## 2 Principles
### Schematic of the Multi-Band Phased Array Radar  
The structure of a multi-band phased array radar based on the proposed method is shown in Figure 2. According to Equation (3), the output signal contains the frequency components of optical pulses. At the transmitter, differential detection is used to eliminate these components and generate pure multi-band signals. A mode-locked laser (MLL1) produces optical pulses with the repetition frequency of \(f_{s1}\), which are provided to all transmitting elements. The baseband LFM signal with a center frequency, \(f_{s1}/4\), generatedby an arbitrary waveform generator (AWG) is phase-shifted using a phase shifter (PS). A variable optical delay line (VODL) introduces a delay to the optical pulses. The delayed optical pulses are then split into two paths. In the upper path, a Mach-Zehnder modulator (MZM) modulates the baseband signal onto the optical pulses, converting baseband signals into multiple frequency bands. Meanwhile, a variable optical attenuator (VOA) controls the power of optical pulses in the lower path to match the power of the upper path. A BPD performs photoelectric conversion and differential detection of the signals from both paths. The transmitted signal can be expressed as:  
\[ \begin{split} S_{t}(t)&=\bigg{\{}T_{M}[s_{0}(t)] \times u_{1}(t+\Delta t_{1})-\frac{1}{2}u_{1}(t+\Delta t_{1})\bigg{\}}*h_{oe}(t )*h_{BPF}(t)\\ &=A\sum\limits_{n=0}^{k_{0}}\cos(2\pi(n\pm\frac{1}{4})f_{s1}t\pm \pi\frac{B_{0}}{T_{0}}t^{2}+2\pi nf_{s1}\Delta t_{1}\pm\varphi_{1})\end{split} . \tag{8}\]  
After differential detection, the output of the BPD is a pure multi-band signal. After band selection by BPF, multi-band signals are amplified and transmitted by the array antenna. At the receiver, MLL2 generates optical pulses with the repetition frequency of \(f_{s2}\). To increase both the number of bands and the bandwidth of each band in the multi-band phased array radar, channel interleaving is employed to expand the instantaneous bandwidth at the receiver [29; 30; 31]. A wavelength division multiplexer (WDM) splits the optical pulses into two channels with different central wavelengths. Optical pulses in the first channel are introduced by a fixed delay of \(1/(2f_{s2})\). The optical pulses from two channels with different central wavelengths are recombined using WDM. This results in optical pulses with doubled repetition frequency, which are then provided to all receiving elements. A VODL introduces a delay in the optical pulses. The BPF filters the multi-band echo signal received by the antenna. The MZM modulates the amplified multi-band signal onto the optical pulses and down-converts it. The optical pulses are then split into two channels again using the same WDM, with the first channel delayed by \(1/(2f_{s2})\) by an  
Figure 2: Architecture of a multi-band phased array radar. MLL, mode-locked laser; AWG, arbitrary waveform generator; MZM, Mach–Zehnder modulator; VOA, variable optical attenuator; PD, photodetector; BPD, balanced photodetector; VODL, variable optical delay line; ODL, optical delay line. optical delay line (ODL). The outputs from the two channels are converted into microwave signals by photodetectors. These microwave signals are phase-shifted by equal amounts through two PSs and then low-pass filtered. The output signals, \(S_{m\_cH1}(t)\) and \(S_{m\_cH2}(t)\), from the two channels in the \(m\)-th receiving element can be expressed as:  
\[ \begin{array}{ll}S_{m\_cH1}(t)=A\sum\limits_{k=0}^{k_{0}}\cos(2\pi(n_{k}f_{s2 }-\frac{2k+1}{4}f_{s1})(t-\frac{1}{2f_{s2}})-\pi\frac{B_{0}}{T_{0}}(t-\frac{1}{ 2f_{s2}})^{2}+2\pi n_{k}f_{s2}\Delta t_{2}+\varphi_{2})\\ S_{m\_cH2}(t)=A\sum\limits_{k=0}^{k_{0}}\cos(2\pi(n_{k}f_{s2}-\frac{2k+1}{4}f_{ s1})t-\pi\frac{B_{0}}{T_{0}}t^{2}+2\pi n_{k}f_{s2}\Delta t_{2}+\varphi_{2}-k\pi) \end{array}  \tag{9}\]  
It is evident that the phase changes induced by VODL and PS remain unchanged after channel interleaving; thus, beamforming is not affected.