# Full-field and wideband surface deformation reconstruction from the Fourier domain by spatio-temporal heterodyne interferometry (STHI)

Olivier Jacquin, Bilal Ketfi, Olivier Hugon, Eric Lacot

► To cite this version:

Olivier Jacquin, Bilal Ketfi, Olivier Hugon, Eric Lacot. Full-field and wideband surface deformation reconstruction from the Fourier domain by spatio-temporal heterodyne interferometry (STHI). Optics Express, 2024, 32 (26), pp.45541. 10.1364/OE.536522. hal-04928898

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Submitted on 4 Feb 2025

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# Full-field and wideband surface deformation reconstruction from the Fourier domain by spatio-temporal heterodyne interferometry (STHI)

OLIVIER JACQUIN,<fnref n="1" /> BILAL KETFI,<fnref n="1" /> OLIVIER HUGON,<fnref n="1" />,<fnref n="2" />
AND ERIC LACOT<fnref n="1" />,*

**Abstract:** In this paper, we propose what we believe to be a new full-field laser vibrometer designed to detect longitudinal deformations of a scattering surface, which are either harmonic, transient, stationary, or progressive (but necessarily repeatable) with a large bandwidth (10kHz – 10 MHz) using a slow camera that has a narrow detection bandwidth (<1kHz). Based on an interferometric setup, this vibrometer combines spatial and temporal beatings to access the deformation characteristics in the frequency domain. In our setup, the exposure time of the camera is used to average a heterodyne signal, giving the amplitude and phase of the deformation at a given frequency for each camera pixel. The surface under investigation is continuously illuminated, making it ideal for the study of transient or progressive vibrations. The principle and performance of our Fourier-domain laser vibrometer will be explained and illustrated by numerical and, above all, experimental examples.

Published by Optica Publishing Group under the terms of the [Creative Commons Attribution 4.0 License](https://creativecommons.org/licenses/by/4.0/).
Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

## 1. Introduction

Both optical metrology and optical non-destructive testing play vital roles in quality control, inspection, and maintenance processes across different industries. They offer non-contact, non-destructive, high-resolution measurement and inspection capabilities, making them valuable tools for ensuring product quality, process efficiency, and structural integrity [1,2]. Moreover, optical metrology and more particularly interferometry has the advantages of high spatial resolution (wavelength resolution) and high sensitivity (quantum noise or laser noise limitation) [3–6].

In the context of optical metrology and more particularly of vibrometry [7–10], we have developed a spatial and temporal heterodyne interferometer (STHI) for full-field imaging. The vibrometer that is presented in this paper to capture rapid movements (10kHz – 10 MHz) is neither a full-field stroboscopic vibrometer using a modulated (or pulsed) illumination, nor a scanning laser vibrometer using a fast photodiode [8]. Indeed, we have used an expanded laser beam that illuminates continuously the full surface under investigation and the longitudinal surface deformation (i.e. the distance variation between the laser and the surface) is finally reconstructed from the Fourier domain [11–14]. The special feature of STHI is the parallel acquisition of all pixels, enabling transient deformations to be measured in 2D with a single frequency sweep (1D). The acquisition time therefore does not depend on the number of pixels in the image, but only on the frequency range required to reconstruct the deformation. The advantage of this continuous method over pulsed illumination is that it is suitable for the study of the transient or progressive deformations. Continuous illumination optimizes the use of optical power compared to pulsed

<footnote n="1">¹Université Grenoble Alpes, CNRS, LIPhy, F 38000 Grenoble, France</footnote>

<footnote n="2">²Grenoble INP – UGA, F 38000 Grenoble, France</footnote>

<footnote>*eric.lacot@univ-grenoble-alpes.fr</footnote>
illumination, which allows to deal with less cooperative targets or to use a less powerful laser or
a less sensitive detection.

## 2. Principle of laser Fourier vibrometry

### 2.1. *Experimental STHI setup*

Figure 1 shows the off-axis heterodyne Mach-Zehnder interferometer used for the characterization of the time evolution of the sample scattering surface deformation that is caused by a piezoelectric transducer.

<figure><img src="image_3.png" /><figcaption>**Fig. 1.** STHI setup constituted by a heterodyne Mach-Zehnder interferometer for the full-field spatio-temporal analysis of the surface deformations induced by the piezo electric transducer. Reference arm (>); Sample arm (>>); He-Ne: linearly polarized helium-neon laser (Thorlabs HNL150 L); S: sample under investigation; PZT: piezoelectric transducer (Prowave 400ET180); AOM1&2: acousto-optic modulators (Intra Action AFM-402A1 @40 MHz); P: polarizer; $\frac{1}{2}$: half-wave plate; $\frac{3}{4}$: quarter-wave plate; PBS1&2: polarizing beam splitters; BS: tilted beam splitter; BE1&2: ×5 beam expanders; L1, L2: lenses; D: diaphragm; M: mirror; CMOS: camera (BASLERace ACA 2000-165-pixel size 5.5 µm × 5.5 µm); PC: personal computer; FG: 3 channels function generator (Tektronics AFG3102).</figcaption></figure>

For this measurement, the interferometer is illuminated with a polarized He-Ne laser ($\lambda_L$ = 632.8 nm, $P_L$ = 15 mW), followed by a half-wave plate and a polarizer that allows to simultaneously control the optical power sent in the setup and the power ratio between the reference and the signal (i.e. sample) interferometer arms. This allows optimizing both the interferometric contrast and the use of the camera dynamic range.

Because of the low coherence length ($\approx$10 cm) of our He-Ne laser, a variable delay line (moving mirror M) has been introduced to balance the length of the reference and the sample arms.
In this interferometric setup, the most important components are the tilted beam splitter and the two acousto-optic modulators, which allow producing an interference patterns with a spatio-temporal modulation in the camera plane (x,y) [11–14,19].

The 2D spatial modulation (x,y) is controlled by the tilted beam splitter in front of the camera. It allows recombining the sample wave and the reference wave with non-colinear propagation directions, producing a 2D interferometric pattern with linear fringes characterized by two spatial frequencies along the x and y-axis respectively:

$$F_{X,R} = \frac{k_{X,R}}{2\pi} = \frac{\sin(\theta_{X,R})}{\lambda_L} \qquad (1)$$

$$F_{Y,R} = \frac{k_{Y,R}}{2\pi} = \frac{\sin(\theta_{Y,R})}{\lambda_L}, \qquad (2)$$

where $\theta_{X,R}$ and $\theta_{Y,R}$ are the angles linked to the propagation direction of the optical reference wave in relation to the propagation direction of the sample wave along the camera optical axis (z axis).

The temporal modulation is controlled by two AOMs that allow the optical laser frequency ($F_L = \frac{c}{\lambda_L}$) to be shifted. The frequency shifts in the reference arm and in the sample arm are respectively noted $F_{AOM,1}$ and $F_{AOM,2}$. The temporal beat frequency produced after recombination of the two arms by the beam splitter (BS) is:

$$F_T = F_{AOM,2} - F_{AOM,1}. \qquad (3)$$

The $F_T$ frequency possible range is $0 \le |F_T| \le 10MHz$ due to the bandwidth of the AOM. In Fig. 1, the ×5 beam expander of the signal arm (BE2) allows illuminating a surface area of approximately 1cm² on the sample (S) and the piezoelectric transducer (PZT) allows controlling the deformation of the sample scattering surface. The L2 lens allows obtaining an image of the sample surface on the camera with a 1:1 magnification. The size of the diaphragm located near the imaging lens (L2) allows controlling the size of the speckle grains ($d_{sp}$) (generated by the back-scattering light from the sample) in the camera plane. The speckle grains must be large enough to contain at least one period of the spatial modulation while the minimum measurable period is limited by the resolution of the camera. In practice, the diaphragm size and the orientation of the tilted beam splitter are adjusted so that a speckle grain covers at least 8 × 8 pixels of the camera and there are a least 2 linear interference fringes in each speckle grain [15,16].

In the reference arm, the beam expander (BE1) and the L1 lens allows controlling the wavefront of the reference wave to avoid distortion of the linear interference fringes. Our interferometer is a polarimetric laser interferometer where the different waveplates allows controlling the polarization direction in order to optimize the interferometric contrast while minimizing the optical losses [17,18].

Moreover, to ensure a good synchronization between the start-up of the temporal modulation (by the way of the AOMs), the sample deformation (by the way of the PZT) and the acquisition of the interference images (by the way of the camera), these different events are triggered with the same computer-generated signal.

It must be underlined that the vibrometer described in this paper is not a stroboscopic vibrometer. In our case, an expanded laser beam illuminates continuously the vibrating surface under investigation. At this point one can notice that our STHI setup is very similar to the off-axis holographic setup developed by M. Gross in [19] and used for example by James et al in [11,12]. The first difference is that the STHI setup (shown in Fig. 1) generates interferometric images of the sample surface instead of holograms which avoids us from having to reconstruct the image. Other important differences come from the signal processing that is explained in the next section.
## 2.2. Signal acquisition and processing

Considering the different frequencies mentioned in the previous section, the complex amplitude of the reference electric field of our Mach-Zehnder interferometer can be written:

$$ \hat{E}_R(x, y, t) = E_R \exp[j(2\pi((F_L + F_{AOM,1})t - F_{X,R}x - F_{Y,R}y) - \varphi_R)], \quad (4) $$

where $E_R$ and $\varphi_R$ are real quantities corresponding to the amplitude and the phase shift of the reference field. The phase term $\varphi_R$ considers the optical length difference between the reference arm and the sample arm of our interferometer, which is controlled by the variable delay line. The complex amplitude of the signal field scattered by the sample is given by:

$$ \hat{E}_S(x, y, t) = E_S(x, y) \exp[j(2\pi(F_L + F_{AOM,2})t - \varphi_S(x, y) - \phi(x, y, t))], \quad (5) $$

where $E_S(x, y)$ and $\varphi_S(x, y)$ are respectively the amplitude and the phase of the speckle field generated by the laser light that is back-scattered from the surface under investigation.

In Eq. (5), $\phi(x, y, t)$ is a phase term related to the altitude $z(x,y,t)$ of the scattering target:

$$ \phi(x, y, t) = \frac{2\pi}{\lambda_L} 2z(x, y, t). \quad (6) $$

The recombination of the sample and the reference fields in the camera plane gives an interference pattern with a spatio-temporal modulation with the following frequencies: $F_{X,R}$, $F_{Y,R}$ and $F_T$. The pattern expression is given by:

$$ I(x, y, t) = \begin{cases} E_S^2(x, y) + E_R^2 \\ +2E_R E_S(x, y) \cos \left( \begin{array}{l} -2\pi(F_{X,R}x + F_{Y,R}y + F_T t) \\ +\phi(x, y, t) + \varphi_S(x, y) - \varphi_R \end{array} \right) \end{cases}. \quad (7) $$

Due to the integration time (i.e. the exposure time) $T_I$ of the camera, the time evolution of the interference pattern is averaged [20,21]:

$$ \langle I(x, y, t) \rangle_{T_I} = \begin{cases} E_S^2(x, y) + E_R^2 \\ +2E_R E_S(x, y) \times \left\{ \cos \left( \begin{array}{l} -2\pi(F_{X,R}x + F_{Y,R}y + F_T t) \\ +\phi(x, y, t) + \varphi_S(x, y) - \varphi_R \end{array} \right) \right\}_{T_I} \end{cases} \quad (8) $$

In the Eq. (8), the time average operator is defined by: $\langle f(x, y, t) \rangle_{T_I} = \frac{1}{T_I} \int_0^{T_I} f(x, y, t) dt$.

After a bandpass filtering (see Fig. 2) around the positive spatial frequencies ($F_{X,R}$, $F_{Y,R}$) in the Fourier space [22], and coming back to the real space (x,y) by the way of an inverse spatial Fourier transform, one obtains the complex interferometric signal:

$$ S(x, y, F_T) = \begin{cases} E_R E_S(x, y) \\ \times \exp[j(-2\pi(F_{X,R}x + F_{Y,R}y) + \varphi_S(x, y) - \varphi_R)] \\ \times \langle \exp[j\phi(x, y, t)] \exp[-j2\pi F_T t] \rangle_{T_I} \end{cases} \quad (9) $$

At this point, one can notice that this filtering method allows obtaining the complex interferometric signal ($S(x, y, F_T)$) and consequently the quantitative interferometric phase ($\arg[S(x, y, F_T)]$) [23] from the acquisition of a single interferometric pattern, by comparison with the four frame
<figure><img src="image_4.png" /><figcaption>Fig. 2. Numerical examples of the extraction of the complex interferometric signal $S(x, y, F_T)$ from the time averaged interferometric pattern $\langle I(x, y, t) \rangle_{T_i}$ through the filtering of the spatial Fourier space. Numerical simulation for a sinusoidal deformation at the ultrasonic frequency $F_a = 40kHz$ of a stationary Gaussian shape with a radius of 2.6 mm (52 pixels) and maximum amplitude of 200 nm ($\approx \frac{\lambda_L}{3}$). Top row (a,b,c): interference patterns: $\langle I(x, y, t) \rangle_{T_i}$; Middle row (d,e,f): square modulus of the 2D spatial Fourier transforms: $|FT[\langle I(x, y, t) \rangle_{T_i}]|^2$, in a logarithmic scale. The border of the applied bandpass filter is described by the black square; bottom row (g,h,i): modulus of the complex interferometric signals: $|S(x, y, F_T)|$, in a logarithmic scale. Left column (a,d,g): $F_T = 0kHz$; Middle column (b,e,h): $F_T = 20kHz$; Right column (c,f,i): $F_T = 40kHz$.</figcaption></figure>

phase-shifting method mentioned and used in Refs. [17,22]. Our signal processing method is also simpler than the use of a Hilbert transform that build an analytic signal from which the phase is extracted [24,25].

Figure 2 shows a numerical example for the extraction of the complex interferometric signal $S(x, y, F_T)$ from the time averaged interferometric pattern $\langle I(x, y, t) \rangle_{T_i}$. In this numerical simulation, we have studied a stationary sinusoidal deformation with an ultrasonic modulation frequency ($F_a = 40kHz$):

$$ \Delta z(x, y, t) = z(x, y, t) - z(x, y, 0) = d(x, y) \times \sin(2\pi F_a t - \varphi_a). \quad (10) $$

The simulated deformation has a Gaussian shape and the corresponding expression of the phase variation $\phi_a(x, y)$ is:

$$ \phi_a(x, y) = \frac{2\pi}{\lambda_L} \times 2d(x, y) = \frac{2\pi}{\lambda_L} \times 2d_a \times \exp \left[ -\frac{x^2 + y^2}{W_a^2} \right]. \quad (11) $$
In the numerical example showed in Fig. 2, $d_a = 210nm \approx \frac{\lambda_L}{3}$, $W_a = 2.6mm$ ($\approx 52pixels$) and $\varphi_a = 0$.

The top row of Fig. 2. shows 3 different averaged interferometric patterns $\langle I(x, y, t) \rangle_{Tl}$ obtained for 3 different beat frequencies: $F_T = 0kHz$, $20kHz$ and $40kHz$.

For $F_T = 0kHz$ and $40kHz$, the linear interference fringes induced by the surface deformation are immobilized by the temporal beating and consequently they are clearly visible inside each speckle grain (see Fig. 2(a) and 2(c)). Conversely, for $F_T = 20kHz$, no interference fringe is visible (see Fig. 2(b)) because the fringes moved across the camera plane during the exposure time and therefore the modulation amplitude is averaged to zero.

Jointly, the middle row of Fig. 2 shows the spatial Fourier transform of the interferometric pattern, which is principally composed of 3 spots located at the frequency positions ($-F_{X,R}$, $-F_{Y,R}$), (0, 0) and ($F_{X,R}$, $F_{Y,R}$) for $F_T = 0kHz$ and $40kHz$ (Figs. 2(d) and 2(f)). In Fig. 2(e), obtained with $F_T = 20kHz$, one can mainly observe the autocorrelation signal (central spot). Although their intensities are negligible, two satellite spots corresponding to the correlation signals between the reference and the sample beam are also visible because of the logarithmic scale.

These results could be easily explained by using the Bessel decomposition of the sinusoidal phase modulation [26]:

$$ \langle \exp[j\phi(x, y, t)] \exp[-j2\pi F_T t] \rangle_{Tl} = \left\{ \left( \sum_{n=-\infty}^{n=+\infty} J_n[\Phi_a(x, y)] \exp[j(2\pi n F_a t - n\varphi_a)] \right) \right\}_{Tl} \quad (12) $$

In Eq. (12), $J_n[\varphi]$ is the Bessel function of the first kind and of order n. Equation (12) shows that the time averaging of the phase modulation made by the camera is different from zero only when the beat frequency is equal to a multiple of the ultrasonic frequency $F_T = nF_a$. Otherwise, the temporal modulation is averaged to zero (i.e. no interference) during the exposure time of the camera and therefore the complex interferometric signal is equal to zero:

$$ \langle \exp[j\Phi(x, y, t)] \times \exp[-j2\pi F_T t] \rangle_{Tl} = \left\{ \begin{array}{ll} J_n[\Phi_a(x, y)] \exp[-jn\varphi_a] & \text{if } F_T = nF_a \\ 0 & \text{if } F_T \neq nF_a \end{array} \right\} \quad (13) $$

After a square band-pass filtering in the vicinity of $F_{X,R}$, $F_{Y,R}$, with a bandwidth ($\Delta F_X = \Delta F_Y \approx \frac{1}{d_{sp}}$) linked to the size of the speckle grains $d_{sp}$ and followed by an inverse Fourier Transform, the bottom row of Fig. 2. (g,h,i) shows the complex signal, where the modulus exhibits the speckle pattern without the fringes. More precisely Fig. 2(g) shows: $|S(x, y, 0kHz)| = E_R E_S(x, y) J_0[\Phi_a(x, y)]$, Fig. 2(h) shows: $|S(x, y, 20kHz)| \approx 0$, while Fig. 2(i) shows $|S(x, y, 40kHz)| = E_R E_S(x, y) J_1[\Phi_a(x, y)]$.

In Fig. 2, the argument of the complex signal $S(x, y, F_T)$ is not shown because the information that we want to extract in this numerical example (i.e. $\varphi_a$) is heavily hidden, especially by the random speckle phase $\varphi_S(x, y)$ but also by the linear phase variation $F_{X,R}x + F_{Y,R}y$, which could be easily removed if necessary by a frequency shifting ($-F_{X,R}$, $-F_{Y,R}$) in the Fourier space.

On Fig. 2(c), the linear interference fringes are visible on the time averaged interferometric pattern, which indicates that they are apparently immobilized. This happens when the temporal beat frequency ($F_T$) is equal to a frequency component of the phase modulation signal and consequently, the complex interferometric signal is different from zero.

Similarly to a detection setup with a lock-in amplifier, Eq. (9) shows that $S(x, y, F_T)$ contains $\langle \exp[j\Phi(x, y, t)] \times \exp[-j2\pi F_T t] \rangle_{Tl}$, which is linked to the Fourier component of $\exp[j\Phi(x, y, t)]$ at the RF frequency. In our case, the camera plays the role of the low-pass filter whose bandwidth is fixed by the integration time of the camera. Therefore, an arbitrary phase $\Phi(x, y, t)$ (and consequently an arbitrary deformation $\Delta z(x, y, t)$) of any shape (transient, stationary, propagative, ...) can be determined by sampling the entire spectrum of the time domain signal at each
position (x,y). To achieve this, one must acquire an odd number (2N + 1) of complex images obtained at different beat frequencies $F_{T,n}$ satisfying the following conditions:

$$-F_{T,max} = -\frac{N}{T_I} \le F_{T,n} = n\Delta F_I = n\frac{1}{T_I} \le F_{T,max} = \frac{N}{T_I}. \quad (14)$$

In Eq. (14), n is an integer with $n \in [-N, N]$. Next one can calculate (for each pixel positions (x,y)) the inverse temporal Fourier transform of $S(x, y, F_T)$ to obtain the complex interferometric signal in the time domain:

$$\hat{S}(x, y, t_m) = \sum_{n=-N}^{N} S(x, y, F_{T,n}) \exp[+j2\pi F_{T,n}t_m], \quad (15)$$

$$\hat{S}(x, y, t_m) = \left\{ \begin{array}{ll} \frac{E_R E_S(x, y)}{T_I} & \times \exp[j(-2\pi(F_{X,R}x + F_{Y,R}y) + \varphi_S(x, y) - \varphi_R)] \\ & \times \exp[j\Phi(x, y, t_m)] \end{array} \right\}. \quad (16)$$

In Eq. (16), $0 \le t_m = m\frac{T_I}{2N} \le t_{max} = T_I$, where $m \in [0, 2N]$. Then, we extract the argument of the time domain complex signal:

$$\Psi_S(x, y, t) = \operatorname{Arg}[\hat{S}(x, y, t)] = \phi(x, y, t) + \Psi_0(x, y), \quad (17)$$

where

$$\Psi_0(x, y) = -2\pi(F_{X,R}x + F_{Y,R}y) + \varphi_S(x, y) - \varphi_R \quad (18)$$

is an unwanted static phase term.

Finally, the deformation is obtained by subtraction with a reference measurement at the time $t=0$ in order to eliminate the unwanted phase term:

$$\Delta z(x, y, t) = \frac{\lambda_L}{2} \times \frac{(\Phi(x, y, t) - \Phi(x, y, 0))}{2\pi} = \frac{\lambda_L}{2} \times \frac{(\Psi_S(x, y, t) - \Psi_S(x, y, 0))}{2\pi}. \quad (19)$$

At this point, one can notice that this part of the signal processing is only possible if the speckle field keep correlated during deformation, which means that it needs to be small compared to the size of the speckle grains in the direction of deformation.

# 3. Laser Fourier vibrometry: experimental results

## 3.1. Static deformation of a thin scattering plate

In order to explain the signal processing on a simple example, we first consider a static deformation before the study of a dynamic case. Our first experimental demonstration is the measurement of a static deformation obtained by means of a tip mounted on a piezoelectric transducer that pushes a thin scattering plate:

$$\Delta z(x, y, t) = d(x, y). \quad (20)$$

The deformation has a unique Fourier component:

$$\langle \exp[j(\Phi(x, y, t) - \Phi(x, y, 0))] \times \exp[-j2\pi F_T t] \rangle_{TI} = \begin{cases} \exp\left[j\frac{2\pi}{\lambda_L} \times 2d(x, y)\right] & \text{if } F_T = 0 \\ 0 & \text{if } F_T \neq 0 \end{cases} \quad (21)$$
In this straightforward situation, the complex interferometric signal obtained from a single interferometric pattern is reduced to:

$$S(x, y, F_T = 0) = \left\{ \begin{array}{l} E_R E_S(x, y) \\ \times \exp[j(-2\pi(F_{X,R}x + F_{Y,R}y) + \varphi_S(x, y) - \varphi_R)] \\ \times \exp[j\frac{2\pi}{\lambda_L} \times 2d(x, y)] \end{array} \right\}. \quad (22)$$

In this case, Eq. (15) gives:

$$\hat{S}(x, y) = S(x, y, F_T = 0). \quad (23)$$

Static deformation is obtained from the argument of this complex signal:

$$\Psi_S(x, y) = \operatorname{Arg}[\hat{S}(x, y)] = \frac{2\pi}{\lambda_L} \times 2d(x, y) + \Psi_0(x, y). \quad (24)$$

Figures 3(a) and (b) show respectively the raw phases of the signal and of the reference where no deformation are directly visible due to high frequency noise induced by a phase wrap effect (phase jumps in the vicinity of $\pi$ and $-\pi$). Finally, Fig. 3(c) shows the raw phase difference ($\Psi_S(x, y) - \Psi_0(x, y)$) extracted directly from our measurements, where we can begin to observe the effect of the deformation with a high frequency noise induced by a phase wrap effect (phase jumps in the vicinity of $2\pi$ and $-2\pi$). After a low-pass filtering in the spatial Fourier space and the application of a 2D unwrapping algorithm (MLBT) [27], one obtains the smooth static deformation $d(x, y)$ shown in Fig. 3(d) where the noise level is linked to the width of the Gaussian smoothing. The choice of the filter width results from a trade-off between the transversal spatial resolution of the deformation and the quality of the phase unwrapping. In the presented example, we used a Gaussian filter with a width of 32 × 32 pixels, corresponding to a spatial resolution of roughly 180 µm.

## 3.2. Transient sinusoidal deformation of the sample surface

Our second experimental demonstration is the measurement of a transient sinusoidal deformation of the surface of a piezoelectric transducer, which is driven by a function generator (see Fig. 1). The deformation can be expressed in the following way:

$$\Delta z(x, y, t) = \frac{\lambda_L}{2} \frac{\phi_a(x, y)}{2\pi} \times \sin(2\pi F_a t - \varphi_a) \times \Pi_{\tau}(t - \tau_c) = d(x, y) \times \sin(2\pi F_a t - \varphi_a) \times \Pi_{\tau}(t - \tau_c), \quad (25)$$

where $d(x, y)$ and $\varphi_a$ are respectively the amplitude and the phase of the deformation at the ultrasonic frequency: $F_a = 40kHz$. $\Pi_{\tau}(t)$ is a gate function with a temporal width of $\tau = 30ms$ shorter than the exposure time $T_I = 50ms$ of the camera. Experimentally, the transient deformation starts with a delay of 5 ms after the beginning of the interference recording and therefore is centered at $\tau_c = 20ms$.

As already explained, our interferometric setup with a spatio-temporal modulation allows obtaining complex signals that are proportional to the temporal Fourier transform (FT) of the phase modulation:

$$S(x, y, F_T) \propto \langle \exp[j\Phi(x, y, t)] \times \exp[-j2\pi F_T t] \rangle_{TI} \propto FT \left[ \exp[j\Phi(x, y, t)] \Pi_{TI}\left(t - \frac{T_I}{2}\right) \right] \quad (26)$$

For two different pixels (among the 1024 × 1024 pixels of the recorded interferograms), one in the center of the interferograms (near the maximum of the deformation), and one near a corner of the interferograms (with a smaller deformation), Fig. 4 shows the complex spectra extracted from the 603 recorded interferograms obtained from different beat frequencies with a step of 1 Hz
<figure><img src="image_5.png" /><figcaption>Fig. 3. Experimental image (512 × 512 pixels) of a static deformation (Δz(x, y, t) = d(x, y)) of the sample surface; (a): raw signal phase (b): raw reference phase (c): raw phase difference (d): static deformation obtained after unwrapping and filtering of the raw phase difference.</figcaption></figure>

($<\frac{1}{T_I} = 20Hz$), which allows oversampling the Fourier spectrum for a better observation of its measured shape.

More precisely, Figs. 4(a), 4(b) and 4(c) show respectively the complex spectra obtained for the 201(×3) beat frequencies in the 3 measurement ranges: 0kHz ± 100Hz, 40kHz ± 100Hz and -40kHz ± 100Hz. The simultaneous observation (on the same plot) of both the envelope (width = 30 ms) and the 40kHz carrier (period = 0.025 ms) is not possible. For visualization purpose, the acquired vibration spectrum has been numerically translated around 201 Hz. By this way, a simultaneous observation of both the envelope and the carrier signal is possible in Fig. 4(d).

The experimental spectra are in good agreement with the theoretical calculation of the Fourier transform of the transient sinusoidal deformation of the sample surface. For simplicity, assuming that $max[\Phi(x, y, t)]<<\pi$ (i.e. $d(x, y)<<\lambda$) and for an integration time $T_I>\tau$, Eq. (26) becomes:

$$FT \left[ \exp[j\Phi(x, y, t)]\Pi_{T_I}\left(t - \frac{T_I}{2}\right) \right] \approx FT \left[ \left[1 + j\Phi(x, y, t)\right]\Pi_{T_I}\left(t - \frac{T_I}{2}\right) \right] \\ \approx \left\{ \begin{array}{l} T_I \sin_c(\pi FT_I) \exp\left(-j2\pi F \frac{T_I}{2}\right) \\ \quad + \frac{\Phi_a(x,y)}{2} \tau \sin_c(\pi(F - F_a)\tau) \exp(-j2\pi(F - F_a)\tau_c) \\ \quad - \frac{\Phi_a(x,y)}{2} \tau \sin_c(\pi(F + F_a)\tau) \exp(-j2\pi(F + F_a)\tau_c) \end{array} \right\} \quad (27)$$
<figure><img src="image_6.png" /><figcaption>Fig. 4. Experimental measurement of a transient sinusoidal deformation of the sample surface for two different pixels. a) 0kHz ± 100Hz, b) 40kHz ± 100Hz, c)–40kHz ± 100Hz, d) reconstructed deformation of the sample surface for two different pixels. Central pixel (512,512): blue dotted line, corner pixel (256,256): red solid line. Intentionally lowered to ±201 Hz for a better visualization of the transient oscillations.</figcaption></figure>

and therefore:

$$|S(x, y, F_T)|^2 \approx [E_R E_S(x, y)]^2 \times \left\{ \begin{array}{l} T_I^2 \sin^2(\pi F T_I) \\ + \left[ \frac{\Phi_a(x,y)}{2} \right]^2 \tau^2 \times \left[ \begin{array}{l} \sin^2(\pi(F - F_a)\tau) \\ + \sin^2(\pi(F + F_a)\tau) \end{array} \right] \end{array} \right\}. \quad (28)$$

In agreement with Eq. (28), one can observe in Fig. 4(a), the oscillations of the sinc function with a spectral width of: $\frac{1}{T_I} = 20Hz$, while Figs. 4(b) and 4(c) also show a sinc function with a wider spectral width: $\frac{1}{\tau} = 33Hz$, because the surface deformation is a transient one ($\tau < T_I$).

The comparison of the maxima of the spectra at 0kHz and at +/- 40kHz (intentionally lowered to +/-201 Hz) shows that the deformation is not the same for the 2 positions corresponding to the 2 pixels analyzed in this example.

Also, Eq. (27) allows to understand that the extraction of the transient information $\tau_c$ linked to the temporal position of the time gate is included in the phase of the Fourier transform and therefore its measurement depends on the phase stability of our Mach-Zehnder interferometer [3,4].

From our 603 (i.e. 201 × 3) interference measurements corresponding to a frequency range intentionally lowered to –301 Hz to 301 Hz with a step of 1 Hz, and by using zero padding to enhance the temporal definition of the reconstructed signal, we built a 4 times extended complex
interferometric signal (i.e. a 4 time extended frequency domain) made up of 2409 frequencies between -1204 Hz (-301 Hz × 4) and 1204 Hz (301 Hz × 4) with a 1 Hz step.

Clearly, Fig. 4(d) shows the transient sinusoidal deformation at the two observed positions, obtained by making an inverse Fourier transform of the complex extended spectra mentioned just above. One can observe that the time gate has been well reconstructed at the right place (30 ms time width with a starting delay of 5 ms). Again, one can understand that the simultaneous observation (in Fig. 4(d)) of both the long-time gate (30 ms) and the oscillations is possible because the vibration frequency (+/-40kHz) has been numerically analyzed as if it was lower frequencies of +/-201 Hz in the Fourier spectrum reconstructed with the experimental measurements.

Coming back to the whole surface analysis (i.e. the analysis of all pixels), our measured complex interferometric signals correspond to a data volume of: 1024 × 1024 pixels × 603 frequencies, which is extended to: 1024 × 1024 pixels × 2409 frequencies by using zero padding. From this data volume, a numerical inverse Fourier Transform is carried out pixel by pixel and finally the time evolution of the surface deformation (Δz(x, y, t)) is obtained by extracting the argument of the time domain complex signals (see Eq. (19)) and the subtraction of a reference phase. In the same way as for the measurement of the static deformation, a low pass filtering (Gaussian smoothing with a lower width of 5 × 5 pixels) and the application of a 2D unwrapping algorithm (MLBT) have been made for each of the time values to obtain a smooth surface deformation with a spatial resolution of roughly 1mm × 1mm.

The transient time evolution of the whole surface experimentally obtained is shown in Fig. 5, where Figs. 5(a), 5(b) and 5(c) correspond respectively to a minimum, a zero and a maximum of the oscillation during the transient time gate with a deformation amplitude of ≈50nm, while Fig. 5(d) shows a zoom of the measured noise that is detected after the transient time gate. The standard deviation on this noisy surface is equal to 0.3nm (≈ λ/f 2000 ). The noise level is lower than

<figure><img src="image_7.png" /><figcaption>**Fig. 5.** Transient temporal evolution of the whole illuminated surface (i.e. of the 1024 × 1024 measured pixels) for 4 particular time values. a) $t_a = 16,7$ms (near the minimum of the oscillation); b) $t_b = 17,9$ms (near the zero of the oscillation); c) $t_c = 19,1$ms (near the maximum of the oscillation); d) $t_d = 49,82$ms (zoom of the measured noise after the transient time gate). The surface oscillation was completely reconstructed over a time of 50 ms with a time step of 300µs (see Visualization 1).</figcaption></figure>
the noise on Fig. 3(b) because of a narrower filtering in the spatial frequency domain, resulting in a weaker transversal spatial resolution of the deformation.

Data processing was carried out using MATLAB and a PC with the following characteristics: 4 Ghz processor with 16 cores and 512Gb RAM. Data processing requires the creation and manipulation of complex 3D matrices with the following dimensions: 1024 × 1024 × 2409, requiring approximately 50Gb of memory. The duration of the whole process for the obtaining of the spatio-temporal evolution of the surface deformation of the sample was roughly equal to 15 minutes, including 3 minutes for the acquisition of the 603 interferograms and 12 minutes for the signal processing (0.5 minute for the reading of the interferometric images, 2 minutes for the extraction of the complex interferometric signals, 4 minutes for the inverse Fourier transform of the extended data volume and 6 minutes for the Gaussian smoothing and the use of the unwrapping algorithm).

## 4. Conclusion and perspectives

In this paper, we have presented the operating principle of a laser vibrometer using a continuous illumination to capture ultrasonic surface vibration (10kHz-10 MHz) with a slow camera (detection bandwidth <1kHz). To capture rapid movement with a slow detection, our laser vibrometer uses spatio-temporal heterodyne interferometer, which is a Mach-Zehnder interferometer with a spatio-temporal modulation. The spatial beating allows extracting (by spatial filtering) the interferometric phase information from a single interferometric image. The temporal modulation averaged by the camera response time allows obtaining (by a lock-in detection principle) frequency by frequency the Fourier spectrum of the phase modulation induced by the deformation at each pixel position. This laser vibrometer allows detecting a surface deformation that is either harmonic or transient, stationary or progressive, but this deformation must necessarily be repeatable.

The performances of this laser vibrometer working in the Fourier domain have been illustrated with the experimental detection of a repeatable transient ultrasonic vibration (40kHz) with a vibration amplitude of 50.0 +/- 0.3 nm during the vibration time gate. The good temporal reconstruction (position and width) of the transient time gate clearly shows that the phase noise or the phase drift of our interferometer are sufficiently weak to allow correct phase measurements of the Fourier components and therefore a good temporal reconstruction of the time evolution of the surface deformation.

In a near future, we will investigate the possibility to reconstruct the transverse (x, y) deformations of a sample with this STHI technique. We also plan to use it to do Photo-Acoustic Tomography (PAT) with optical detection [28-30]. In this imaging modality, the accurate detection of the time evolution of surface deformations induced by acoustic pressure waves, which have been generated by optical absorption inside a sample, is very important for the 3D reconstruction of the absorption zone by a triangulation technique [31,32].

**Funding.** Agence Nationale de la Recherche (ANR-22-ASTR-0026).

**Acknowledgment.** The authors thank Leo Djevahirdjan for technical assistance.

**Disclosures.** The authors declare no conflicts of interest.

**Data availability.** Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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