## 2 Photonic Microwave Distance Interferometer  
### Interferometer Configuration  
Figure 1 describes the distance interferometer devised in this investigation. The light source is an Er-doped fiber mode-locked laser emitting 120 fs short pulses at a repetition rate (\(f_{r}\)) of 100 MHz. When the laser pulses are observed using a photodetector over a time period, the resulting electrical signal exhibits an RF comb structure in the frequency domain. As illustrated in Figure 1a, the resulting RF comb is in fact a down-converted version of the optical comb of the mode-locked source. Each RF comb mode element represents an integer-multiple harmonic frequency of the pulse repetition rate \(f_{r}\), which can be isolated from other modes through an electrical bandpass filter. An N-th harmonic frequency has the wavelength (\(\Lambda_{\text{N}}\)), expressed as \(\Lambda_{\text{N}}=\frac{c}{\text{N}f_{r}}\), with \(c\) being the speed of light in a vacuum. Lower harmonic frequencies, i.e., longer wavelengths, are preferable as the interferometer source to enlarge the non-ambiguity range of distance measurement, whereas higher harmonic frequencies offer finer measurement resolutions.  
The source beam is launched into free space and divided into the reference beam (green) and measurement beam (blue) with a 50:50 beam splitter as shown in Figure 1b. Both the reference beam and measurement beam are converted into electrical signals and mixed with a local oscillator signal (1 GHz-100 kHz) stabilized to the Rb atomic clock by phase-locked loop (PLL) control as shown in Figure 1c. The reference beam directly goes to a reference photodetector, while the measurement beam reaches a measurement photodetector after reflecting from a target mirror installed on a motorized stage with a 2.5 m travel range. In this study, the 10 th harmonic of 1 GHz is selected to be used as the microwave source. The phase delay (\(\theta_{\text{N}}\)) of the 1 GHz source microwave signal is measured by down-shifting to a 100 kHz signal using an electrical local oscillator set to operate near 1 GHz. The target distance (\(L\)) is then determined as:\[L=\frac{\Lambda_{\rm N}}{2n_{\rm air}}(m_{\rm N}+\theta_{\rm N}) \tag{1}\]  
where \(n_{\rm air}\) denotes the refractive index of air and \(m_{\rm N}\) is an integer. The phase delay \(\theta_{\rm N}\) between the measurement and measurement paths is measured with respect to the particular photonic wavelength \(\Lambda_{\rm N}\). Finally, the super-heterodyned electrical signals of 100 kHz are processed through a lock-in amplifier that is used as a phase meter to determine the interferometric phase delay \(\theta_{\rm N}\). The phase measurement resolution is found to be 0.01\({}^{\circ}\) with an accuracy of 0.02\({}^{\circ}\), from which the measurement precision is estimated to be 4.2 \(\upmu\)m.  
Figure 1: Photonic microwave distance interferometer. (**a**) Optical spectrum of a mode-locked laser and its radio frequency (RF) synthetic wavelengths. (**b**) Optical configuration for absolute distance measurement. (**c**) RF electrical circuit for interferometric phase detection. BM: balanced mixer, BS: beam splitter, c: speed of light in air, CC: corner cube, HPF: high pass filter, L: lens, LPF: low pass filter, PC: personal computer, PD: photodetector, _ref_: reference, _ma_: measurement, \(\Delta f\): beat frequency, \(f_{r}\): pulse repetition rate, N/\(f_{r}\): N-th harmonics of \(f_{r}\), \(\Lambda\): synthetic wavelength, \(\theta\): interferometric phase.