## 2 Principle  
The schematic diagram of the proposed microwave frequency measurement system is shown in Figure 1. The main part in the proposed structure is the LFM signal generation by heterodyning of the monolithic integrated laser array, which consists of four distributed feedback (DFB) lasers and an array waveguide grating, is demonstrated in our reported reference [12]. The laser array is beaten at a photodetector after amplifying via the erbium-doped optical fiber amplifier. As reported in [13], the performance of the laser array can be realized by loading the bias current to ground-signal-ground electrode port as well as direct current (DC) to thermal tuners electrode port. In [12], DFB1 and DFB2 in one integrated transmitter module are used to generate the microwave signal. When a saw-tooth signal is injected to modulate the DC1, DFB1 can realize the wavelength linear scanning. After detection at a photodiode, the LFM signal is generated, of which the duration time and bandwidth are totally determined by the repeat frequency and amplitude of the saw-tooth signal. The frequency-scanning performance of the LFM signal can be obtained by calculating the real-time frequency distribution of the generated LFM signal using short-time Fourier transform (STFT). After down-conversion with an unknown microwave signal received by the antenna, a pair of pulses is recorded in the oscilloscope after passing through the low-pass filter. Figure 1: Experimental setup. CS: current source, SG: signal generator, DFB: distributed feedback laser, BT: bias-tee, BC: bias current; TEC: thermoelectric coolers; AWG: array waveguide grating, EDFA: erbium-doped optical fiber amplifier, PD: photodetector, RA: received antenna, LPF: low-passband filter, OSC: oscilloscope. As shown in Figure 1, the received unknown microwave signal in Figure 2a and the frequency-to-time relationship curve of the generated LFM signal based on the monolithic integrated laser array in Figure 2b are passed into the mixer to generate the frequency down-conversion signal. A portion of down-conversion signal between the received unknown microwave signal \(S_{f_{i}}\) (\(i=1,2\)) and the frequency-scanning signal \(S_{f}\) is filtered by the low-pass filter. Therefore, a pair of optical pulses can be observed at the output of low passband filter shown in Figure 2c. The principle of the proposed microwave signal frequency measurement system based on FTM is shown in Figure 2d when a pair of beat-notes is filtered. The beat-note between the unknown microwave signal and the frequency scanning signal is matched with the passband of the low passband filter. The relationship between the frequency \(f_{i}\) (\(i=1,2\)) of input microwave signal in the frequency domain and the time difference \(\Delta T\) of optical pulses in the time domain is established, which is used to achieve the FTM. For a single-tone microwave signal, when a portion of the beat-note frequency signal is recorded in the oscilloscope, the beat-note frequency is expressed as  
\[f_{\text{beat}}=f-f_{1} \tag{1}\]  
where \(f\) is the frequency of the generated LFM signal. As shown in Figure 2d, a pair of pulses can be observed at the output of the low passband filter in one sweep period, where one pulse is the beat-note signal between the input microwave signal and the positive chirp rate section of the generated LFM signal, the other comes from the beat-note signal between the input microwave signal and the negative chirp rate section of the generated LFM signal. The down-conversion frequency at the output of the low passband filter can be written as  
\[f_{\text{beat}}\big{|}_{t=t_{1}}=\left.f_{\text{beat}}\right|_{t=t_{2}}=f_{LPF} \tag{2}\]  
where \(t_{1}\) and \(t_{2}\) is the exact time when the filtered frequency is measured by the oscilloscope. Therefore, the FTM relationship is established as  
\[f_{1}\propto\Delta T \tag{3}\]  
where \(\Delta T=t_{2}-t_{1}\) is the time difference between double pulses.