## 3.1 Innovations

To enhance the clock synchronization performance for the core level of telecom networks, we carry out frequency signal stable transmission through a branching optical fiber, where the performance of the GSNN-based synchronization scheme is limited. After the frequency signal is transmitted through the optical fiber, the phase of the signal is jittered due to the transmission delay variation, i.e., \(\delta \varphi =\omega \cdot \delta \tau\). If the transmission frequency \(\omega\) is increased, it is equivalent to providing a higher gain for the measurement of phase fluctuations. In addition, the increase in gain by increasing the transmission frequency can in turn weaken the influence of other noises, such as laser phase noise, and indirectly reduce the system's requirements for other noises, thereby increasing the sensitivity of signal phase detection.

Phase detection and phase correction are two key technologies to realize stable signal transmission. However, the traditional methods based on intensity modulation and direct detection have low sensitivity, and the frequency supported by the phase detector is limited to tens of GHz, which cannot meet the phase detection requirements of high-frequency signals. On the other hand, high-speed photoelectric conversion can also cause amplitude-phase conversion noise to deteriorate the accuracy of detection. To solve this problem, we proposed a dual-heterodyning phase error transfer (DHPT) scheme to detect the phase error of the millimeter-wave signal induced by the fiber delay variation and applied an acousto-optic frequency shifter (AOFS) to cancel the phase noise, respectively [20]. The terminal user only needs to convert the received high-frequency signal into the required frequency range through down-conversion and other means. Furthermore, the theoretical analysis reveals the relationship between the system instability and the frequency of the transmitted signal, which testifies to the potential high stability obtained thanks to the higher frequencies of the transmitted signals [21].

In this article, we present the simultaneous dissemination of the terahertz signals to multiple independent remote sites on a branching optical-fiber network. The transmitted terahertz wave signal over the fiber link is obtained by extracting two optical carriers from an optical frequency comb. The phase fluctuation due to the optical carrier separation link and fiber link is compensated by a feedback network, which includes a phase-locked loop (PLL) and a fast response AOFS. The phase noise within the loop bandwidth is effectively suppressed, thus, the high phase-stable terahertz signal is achieved at the remote end. The results obtained show that the frequency transmission based on optical fiber can achieve high precision clock synchronization, which can be used for accurate clock synchronization for the core level of telecom networks. Besides, the factors that cause the performance limitations of the photonic terahertz signal distribution system are further analyzed.

## 3.2 The proposed stable terahertz signal distribution system

In this section, we will introduce the proposed terahertz signal stable distribution on branching optical-fiber networks in detail. The schematic diagram of the terahertz wave distribution on branching optical-fiber networks is shown in Fig. 2. An optical frequency comb generator (OFCG) based on a Fabry–Perot electro-optic modulator with 2.5 GHz free spectral range is driven by a 25 GHz microwave synthesizer, which produces a low phase noise OFC with a 25 GHz frequency interval and more than a 10 THz spectral span. The microwave synthesizer is synchronized to a Rubidium reference. The rubidium clock is synchronized with the standard time scale assigned by UTC.

The OFC is divided into three branches by passing through polarization-maintained couplers (PMCs). The two branches are the reference for detecting the phase error induced by the optical carrier separation link and transmission fiber link. The last branch is frequency shifted by AOFS1, which is used to obtain two phase-locked optical carriers with terahertz frequency spacing. Then the OFC shifted by AOFS1 is subdivided into three paths. Each optical carrier that generates a terahertz signal can be filtered out by an optical filter (OF), or replaced by a polarization-maintaining arrayed waveguide grating. To avoid homodyne mixing in the phase measurement of the two terahertz signals, the two optical carriers are shifted to different frequencies by AOFS2 and AOFS3, respectively.

The selected optical carriers from the OFC shifted by AOFS1 can be written as,

$${E_a}\left( t \right)=\exp \left\{ {j\left[ {\left( {{\omega _1}+{\omega _{IF1}}} \right)t+{\varphi _1}} \right]} \right\}$$

1

$${E_b}\left( t \right)=\exp \left\{ {j\left[ {\left( {{\omega _2}+{\omega _{IF1}}+{\omega _{IF2}}} \right)t+{\varphi _2}+{\varphi _{IF2}}\left( t \right)} \right]} \right\}$$

2

$${E_c}\left( t \right)=\exp \left\{ {j\left[ {\left( {{\omega _2}+{\omega _{IF1}}+{\omega _{IF3}}} \right)t+{\varphi _2}+{\varphi _{IF3}}\left( t \right)} \right]} \right\}$$

3

where \({\omega _1}\) is the selected optical carriers’ angular frequency, \({\varphi _1}\) is the initial phase. \({\omega _{IF1}}\) is the angular frequency of the AOFS1 drive signal. The AOFS1 is driven by the rubidium reference clock. The \({\omega _2}\) is selected optical carriers’ angular frequency, \({\varphi _2}\) is the initial phase. \({\omega _{IF2}}\)and \({\varphi _{IF2}}\left( t \right)\)are the angular frequency of the AOFS2 drive signal and its phase, respectively. \({\omega _{IF3}}\)and \({\varphi _{IF3}}\left( t \right)\)are the angular frequency of the AOFS3 drive signal and its phase, respectively. Both AOFS2 and AOFS3 are driven by the VCO and are used as part of the PLL to compensate for the phase fluctuations induced by the optical path. Since signal amplitude has a limited impact on the system, it is omitted for the sake of simplicity.

Since the frequency distribution of the proposed terahertz signal in the branch optical fiber network is two relatively independent systems, the working principle can be introduced by taking one of them as an example. The photonic terahertz signal is formed by coupling the filtered optical carriers through two different separate links and then transmitted to the remote via a fiber optic link. At the remote nodes, the photonic terahertz signal passing through the fiber link can be expressed as,

$${E_{THz}}\left( t \right)=\cos \left\{ {\left( {{\omega _2} - {\omega _1}+{\omega _{IF2}}} \right)\left( t \right)+{\varphi _{IF2}}\left( t \right) - {\varphi _v}\left( t \right)} \right\}$$

4

where \({\varphi _v}\left( t \right)=({\omega _2}+{\omega _{IF1}}+{\omega _{IF2}}){\tau _b}(t) - ({\omega _1}+{\omega _{IF1}}){\tau _a}(t)+({\omega _2} - {\omega _1}+{\omega _{IF2}}){\tau _{link}}\left( t \right)\). The\({\tau _{link}}\left( t \right)\)is the delay change of the transmitted optical fiber link due to external temperature and pressure. The\({\tau _a}\left( t \right)\)and\({\tau _b}\left( t \right)\) are time-varying transmission delays of the optical carriers due to the different separated paths. Therefore, it is necessary to compensate for the phase fluctuations caused by the optical fiber link and the carrier separation link to achieve stable transmission of the photonic terahertz signals.

In this paper, the round-trip correction mechanism is adopted for phase compensation. Then the remote terahertz signal is power split into two branches by PMC. One is down-converted after being converted by the photo-detector (PD) and provided to the user, the other is frequency shifted by an AOFS4 to avoid the Rayleigh backscattering and transmitted back to the local end through the same fiber link. Since the fiber delay changes slowly, the forward transmission time and the backward transmission time are the same. Then the returned optical carriers exhibit double the one-way fiber-induced phase noise. Based on the proposed DHPT scheme, the phase fluctuation induced by the separated path and the optical fiber transmission delay variations is mapped onto an intermediate frequency (IF) signal \({E_{IF}}(t)\),

\({E_{IF}}\left( t \right)=\cos \left\{ {\left( {2{\omega _{IF2}} - {\omega _{Rb}}} \right)t+2{\varphi _{IF2}}\left( t \right) - 2{\varphi _v}\left( t \right) - {\varphi _{Rb}}} \right\}\) (5)

where, \({\omega _{Rb}}\) is the angular frequency of the rubidium oscillator, and\({\varphi _{Rb}}\)is its initial phase which is considered a constant. It should be noted that the phase of the IF\({E_{IF}}(t)\) signal and the phase of the remote terahertz signal \({E_{THz}}\left( t \right)\) are coherent.

Based on the phase-locked loop theory [22], the \({E_{IF}}(t)\) signal is discriminated by a digital phase and frequency detector compared with the rubidium oscillator. Then the error signal is integrated into a loop filter to control the phase of VCO. When the loop is locked, the steady-state error is zero. Then the locked remote node's terahertz wave signal can be expressed as,

$${E^{\prime}_{THz}}\left( t \right)=\cos \left[ {\left( {{\omega _2} - {\omega _1}+{\omega _{IF2}}} \right)t+N{\varphi _{Rb}}} \right]$$

6

where the number is determined by the frequency of the AOFS2 drive signal and the frequency of the rubidium clock reference. It can be seen that \({E^{\prime}_{THz}}(t)\)is independent of the phase fluctuation induced by the separated path and the optical fiber transmission delay variations. Thus, a high phase-stable terahertz signal is obtained at the remote core-level nodes.