## 2 Scheme of Multiple-Access Time and Frequency Transfer
Sites B and C modulate the frequency signals to the lasers through the electro-optic modulator (EOM) and then enter the optical link after passing through the circulator. After the modulated laser signal reaches master site A, it is sent to the photodetector PD2 through the circulator. The optical signal output by PD2 obtains the electrical signal with the frequencies of \(F_{\text{rep}}\) and \(2F_{\text{rep}}\) transmitted from slave sites B and C. At the same time, slave sites B and C will output the optical pulse signal sent from master site A to photodetectors PD3 and PD4, respectively, through the circulators and use the filter to extract the pure electrical signal with the frequencies of \(F_{\text{rep}}\) and \(2F_{\text{rep}}\). Next, the phase comparison of the signals received by each site is performed to obtain the two-way difference value so as to realize the two-way time and frequency transfer between master site A and slave site B and between master site A and slave site C.  
At master site A, the signal output by PD1 can be filtered to obtain the local electrical signals with frequencies of \(F_{\text{rep}}\) and \(2F_{\text{rep}}\)  
differences \(\varphi_{\rm CA}\) between the local time-frequency reference signals of slave station C and the time-frequency signal sent from master station A. The timing diagram of the multiple-access time and frequency transfer is shown in Figure 2. The calculation of the two-way time-frequency comparison of the three sites is performed as follows. For transfer between sites A and B, we assume that the phase difference between sites A and B is \(\Delta\varphi_{\rm AB}\). \(\tau_{\rm AB}\) and \(\tau_{\rm BA}\) are the timing delays of the laser beams from site A to B and site B to A in the fiber and freespace link, respectively. \(\tau_{\rm TA}\) and \(\tau_{\rm RA}\) are the fixed transmitting timing delay and receiving timing delay caused by the transmission link on site A, respectively. \(\tau_{\rm TB}\) and \(\tau_{\rm RB}\) are the fixed transmitting timing delay and receiving timing delay caused by the transmission link on site B, respectively. Similarly, for transfer between sites A and C, we assume that the phase difference between sites A and C is \(\Delta\varphi_{\rm AC}\). \(\tau_{\rm AC}\) and \(\tau_{\rm CA}\) are the timing delays of the laser beams from site A to C and site C to A in the fiber and free-space link, respectively. \(\tau_{\rm TC}\) and \(\tau_{\rm RC}\) are the fixed transmitting timing delay and receiving timing delay caused by the transmission link on site C, respectively. Considering the relations of the abovementioned timing differences, for sites A and B, \(\varphi_{\rm AB}\) and \(\varphi_{\rm BA}\) are written as follows. \[\boldsymbol{\varphi}_{\rm AB}=\boldsymbol{\tau}_{\rm TB}+\boldsymbol{\tau}_{ \rm BA}+\boldsymbol{\tau}_{\rm RA}+\Delta\boldsymbol{\varphi}_{\rm AB} \tag{1}\]  
\[\boldsymbol{\varphi}_{\rm BA}=\boldsymbol{\tau}_{\rm TA}+\boldsymbol{\tau}_{ \rm AB}+\boldsymbol{\tau}_{\rm RB}-\Delta\boldsymbol{\varphi}_{\rm AB} \tag{2}\]  
Based on Equations (1) and (2), the relative phase difference between sites A and B can be calculated by  
\[\Delta\boldsymbol{\varphi}_{\rm AB}=\frac{\boldsymbol{1}}{2}[(\boldsymbol{ \varphi}_{\rm AB}-\boldsymbol{\varphi}_{\rm BA})-(\boldsymbol{\tau}_{\rm BA}- \boldsymbol{\tau}_{\rm AB})-(\boldsymbol{\tau}_{\rm TB}-\boldsymbol{\tau}_{\rm RB })-(\boldsymbol{\tau}_{\rm RA}-\boldsymbol{\tau}_{\rm TA})] \tag{3}\]  
where \(\varphi_{\rm AB}\) and \(\varphi_{\rm BA}\) can be measured by a phase discriminator. \(\tau_{\rm AB}\) and \(\tau_{\rm BA}\) are identical as the two-way links are reciprocal. The other two fixed terms, \((\tau_{\rm TB}-\tau_{\rm RB})\) and \((\tau_{\rm RA}-\tau_{\rm TA})\), can be calibrated by a high-precision counter. Finally, \(\Delta\varphi_{\rm AB}\) can be calculated to determine the real timing difference between sites A and B. For sites A and C, the calculation is almost the same. \(\varphi_{\rm AC}\) and \(\varphi_{\rm CA}\) are written as follows.