IMAGING OF SURFACE ACOUSTIC WAVES AND MEMS WITH PICOMETER DISPLACEMENTS USING HOLOGRAPHIC VIBROMETRY

Holographic vibrometry is an interferometric method for the fast and spatially resolved characterization of surface vibrations on the nano- and picoscale [1]. Similar to laser Doppler vibromery (LDV), it exploits the Doppler shift light experiences when it is reflected from a moving object. It allows for a full-field measurement instead of relying on a point-by-point scanning process and thus offers very fast measurements. This is crucial when measurements are to be performed in production or quality control.
In the following, we assume light with frequency \(f_L\) which is reflected from an object surface harmonically vibrating with frequency \(f_{obj}\) and displacement amplitude \(\hat{z}_{obj}\). This corresponds to a harmonic phase modulation which generates sidebands at \(f_L\pm m\cdot f_{obj}\) as illustrated in equation (1) using the Jacobi Anger identity. The object displacement is thus encoded in the amplitude factors of the different spectral components which are given by the Bessel functions \(J_m\). \[e^{i 2\pi f_L t}\cdot e^{i\frac{4 \pi \hat{z}_{obj}}{\lambda}\sin{(2 \pi f_{obj} t)}}\ = \sum_m J_m \left(\frac{4 \pi \hat{z}_{obj}}{\lambda}\right) \cdot e^{i \left(2 \pi (f_L + m f_{obj}) t\right)}, \ \ m \in Z\] For displacements smaller than \(\approx\) 10 nm, \(J_0\) is close to unity and \(J_n \approx 0\) for \(n\geq2\) but \(J_1\) is approximately proportional to the displacement \(\hat{z}_{obj}\). By recording a hologram of this spectral component at \(f_L\pm f_{obj}\) and reconstructing its image in the object plane by numerical Fresnel backpropagation, the spatial distribution of the sideband amplitude and thus the vibration displacement across the object’s surface is obtained.
In practice, this is achieved by shifting the frequency of the light in the reference arm by \(\Delta f = f_{obj}\) using acousto-optic modulators, such that it matches the frequency of the sideband. A frequency shift of \(\Delta f = f_{obj}+f_b\) leads to a beat frequency \(f_b\) between the reference arm and the first sideband in the object arm. If a sequence of images is recorded, narrow-band filtering can be employed in the frequency domain to boost the SNR according to the working principle of a lock-in amplifier.

We present measurement results of a MEMS cantilever at its resonance in the kHz-regime, shown in Fig 1 (a-b). At 100 fps, 12800 frames were recorded in just above 2 min, resulting in a noise floor on the order of 10 pm. A measurement of a standing surface acoustic wave field between two interdigital transducers at 25.7 MHz is shown in (c). It was reconstructed from 400 frames, corresponding to a measurement time of 10 s at 40 fps. Similar measurements using LDV can take up to several days [2]. We sincerely express our gratitude to Prof. Dr. Claudia Lenk and Dr. Jörg König for providing us with the MEMS- and SAW-samples, respectively.