A. Dual-comb light source

We used two Er-doped-fiber-based pulsed lasers that are amplified by the Er-doped fiber amplifiers as the L-comb and the S-comb. There were no optical isolators at the output ports of the L-comb and the S-comb. The output power, spectral bandwidth, and the pulse duration of each comb were about 100 mW, 100 nm, and a few picoseconds, respectively. The repetition frequency of the S-comb (frepS) was phase-locked to the radio frequency (RF) reference signal generated by a function generator. The carrier-envelope offset frequencies of the L-comb and the S-comb detected by f − 2f self-referencing interferometers were also phase-locked to the RF sources. The two optical beat frequencies between a reference cw laser with a wavelength of about 1550.1 nm and a comb tooth of each OFC adjacent to the frequency of the cw laser were also stabilized to the RF source. The stabilization of two beat frequencies realizes the mutual phase locking of the two OFCs. As a result, the relative timing jitter between the L-comb and the S-comb was about 50 fs.2222. M. Okano and S. Watanabe, arXiv:2208.00764 (2022). The typical experimental conditions were frepS ∼ 61.531 MHz and the difference between the repetition frequencies, Δfrep ∼ 300 Hz. To achieve a coherent averaging condition, frepS/Δfrep was set to the exact integer value by setting the beat frequency of the S-comb appropriately. More details of the frequency stabilization procedure are described in Ref. 2323. T. Fukuda, M. Okano, and S. Watanabe, Opt. Express 29(14), 22214–22227 (2021). https://doi.org/10.1364/oe.431104. From the separate measurement, we consider that the laser relative intensity noise possibly limits the signal-to-noise ratio of the measurement, which is consistent with the previous study.2424. N. R. Newbury, I. Coddington, and W. Swann, Opt. Express 18(8), 7929–7945 (2010). https://doi.org/10.1364/oe.18.007929

B. Modified Sagnac geometry and sample structure

The modified Sagnac geometry was used to realize a time-domain analysis of the S-comb. The power and polarization of the S-comb were controlled by a polarizer (POL) and half-wave plate (HWP) located immediately after the exit port of the S-comb light source. After passing through a beam splitter (BS), the S-comb entered the Sagnac geometry unit [Fig. 1(a); inside the green dashed border line]. A polarizing beam splitter (PBS1) was used to divide the S-comb into P-polarized and S-polarized beams, and hereafter we refer to these beams as Beam 1 and Beam 2, respectively. Beam 1 passed through PBS1 and a quarter-wave plate (QWP1), and then was reflected by a mirror, and by passing through QWP1 again, Beam 1 was converted to S-polarized light. As a result, Beam 1 was reflected by PBS1 and reached the sample by passing through QWP2 and an objective lens (OL). Meanwhile, Beam 2 was reflected by PBS1 and traveled through a polarization-maintaining (PM) fiber-based polarization rotation unit [Fig. 1(a); inside the gray dashed border line].The polarization rotation unit used in this work consists of a PM fiber (PM1017-C, Yangtze Optical Fiber and Cable) with a length of about 100 m and a fused fiber polarization combiner/splitter (PFC1550A, Thorlabs), which has three ports: the blue port for both polarization directions, the red port for light polarized along the fast axis, and the light gray port for light polarized along the slow axis. Beam 2 entered the PM fiber through a fiber collimator (CL) where the polarization of Beam 2 is along the slow axis of the PM fiber. We connected the end of the PM fiber to the blue port of the fiber polarization splitter, and Beam 1 was directed to the light gray port. To adjust the total fiber length, an additional PM fiber (indicated as a black line) with a length of about 88 cm was connected to the light gray port. We further connected the exit of the additional PM fiber to the red port in such a way that the slow-axis output couples to the fast axis [Fig. 1(a); “connector part”]. As a result, Beam 2 traveled back to the PM fiber with a polarization along the fast axis, i.e., its polarization was rotated by the polarization unit. After the rotation, Beam 2 passed through PBS1, QWP2, and OL in front of the sample. Because of the longer traveling distance of Beam 2, the reflection of the pulse of Beam 2 at the sample surface is delayed by Δt with respect to the reflection of the pulse of Beam 1. From the total fiber length of the polarization rotation unit and its refractive index (≈1.44),2525. I. H. Malitson, J. Opt. Soc. Am. 55(10), 1205–1209 (1965). https://doi.org/10.1364/josa.55.001205 we estimated a Δt of about 0.98 µs. We placed the CL on a mechanical delay stage to manually tune Δt.The inset on the right upper side of Fig. 1(a) shows a microscopic image of the SAW device. The SAW device consists of patterned titanium and gold layers on a 128° Y-cut 0.5 mm-thick LiNbO3 single crystal substrate fabricated using photolithography and electron-beam evaporation. The thicknesses of the titanium and gold layers were ∼3 and ∼70 nm, respectively, and interdigitated electrodes with a period of ≈21.3 μm were formed on the substrate. We applied a sinusoidal voltage with a frequency fSAW ≈ 180 MHz to the electrodes to excite Rayleigh-type SAWs on the surface of the LiNbO3 substrate. In this device, the SAW propagation direction is perpendicular to the x-axis of the crystal, and the phase velocity of the SAW is vSAW ≈ 3.67 km/s.2626. A. Holm, Q. Stürzer, Y. Xu, and R. Weigel, Microelectron. Eng. 31(1–4), 123–127 (1996). https://doi.org/10.1016/0167-9317(95)00334-7 The SAW propagates through the rectangular-shaped titanium/gold pad of size 400 × 360 μm2, resulting in a sinusoidal displacement of the surface along the z-direction with a vibration frequency fSAW. The wavelength of the SAW was λSAW = vSAW/fSAW ≈ 20 μm. Beams 1 and 2 were focused on the titanium/gold pad by the OL, and since a spot diameter of about 3 µm is sufficiently small with respect to λSAW, the local surface vibration at the spot can be probed.Figure 1(b) illustrates the experimental condition at two different time instants. At times t = t1 and t = t1 + Δt, the wave packets of Beams 1 and 2, respectively, are reflected from the sample surface, while the surface at the probe spot continuously moves along the z-direction because of the SAW propagation on the LiNbO3 substrate. The difference between the z-positions of the surface at t1 + Δt and t1 is defined as displacement D. The value of D reaches its maximum if t1 corresponds to a time when the z-position of the surface is at its minimum, and t1 + Δt corresponds to a time when the z-position of the surface is at its maximum. This situation is realized if the following three conditions are satisfied: (i) The repetition period of the S-comb is perfectly synchronized with the period of the SAW vibration (or an integral multiple of it), (ii) the interval Δt is equal to half the period of the SAW vibration (or an odd multiple of the half period), and (iii) the reflection of Beam 1 occurs when the z-position of the sample surface is at its minimum. Under these conditions, the displacement D is equal to twice the SAW amplitude A0, i.e., D = 2A0. The strategy used in our experiment to fulfill the above three conditions, is described in Sec. .When Beams 1 and 2 reached the sample, they were circularly polarized with opposite helicities. Therefore, after reflection at the sample, the two beams traveled through the opposite paths owing to the polarization management2121. T. Tachizaki, T. Muroya, O. Matsuda, Y. Sugawara, D. H. Hurley, and O. B. Wright, Rev. Sci. Instrum. 77(4), 043713 (2006). https://doi.org/10.1063/1.2194518 as follows: After reflection at the sample, each beam passed through QWP2. Since Beam 1 became P-polarized, it passed through PBS1, traveled through the polarization rotation unit, and returned to PBS1 as an S-polarized beam. Then, Beam 1 was reflected at PBS1 to the exit of the Sagnac geometry unit. Since Beam 2 became S-polarized, it was reflected at PBS1, and passed two times through QWP1 due to the reflection at the mirror behind QWP1. Therefore, it was finally transmitted through PBS1 and left the Sagnac geometry unit as a P-polarized beam. We stress that Beam 1 and Beam 2 travel through exactly the same optical path except for the small path difference due to the sample vibration.Now, we briefly consider the signal obtained in such a system. If the sample is static (D = 0), the optical phases of the two beams are almost equal at the exit of the Sagnac geometry unit, because the two beams travel the same optical path in opposite directions. On the other hand, when the sample exhibits a vibration (D ≠ 0), the optical phases of the two beams differ by 2δϕ as illustrated in Fig. 1(c). The relation between D and δϕ is expressed as follows:2δϕ=2D⋅nairλ×2π,(1)where λ is the wavelength of the considered frequency component of the S-comb and nair is the refractive index of air. If the phase difference δϕ is precisely measured by DCS, the displacement D can be obtained from Eq. (1).

C. Dual-comb spectroscopy

To perform the DCS measurement, we first combined the L-comb beam and Beams 1 and 2 exiting from the Sagnac geometry unit by using the BS. The power and polarization of the L-comb beam were controlled by a POL and a HWP as shown in Fig. 1(a). After combining them, they passed through an optical bandpass filter (BPF) with a bandwidth of 12 nm to reduce the bandwidth to such a level that an aliasing effect1616. I. Coddington, N. Newbury, and W. Swann, Optica 3(4), 414–426 (2016). https://doi.org/10.1364/optica.3.000414 is avoided when mapping the signal to the RF domain in the DCS measurement [the dual-comb interferometer unit is shown by the purple dashed border in Fig. 1(a)]. Two interference signals were measured: that between Beam 1 and the L-comb and that between Beam 2 and the L-comb. To separately obtain the two interference signals in the RF domain by two photodetectors (PD1 and PD2), we split the combined beam by PBS2. The detected optical power was typically a few mW. The electric signals detected by the PDs were filtered by electrical bandpass filters (3–30 MHz) and were sampled by a digitizer (M2p.5962-x4, Spectrum) in a personal computer with channel 1 (Ch1) and channel 2 (Ch2). In our experiment, because the interference signal is recorded with a sampling frequency frepS and the update frequency of the signal is Δfrep, the interference signal repeats with a sampling point interval frepS/Δfrep when there are no vibrations on the sample surface. Under the coherent averaging condition where frepS/Δfrep is set to be an integer value, the observed time-domain interference signals can be coherently accumulated within about a few seconds.In this paper, three kinds of experiments were conducted by changing the voltage amplitude, the phase, and the sample position (see Sec. for details). These experiments were conducted on different days and thus had slightly different experimental conditions, such as different frepS, different Δfrep, and different ratios of frepS/Δfrep.

D. Frequency synchronization

To determine the actual SAW amplitude A0 from the measured displacement D, it is crucial to synchronize fSAW and frepS as mentioned in Sec. . We performed the following procedures to achieve this synchronization: First, we set the frequency of the voltage applied to the SAW device to exactly three times the repetition frequency of the S-comb (fSAW = 3frepS ∼ 184.6 MHz). Second, the interval Δt was optimized by changing the optical path length of the polarization rotation unit by moving the mechanical delay stage. Finally, we tuned the phase of the voltage applied to the SAW device to a value where Beam 1 is reflected from the sample at a time when the z-position of the sample surface is at its minimum.The vibration of the sample surface after the synchronization procedure is shown schematically in Fig. 2(a). We consider the z-position of the sample surface at the y-position of the beam spot. Under ideal conditions, Beam 1 is reflected at times when the z-position of the sample surface is at its minimum (z = −A0). Because we consider fSAW = 3frepS, all pulses of Beam 1 are reflected from the sample surface at the same z-position. As explained in Sec. , we also need to tune Δt to a value that fulfills the following equation:where TSAW is the period of the SAW vibration defined as TSAW = 1/fSAW, and k is an integer. If Eq. (2) is fulfilled, Beam 2 is always reflected from the sample surface at times when the z-position of the sample surface is at its maximum (z = A0). The integer k is determined by the total optical path length of the polarization rotation unit, and it is around 181 in our experiment. We modulated the sinusoidal voltage applied to the SAW device by a square wave with a modulation frequency Δfrep/2. Therefore, as shown on right-hand side of Fig. 2(a), the vibration of the sample surface is stopped during the second half of the 2/Δfrep interval (see the gray arrow at the top of the figure).In our setup, the interferogram originating from the interference between Beam 1 (Beam 2) and the L-comb is recorded on channel 1 (channel 2), and the result is schematically shown by the red (green) curve in Fig. 2(b). Because the update frequency of the interferograms is Δfrep, and the pulse modulation frequency of the voltage applied to the SAW device is Δfrep/2, the measured time-domain profiles of each interferogram are repeated with a period 2/Δfrep. During the first half of the 2/Δfrep interval, the time-domain data contain the optical phases of Beams 1 and 2 reflected from the vibrating surface, ϕ11 and ϕ21, respectively. During the second half of the 2/Δfrep interval, they contain the optical phases of Beams 1 and 2 in the case of no vibration, ϕ12 and ϕ22, respectively. The phase difference α = ϕ12 − ϕ22 is the residual phase difference between Beams 1 and 2, which even exists when the vibration is stopped. On the other hand, the phase difference ϕ11 − ϕ21 is the sum of α and the phase difference 2δϕ induced by the SAW vibration [as shown in Fig. 1(c)]. The sample displacement D is determined using Eq. (1) and 2δϕ = (ϕ11 − ϕ21) − (ϕ12 − ϕ22). The advantage of the present method is the direct comparison of the optical phases of the two pulses of a pulse pair, which are reflected from the sample surface at different times, but have propagated through nearly the same optical path (with the same optical components) when they leave the Sagnac geometry unit. Therefore, this setup minimizes the effect of phase jitter. The resulting improved precision is experimentally shown and discussed in Sec. .