## 2 Principles
### Transmitting  
The proposed method enables multi-band signal generation and beamforming through joint phase-frequency distribution manipulation, as illustrated in Figure 1b,c. The optical pulses with a repetition frequency of \(f_{s1}\) can be expressed as:  
\[u_{1}(t)=\sum\limits_{l=-\infty}^{+\infty}\delta(t-lT_{s1}+\Delta t_{1})\approx \sum\limits_{n=0}^{+\infty}\cos(2\pi nf_{s1}(t+\Delta t_{1})), \tag{1}\]  
where \(T_{s1}\) is the time interval of optical pulses and \(\Delta t_{1}\) is the introduced delay. \(n\) refers to the frequency component of the optical pulses, which is served as multiple LOs. The baseband linear frequency modulated (LFM) signal after electronic phase-shifting can be expressed as:  
\[S_{0}(t)=\cos(2\pi f_{0}t+\pi\frac{B_{0}}{T_{0}}t^{2}+\varphi_{1}), \tag{2}\]  
where \(f_{0}\), \(B_{0}\), and \(T_{0}\) are the center frequency, bandwidth, and duration of the baseband LFM signal, respectively. \(\varphi_{1}\) is the phase introduced by the electronic phase shifter. An intensity modulator modulates this signal onto the optical pulses. The photodiode then converts the optical outputs into microwave signals. After bandpass filtering, ignoring the amplitude coefficient, the output signal can be represented as:  
\[\begin{array}{ll}S_{1}(t)&=\{T_{M}[S_{0}(t)]\times u_{1}(t+\Delta t_{1})\}*h _{oe}(t)*h_{BPF}(t)\\ &\approx\left\{\frac{1}{2}[1+\alpha\cos(2\pi f_{0}t+\pi\frac{B_{0}}{T_{0}}t^{2 }+\varphi_{1})]\sum\limits_{n=0}^{+\infty}\cos(2\pi nf_{s1}(t+\Delta t_{1})) \right\}*h_{oe}(t)*h_{BPF}(t)\\ &=A\sum\limits_{n=0}^{k_{0}}\cos(2\pi(nf_{s1}\pm f_{0})t\pm\pi\frac{B_{0}}{T_{ 0}}t^{2}+2\pi nf_{s1}\Delta t_{1}\pm\varphi_{1})+\frac{1}{2}u_{1}(t+\Delta t_{ 1})*h_{oe}(t)*h_{BPF}(t)\\ &\end{array}, \tag{3}\]  
where \(T_{M}\) is the transfer function of the intensity modulator and \(\alpha\) is a coefficient related to the modulation depth. \(A\) is the amplitude coefficient. The harmonics generated by the modulator nonlinearity are neglected. \(h_{oe}\) and \(h_{BPF}\) are the impulse responses of the photodiode and band pass filter (BPF), respectively. \(k_{0}\) is a coefficient determined by the bandwidth of the BPF, which represents the total band number of the multi-band signal.  
Equation (3) demonstrates that introducing delays, \(\Delta t_{1}\), in optical pulses imparts distinct phases, \(2\pi nf_{s1}\Delta t_{1}\), to signals of different frequency bands, \(nf_{s1}\pm f_{0}\), which are generated through mixing the baseband signal with various optical pulse frequency components, \(nf_{s1}\). This tightly couples time, frequency, and phase, enabling simultaneous multi-band signal generation and beamforming. The output signal, \(S_{1}(t)\), contains multiple frequency bands. Multi-band beamforming can be achieved by adjusting the delay, \(\Delta t_{1}\), and electronic phase shift, \(\varphi_{1}\), as shown in Figure 1c. For any aperture fill time, \(t_{d1}\), ideal multi-band beamforming requires the phase compensation for all bands, which means \(2\pi nf_{s1}\Delta t_{1}\pm\varphi_{1}=2\pi(nf_{s1}\pm f_{0})t_{d1}+2\pi\) (where \(z\) is an integer). For \(n\) taking any natural number, the above equation has the following solution:  
\[\Delta t_{1}=t_{d1}-\frac{[f_{s1}t_{d1}]}{f_{s1}};\varphi_{1}=2\pi f_{0}t_{d1 }-2\pi[f_{0}t_{d1}]. \tag{4}\]  
Multi-band beamforming can be achieved by realizing the correct phase adjustment of the multi-band signal. It is important to note that the max adjustment range of \(\Delta t_{1}\) and \(\varphi_{1}\) is \(1/f_{s1}\) and \(2\pi\), respectively.