1. Introduction

Imaging the microscale vibrational fields of both amplitude and phase, is essential for investigating resonant microelectromechanical systems (MEMS), which have been widely applied in sensing [1], quantum information [2], and wireless communication [3]. Traditional electrical measurements only reveal electrical characteristics like frequency response and impedance curve, but cannot unveil dynamic characteristics, including mode shapes, energy dissipation pathways, spurious modes, and mode coupling and transfer. To better obtain this critical information and help optimize the dynamics of micro-scale resonant structures, an alternative vibrational imaging method is demanded. Optical imaging method is not only non-contact, but also easily reaching the sub-picometer vibrational amplitude range. It has thus been a critical tool for imaging MEMS for a wide frequency range, including micro mirrors (∼ 1 kHz) [4], inertia sensors (∼ 100 kHz) [5], piezoelectric micromachined ultrasonic transducers (PMUT, ∼ 10 MHz) [6], surface acoustic wave devices (SAW, 100 MHz ∼ 1 GHz) [7,8], bulk acoustic wave devices (BAW, ∼ 1 GHz) [9,10], thin bulk acoustic wave devices (FBAR, ∼ 1 GHz) [11], etc. Notably, the recent trend is pushing a number of important MEMS devices to the super high frequency (SHF, between 3 GHz and 30 GHz) or even higher range because of the strong demand in quantum phononics and radio frequency systems [12,13]. It is a known challenge to capture the small vibrations at SHF from the large detector noise, as the vibrational amplitude drops precipitously to the femtometer level at normal driven power levels. Therefore, it is desired to develop an optical imaging technique that can cover the ultra-wide frequency range of various resonant MEMS devices spanning from low frequency to SHF.

The earliest optical imaging methods for micro-vibrations relied on stroboscopic videography [14] and later stroboscopic interferometry [15,16]. Both of them use light strobes to freeze the MEMS motions followed by image processing. The former could only detect in-plane motions and the latter could capture both in-plane and out-of-plane motions, while both are capped at about 1 MHz and nanometer resolution. The stroboscopic interferometry method obtains the displacements using phase-shift reconstruction processing multiple interferograms [15,16]. It could be regarded as one special configuration of the stroboscopic digital holographic microscopy (DHM), which includes many variations such as off-axis configuration, phase-shift holography, frequency-shift holography and so on. The well-known off-axis configuration processes only one interferogram by sequential spatial Fourier transform, sideband filtering and inverse Fourier transform [17]. The DHM has been a popular method with continuous research efforts and until recent time, a common-path holography is developed to involve illumination from two directions, leading to the generation of multiplexed holograms to extract both phase and amplitude information [18]. Stroboscopic DHM is now limited at tens of MHz and a resolution of sub-nanometer. White-light interferometry was also investigated as another full-field technique by combining with stroboscopic light pulses to freeze the vibrational motion to reconstruct the dynamic mode shapes [19]. It could reach several MHz with a displacement resolution in the deep sub-nanometer range.

The above full-field dynamic imaging techniques are known for their fast capturing and reconstruction speed, but are difficult to extend to the SHF range and femtometer amplitude resolution. Scanning imaging techniques, on the other hand, focus the laser beam to a single spot instead of illuminating the whole device, and rely on photodetectors to record the interferometric signal as voltages, rather than processing the grey scale of a camera image [20–34]. Optical knife-edge method is designed to distinguish small in-plane displacements at the edges of microscale devices based on the change in reflected intensity inside a micron-size laser spot [20]. It could reach a sub-picometer resolution around tens of MHz, but is clearly restricted to in-plane vibrations at edges of high brightness contrast. X-ray diffraction microscopy uses Bragg diffraction contrast as a scattering mechanism to image local dynamic lattice perturbation induced by SAW or other resonant devices [21,22]. Although it could reach 1 GHz and discriminate out-of-plane and in-plane motions, it usually only works for nanometer-scale displacements and is difficult to generate SI-traceable results. Pump-probe techniques involving pulsed lasers have been also extensively studied to image micro-acoustic systems, and they work particularly well for those devices not possible to excite electrically, but their frequency resolution is limited to the MHz range due to the length of the delay line [23,24,25]. Recently, Tang developed an optomechanical spectroscopy for measuring the natural vibrations of mesoscopic particles which are stimulated photoacoustically and then detected acoustically by an optomechanical resonance [26]. This provides a novel technique to characterize micro-organisms and cells, which possess fingerprinting frequencies from tens of MHz to 1 GHz.

In addition to these scanning techniques, scanning interferometry is still the most popular method, including homodyne and heterodyne, both of which superpose a vibration-modulated laser beam with a reference split. In laser homodyne interferometry, the extracted signal is at the same frequency with the device-driven frequency (where the prefix homo- coming from), which leads to an electro-magnetic interference (EMI) at high frequencies. Therefore, the detection bandwidth of homodyne interferometry is usually limited around hundreds of MHz [27,28,29]. On the other hand, heterodyne interferometry introduces a small frequency shift in its reference beam, leading to a mixed signal shifted from the device driven frequency, and this guarantees immunity to EMI, thus achieving a higher frequency bandwidth up to several GHz [30–34], making it the currently incumbent imaging technique. It has also been investigated recently to combine with non-scanning full-field techniques to achieve a much better imaging efficiency and robustness [35,36]. Although variously modified heterodyne methods have been developed, many of these methods rarely exceed 3 GHz, with one exception [34] reaching 10 GHz but showing a noise 10 times higher than those capped at 3 GHz, due to the large classical noise from the photodetector. Therefore, it remains a challenge to image micro-vibrations from DC to the SHF range at a high signal-to-noise ratio (SNR).

In our previous work, pulsed laser interferometry (PLI) is proposed to optically mix down the SHF vibrations to a low-frequency beat note using a femtosecond laser, reaching a 12-GHz bandwidth while maintaining a femtometer amplitude resolution (55 fm ∕√Hz) [37–39]. Although this method shows the first femtometer amplitude imaging beyond 10 GHz, it is not applicable to vibrations lower than approximately 1 GHz. This is limited by the nature of the pulsed laser stroboscopic sampling, which could only work for ultrahigh- or superhigh-frequency vibrations. In this work, we report PLI with an adaptive switch which images ultrahigh- and superhigh-frequency vibrations by stroboscopic mixing while measuring lower-frequency vibrations based on the homodyne scheme. Such a switchable PLI achieves the imaging of resonant MEMS from DC up to 10 GHz with a femtometer amplitude resolution down to 30 fm/√Hz above MHz frequencies, which has not been shown in all previous works. We then demonstrate the successful imaging of the dynamic vibrational modes of a comb-drive resonator (∼ 170 kHz), a BAW resonator (∼ 2.6 GHz), and an FBAR device (up to 6.2 GHz). This method provides insights into mode profiles, k-space analysis, mode identification, etc., and it is also the first time that the infamous spurious modes of a pentagon-shaped FBAR are visualized, benefiting the design and optimization of their SHF dynamics.

2. Setup and principles

The setup of the proposed switchable PLI is illustrated in Fig. 1. Optical pulses of duration ∼100 fs, wavelength centered at 780 nm, and tunable repetition rate, ${\textrm{f}_\textrm{p}}$, from 50.0 MHz to 52.5 MHz, is launched from a femtosecond fiber laser. After beam expanding and collimation, a larger portion of the laser is directly measured by a high-speed photodetector (PD1, EOT). The other portion is guided to a typical Michelson interferometer to be split into two arms, with one arm focused on a device under test (DUT) through an objective lens, while another arm reflected from a reference mirror fixed on piezoelectric stage. The stage is controlled for aligning the two splits to overlap temporally and interfere at a slow low-noise photodetector (PD2, Menlo Systems). The vibration of the DUT is driven by a signal generator (SG, Rohde & Schwarz) at frequency ${\textrm{f}_{\textrm{vib}}}$. By means of analog proportional-integral-derivative control, the position of the reference mirror is actively stabilized at the quadrature point. The PID stabilization has two benefits, one being immune to environmental disturbance, while the other achieving the highest sensitivity at the quadrature point [7,28].

 

Fig. 1. Experimental setup of PLI with an adaptable switch to opt between low-frequency vibrations and high-frequency vibrations. BS: beam splitter, PD: photodetector, PBS: polarized beam splitter, OL: objective lens, DUT: device under test, RM: reference mirror, SG: signal generator, LIA: lock-in amplifier, PID: proportional-integral-derivative controller.

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For micro-vibrations well above the laser repetition rate, ${\textrm{f}_\textrm{p}}$, the femtosecond laser pulses stroboscopically mix with the vibrations to generate a slow beat note signal with its frequency at ${\textrm{f}_\textrm{b}} = |{{\textrm{f}_{\textrm{vib}}} - \textrm{n} \cdot {\textrm{f}_\textrm{p}}} |$, in which n is an integer making the beat note frequency the lowest possible value. The frequency of this beat note could be obtained by the calculation provided in the Supplement 1 [40,41]. The beat note signal carries the amplitude and phase information of the super high frequency vibrations, and could be demodulated using a lock-in amplifier in the end. The pulsed laser repetition rate ${\textrm{f}_\textrm{p}}$ could be tuned in a way that the beat note frequency falls within the bandwidth of the low-pass filter following the PD2 (${\textrm{f}_{\textrm{BW}}}$ = 25 MHz). Therefore, we must guarantee that $|{{\textrm{f}_{\textrm{vib}}} - \textrm{n} \cdot {\textrm{f}_\textrm{p}}} |< {\textrm{f}_{\textrm{BW}}}$, which means that this stroboscopic mixing could cover all vibrational frequency ranges in ($\textrm{n} \cdot {\textrm{f}_\textrm{p}} - {\textrm{f}_{\textrm{BW}}}$, $\textrm{n} \cdot {\textrm{f}_\textrm{p}} + {\textrm{f}_{\textrm{BW}}}$) where $\textrm{n} = 1,2,3, \ldots $. Given the tuning range of ${\textrm{f}_\textrm{p}}$, the frequency range that could be measured by the stroboscopic scheme is above 25 MHz. Below this frequency, it could not generate a beat note signal and this method could not apply to image micro-vibrations. Fortunately, traditional homodyne measurement could take care of micro-vibrations below this frequency very well. In order to cover the frequencies of various resonant MEMS from DC to 10 GHz, we use an adaptive switch to choose the stroboscopic mixing detection method above 25 MHz (switch in bottom position in Fig. 1), while choosing the traditional homodyne scheme below 25 MHz (switch in top position in Fig. 1).

It is also important to be able to demodulate the interfered optical signal in a high signal-to-noise ratio. For the DUT above 25 MHz and the switch steered to the bottom position, the pulsed laser detected by the PD1 is converted to an electrical signal, and enters into an RF mixer as a signal ${\textrm{f}_{\textrm{RF}}}$ with repetition rate ${\textrm{f}_\textrm{p}}$. The local oscillation (LO) signal ${\textrm{f}_{\textrm{LO}}}$ of the mixer is from the same signal generator which also drives the DUT, and thus ${\textrm{f}_{\textrm{LO}}} = {\textrm{f}_{\textrm{vib}}}$. The mixer produces a lower side band intermediate frequency (IF) ${\textrm{f}_{\textrm{IF}}} = |{{\textrm{f}_{\textrm{LO}}} - \textrm{n} \cdot {\textrm{f}_\textrm{p}}} |$, in which $\textrm{n} = 1,2,3, \ldots $. This signal is exactly the same frequency as the optical beat note ${\textrm{f}_\textrm{b}}$, and is imported to a lock-in amplifier (LIA, Zurich Instrument) as the reference signal. Clearly, the stroboscopic beat note is always at the same frequency of this lock-in reference, and we could obtain the vibrational amplitude and phase by lock-in demodulation in a significantly improved SNR.

On the other hand, when the DUT is below 25 MHz and the switch steered to the top position (Fig. 1), the signal generator directly drives the DUT and connects to the reference of the LIA. In such a way, the useful signal carried in the interference beam is at the same frequency as the drive signal, manifesting itself as the traditional homodyne detection scheme. Therefore, the design of the PLI with an adaptable switch ensures successfully covering an ultra-wide bandwidth from DC to more than 10 GHz.

3. Measurement of noise equivalent power

With stroboscopic mixing, quadrature point stabilization, and lock-in detection, the sensitivity-to-noise ratio of the proposed switchable PLI system is promoted to a great extent. We record the one-sided noise power spectral density (PSD) of the system when the RF power provided to the DUT is disconnected. The noise PSD represents the minimum detectable displacement since any vibrational signal below it would be submerged in noise. For frequencies below 50 MHz, the switch is steered to the top position, the root mean square (RMS) of the lock-in amplitude is recorded from 100 Hz to 50 MHz, which is the bandwidth of our LIA, as shown in Fig. 2(a). There is a clear flicker noise at low frequencies and it drops down to 33 fm/√Hz around MHz frequencies. For frequencies above 50 MHz, the switch is steered to the bottom position, the average noise PSD up to 8 GHz is 30.8 fm/√Hz, as shown in Fig. 2(b). We average the noise at each frequency for 6 times and plot the error bar as one standard deviation. Such a noise at GHz frequencies is more than 10 times smaller than that obtained using laser heterodyne measurement in Ref. [34].

 

Fig. 2. Amplitude noise PSD measurement. (a) Noise PSD from DC to 50 MHz. (b) Noise PSD from 50 MHz to 8 GHz.

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The noise of this setup includes shot noise, classical noise and flicker noise. The shot noise arises from the particle-like nature of photons and photoelectrons in the detection process. The classical noise, also known as the thermal noise, is related to the thermal motion of electrons in the photodetector, and is quantified as the noise equivalent power (NEP) causing voltage fluctuations. The flicker noise has a frequency bandwidth of 1/f and only appears in the low-frequency range where a homodyne detection scheme is used. It does not exist in the stroboscopic detection range for frequencies higher than 25 MHz, because all higher frequencies are converted to similar beat note frequencies for signal extraction. Systematic errors also influence the noise floor such as optical cross talk between the measurement and reference channels due to imperfect correlation of intensity and polarization. We note that the low noise approaching 10 GHz of our setup is mainly due to the elimination of those high-noise GHz photodetectors which are unavoidable in most other techniques. Noting that in the noise measurement and subsequent experiments at the SHF range, we adjust the repetition frequency ${\textrm{f}_\textrm{p}}$ to keep the beat signal ${\textrm{f}_\textrm{b}}$ around 2 MHz consistently. In this way, the average noise floor up to 8 GHz (30.8 fm/√Hz) is nearly the same with that at 2 MHz (33 fm/√Hz), which is due to the stroboscopic sampling that converts high-frequency vibrations at ${\textrm{f}_{\textrm{vib}}}$ to a low-frequency beat note at ${\textrm{f}_\textrm{b}}$.

4. Experimental results

Based on the proposed switchable PLI, we image the vibrational field of three representative resonant MEMS devices ranging from low to high frequencies, including an electrostatic comb-drive resonator (∼ 170 kHz), a high-overtone bulk acoustic wave resonator (∼ 2.6 GHz), and a film bulk acoustic resonator (up to 6.2 GHz).

The first device is a comb-drive resonator fabricated from silicon-on-insulator (SOI) wafers, with its shuttle mass supported by four micron-thin beams and actuated electrostatically by a pair of comb-shaped electrodes, as shown in Fig. 3(a). A DC and an AC voltage are applied across the left fixed comb and the shuttle mass at the torsional mode frequency, and as a result, a seesaw-like out-of-plane resonance is excited for the rectangular shaped resonant body. Since its resonance frequency is far below 50 MHz, the adaptable switch is steered to the top position so the reference signal of the LIA is just a split of the signal generator excitation signal.

 

Fig. 3. Measurement results of an electrostatic comb-drive resonator. (a) Microscopic image of the resonator, with electrical excitation signals. (b) Frequency response from 167 kHz to 171 kHz showing both amplitude and phase. (c) Phase and (d) Amplitude mapping of the com-drive resonator at 170 kHz. The scanning area is 40 µm × 90 µm. (e) 3D view of the reconstructed mode shape. An animated motion is shown in the supplementary Visualization 1.

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As shown in Fig. 3(b), we first measure the amplitude and phase with the laser spot kept at one side of the shuttle mass. The amplitude shows a Lorentz-shaped peak slightly higher than 169 kHz and a phase drop centered at the resonance for 180°, manifesting the generic frequency response of a resonant system. The amplitude drops as the driving voltage reduces due to the drop in the electrostatic force. We then set the DC and AC voltage as constant at 24 V and 3 V, respectively, and map the vibrational phase and amplitude at the frequency of 170 kHz. During the mapping, the laser spot is raster scanning across the 40× 90 µm² rectangular-shaped shuttle mass with a step size of 1 µm. The extracted phase and amplitude maps as well as the 3D view of reconstructed mode shape are shown in Figs. 3(c) to (e). The animated displacement field is provided as in the supplementary Visualization 1. It could be observed that the phase flips 180° across the two sides of the shuttle mass, exhibiting an expected seesaw-like motion. We note that there is apparent asymmetry at the two sides, showing an imperfect torsional rotation.

The second MEMS device is a type of BAW resonator [42]. Specifically, it is an AlN piezoelectric transducer (75 µm × 75 µm) fabricated on a silicon substrate, forming a so-called high-overtone bulk acoustic resonator, as shown in Fig. 4(a). An electric field is applied across the thickness of the AlN layer through the top and bottom metals, and out-of-plane vibration along the thickness is excited by the piezoelectric d₃₃ coupling. The demonstrated frequency is set at 2.562 GHz while the repetition rate of the pulsed laser is tuned to obtain a beat note around 2 MHz. This frequency is near its resonance peak, which is determined by sweeping through its resonance (not shown here). The adaptable switch is steered to the bottom position to enable stroboscopic mixing detection for GHz micro-vibrations. The laser pulse is focused on the surface of the BAW and raster scanning a 70 µm × 70 µm area with a step size of 1 µm (Fig. 4(b)). The filter bandwidth of the LIA is set to 1 Hz, and the dwelling time at each scanning position is set to 1.2 s, leading to ∼1.6 hour for one complete acquisition of a mode map.

 

Fig. 4. The BAW resonator and its imaged mode. (a) Schematic of the BAW resonator, with a probe beam irradiated on its surface. (b) Microscopic photograph of the BAW resonator [42] with a laser spot at its center. (c) Absolute phase and (d) amplitude mappings of a resonance mode at 2.562 GHz. (e) Mode shape calculated from the amplitude and phase mappings of the resonance mode, with an animated version available in the supplementary Visualization 2. (f) Spatial 2D-FFTs of the amplitude mapping.

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Figures 4(c) and (d) show the vibrational phase and amplitude, respectively, where superposition of acoustic standing waves in the horizontal directions upon the main longitudinal mode is clearly visible. We then combine the amplitude and phase mappings to produce a 3D mode shape as shown in Fig. 4(e) and an animated mode is presented as the supplementary Visualization 2. Figure 4(f) shows the 2D fast Fourier transform on the amplitude map, in which two horizontal modes are clearly visible (as the two concentric slowness rings) and appeared to be isotropic. It also shows that all traveling waves are equal in magnitude along each direction, which means a standing wave inside the resonator cavity. We note that the achieved detection resolution would enable us to characterize even higher-frequency vibrations, however, the spatial resolution is limited by optical diffraction, resulting in undistinguishable patterns as the periodicity of the standing waves is too small at higher frequencies.

The third device is an FBAR which is the key component for 5 G bandpass filters and sensors [43,44]. A schematic view of this FBAR is illustrated in Fig. 5(a), showing a AlN piezoelectric layer sandwiched between a set of top and bottom Mo electrodes, which is released from the silicon substrate. An RF power is applied across the electrodes to excite out-of-plane vibrations along the thickness direction due to piezoelectric coupling. The shape of the pentagon is irregular in order to restrain the generation of those spurious modes. Figure 5(b) shows an FBAR chip with its co-planar ground-signal-ground electrodes bonded for one-port excitation. An S₁₁ measurement using a network analyzer is performed to find the rough frequency range of resonance (not shown here). Before mode mapping by our PLI, we use the scanning stage to obtain the coordinates of five vertices of the FBAR as shown in Fig. 5(c).

 

Fig. 5. The FBAR resonator and its mapped mode shape. (a) Schematic of the FBAR resonator [43,44]. (b) Photograph of the FBAR with bonded electrodes. (c) Microscopic image of the FBAR with the coordinates of the five vertices. (d) Absolute phase and (e) amplitude through scanning the surface of the FBAR at 2.570 GHz. (f) Reconstructed mode shape.

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According to the obtained coordinates, a meander pathway is adopted to scan the laser spot across the FBAR surface. The filter bandwidth of the LIA is set to 1 Hz, while the dwelling time interval at each scanning location is set to 1.2 s, and the scanning step size being 1 µm and total points being 10033 points, respectively. Thus, it took about 3.3 hours to obtain the absolute phase and amplitude maps (frequency at 2.570 GHz), as shown in Figs. 5(d) and (e). The reconstructed 3D mode shape is shown in Fig. 5(f) and an animated mode is presented in the supplementary Visualization 3. Such a five-lobed mode shape is categorized as one of the spurious modes with its amplitude in the single-digit picometer, which will require a femtometer amplitude resolution for high-SNR mapping.

Another spurious mode at a slightly different frequency is also found at 2.635 GHz with its displacement field shown in Fig. 6(a). It shows various intersected stripes parallel to the pentagon edges, featuring shear horizontal modes, which is distinct to the mode at 2.570 GHz. The existence of these rich spurious modes is experimentally demonstrated for a typical FBAR resonator for the first time. We note that a complete analysis on its modes will be presented in another future paper. A 2D-FFT of its amplitude map at 2.635 GHz is plotted in Fig. 6(b), where superposition of a number of in-plane modes are clearly visible in the directions perpendicular to each edge of pentagon. It also indicates that the traveling waves in each direction are equal in magnitude, confirming a standing wave inside the FBAR.

 

Fig. 6. Mode shapes and their characterization. (a) Mode shape and (b) Spatial 2D-FFTs in the k-space at 2.635 GHz. (c) Mode shape at 6.2305 GHz. Animated mode shapes are available in the supplementary Visualization 3. (d) Frequency response in black and the peak amplitude in blue as a function of power attenuation, where the red double-sided arrow shows the same amplitude for the same driving conditions.

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Figure 6(c) presents a high-order mode found at 6.2305 GHz showing a dome-shaped profile but offset to one edge of the pentagon. Figure 6(d) presents the frequency response in a black color, and the peak amplitude as a function of attenuation in the excitation power in a blue color. Although the maximum vibration amplitude drastically decreases below 1 pm, the resonance is still clearly visible because of the femtometer-level noise PSD at such high frequencies. We then record the peak amplitude as a function of the attenuation in the applied RF power. Obviously, with the RF power attenuation from 0 dB to 30 dB, the peak amplitude gradually drops and finally levels down to the noise floor, which confirms that this mode is truly a mechanical vibration in nature. It is the first time that a clear vibrational profile which is below 1 pm and over 6 GHz is experimentally mapped.

5. Discussion

We summarize a number of currently representative dynamic imaging techniques with this work in Table 1. Compared with other vibrational imaging methods, our method has outstanding advantages as an extremely low noise floor and an ultra-wide bandwidth.

Table 1. Comparison of optical dynamic imaging methods

View Table

First, we note that the noise floor is a metric that changes significantly with frequency, and not all papers reported the associated frequency of their noise floor. We assume that each reported metric is the lowest achievable value, and should typically occur at relatively high frequency to avoid the flicker noise. Secondly, it is well known that the laser power plays a significant role in the noise floor, but unfortunately most cited reference does not report the used laser power level. It is therefore not easy to conduct a rigorous comparison. In our experiment, the laser power at the detector is about 300 µW, which would lead to a theoretical shot noise limited resolution $\textrm{d}x = \frac{1}{{4\mathrm{\pi }}}\sqrt {\frac{{2\mathrm{hc\lambda B}}}{{\mathrm{\eta P}}}} \cong 3\; \textrm{fm}/\sqrt {\textrm{Hz}} $, where h is the Planck constant, c is the light speed, $\mathrm{\lambda }$ is the light wavelength (780 nm), B is the detection bandwidth (1 Hz), $\eta $ is the quantum efficiency of our photodetector (0.8), and P is the laser power [45].

Third, the disadvantage of our stroboscopic detection technique is that our system could not measure the vibrational frequency, but only the vibrational amplitude and phase. This is because that we need to know the applied driven frequency to actively tune the pulsed laser to obtain a beat note signal at suitable frequencies. On the other hand, other methods like homodyne and heterodyne interferometry directly measure the spectrum and therefore they could help reveal the unknown vibrational frequencies.

With the proposed switchable PLI system, we have achieved dynamic imaging of three MEMS devices including a comb-drive resonator, a BAW resonator, and a FBAR. Spatial 2D-FFTs of the amplitude mappings of the GHz BAW and FBAR devices are presented to show standing waves localized inside the MEMS resonant cavities. The propagation direction of spurious as well as acoustic leakage could be extracted and analyzed to help us better understand the design and manufacturing defects. The experimental result is shown for the first time and is expected to compare with finite-element simulation in the future, particularly for the technologically-important pentagon-shaped FBAR. Further study also includes analysis on dispersion curves, group and phase velocity, etc.

6. Conclusions

In conclusion, we have proposed a pulsed laser interferometry with an adaptive switch to image the micro-vibrations from DC to 10 GHz with an amplitude resolution around 30 fm/√Hz above MHz frequencies. The switch enables a stroboscopic detection of vibrations above 25 MHz and a traditional homodyne detection of vibrations below 25 MHz. Such a design leads to a low-noise imaging for ultrahigh- and superhigh-frequency vibrations and at the same time retaining excellent imaging capability of lower-frequency vibrations as well. To further demonstrate our proposed method, vibrational field patterns of a kHz comb-drive resonator, a GHz BAW and a GHz FBAR are imaged and analyzed. For the first time, dynamic imaging of an FBAR is presented showing interesting spurious modes and k-space analysis. A clear mechanical mode above 6 GHz with amplitude as low as 100 fm is also shown and confirmed. This work has demonstrated the best vibrational displacement resolution to date for frequencies up to 10 GHz. Future study will focus on further broadening the detection bandwidth beyond 15 GHz, to cover the millimeter-wave range. We believe that the proposed method can emerge as a new paradigm for imaging micro-vibrations and to advance our knowledge of new physics in MEMS resonators.