DPLL being unable to track the signal with frequency change. According to Eq. (4), the phase of the sideband beatnote signal contains the same part (2πfcbeat1t+φcbeat1) as that of the carrier beatnote signal, and thus the frequency tracked by the carrier DPLL can be fed forward to the sideband DPLL to track the signal of frequency variation. This design method was presented in Ref. [17,34].

Thus, the optimized design scheme of clock sideband DPLL is shown in Fig. 5, where the carrier PIR output plus or minus 1 MHz is fed forward to two sideband DPLLs. Based on this scheme, a phasemeter is implemented in the FPGA electronic board and a test scheme is established to perform electrical verification of the phasemeter. Signals with the same frequency reference and three frequency dots are combined by a power splitter/combiner and fed into the phasemeter for verification measurement, as shown in Fig. 6. The combined signal in time domain can be written as

$$U(t) = {U_0}\sin ({2\pi \cdot {f_h} \cdot t} )+ {U_1}\sin ({2\pi \cdot {f_L} \cdot t} )+ {U_1}\sin ({2\pi \cdot {f_U} \cdot t} ). $$

 figure: Fig. 6.

Fig. 6. Electrical verification scheme of the phasemeter. The couplers and the power splitter are used to divide and combine the input signals (9 MHz, 10 MHz, 11 MHz) to generate reference signals and three-frequency mixed signals, whose phases are measured through different DPLLs.

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Here, U0 (200 mV) is the amplitude of the carrier signal, U1 (14 mV) is the amplitude of the clock sideband signal, which is used to simulate the weak signal measurement of the clock side beatnote. The fh (10 MHz) is the frequency of the carrier signal, fL (9 MHz) and fU (11 MHz) are the frequencies of the lower and the upper clock sideband signals, respectively.

The noise floor of the multi-channel phasemeter for the multiple-frequency signals can be obtained by performing phase difference measurement on the input signal U(t). As shown in Fig. 7, the differential phase noise floor of clock sideband DPLLs, represented by the orange curve (a), meets the requirement of inter-satellite clock transfer, and is limited by ADC quantization noise (∼5.6 × 10−7 cycles/Hz1/2) at high frequency range. The phase noise floor of the carrier DPLL in the multi-channel phasemeter is consistent with that of the single channel carrier phasemeter, which indicates that the phase measurement of the carrier DPLL in the multi-channel phasemeter will not be affected under the condition of multi-frequency signals input. In other words, clock sideband modulation signals will not affect carrier phase measurements in gravitational wave detection. Note that pilot signals are not added to correct the timing jitter of the ADC in the clock sideband DPLL. Therefore, these measurement results also contain the jitter noise of the ADC.

 figure: Fig. 7.

Fig. 7. The noise floor measurement results of the multi-channel phasemeter. (a) Differential phase noise floor of clock sideband DPLLs, which is limited by ADC quantization noise at high frequency range. (b) Phase noise floor of carrier DPLL. (c) Requirement of clock transfer. (d) Phase noise floor of single-channel phasemeter, which is only used to measure the phase of single frequency input signal.

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3. Experimental setup of weak-light clock transfer and differential clock noise measurement

An experimental setup of bi-directional weak-light clock transfer with an optical fiber of 10 km is established. As shown in Fig. 8, two optical benches (bench1 and bench 2) are used to simulate the two satellites SC1 and SC2, which can be connected using a short optical fiber or an optical fiber of 10 km long. The USO1 is an oven-controlled crystal oscillator (OXCO), while the USO2 is replaced with a rubidium atomic clock. One laser source is split into two beams with an optical fiber coupler, and the laser beams access different optical paths for frequency shift and phase modulation. To simulate the heterodyne frequency in the process of inter-spacecraft weak-light phase locking, the driving radio frequency sources of the AOMs are different, whose center frequencies are 49 MHz and 57 MHz, respectively. Two signal generators with high performance (R&S SMB100B), frequency reference to the two USOs, are used as the frequency synthesizers to produce 2 GHz and 1.999 GHz signals to drive the two EOMs. The phase modulation factor of the laser is set as 0.51 rad, which corresponds to an optical power of 7% in the 1st order clock sideband. An optical circulator is used to isolate the incoming laser from the local laser and realize bi-directional propagation of the two laser beams. The optical power is attenuated by attenuator to produce interference between the local laser (strong light) and the incoming laser (weak light). The optical power of the local laser is set as 0.1 mW, while the optical power of the incoming laser is only 1 nW, among which 70 pW is used for single clock sideband modulation. The PD output is fed into the phasemeter for phase measurement. The external frequency references of the two FPGA electronic boards are also the two USOs, respectively. An electronic mixing is simultaneously performed to measure the noise contribution of the both the USOs and the frequency synthesizers directly. A third FPGA electronic board is used to monitor the differential clock noise of the two USOs. In addition, in order to simulate the Doppler frequency shift due to the relative motion between the spacecrafts, the laser beam in SC2 is performed a frequency modulation with 4sin(2π·(2 × 10−3)·t) MHz.

 figure: Fig. 8.

Fig. 8. Bi-directional clock transfer experimental setup. The clock signal is transmitted through both the optical link and the electrical link (red dotted line), and the electronic mixing is used to compare with the weak-light clock transfer link.

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4. Experimental results

4.1 Weak-light clock transfer system noise floor

The typical frequency spectrum of the PD output signal is shown in Fig. 9. It can be seen that the power ratio of the clock sideband beatnote to the carrier beatnote is about -23 dB, which is consistent with the relative optical power of 7%. When all clock frequencies are referenced to the same USO, the clock sideband beatnote signals mainly contain additional clock jitter noise from the clock transfer link.

 figure: Fig. 9.

Fig. 9. The frequency spectrum of the PD output signal. The frequency spectrum includes the carrier beatnote and two clock sideband beatnote signals.

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The noise floor of clock transfer link is measured without the optical fiber of 10 km, which is shown in Fig. 10(a). It can be seen that the clock jitter noise of the weak-light clock transfer link is consistent with that measured by electronic mixing in the frequency range below 50 mHz frequency range, indicating that the noise floor of the weak-light clock transfer link is limited by the frequency synthesizer noise in this frequency range. In the frequency range higher than 50 mHz, the clock jitter noise of weak-light clock transfer is worse than that in the electronic mixing. According to the analysis, this could be attributed to the effects of the PD readout noise, EOM and optical fiber noise in this frequency range. The difference of the clock jitter noise between weak-light clock transfer and electrical transfer is shown in green curve in Fig. 10(a), which approximately follows a white phase noise of 40 fs/Hz1/2. Thus, after removing the noise contribution of the frequency synthesizer, the additional noise of the weak-light clock transfer chain can meet the requirement of inter-satellite clock transfer in the full frequency band of space-gravitational wave detection. This also shows that the clock noise introduced by the optical path is below the clock transfer requirement. Since the phase noise can be converted to the frequency noise by taking the first order differentiation of time, the Allan deviation is obtained in Fig. 10(b). At the average time shorter than 100 s, the frequency instability of the weak-light clock transfer chain is worse than that of electronic mixing, and a common frequency drift dominates at the longer average time. The Allan deviation of the differential noise is also given in green curve in Fig. 10(b), which approximately follows a white phase noise curve of 7 × 10−14τ -1, which reaches 6.7 × 10−17 at the average time of 1000 s. At present, a higher-performance space-borne frequency synthesizer is being customized to meet the clock transfer requirements for TianQin.

 figure: Fig. 10.

Fig. 10. The noise floor of clock transfer link. (a) Amplitude spectral density of clock jitter noise. Red curve: the clock jitter noise of the weak-light clock transfer link, which contain GHz frequency synthesizer noise and optical path noise. Blue curve: the clock jitter noise of the electrical link, mainly limited to GHz frequency synthesizer noise. Green curve: the difference of clock jitter noise between weak-light and electrical clock transfer link. (b) The fractional frequency instability of the clock transfer link.

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4.2 Frequency difference measurement

Under the conditions that the USOs of the two benches in Fig. 8 are free running, an optical fiber with the length of 10 km is introduced to simulate the long-baseline inter-satellite light propagation and the frequency of the incoming laser is modulated by fcenter + 4sin(2π·(2 × 10−3)·t) MHz. The beatnote signal with frequency variation and low SNR is fed into the multi-channel phasemeter for phase measurement. The measurement results of the frequencies of the beatnote signal are shown in Fig. 11, where the center frequencies of measured signal are 7 MHz, 8 MHz and 9 MHz. These measured frequencies vary in the range of -4 MHz to 4 MHz around each center frequency, and the variation period is 500 s. All the measured values are consistent with the setting parameters.

 figure: Fig. 11.

Fig. 11. Frequency measurement of the beatnote signal. The center frequencies are 7 MHz, 8 MHz and 9 MHz, respectively. (a) Frequency of the lower clock sideband beatnote. (b) Frequency of the carrier beatnote. (c) Frequency of the upper clock sideband beatnote.

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The frequency difference between the two USOs can be calculated from the three measured frequency values, and the result is shown in Fig. 12. It can be seen that the fractional frequency difference of the USOs is 1.10(1) × 10−8. There are several frequency bad data points in all measurement results of the frequency differences because the USO1 is in abnormal working condition. Nevertheless, the difference between the measurement results of weak-light clock transfer and electronic mixing shows effective suppression of the USO noise in Fig. 12(d), with a standard deviation of 8 mHz. In addition, the frequency difference fluctuates either significantly or subtly at different times in Fig. 12(d), which is mainly affected by temperature fluctuation and airflow disturbance around the long optical fiber. The Allan deviation of the fractional frequency difference of the two USOs is shown in Fig. 12 (e). Due to the environmental influence on the 10 km optical fiber, the additional noise of optical transfer chain is significantly greater than that without the optical fiber. In fact, it reaches a value of 2.7 × 10−14 at an average time of 1000s, as shown in the green curve in Fig. 12(e).

 figure: Fig. 12.

Fig. 12. (a)-(d) The frequency difference between the two USOs. (a) The frequency difference based on weak-light clock transfer. (b) The frequency difference based on electrical clock transfer. (c) The direct measurement of frequency difference between the two USOs (×199.9). (d) The frequency difference between weak-light and electrical clock transfer. (e) The Allan deviation of the fractional frequency difference.

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4.3 Differential clock jitter noise measurement

The differential clock jitter noise between the two USOs is measured based on weak-light and electrical clock transfer link, and the measurement results are shown in Fig. 13. Figure 13(a) shows the differential clock jitter noise obtained by the weak-light clock transfer, where the black curve represents the clock difference reached 800 ms within 20 hours. After removing the linear drift of the clock difference, the differential clock jitter noise (represented by the red curve) varies between -2 µs and 2 µs within 20 h. Figure 13(b) shows the differential clock jitter noise measured by the electronic mixing, which is extremely consistent with the results obtained by the weak-light clock transfer. The difference of the differential clock jitter noise between weak-light clock transfer and electronic mixing is shown in the red curve in Fig. 13(c), which eliminates the noise introduced by the frequency synthesizer and leaves the additional noise of the optical transfer chain. At the same time, it can be seen that the additional noise of the optical transfer chain has a strong coupling with the ambient temperature fluctuation (represented by the black curve). Based on the noise floor results of clock transfer link without 10 km optical fiber shown in Fig. 10, here the source of optical path noise is generated due to the temperature effect on the 10 km optical fiber.

 figure: Fig. 13.

Fig. 13. The measurement results of differential clock jitter noise between two USOs. (a) The clock difference and clock jitter between the USOs are measured by weak-light clock transfer. (b) The clock difference and clock jitter between the USOs are measured by electronic mixing. (c) The difference between weak-light and electrical clock transfer, which has a strong correlation with the ambient temperature.

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The amplitude spectral density of the differential clock jitter noise between two free running USOs is given in Fig. 14. It can be seen that the relative clock jitter noise is extremely high, which needs to be suppressed through data post-processing in space gravitational wave detection. The difference between the measured results obtained by the two methods is given by green curve, which shows the presence of a series of harmonics of 2 mHz. It is inferred that the harmonics is induced by the coupling of the modulation frequency of 4sin(2π·(2 × 10−3)·t) MHz with the time delay of the phasemeter. the clock jitter noise of the optical path is below 4.1 ps/Hz1/2 above 6 mHz after removing the harmonics, which is mainly contributed by the 10 km optical fiber.

 figure: Fig. 14.

Fig. 14. Amplitude spectral density of differential clock jitter noise between two free running USOs. Red curve: the clock jitter noise obtained by weak-light clock transfer. Blue curve: the clock jitter noise obtained by electrical clock transfer. Green curve: the difference between the two measurement methods.

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4.4 Clock noise correction for dual one-way carrier phase measurement

In Fig. 8, the beatnote signal is sampled by the ADC driven by USO and then fed into the DPLL for phase measurement at each optical bench. The USO noise will enter into the phase measurement in this process. For carrier phase measurement, the output of the carrier DPLL in bench 1 and bench 2 can be described separately as

$$\left\{ \begin{array}{l} \phi_1^{carrier}(t) = 2\pi {f_h}t - 2\pi f_1^{carrier}{\tau_{21}} + 2\pi {f_h} \cdot \delta t_1^{USO} + N_1^{noise}\\ \phi_2^{carrier}(t) = 2\pi {f_h}t - 2\pi f_2^{carrier}{\tau_{12}} + 2\pi {f_h} \cdot \delta t_2^{USO} + N_2^{noise} \end{array} \right\},$$
where fh is the carrier beatnote frequency, $f_i^{carrier}$ represents the frequency of laser i in bench i, τ represents the optical delay of the link, ${N}_{i}^{{noise}}$ represents other noise terms, ${\delta t}_{1}^{{USO}}$ and ${\delta t}_{2}^{{USO}}$ are the clock jitter noise of the USO in SC1 and SC2 that are introduced during ADC sampling, respectively. Taking the difference of the measured carrier phase from two benches in above equation can obtain the distance variation information between spacecraft, but the differential clock jitter noise is also retained. The dual one-way carrier phase difference eliminates the phase ramp with constant slope fh and can be described as
$$\phi _{12}^{carrier}(t) = 2\pi (f_1^{carrier}{\tau _{21}} - f_2^{carrier}{\tau _{12}}) + 2\pi {f_h} \cdot (\delta t_1^{USO} - \delta t_2^{USO}) + {N^{noise}},$$
where the first term in the above equation represents the phase change of the optical carrier introduced by the distance change, and the second term represents the phase noise introduced by the coupling of the inter-spacecraft differential clock jitter noise and the laser heterodyne frequency. Due to the effect of temperature change, the phase noise introduced by the 10 km optical fiber in the measured optical carrier beatnote phase difference is much larger than the differential clock jitter noise of the two USOs, so that we cannot see the effect of clock jitter noise on phase measurement. To verify the coupling of differential clock jitter noise with heterodyne frequency and correct the effect of clock noise using the measured differential clock jitter, we replace the optical carrier phase measurement with a radio frequency (RF) carrier phase measurement with a frequency of 10 MHz to exclude the effect of the optical noise in Fig. 8.

The measurement results of the RF carrier phase for both test benches are shown in Fig. 15(a) and (b), which are produced by the accumulation of 10 MHz frequency. The RF carrier phase difference between benches is shown in Fig. 15(c), which demonstrates a strong correlation with the measured differential clock jitter shown in Fig. 13(a) and (b). This indicates that the main noise in the carrier phase difference is the differential clock jitter of the two USOs, as shown in Eq. (22). The relationship between the carrier phase difference $\phi _{\textrm{12}}^{\textrm{carrier}}$ and measured differential clock jitter ${\delta t}_{{12}}^{{USO}}$ is shown in Fig. 15(d), and the slope of the fitting curve is 2π·10 MHz. This indicates that the differential clock jitter noise between USOs is coupled into the carrier phase measurement. Therefore, the measured differential clock jitter can be used to correct the clock noise in dual one-way carrier phase measurement.

 figure: Fig. 15.

Fig. 15. Radio frequency carrier phase measurement. (a) Carrier phase measurement at bench1. (b) Carrier phase measurement at bench2. (c) Carrier phase difference between two benches. (d) The linear relationship between carrier phase difference and measured differential clock jitter.

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The clock noise correction results of carrier phase measurement are shown in Fig. 16. The red curve is the carrier phase difference, which is mainly affected by differential clock jitter. The blue curve shows that the clock noise is corrected by weak-light clock transfer. The reason for the frequency peak at the 2 mHz harmonic in the blue curve has been explained in Fig. 14. The blue curve and the green curve are inconsistent at low frequencies, which is mainly due to the weak-light clock transfer affected by the coupling noise between fiber and temperature. After clock noise correction, the residual noise is less than 1 × 10−4 cycles/Hz1/2 above 6 mHz. The remaining noise here includes ADC jitter noise and ADC front-end transformer noise at low frequency.

 figure: Fig. 16.

Fig. 16. Clock noise correction in dual-one-way carrier phase measurement. (a) The carrier phase difference between the two benches. The clock noise is not corrected. (b) The clock noise is corrected using measured differential clock jitter by optical clock transfer. (c) The clock noise is corrected using measured differential clock jitter by electrical clock transfer. (d) The noise floor between two phasemeters. PM: Phasemeter.

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5. Conclusion

USO noise is one of the main noise sources in space-based gravitational wave detection. The TDI data post-processing algorithm and inter-spacecraft clock transfer technology are used to suppress clock noise in gravitational wave signals. However, in the currently reported research, the performance of differential clock jitter readout has not been verified under extreme conditions for ground-based simulated experiments of inter-spacecraft clock transfer. In this work, we derive the additional noise requirements of the clock transfer link for TianQin based on TDI algorithm and give the clock transfer requirement curve that is in well agreement with 77 fs/Hz1/2·(1 + (6 mHz/f)4)1/2 and is used to evaluate the performance of experimental setup for clock transfer chain. Furthermore, we perform bi-directional clock transfer experiment, and an optical fiber with 10 km is utilized to simulate the long-baseline propagation of the laser source. To simulate the actual on-orbit conditions, the optical power used for single clock sideband modulation is only 70 pW, and the frequency of laser is modulated with 4sin(2π·(2 × 10−3)·t) MHz Doppler frequency. Experimental results indicate that the GHz frequency synthesizer is the main noise source below 50 mHz frequency range in the inter-spacecraft clock transfer chain. In subsequent experiments, we will customize high-performance frequency synthesizers that meet the requirements of inter-spacecraft clock transfer for TianQin. After removing the noise of the frequency synthesizer, the additional noise of the optical link is below 40 fs/Hz1/2 above 6 mHz frequency range, and the fractional frequency instability is less than 6.7 × 10−17 at the average time of 1000 s, which meet the demand of clock transfer for TianQin. The relative clock jitter noise and frequency difference of the two free-running USOs have been measured, which verifies the coupling characteristics of the clock noise with the internal devices and the ambient environment. The measured differential clock jitter noise is used to eliminate the clock noise in the carrier phase measurement. The residual phase noise is 1 × 10−4 cycles/Hz1/2 at 6 mHz after clock noise correction. The development of clock jitter readout scheme base on clock transfer in this paper is applicable to space-based gravitational wave detection and inter-satellite clock synchronization for autonomous navigation satellites.

Funding

National Key Research and Development Program of China (2020YFC2200200); National Natural Science Foundation of China (11804108, 12105375).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (16)
Fig. 1.
Fig. 1. Schematic TianQin configuration. Each of the three spacecrafts (SC) is equipped with the same optical bench, which is labeled by 1, 2, 3 ($\mathrm{1^{\prime}}$, $\mathrm{2^{\prime}}$, $3^{\prime}$). The three spacecrafts are connected through laser, which forms six laser links. The armlength is denoted by Li, and i is primed or unprimed depending on the direction of the light propagation. Each satellite carries a free-running USO for frequency reference, and its clock noise is q.

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Fig. 2.
Fig. 2. (a) Scheme diagram of inter-spacecraft clock transfer for an interferometric arm in TianQin. The USO provides the frequency reference for phase measurements on each satellite. The USO frequency is scaled up to GHz and drives the EOM to modulate the phase of the laser source. The beatnote signals are measured in a phasemeter whose frequency reference is a USO. (b) The spectrum structure of the two lasers after phase modulation.

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Fig. 3.
Fig. 3. The requirement curve in the clock transfer chain. The red curve is derived based on TDI combination X. The blue curve is described as 77 fs/Hz1/2·NSF. The NSF represents the noise shape function, which is (1 + (6 mHz/f)4)1/2.

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Fig. 4.
Fig. 4. The block diagram of the DPLL. It mainly contains a phase discriminator (PD), a proportional-integral regulator (PIR), a numerically controlled oscillator (NCO), and a decimation filter.

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Fig. 5.
Fig. 5. (a) The designed scheme of the carrier DPLL and the clock sideband DPLLs. The PIR of the carrier DPLL is ±1 MHz offset and fed back to the two sideband DPLLs. The clock sideband DPLLs are used to track the clock sideband beatnote signals. (b) The frequency spectrum of the PD output signal. The sideband beatnote signal has the characteristics of low SNR and its center frequency varies in the ∼MHz range.

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Fig. 6.
Fig. 6. Electrical verification scheme of the phasemeter. The couplers and the power splitter are used to divide and combine the input signals (9 MHz, 10 MHz, 11 MHz) to generate reference signals and three-frequency mixed signals, whose phases are measured through different DPLLs.

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Fig. 7.
Fig. 7. The noise floor measurement results of the multi-channel phasemeter. (a) Differential phase noise floor of clock sideband DPLLs, which is limited by ADC quantization noise at high frequency range. (b) Phase noise floor of carrier DPLL. (c) Requirement of clock transfer. (d) Phase noise floor of single-channel phasemeter, which is only used to measure the phase of single frequency input signal.

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Fig. 8.
Fig. 8. Bi-directional clock transfer experimental setup. The clock signal is transmitted through both the optical link and the electrical link (red dotted line), and the electronic mixing is used to compare with the weak-light clock transfer link.

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Fig. 9.
Fig. 9. The frequency spectrum of the PD output signal. The frequency spectrum includes the carrier beatnote and two clock sideband beatnote signals.

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Fig. 10.
Fig. 10. The noise floor of clock transfer link. (a) Amplitude spectral density of clock jitter noise. Red curve: the clock jitter noise of the weak-light clock transfer link, which contain GHz frequency synthesizer noise and optical path noise. Blue curve: the clock jitter noise of the electrical link, mainly limited to GHz frequency synthesizer noise. Green curve: the difference of clock jitter noise between weak-light and electrical clock transfer link. (b) The fractional frequency instability of the clock transfer link.

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Fig. 11.
Fig. 11. Frequency measurement of the beatnote signal. The center frequencies are 7 MHz, 8 MHz and 9 MHz, respectively. (a) Frequency of the lower clock sideband beatnote. (b) Frequency of the carrier beatnote. (c) Frequency of the upper clock sideband beatnote.

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Fig. 12.
Fig. 12. (a)-(d) The frequency difference between the two USOs. (a) The frequency difference based on weak-light clock transfer. (b) The frequency difference based on electrical clock transfer. (c) The direct measurement of frequency difference between the two USOs (×199.9). (d) The frequency difference between weak-light and electrical clock transfer. (e) The Allan deviation of the fractional frequency difference.

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Fig. 13.
Fig. 13. The measurement results of differential clock jitter noise between two USOs. (a) The clock difference and clock jitter between the USOs are measured by weak-light clock transfer. (b) The clock difference and clock jitter between the USOs are measured by electronic mixing. (c) The difference between weak-light and electrical clock transfer, which has a strong correlation with the ambient temperature.

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Fig. 14.
Fig. 14. Amplitude spectral density of differential clock jitter noise between two free running USOs. Red curve: the clock jitter noise obtained by weak-light clock transfer. Blue curve: the clock jitter noise obtained by electrical clock transfer. Green curve: the difference between the two measurement methods.

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Fig. 15.
Fig. 15. Radio frequency carrier phase measurement. (a) Carrier phase measurement at bench1. (b) Carrier phase measurement at bench2. (c) Carrier phase difference between two benches. (d) The linear relationship between carrier phase difference and measured differential clock jitter.

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Fig. 16.
Fig. 16. Clock noise correction in dual-one-way carrier phase measurement. (a) The carrier phase difference between the two benches. The clock noise is not corrected. (b) The clock noise is corrected using measured differential clock jitter by optical clock transfer. (c) The clock noise is corrected using measured differential clock jitter by electrical clock transfer. (d) The noise floor between two phasemeters. PM: Phasemeter.

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Equations (22)

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$$\begin{array}{c} {E_1}(t) = {E_1}{e^{i[{2\pi f_1^ct + \varphi_1^c(t )} ]+ im\sin [{2\pi f_1^{Mod}t\textrm{ + }{N_1} \cdot \varphi_1^{USO}(t )+ \varphi_1^{FS}(t )+ \varphi_1^{EOM}(t )} ]}}\\ \approx {E_1}{J_0}(m ){e^{i[{2\pi f_1^ct + \varphi_1^c(t )} ]}}\textrm{ + }{E_1}{J_1}(m ){e^{i[{2\pi ({f_1^c + f_1^{Mod}} )t\textrm{ + }{N_1} \cdot \varphi_1^{USO}(t )+ \varphi_1^{FS}(t )+ \varphi_1^{EOM + }(t )+ \varphi_1^c(t )} ]}}\\ - {E_1}{J_1}(m ){e^{i[{2\pi ({f_1^c - f_1^{Mod}} )t - {N_1} \cdot \varphi_1^{USO}(t )- \varphi_1^{FS}(t )- \varphi_1^{EOM - }(t )+ \varphi_1^c(t )} ]}}, \end{array}$$
$$\phi _2^c(t) = 2\pi f_1^c \cdot [{t - {\tau_{12}}(t )} ]+ 2\pi {f_h}t + {\varphi _h}(t) + \varphi _1^c[{t - {\tau_{12}}(t )} ]+ {\varphi _{OPLL}}(t )+ 2\pi \int_0^{t - {\tau _{12}}(t)} {{f_D}(t )dt}. $$
$$\begin{array}{c} {E_2}(t )\approx {E_2}{J_0}(m ){e^{i\phi _2^c(t )}}\textrm{ + }{E_2}{J_1}(m ){e^{i[{2\pi f_2^{Mod}t\textrm{ + }{N_2} \cdot \varphi_2^{USO}(t )+ \varphi_2^{FS}(t) + \varphi_2^{EOM + }(t )+ \phi_2^c(t )} ]}}\\ - {E_2}{J_1}(m ){e^{i[{ - 2\pi f_2^{Mod}t - {N_2} \cdot \varphi_2^{USO}(t )- \varphi_2^{FS}(t )- \varphi_2^{EOM - }(t )+ \phi_2^c(t )} ]}}. \end{array}$$
$$\scalebox{0.9}{$\begin{aligned} &V_{PD1}^{AC} = 2{R_{pd}}{G_{TIA}}\sqrt {{\eta _{het}}{P_{Local}}{P_{rec}}} J_0^2(m )\times \\ &\left\{ \begin{array}{@{}c@{}} \cos ({2\pi f_{beat1}^ct + \varphi_{beat1}^c} )\\ + \dfrac{{J_1^2(m )}}{{J_0^2(m )}}\cos [{2\pi ({f_{beat1}^c + f_2^{Mod} - f_1^{Mod}} )t + {N_2} \cdot \varphi_2^{USO}[{t - {\tau_{21}}(t )} ]- {N_1} \cdot \varphi_1^{USO}(t )+ \varphi_{SB\_beat1}^{noise} + \varphi_{beat1}^c} ]\\ + \dfrac{{J_1^2(m )}}{{J_0^2(m )}}\cos [{2\pi ({f_{beat1}^c - f_2^{Mod} + f_1^{Mod}} )t - {N_2} \cdot \varphi_2^{USO}[{t - {\tau_{21}}(t )} ]+ {N_1} \cdot \varphi_1^{USO}(t )- \varphi_{SB\_beat1}^{noise} + \varphi_{beat1}^c} ]\end{array} \right\} + {N_0}, \end{aligned}$}$$
$$\left\{ \begin{array}{l} f_{beat1}^c(t) = 2{f_D}(t) + {f_h},\\ \varphi_{beat1}^c = \varphi_1^c[{t - {\tau_{12}}(t) - {\tau_{21}}(t)} ]- \varphi_1^c(t) - 2\pi {f_h}{\tau_{21}}(t) - 2\pi (f_1^c + 2{f_D}) \cdot [{{\tau_{12}}(t) + {\tau_{21}}(t)} ]\\ + {\varphi_h}[{t - {\tau_{21}}(t)} ]+ {\varphi_{OPLL}}[{t - {\tau_{21}}(t)} ],\\ \varphi_{SB\_beat1}^{noise} ={-} 2\pi f_2^{Mod}{\tau_{21}}(t) + \varphi_2^{FS}[{t - {\tau_{21}}(t)} ]- \varphi_1^{FS}(t) + \varphi_2^{EOM + }[{t - {\tau_{21}}(t)} ]- \varphi_1^{EOM + }(t). \end{array} \right.$$
$$\scalebox{0.86}{$\begin{aligned} \delta {t_{21}} &= \frac{{{\phi _{lower\_SB}} - {\phi _{upper\_SB}}}}{{4\pi {f^{Mod}}}}\\ &= \frac{{{N_1} \cdot \varphi _1^{USO}(t) - {N_2} \cdot \varphi _2^{USO}(t - {\tau _{21}}(t)) + [\varphi _1^{FS}(t) + \varphi _1^{EOM - }(t)] - [\varphi _2^{FS}(t - {\tau _{21}}(t)) + \varphi _2^{EOM + }(t - {\tau _{21}}(t))]}}{{\textrm{ }2\pi {f^{Mod}}}}\\ &\quad + \frac{{2\pi f_2^{Mod}{\tau _{21}}(t) + {N_0}}}{{\textrm{ }2\pi {f^{Mod}}}}, \end{aligned}$}$$
$$\begin{aligned} {X_1} &= X_1^q - \frac{{{b_{1^{\prime}}}}}{2}[(I - {D_3}{D_{3^{\prime}}})({r_{1^{\prime}}} + {D_{2^{\prime}}}{r_3}) + (I - {D_{2^{\prime}}}{D_2})({r_1} + {D_3}{r_{2^{\prime}}})] + {a_1}[{r_{1^{\prime}}} + {D_{2^{\prime}}}{r_3}]\\ &\quad - {a_{1^{\prime}}}[{r_1} + {D_3}{r_{2^{\prime}}}] + {a_{2^{\prime}}}[{r_{1^{\prime}}} - (I - {D_{2^{\prime}}}{D_2}){r_1} + {D_{2^{\prime}}}{r_3}] - {a_3}[{r_1} - (I - {D_3}{D_{3^{\prime}}}){r_{1^{\prime}}} + {D_3}{r_{2^{\prime}}}]. \end{aligned}$$
$$\begin{aligned} X_1^q &= {b_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}})(I - {D_{2^{\prime}}}{D_2}){q_1} + {a_1}(I - {D_{2^{\prime}}}{D_2}){q_1} - {a_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}}){q_1}\\ &\quad + {a_{2^{\prime}}}{D_3}(I - {D_{2^{\prime}}}{D_2}){q_2} - {a_3}{D_{2^{\prime}}}(I - {D_3}{D_{3^{\prime}}}){q_3}, \end{aligned}$$
$${b_{1^{\prime}}} = {v_1} - {v_{1^{\prime}}}\textrm{ , }{a_1} = {v_{2^{\prime}}} - {v_1}\textrm{ , }{a_{1^{\prime}}} = {v_3} - {v_{1^{\prime}}}\textrm{ , }{a_{2^{\prime}}}\textrm{ = }{v_1} - {v_{2^{\prime}}}\textrm{ , }{a_3} = {v_{1^{\prime}}} - {v_3}, $$
$${r_1} = {q_1} - {D_3}{q_2}\textrm{ + }N_1^c - {D_3}N_2^c, $$
$${q_1} = \frac{{{N_1} \cdot \varphi _1^{USO}(t)}}{{2\pi {f^{Mod}}}},\textrm{ }{q_2} = \frac{{{N_2} \cdot \varphi _2^{USO}(t)}}{{2\pi {f^{Mod}}}},\textrm{ }N_1^c = \frac{{\varphi _1^{FS}(t) + \varphi _1^{EOM - }(t)}}{{2\pi {f^{Mod}}}},\textrm{ }N_2^c = \frac{{\varphi _2^{FS}(t) + \varphi _2^{EOM + }(t)}}{{2\pi {f^{Mod}}}}. $$
$${r_{1^{\prime}}} = {q_1} - {D_{2^{\prime}}}{q_3} + N_1^c - {D_{2^{\prime}}}N_3^c\textrm{, }{r_{2^{\prime}}} = {q_2} - {D_{3^{\prime}}}{q_1} + N_2^c - {D_{3^{\prime}}}N_1^c\textrm{, }{r_3} = {q_3} - {D_2}{q_1} + N_3^c - {D_2}N_1^c. $$
$$\begin{array}{c} X_1^{residual} ={-} [{b_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}})(I - {D_2}{D_{2^{\prime}}}) + {a_1}(I - {D_2}{D_{2^{\prime}}}) - {a_{1^{\prime}}}(I - {D_3}{D_{3^{\prime}}})]N_1^c\\ + {a_{2^{\prime}}}{D_3}(I - {D_2}{D_{2^{\prime}}})N_2^c - {a_3}{D_{2^{\prime}}}(I - {D_3}{D_{3^{\prime}}})N_3^c. \end{array}$$
$$S_{{X_1}}^{residual}(f) = 4{\sin ^2}(2\pi fL)[({b_{1^{\prime}}} + {a_1} - {a_{1^{\prime}}})4{b_{1^{\prime}}}{\sin ^2}(2\pi fL) + {({a_1} - {a_{1^{\prime}}})^2} + {a_{2^{\prime}}}^2 + {a_3}^2]{S_{{N^c}}}(f). $$
$$S_X^{secondary}(f) = [8{\sin ^2}(4\pi fL) + 32{\sin ^2}(2\pi fL)]{S_{acc}}(f) + 16{\sin ^2}(2\pi fL){S_{opt}}(f). $$
$${S_{acc}}(f) = {[\frac{1}{\lambda } \cdot \frac{{{s_a}}}{{{{(2\pi f)}^2}}}]^2} = \frac{{5.7 \times {{10}^{ - 22}}}}{{{f^4}}}\textrm{ }\frac{{\textrm{cycle}{\textrm{s}^2}}}{{\textrm{Hz}}}\textrm{ , }{S_{opt}}(f) = {[\frac{1}{\lambda } \cdot {s_x}]^2} = 8.8 \times {10^{ - \textrm{ }13}}\textrm{ }\frac{{\textrm{cycle}{\textrm{s}^2}}}{{\textrm{Hz}}}, $$
$$\sqrt {{S_{{N^c}}}(f)} \le \sqrt {\frac{{[8{{\sin }^2}(4\pi fL) + 32{{\sin }^2}(2\pi fL)]{S_{acc}}(f) + 16{{\sin }^2}(2\pi fL){S_{opt}}(f)}}{{4{{\sin }^2}(2\pi fL)[({b_{1^{\prime}}} + {a_1} - {a_{1^{\prime}}})4{b_{1^{\prime}}}{{\sin }^2}(2\pi fL) + {{({a_1} - {a_{1^{\prime}}})}^2} + {a_{2^{\prime}}}^2 + {a_3}^2]}}}. $$
$$G(s) = \frac{{{\varphi _{nco}}(s)}}{{{\varphi _e}(s)}} = \frac{{{G_{PD}} \cdot {G_{PA}} \cdot {K_P} \cdot T \cdot s + {G_{PD}} \cdot {G_{PA}} \cdot {K_i}}}{{{s^2}{T^2}}},\textrm{ }H(s) = \frac{{{\varphi _{nco}}(s)}}{{{\varphi _{in}}(s)}} = \frac{{2\zeta {w_n}s + w_n^2}}{{{s^2} + 2\xi {w_n}s + w_n^2}}, $$
$${\left( {\frac{S}{N}} \right)_o} = {\left( {\frac{S}{N}} \right)_{in}} \cdot \frac{{{B_i}}}{{{B_L}}} = {\left( {\frac{S}{{{N_0}}}} \right)_{in}} \cdot \frac{1}{{{B_L}}}, $$
$$U(t) = {U_0}\sin ({2\pi \cdot {f_h} \cdot t} )+ {U_1}\sin ({2\pi \cdot {f_L} \cdot t} )+ {U_1}\sin ({2\pi \cdot {f_U} \cdot t} ). $$
$$\left\{ \begin{array}{l} \phi_1^{carrier}(t) = 2\pi {f_h}t - 2\pi f_1^{carrier}{\tau_{21}} + 2\pi {f_h} \cdot \delta t_1^{USO} + N_1^{noise}\\ \phi_2^{carrier}(t) = 2\pi {f_h}t - 2\pi f_2^{carrier}{\tau_{12}} + 2\pi {f_h} \cdot \delta t_2^{USO} + N_2^{noise} \end{array} \right\},$$
$$\phi _{12}^{carrier}(t) = 2\pi (f_1^{carrier}{\tau _{21}} - f_2^{carrier}{\tau _{12}}) + 2\pi {f_h} \cdot (\delta t_1^{USO} - \delta t_2^{USO}) + {N^{noise}},$$
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