## 3 Optical Controlled Phased Sub-Array in Multi-Hop Bent-Pipe Payload
### Optically Controlled Beam Steering  
The overall phase-shifting network structure is a two-layer phase regulation, with the first layer consisting of \(M\) AWGRs, which introduce OTTDs for \(M\) subarrays. Each AWGR can generate a different true time delay controlled by the wavelength. The second layer consists of \((M-1)m\) digital phase shifters, who provide the exact phase shift. The time difference between two adjacent array elements is \(\tau=dsin\theta/c\), where \(d\) is the distance between two antenna array elements, \(c\) is the light speed, and \(\theta\) represents the beam angle. The phase difference is \(\varphi=2\pi dsin\theta/\lambda_{0}\), where \(\lambda_{0}\) is the center frequency of the signal. For example, when the beam is pointing at \(\theta_{B}\), the time delay and phase difference can be expressed as \(\tau_{B}=dsin\theta_{B}/c\) and \(\varphi_{B}=2\pi dsin\theta_{B}/\lambda_{0}\), respectively. Based on beamforming theory [41], the far field pattern (FFP) of such an array could be expressed as:  
\[AF(\theta)=\sum_{i=0}^{M-1}\sum_{l=0}^{m-1}[exp(-j((im+l)kdsin(\theta))(\alpha _{\lambda_{0}}(i)+\alpha_{B}(l)))] \tag{1}\]  
Every subarray element phase difference of the digital phase shifter can be expressed as:  
\[\alpha_{B}(l)=[0,\varphi_{B},2\varphi_{B},\ldots,(m-1)\varphi_{B}],l=0,1,2, \ldots,m \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Example** & **No. Beam** & **Angle** & **Beam Width** & **Antenna Type** \\ \hline Starlink & 16 & 56.55\({}^{\circ}\) & 1.3\({}^{\circ}\) & Phase array \\ OneWeb & 16 & 40\({}^{\circ}\) & TX:3\({}^{\circ}\); RX:2\({}^{\circ}\) & Phase array \\ Telesat & 16 & 45\({}^{\circ}\) & 1.55\({}^{\circ}\) & Phase array \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Three large constellation beam coverage designs. Figure 4: Principle of AWGR Assisted Mm-wave Beam-hopping System. To be more compatible with satellite communications, the delay design introduced by the AWGR for each sub-array needs to be predefined. For a optical carrier, wavelength \(\lambda_{0}\), the \(i^{th}\) subarray true time delay introduced by AWGR is:  
\[\alpha_{\lambda_{0}}(i)=2\pi c[0,m\tau_{B},2m\tau_{B}\dots,\tau_{B}(M-1)m]/ \lambda,i=0,1,2,\dots M \tag{3}\]  
Such a delay eliminates the delay difference between sub-arrays. In addition, since the delay time of the AWGR is fixed for a specific wavelength, WDM can be converted to space division multiplexing in optical network switching by this structure. And according to our previous research, AWGR allows for delay control of vector synthesis [42]. That is, the delay introduced by \(\lambda_{0}\) can be achieved by simultaneously loading the RF signal into \(\lambda_{1}\) and \(\lambda_{2}\), and the combination of vector synthesis of delay quantities allows for a more refined OTTD. The scanning angle of the beam is determined by the amount of phase shift of the array. The minimum phase shift of the digital phase shifter is indicated by \(2\pi/2^{K}\). The angle at which the antenna beam is pointed can therefore be expressed as \(p\) digital phase shifts and \(q\) true time delays  
\[\varphi=\frac{2\pi dsin\theta}{\lambda}=p\times 2\pi/2^{K}+q\times 2\pi\tau_{ min}c/\lambda_{0} \tag{4}\]  
Using a differential solution, the minimum beam accuracy can be obtained  
\[\frac{d\varphi}{d\theta}=2\pi dcos\theta/\lambda \tag{5}\]  
\[\Delta\theta=\frac{\Delta\varphi\lambda}{2\pi dcos\theta}=\frac{\lambda}{2^{K} dcos\theta} \tag{6}\]  
The wavelength does not need to be adjusted when fine-tuning the scanning angle, so the phase delay caused by the true time delay generated by AWGR can be ignored when performing high-precision scanning.