Temporal-offset dual-comb vibrometer with picometer axial precision

A quantitative evaluation of extremely small mechanical vibrations of surfaces of solids is important for fundamental physics as well as device applications. For example, the mechanism of interconversion between spin and mechanical angular momentum by surface acoustic waves (SAWs) is a novel route to generate spin currents in nonmagnetic metals^1–4 and opens possibilities for research in the area of spin-mechatronics.⁵ In order to estimate the conversion efficiency of this mechanism, the evaluation of the surface vibration amplitude is crucial. The evaluation of the vibration amplitude is also useful for the design of SAW filters and micro-electromechanical resonators with better efficiencies.^6,7

Amplitudes of extremely small surface vibrations are usually determined by optical interferometry. For example, homodyne interferometry enables us to measure absolute vibration amplitudes down to picometers (the phase information can also be obtained).^6,8,9 A sensitivity of the order of femtometers (7.1 fm/Hz^1/2 at the vibration frequency of 21 MHz) can be achieved¹⁰ by using a heterodyne interferometer.^11,12 Very recently, a single pulsed laser interferometer with a combination of a stroboscopic optical sampling that enables high precision of the vibration amplitude measurement (54.9 fm/Hz^1/2 up to 12 GHz) has been reported.¹³ Although precision of most interferometric experiments based on continuous-wave (cw) lasers is very high, spurious reflections can cause additional interferometric signals, which is often troublesome, since this degrades the accuracy of optical-phase measurements.¹⁴ In addition, the cw-laser-based approach limits the measurable range, or the so-called ambiguity range, to less than the wavelength of the laser.¹⁵ 

Multiwavelength interferometry, based on two femtosecond-frequency-combs,¹⁶ is a novel approach to determine the distance to objects with large ambiguity range and high precision, which is sometimes referred to as dual-comb ranging. By using two pulsed lasers, time-of-flight distance measurements are possible, which allow us to easily distinguish between signals from the sample and spurious reflections, which improves the accuracy of the distance measurement.¹⁷ Such time-of-flight measurements also lead to a dramatic extension of the ambiguity range up to kilometers.^17,18 Note that in dual-comb ranging, the optical phase of a pulse reflected from the sample is compared with that of a reference pulse, and this measurement scheme can also be used to perform vibration amplitude measurements with a large ambiguity range.¹⁹ However, due to the phase jitter of the laser, the precision of such dual-comb vibrometers (which is of the order of sub-nanometers)¹⁹ is still insufficient to probe the amplitude of a SAW vibration, since the latter is of the order of sub-angstroms.

In this article, we present a dual-comb vibrometer with picometer axial precision, which enables us to accurately measure SAW-vibration amplitudes. To achieve this, we first divided one of the frequency combs into two beams by a polarizing beam splitter, and thus each pulse of the original beam was split into a pulse pair with a tunable delay. Due to the delay, the two pulses of a pulse pair were reflected at the vibrating sample at different times corresponding to different axial positions (e.g., maximum and minimum) of the sample surface. Therefore, the difference between the optical phases of the two pulses can be used to determine the SAW amplitude. In our setup, the polarizing beam splitter and the sample are part of a modified Sagnac geometry,^20,21 and thus the two pulses traveled through the same optical components (but in reverse order) before their phase difference was determined by dual-comb spectroscopy (DCS). Note that this geometry dramatically decreases the phase jitter between the two beams. In addition, because of the time-of-flight detection scheme used in DCS, the signal reflected from the sample can be easily separated from other signals, which provides an accurate result free from spurious reflections. We succeeded in determining the absolute vibration amplitude of a SAW with a precision of 4 pm. The SAW propagation was clearly monitored by changing the phase of the SAW with respect to the timing of the laser pulse. This accurate optical measurement technique allows us to determine extremely tiny vibration amplitudes of surfaces, which is useful for investigations in fundamental physics and device design.

A schematic of the experimental setup is shown in Fig. 1(a). On the left lower side, two frequency-comb light sources are shown; the two optical frequency combs (OFCs), which are referred to as the local comb (L-comb) and the signal comb (S-comb), allow us to determine small optical phase shifts of the S-comb pulses that are reflected from the sample. In this work, the phase shift was induced by an extremely tiny surface displacement due to a SAW.

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 (f_repS) 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.²² The typical experimental conditions were f_repS ∼ 61.531 MHz and the difference between the repetition frequencies, Δf_rep ∼ 300 Hz. To achieve a coherent averaging condition, f_repS/Δf_rep 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. 23. 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.²⁴ 

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),²⁵ 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 f_SAW ≈ 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 v_SAW ≈ 3.67 km/s.²⁶ 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 f_SAW. The wavelength of the SAW was λ_SAW = v_SAW/f_SAW ≈ 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 λ_SAW, 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. II D.

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 management²¹ 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:

where λ is the wavelength of the considered frequency component of the S-comb and n_air 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).

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 effect¹⁶ 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 f_repS and the update frequency of the signal is Δf_rep, the interference signal repeats with a sampling point interval f_repS/Δf_rep when there are no vibrations on the sample surface. Under the coherent averaging condition where f_repS/Δf_rep 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. III B for details). These experiments were conducted on different days and thus had slightly different experimental conditions, such as different f_repS, different Δf_rep, and different ratios of f_repS/Δf_rep.

To determine the actual SAW amplitude A₀ from the measured displacement D, it is crucial to synchronize f_SAW and f_repS as mentioned in Sec. II B. 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 (f_SAW = 3f_repS ∼ 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 f_SAW = 3f_repS, all pulses of Beam 1 are reflected from the sample surface at the same z-position. As explained in Sec. II B, we also need to tune Δt to a value that fulfills the following equation:

where T_SAW is the period of the SAW vibration defined as T_SAW = 1/f_SAW, 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 Δf_rep/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/Δf_rep 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 Δf_rep, and the pulse modulation frequency of the voltage applied to the SAW device is Δf_rep/2, the measured time-domain profiles of each interferogram are repeated with a period 2/Δf_rep. During the first half of the 2/Δf_rep 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/Δf_rep 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. III.

Figures 3(a) and 3(b) show the interferogram data recorded on channel 1 and channel 2 during a single 2/Δf_rep interval. Because the update frequency of the interferograms is Δf_rep, the main signals due to the reflection at the sample (the four signals inside the boxes with gray dashed lines) were observed with the time interval 1/Δf_rep. In addition to the data originating from the reflection at the sample, we observed several spurious signals that originate from reflections at other optical components in the experimental setup. Because of the low transmittance of the objective lens and the limiting coupling efficiency at the polarization rotation unit, the optical power of the beam reflected from the sample surface becomes very weak at the detector. Therefore, the intensity of the main signals is comparable with those of the spurious signals. However, despite the large spurious signals, we can easily eliminate these signals by time gating with an appropriate filter window, ensuring an accurate determination of the displacement. For each of the four signals, we used a 0.06 ms-wide cosine-tapered window (or Tukey window) in real time, which corresponds to a width of 0.30 ns in terms of the effective time. Then, we calculated the Fourier transforms of the four time-gated interferograms to obtain the wavelength-dependent phases ϕ₁₁(λ), ϕ₁₂(λ), ϕ₂₁(λ), and ϕ₂₂(λ).

Figure 3(c) shows the wavelength dependence of 2δϕ(λ) [=ϕ₁₁ − ϕ₂₁ − ϕ₁₂ + ϕ₂₂] in the range 1534.33 ≤ λ ≤ 1540.71 nm, where the signal intensity is sufficiently large. The wavelength resolution determined by the time-gating window width is 0.026 nm, resulting in N = 247 data points in this wavelength range. Using Eq. (1), we then converted the measured 2δϕ(λ) values to D(λ) values. In this experiment, we recorded 49 000 intervals of length 2/Δf_rep (effective measurement time less than 3 min; In this work, an effective measurement time is defined as the time during which the SAW device is vibrated.) and calculated the average of D(λ), $D̄$(λ), and its standard deviation from the mean, $σD̄λ$⁠. Because the value of $D̄$ is, in principle, independent of the wavelength, we calculated the final result D₀ by averaging $D̄$(λ) over the whole wavelength range. Figure 3(d) shows the wavelength dependence of $D̄$(λ), and D₀ is shown by the solid orange line. The standard deviation from the mean of D₀, defined as $σD0$⁠, was calculated according to the law of propagation of uncertainty using the 247 values of $σD̄λ$⁠. Standard deviation of the 247 values of $D̄$(λ) (defined a $σD̄λ$ = $N×σD0$⁠) is also shown as dashed lines. We determined that the displacement of the SAW at 0.50 V is (D₀ ± $σD0$⁠) = (42.3 ± 3.6) pm. Because the time of reflection of Beam 1 coincided with the time when z = −A₀, the absolute value of the SAW amplitude is (A₀ ± $σA0$⁠) = (21.2 ± 1.8) pm.

To discuss the higher precision of the proposed temporal-offset-based multiheterodyne technique, we compare the precisions of the displacement obtained by two different analysis methods using the same experimental data as in Figs. 3(a) and 3(b). The first analysis method was used to derive D₀ in a similar manner as discussed in the previous paragraph using the pulse pair. Furthermore, we used the Allan deviation D₀ as an indicator of the degree of precision as follows: After the calculation of D(λ) of the i-th data interval (1 ≤ i ≤ 49 000), we calculated the value of D₀ for each data interval, $D0i$⁠, by averaging D(λ) of the i-th interval over the whole wavelength range. Then, we calculated the fully overlapping Allan variance of D₀ from $D0i$ according to Eq. (10) in Ref. 27 as a function of the averaging time. The Allan deviation was calculated by taking the square root of the fully overlapping Allan variance. The second analysis method was used to derive a displacement of the surface by comparing the optical phases of the pulse of Beam 1 that were obtained in the cases with and without the SAW vibration. We denote this displacement D₁, which is the displacement that can be obtained by comparing the optical phases of the interferograms with and without the SAW vibration using a single channel. To derive D₁, we determined the difference between the phases of Beam 1 with and without the applied voltage [δϕ₁(λ) = ϕ₁₁(λ) − ϕ₁₂(λ)], converted δϕ₁(λ) to D₁(λ) for each data interval using Eq. (1), and calculated $D1i$ by averaging D₁(λ) of the i-th data interval over the whole wavelength region. Then, we calculated the Allan deviation of D₁ from $D1i$ as a function of the averaging time.

Figure 4 shows the averaging-time dependence of the Allan deviation of D₀ (blue closed circles) and that of D₁ (red open squares). Note that the actual recording time is twice of the averaging-time because of the square wave modulation. We observe a remarkable improvement in the precision when the distance is estimated using two beams with a temporal offset Δt instead of a single beam. At short averaging times (<40 s), the Allan deviations of both D₀ and D₁ decrease as a function of time, while the Allan deviation of D₀ is about two to three orders of magnitude smaller than that of D₁. The improved precision in D₀ was achieved owing to the comparison of the phases of two pulses generated from the same optical wave packet of the S-comb and the fact that both pulses had propagated through nearly the same optical path when we measured their phases by DCS. This reduces the effect of the phase jitter between Beams 1 and 2 of the S-comb beams. In addition, the two pulses of the pulse pair pass through the same optical components in the experimental setup, which suppresses the effect of subtle fluctuations of the positions of individual optical components. Moreover, the effect of the optical phase fluctuations between the two L-comb beams, which interferes with Beams 1 and 2 of the S-comb beams, respectively, was reduced because both pulses interfere with the S-comb at nearly the same time. At longer averaging times (>40 s), the Allan deviation of D₁ starts to increase. This is probably due to a temporal variation of the phase jitter of the laser pulses between the two successive measurements of the interferogram due to a change in the environmental condition, which results in a drift of the measured value of D₁. On the other hand, the Allan deviation of D₀ continuously decreases for longer averaging times, and we can confirm a precision of several picometers at 170 s. The precision can become even smaller at longer averaging times; we achieved a 2.5 pm precision using an averaging time of 336 s in a different experimental run. This result also suggests that our experimental scheme is insensitive to temporal variations of the phase jitter, because we compare the optical phases of S-comb pulses that have propagated through nearly the same optical path.

Note that in our setup, the mutual coherence between two OFCs does not maintain for a long time. However, a short-time mutual coherence of the two OFCs within the time interval 2/Δf_rep (∼6 ms) is enough to accumulate the data to improve the precision down to the picometer level.

The high precision of the temporal-offset-based dual-comb ranging technique enables us to determine the absolute surface displacement of a vibrating object with extremely tiny vibration amplitude. In the following, we determine the surface displacement due to a SAW as functions of the amplitude and phase of the applied voltage, and as a function of the y-position. Note that the measured displacement D₀ depends on the timing of the reflection of Beam 1, since this controls the probed z-position of the sample surface; in Fig. 2(a) we chose a timing corresponding to z = −A₀. The timing can be changed by either changing the phase of the applied voltage or by moving the sample along the SAW propagation direction.

In Fig. 5(a), we plot D₀ as a function of the phase of the applied voltage for a voltage amplitude of 0.50 V, and we can confirm that D₀ exhibits a sinusoidal change. At a phase of 150° (330°), reflection occurs at a time when the z-position of the sample surface is close to its minimum (maximum). The corresponding maximum and minimum values of D₀ are 42.3 and −41.9 pm, respectively. The absolute values of these extrema are the same within the uncertainty of ±3.6 pm, proving that an offset-free determination of the displacement is achieved by this technique.

Figure 5(b) shows the results obtained by changing the sample position along the y-direction for 0.50 V. The displacement shows a sinusoidal oscillation along the y-direction and by fitting the data to the function $D0=A0⋅cos2πy−y0/λ$⁠, we obtained A₀ = 46.8 pm, y₀ = −1.0 μm, and λ = 19.2 μm. This result indicates that the SAW is propagating along the y-direction with a wavelength of λ that is close to λ_SAW. The observed value of λ is slightly smaller than λ_SAW because we performed the measurement at a position where the thin titanium/gold layers are deposited on the LiNbO₃ substrate; it is known that the phase velocity of the SAW is changed by the deposited layers.^28,29 We found that the amount of displacement at y = 38 μm is slightly smaller than that at y = 18 μm. This may be due to a slight change in the beam spot diameter on the sample surface when we moved the sample because a larger spot size degrades the spatial resolution and leads to a smaller numerical value of the measured displacement.

Figure 5(c) shows D₀ as a function of the amplitude of the applied voltage. A linear dependence of D₀ on the applied voltage is clearly observed. The D₀ at 0.50 V is smaller than the amplitude determined in Figs. 5(b) and 5(c). This could be due to an imperfect optimization of the phase of the SAW and/or a degradation of the electric contact. In addition, a small data deviation from the fitting line at low voltages in Fig. 5(c) also originates from the changes of the sample condition during the experiments rather than the instrumental uncertainty.

We have demonstrated a new type of dual-comb ranging that is capable of determining the absolute displacement of a vibrating object with picometer axial precision. The comparison of the optical phases of the two pulses of a pulse pair that propagate through nearly the same optical path (with the same optical components) but are reflected at different times from the vibrating sample surface, dramatically improves the axial precision. We determined the amplitude of a surface vibration on a SAW device with an uncertainty of 4 pm. Measurements of the surface displacement as functions of the amplitude and phase of the voltage applied to the SAW device and the sample position clearly prove that a quantitative characterization of surface vibrations can be achieved by this technique.

We would like to point out some applications using our technique. Our technique is capable of sensing a wide range of surface vibrations with different vibration frequencies and amplitudes. For instance, our scheme can also be applied to the optically-pumped SAW excitation experiments²⁰ where several frequency components are simultaneously excited. In our measurement protocol, because the measurable f_SAW should be integer multiples of f_repS, other frequency components of the SAW vibration are smoothed out as zeros. Therefore, we could uniquely estimate the vibrating amplitude of the optically-pumped SAW with the frequency component of integer multiples of f_repS when the frequency bandwidth of the SAW is less than f_repS.

The achieved precision is the highest among the displacement measurements using dual-comb ranging, and therefore, this new type of dual-comb ranging opens an avenue for ultra-precise and accurate length metrology of extremely tiny displacements. Here, we briefly mention a further potential of our presented method in terms of precision. Since the Allan deviation is inversely proportional to the square root of the measurement time within the presented time range as shown in Fig. 4, it is expected that the precision becomes better as the measurement time becomes longer, if the dependence of Allan deviation on the measurement time maintains even at longer times. However, during such a long measurement time, it is expected that the sample’s conditions, such as the sample’s axial position, change, which reduces the precision. One possible solution to achieve better accuracy within a short measurement time is to use frequency combs with high-repetition frequency such as soliton microcombs, which help increase the update frequency used in interferogram measurements.³⁰ 

S.W. would like to thank Professor O. B. Wright and Professor O. Matsuda at Hokkaido University for discussions on the modified Sagnac geometry. This work was partially supported by JST CREST (Grant No. JPMJCR19J4) and MEXT Quantum Leap Flagship Program (MEXT Q-LEAP) (Grant No. JPMXS0118067246). A part of this work was conducted at the AIST Nano-Processing Facility supported by the “Nanotechnology Platform Program” of the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan (Grant No. JPMXP09-F-21-AT-0085).

The authors have no conflicts to disclose.

A. Iwasaki: Conceptualization (equal); Data curation (lead); Formal analysis (lead); Investigation (lead); Methodology (equal); Resources (supporting); Software (equal); Validation (lead); Visualization (lead). D. Nishikawa: Methodology (supporting). M. Okano: Methodology (supporting). S. Tateno: Resources (supporting). K. Yamanoi: Resources (supporting). Y. Nozaki: Funding acquisition (equal); Project administration (supporting); Supervision (supporting). S. Watanabe: Conceptualization (lead); Funding acquisition (equal); Methodology (equal); Project administration (lead); Resources (lead); Software (equal); Supervision (lead); Writing – original draft (lead); Writing – review & editing (lead).

The data that support the findings of this study are available from the corresponding author upon reasonable request.