## 2 Methods and Experiments  
A schematic of the OFC heterodyne interferometer used in this study is provided in Figure 1. The light source (OFC) was a homemade mode-locked erbium-doped fiber femtosecond laser. The central wavelength of the OFC was 1560 nm, the full width at half maximum (FWHM) of the spectrum was 55 nm, the output power was approximately 8 mW, and the repetition frequency was stabilized to a frequency synthesizer (78 MHz, 33250A, Agilent, Santa Clara, CA, USA), which was referenced to an atomic clock (5071A, Symmetricom, San Jose, CA, USA). The output laser beam from the OFC passed through an acousto-optic modulator (AOM, MGAS80-A1, AA Opto Electronic, Orsay, France) driven by a sinusoidal signal at a constant frequency \(f_{\text{AOM}}\) = 80 MHz. After the AOM, the zero-order beam travels along the original direction with the original frequency, while the first-order beam spreads in another direction due to optical diffraction, and its frequency is shifted by \(f_{\text{AOM}}\). The zero-order beam travels through a beam splitter (BS) and arrives at mirror M\({}_{3}\), then reflects back to BS and overlaps with the first-order beam which is adjusted by mirror M\({}_{2}\). Two beams were coupled into an optical fiber together by a collimating lens (CL\({}_{2}\)). The optical lengths of the two beams were set to be equal. The optical spectra of the two beams were then individually measured by an optical spectrum analyzer (OSA, AQ6370C, Yokogawa Electric, Musashino, Tokyo, Japan). The first-order beam is referred to as the reference arm, while the zero-order beam is called the measurement arm. Figure 2a illustrates the optical spectrum of the two arms. Note that the optical spectra of the two beams are different, due to the diffraction of the AOM. Figure 1: Schematic of the optical frequency comb (OFC) heterodyne interference system. Gray lines: optical fibers; red and green lines: optical paths in free space; black lines: electrical connections. OFC: optical frequency comb; CL: collimating lens; M: mirror; AOM: acoustic optical modulator; BS: beam splitter; DM: dichroic mirror; PZT: piezo-electric transducer; OSA: optical spectrum analyzer; PD: photodetector; BPF: bandpass filter; He-Ne: commercial interferometer; PC: computer. Because the frequency of the first-order beam was shifted by \(f_{\text{AOM}}\), a heterodyne interference signal was detected by the photodetector (PD, Model 1811, New Focus, CA, USA), and the frequency of the heterodyne interference signal was \(f_{\text{beat}}\) = \(f_{\text{AOM}}\) - \(f_{\text{rep}}\) = 2 MHz. The beat signal was extracted by a bandpass filter and then sent into a lock-in amplifier (SR844, Stanford Research System, Sunnyvale, CA, USA) to measure the phase compared with the reference signal. The beam of the commercial interferometer overlapped with the beam of OFC at the dichroic mirror (DM). The target mirror M\({}_{3}\) was driven by a piezo-electric transducer (PZT). The OFC heterodyne interferometer and the commercial interferometer were both used to measure the displacement of M\({}_{3}\) simultaneously, see Section 3 for details. According to the comparison results, the effective center wavelength of the OFC heterodyne interference signal can then be estimated. In the following section, a theoretical method to calculate the effective center wavelength of the OFC heterodyne interferometer was introduced. The complex amplitude of the electric field of two beams can be expressed as:  
\[E_{1}(t)=\sum_{m}A_{1}e^{i[2\pi f_{m}(t-t_{1})+\beta_{m}]}, \tag{2}\]  
\[E_{2}(t)=\sum_{m}A_{2}e^{i[2\pi(f_{m}+f_{beat})(t-t_{2})+\beta_{m}]}, \tag{3}\]  
where \(A_{1}(f_{m})\propto\sqrt{P_{1}(f_{m})}\), \(A_{2}(f_{m})\propto\sqrt{P_{2}(f_{m})}\), \(P_{1}\), and \(P_{2}\) are the optical spectral intensity of the two beams as shown in Figure 2a. The imaginary unit is \(i\), \(t_{1}\), and \(t_{2}\) are the time delay of the two beams, and \(\beta_{m}\) is the initial phase of the \(m^{\text{th}}\) mode.