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Temporal-offset dual-comb vibrometer with picometer axial precision

We demonstrate a dual-comb vibrometer where the pulses of one frequency-comb are split into pulse pairs. We introduce a delay between the two pulses of each pulse pair in front of the sample, and after the corresponding two consecutive reflections at the vibrating sample surface, the initially introduced delay is cancelled by a modified Sagnac geometry. The remaining phase difference between the two pulses corresponds to the change in the axial position of the surface during the two consecutive reflections. The Sagnac geometry reduces the effect of phase jitter since both pulses propagate through nearly the same optical path (in opposite directions), and spurious signals are eliminated by time gating. We determine the amplitude of a surface vibration on a surface-acoustic-wave device with an axial precision of 4 pm. This technique enables highly accurate determination of extremely small displacements.


Introduction
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][2][3][4] and opens possibilities for the research 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 microelectromechanical 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].
A sensitivity on the order of femtometers can be achieved [9] by using a heterodyne interferometer [10,11]. Although the precision of most interferometric experiments based on continuous-wave (cw) lasers is very high, the cw-laser-based approach limits the measurable range, or the so-called ambiguity range, to less than the wavelength of the laser [12]. In addition, spurious reflections can cause additional interferometric signals, which is often troublesome since this degrades the accuracy of the optical-phase measurement [13].
Multiwavelength interferometry based on two femtosecond-frequency-combs [14] is a novel approach to determine the distance to objects with a 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 leads to a dramatical extension of the ambiguity range up to kilometers [15,16]. Such time-of-flight measurements also allow us to easily distinguish between signals from the sample and spurious reflections, which improves the accuracy of the distance measurement [15]. 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 [17]. However, due to the phase jitter of the laser, the precision of such dual-comb vibrometers (which is on the order of sub-nanometers) [17] is still insufficient to probe the amplitude of a SAW vibration, since the latter is on 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 [18,19], 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 of fundamental physics and device design.

Experimental setup
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, 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.

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), then was reflected by a mirror, and by passing through QWP1 again, Beam 1 was converted to S- 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 nm 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 of 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 [21]. The SAW propagates through the rectangular-shaped titanium/gold pad with a size of 400 µm × 360 µm, resulting in a sinusoidal displacement of the surface along the z-direction with a vibration frequency of 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 the spot diameter of about 3 μm is sufficiently small with respect to λSAW, the local surface vibration at the spot can be probed.  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 management [19] 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. 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 2δ as illustrated in Fig. 1(c). The relation between D and δ is expressed as follows: 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 is precisely measured by DCS, the displacement D can be obtained from Eq. (1).

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 Lcomb 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) to reduce the bandwidth to such a level that an aliasing effect [14] is avoided when mapping the signal to the radio frequency (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 Beam1 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 photodiodes (PD1 and PD2), we split the combined beam by PBS2. Finally, the electric signals of PD1 and PD2 were recorded using a digitizer in a personal computer with channel 1 (Ch1) and channel 2 (Ch2).
In this paper, three kinds of experiments were conducted by changing the voltage amplitude, ratio frepS/Δfrep was chosen to satisfy the coherent averaging condition [22]. The method of the frequency stabilization and the calibration procedure are described in Ref. [23].

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 Section 2.1. We performed the following procedures to achieve this synchronization: Firstly, 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).
Secondly, 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 Section 2.1, 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 91 in our experiment. We modulated the sinusoidal voltage applied to the SAW device by a square wave with a modulation frequency of Δfrep/2. Therefore, as shown on the 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  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 the next section.

Characterization of the SAW vibration with picometer precision
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 an 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)  wavelength of λ that is close to λSAW. The observed value of λ is slightly smaller than λSAW, 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 [25,26]. 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.

Conclusion
We have demonstrated a new type of dual-comb ranging that is capable of determining the 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.