## 2 Experimental Setup
\tag{1}\]  
where \(f_{i}(t)\) is the instantaneous frequency (\(i=R\) or \(S\)), in which \(\beta_{2}\) and \(L\) are the group velocity dispersion parameter and length of the DCF, respectively. In addition, \(a_{i}(t)\) is the envelope of the dispersed pulse in the time domain, which is equal to the Fourier transform result of the spectral amplitude that was determined by the responsivity of the OF. It is worth mentioning that Equation (1) is established only when the femtosecond laser pulse width, \(t_{P}\), and dispersion value meet the Fraunhofer condition of \(|i_{P}^{2}/\beta_{2}L|<<1\). Therefore, the output microwave signal from the PD is given by [27]:  
\[I(t)\propto\mathrm{R_{e}}|a_{S}(t)|^{2}+\mathrm{R_{e}}|a_{R}(t)|^{2}+2\mathrm{ R_{e}}|a_{S}(t)||a_{R}(t)|\cos\{2\pi[f_{R}(t)-f_{S}(t)]t\} \tag{2}\]  
where \(\mathrm{R_{e}}\) is the responsivity of the PD. Additionally, \(A_{S}(t)=kA_{R}(t-\Delta\tau)\), where \(k\) is the amplitude ratio of \(A_{S}\) to \(A_{R}\). In addition, assuming that the transmission curve of the OF is a standard rectangular profile in the optical frequency domain, and considering that the maximum time delay of the two laser pulse edges was 0.6 ns, which is much less than the stretched laser pulse width of 24 ns, it is considered that the time variation of \(a_{i}(t)\) hardly affects the result of \(I(t)\).Then, Equation (2) was simplified as follows:  
\[I(t)\propto\mathrm{R_{e}}{a_{R}}^{2}+\mathrm{R_{e}}{a_{R}}^{2}V\cos[2\pi( \Delta t/\beta_{2}L)t] \tag{3}\]  
where \(V=2k/\left(1+k^{2}\right)\) is the visibility of the temporal interferogram. In addition, the first term on the right-hand side can be filtered by the EF in our experiment. Thus, according to Equation (3), the maximum amplitude of the generated microwave signal depends on the power of the input laser and the responsivity of the PD. The value of \(V\) should approximate to 1 in order to make the generated microwave waveform have a good fringe contrast. In our experiment, since the efficiency of the PC pin receiving energy from the reflected laser, \(A_{R}\), was about 0.1%, the PC pin was coated with a 30 dB reflectivity for 1550 nm to balance the signal intensity between \(A_{R}\) and \(A_{S}\). Additionally, the frequency of the generated microwave signal, \(f_{I}=\Delta\tau/\beta_{2}L\), is independent of time but is determined by the parameters of the GDS and DCF. Specifically, the dispersion constant \(D\) of the DCF is equal to \(-c\beta_{2}/\lambda^{2}\), which is related to the group velocity dispersion parameter \(\beta_{2}\) and central frequency \(\lambda\) of the femtosecond laser. In this situation, \(D\) and \(L\) should be considered as constants. In addition, group delay time \(\Delta\tau\) (\(\Delta\tau=2d/c\)) was determined by the optical path length difference, \(d\), in the GDS. Hence, the frequency of the generated microwave signal is further expressed as:  
\[f_{I}=-2d/DL\lambda^{2} \tag{4}\]  
In our experiments, the dispersion constant, relative dispersion slope and length of the DCF were fixed at \(-190\) ps\(/\)(nm\(\cdot\)km), 0.0028 nm\({}^{-1}\) and 20 km, respectively. According to Equation (4), it is clear that the frequency of the microwave pulse is proportional to the optical path length difference. Therefore, frequency tuning can easily be achieved by adjusting optical path length difference, \(d\), in the GDS. Furthermore, the calculated result of the conversion constant between \(f_{I}\) and \(d\) is 0.219 GHz/mm, which will be compared with the experimental results below.