Multi-point vibrometer based on high-speed digital in-line holography Julien Poittevin,1,2 Pascal Picart,1,3,* Charly Faure,1 François Gautier,1,3 and Charles Pézerat1,3 1

Université du Maine, CNRS UMR 6613, LAUM, Avenue Olivier Messiaen, 72085 Le Mans, Cedex 9, France 2

IRT Jules Verne, Chemin du Chaffault, 44340 Bouguenais, France

3

ENSIM, Ecole Nationale Supérieure d’Ingénieurs du Mans, rue Aristote, 72085 Le Mans, Cedex 09, France *Corresponding author: pascal.picart@univ‑lemans.fr Received 22 December 2014; revised 6 March 2015; accepted 7 March 2015; posted 9 March 2015 (Doc. ID 231309); published 6 April 2015

This paper describes a digital holographic setup based on in-line holography and a high-speed recording to get a multipoint vibrometer. The use of a high-speed sensor leads to specificities that enable the in-line configuration to be used. The case of transient vibrations is investigated through a full simulation of the holographic process. The simulation shows that the first instants are critical since distortion may occur, resulting in errors in the phase measurement. Experimental results are provided by exciting an aluminum beam with a transient signal. A comparison with the velocity measured by a pointwise vibrometer is provided. Frequency response functions are extracted and the experimental results confirm the ability of the method to provide full-field contactless measurements at the high-speed time scale evolution of the vibration. © 2015 Optical Society of America OCIS codes: (090.0090) Holography; (090.1995) Digital holography; (090.1760) Computer holography; (100.3010) Image reconstruction techniques; (120.7280) Vibration analysis; (120.7250) Velocimetry. http://dx.doi.org/10.1364/AO.54.003185

1. Introduction

In the domains of acoustics, vibro-acoustics, vibrations of structures, or flow-induced vibrations, laser Doppler vibrometer (LDV) is the most favored instrument for dynamics measurements [1–4]. This technique is based on heterodyne interferometry or self-mixing in a laser diode and may include fiber optics to compact the system [1,5–14]. The use of a laser beam to probe a dynamic object provides noncontact, highaccuracy, and high-resolution measurements. In addition, it allows remote measurement of displacement, velocity, and acceleration of vibrating objects to be obtained [1]. With such instruments, displacements on the order of less half the wavelength of the laser source are measured. A large variety of applications, such as modal analysis, on-line quality control, vibration analysis of operating machines, and biomedical 1559-128X/15/113185-12$15.00/0 © 2015 Optical Society of America

applications were demonstrated [2–4]. The main drawback of LDV is that it can only yield the dynamic data at a particular location on the inspected surface. In order to get full-field data, a laser-scanning mechanism is required and calibration must be carried out [15–17]. Currently, motor-driven scanning mirrors are used to move the measurement probe from one point to another. This operation needs a long time and the dynamic specimen of interest must be quite stationary (i.e., highly controlled excitation). To get simultaneously a collection of data points at the surface of the inspected vibrating object, multipoint vibrometers were developed [18–22]. Such techniques are based on line scanning [18,19] (typically 256 points along a line at up to 80 kHz in [18]), on holographic optical elements associated with a complementary metal-oxide semiconductor (CMOS) sensor (vibration at up to 100 Hz measured in [20]), on frequency multiplexing (20 points with 5 × 4 beams in [21]), or also on three acousto-optic devices and a single high-speed photodetector (5 × 4 beams with a 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS

3185

rate at 500 million samples/s in [22]). Although these techniques are useful to give a set of measurements at several independent surface points, the number of simultaneous measurements at “one shot” is relatively low. Full-field evaluation can be obtained with holographic and speckle interferometry [23]. Both approaches are well-suited because of the high density of measuring points and the reduced measurement time. Vibration analysis with optical holographic interferometry began with the works of Powell and Stetson [24,25] who first established the principle of time-averaging. The use of time-averaging and quasitime-averaging in digital Fresnel holography was then discussed [26–28]. In time-averaging related type recordings, the phase of the vibration signal is lost. The phase relation between points can be obtained using a sinusoidal modulation of the reference wave, leading to a full mapping of the vibration amplitude [29]. Time-averaging is a useful tool for studying vibrations and some interesting examples can be found in Refs. [30–35]. In the past, speckle interferometry (also similar to “digital image-plane holography” [36]), was demonstrated to provide an efficient tool in combination with time-averaging and digital image processing [37], fiber optics and acousto-optic stroboscopic illumination [38], laser-diode modulation [39], and pulsed laser coupled to a classical vibrometer [40]. In the past, the use of stroboscopic [41–43] or pulsed light [44–48] resulted in the direct measurement of the vibration amplitude. Such methods are quite useful for modal analysis and the determination of structural intensity [49,50], to measure 3D vibrations [40], surface acoustic wave [51], the high amplitude autooscillations of a clarinet reed [52], or 3D displacement on a cat’s post-mortem tympanic membrane [53]. Hybrid techniques combining heterodyne holography with time-averaging or frequency shifting were also proposed recently [54–56]. However, the stationary regime is a particular case for investigating the structure vibration behavior, and the characterization of structures under operational or real functioning conditions requires analysis in the time domain. Then, providing a real-time follow-up of the vibration amplitude, whatever the excitation condition is a challenge for full-field optical metrology. As examples, problems that cannot be addressed by a stationary approach are: vibrations of panels induced by hydro or aero-acoustic sources, and structural vibration induced by squeak and rattle noise. A full-field and contactless method having the capability of measuring vibration simultaneously at several locations, and at its time scale then must be developed. Performances of high-power continuous wave lasers (>6 W) and high-speed CMOS sensors (rate up to 1 MHz) have been improved significantly these past years. Such technologies give opportunity of merging holographic interferometry and vibration analysis to develop an adapted approach for a real-time and multi-point recording of transient phenomena. The use of high-speed recordings was 3186

APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015

reported in [57–63]. The optical setup is then considerably simplified since it does not require a pulsed (or double-pulsed) laser, or the stroboscopic generation of light pulses. Demonstrated applications concern modal analysis [63], life science [64], and acoustic wave inspection [65,66]. In [60–64], the digital processing of holograms is based on off-axis holography. A spatial filtering in the Fourier domain is required to get the useful optical phase after inverse Fourier transforming the filtered hologram spectrum. In this paper, we propose an alternative approach based on in-line highspeed digital holography that does not require any Fourier filtering and that can give the optical phase by numerically propagating the digital hologram into the object plane. Since the holograms are recorded with a high-speed matrix sensor, the real-time recording of transient phenomena, at their time-scale evolution, is made possible. The proposed approach is then not limited to stationary phenomena. The multipoint quantitative vibration analysis is demonstrated for a wideband excitation signal and a comparison with a single-point vibrometer is proposed. The use of a highspeed sensor leads to specificities that enable the inline configuration to be quite operational. Particularly, high-speed sensors use large pixels (15–28 μm) which considerably influence the recording conditions. We show that large pixels lead to the in-line configuration, and thus to an optimization of the spatial bandwidth of the sensor, without any requirement for filtering in the Fourier domain of the hologram. The dynamic temporal evolution of the phenomena under interest requires an evaluation of the accuracy at the first instants of the recording. Indeed, as will be shown in the paper, the strong slope of the transient phase at the first instants may be at the origin of measurement errors. Such errors are investigated by using a full numerical simulation of the recording/reconstruction process. So, this paper is organized as follows: Section 2 gives the theoretical background of digital holography and Section 3 discusses the specificities induced by the use of a high-speed sensor. In Section 4, a full numerical simulation of the holographic process is proposed to evaluate the accuracy of the measurement at the first instants of the transient phenomenon. In Section 5, experimental results are provided to demonstrate the use of in-line holography as a multipoint vibrometer. Comparison with results obtained using a single point vibrometer, for the same wideband excitation conditions, exhibit a very good agreement, thus validating the proposed method. 2. Theoretical Background A. Holographic Recording

Let us consider the interference between a smoothplane reference wave and a diffracted object wave. In the recording plane, the hologram is expressed as [67] H
jRj2  jOj2  R O  RO ; 0

0

0

0

(1)

where Rx ; y 
aR exp−2iπu0 x  v0 y  is the reference plane wave, with spatial frequency u0 ; v0 , and O

is the wave diffracted by the object located at the distance d0 from this plane. In the in-line configuration, we have u0 ; v0 
0; 0 (no tilt between reference and object waves), meaning that the three diffraction orders of Eq. (1) are superposed in the reconstructed plane. The object wave at the recording plane is given by a convolution formula (the asterisk means convolution) Ox; y; d0 
A  hx; y;

(2)

where Ax; y
A0 x; y expiψ 0 x; y is the object wave front at the object plane and hx; y; d0  is the impulse response of free-space propagation along distance d0 . Accordingp to [68], we have (λ is the wavelength and i
− 1) qi h exp 2iπ∕λ d20  x2  y2 id0 : hx; y; d0 
− λ d20  x2  y2

(3)

3. Specificities of Very High-Speed Recording

The digital hologram given in Eq. (1) is sampled by M × N sampling points corresponding to the number of pixels of the recording sensor, having pixel pitches px and py . B.

Reconstruction of the Holographic Image

The main way to compute the reconstruction of digitally recorded holograms is based on Eq. (2). At any reconstruction distance dr , which may be different from d0 , the numerically reconstructed hologram corresponds to the discrete version of the equation (discrete Fresnel transform) given in Eq. (4) [67,69–71]. The algorithm uses K; L data points, with generally K; L ≥ M; N. The pixel pitches of the reconstructed image in the x and y directions are given, respectively, by Δη
λdr ∕Lpx and Δξ
λdr ∕Kpy . The spatial resolution in the reconstructed plane are ρx
λdr ∕Npx and ρy
λdr ∕Mpy , respectively, for the x, y directions of the set of reference axis attached to the object     i 2iπdr iπ 2 exp exp x  y2  Ar x; y; dr 
− λ λdr λdr ×

l
L∕2 X k
K∕2 X

Hlpx ; kpy 

l
−L∕2 k
−K∕2

 iπ 2 2 2 2 l px  k py  × exp λdr   2iπ lp x  kpy y : × exp − λdr x 

(4)

From the numerical computation, the amplitude and phase of the diffracted field can be evaluated. C.

the dimension of the experimental setup is directly proportional to the size of the object. Considering the in-line configuration, the reconstructed object may occupy a large part of the reconstructed field. For example, using a wavelength at 532 nm, to study an object size 15 cm × 15 cm, with pixels 20 μm, the distance must be at least 5.6 m [70]. In order to respect the Shannon criteria with an acceptable size of the setup, and to maintain an acceptable illumination (a 10 m long setup will require a very high-power laser), the use of a negative lens was introduced by Schnars et al. [72], then generalized to a lens assembly by Mundt and Kreis [73]. The negative optical system provides a virtual image of the object at a closest distance from the sensor. In the following we note d0 the physical distance between the object and the sensor and d00 the distance between the virtual object produced by the negative lens and the sensor.

Case of Large Objects

The Shannon conditions impose a general rule [69,70]: the reconstructed object must be included in the reconstructed field. In the Fresnel approach, the width of the reconstructed field is equal to λdr ∕px . So,

A. Ratio between the Reference and the Object Waves

We must point out that the use of very high-speed recording leads to very short exposure time [i.e., a few microseconds (μs) or hundreds of nanoseconds (ns)]. Furthermore, when considering a specimen with a rough surface (typically noncooperative target), the photometric efficiency is very low since a few number of photons are really contributing to the object wave in the recording plane. Thanks to the coherent mixing by heterodyning with the reference wave, the object wave is amplified by the reference wave [term R O in Eq. (1)]. So, a weak object wave is balanced by a strong reference wave. This means that the autocorrelation of the object wave [70], given by the discrete Fresnel transform of term jOj2 in Eq. (1), is quite irrelevant compared to the other terms, especially compared to jRj2 . B. Energy Dilution in the −1 Order

When dr ≠ −d00 , the 1 order [term R O in Eq. (1)] is not in focus, meaning that the image of the object is blurred. When dr
−d00 , the 1 order is in focus, and the −1 order [term RO in Eq. (1)] is completely blurred. Reference [70] provided the expression giving the contribution of the focus error to the blurring of the image. This contribution is the width of the point spread function of the object-to-image relationship when defocus occurs, and is given by Eq. (5)    dr   (5) ρxf
Npx 1  0 : d0 A similar relation holds for the y direction. Equation (5) only depends on the sensor width in the considered direction and on ratio dr ∕d00 . It vanishes in the case of perfect focus for 1 order (dr
−d00 ). However, the −1 order is quite blurred and each point of the initial object is spread over a width equal to 2Npx [Eq. (5) with dr
d00 ]. If we consider a point source at the object surface emitting a power P0, then 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS

3187

the illumination produced by this point in the reconstructed plane is I 1
P0 ∕ρx ρy  for the 1 order and I −1
P0 ∕ρxf ρyf  for the −1 order. The ratio I 1 ∕I −1 is then equal to I 1 4N 2 M 2 p2x p2y
: I −1 λ2 d2r

(6)

This ratio is proportional to the square of the pixel pitches and inversely proportional to the distance dr . With the use of a high-speed sensor having large pixel pitches and for which distance dr shall be smaller because of the use of the negative lens, then the ratio is quite larger compared to the use of a classical sensor with small pixels. This means that the out-of-focus −1 order can be superposed to the useful 1 order without contributing to a significant perturbation to the reconstructed image. For example, consider a high-speed sensor with px
py
20 μm and M
N
512, and a “slow” sensor with px
py
3 μm and M
N
1024, for λ
532 nm and dr
1000 mm. Then, we have I 1 ∕I −1
1.55 × 105 for the high-speed sensor and I 1 ∕I −1
1.25 × 103 for the “slow” sensor; that is, a factor 123.4. As a conclusion, an in-line holographic configuration can be considered when using a high-speed sensor with large pixels. C.

Pre-Processing of Digital In-Line Holograms

According to Sections 3.A and 3.B, the main contributors to the hologram are the reference wave jRj2 and the 1 order, the −1 order being very weak. Generally, sensors include a protection window facing the pixel matrix. Unfortunately, and despite a specific coating, a fixed noise pattern due to interferences of the reference wave may exist. It follows that this noise pattern induces pollution in the reconstructed image since it is in the same order of magnitude as that of the 1 order. So as to remove such perturbation, the reference wave must be recorded separately and removed in a pre-processing of the holograms. The processed hologram is written according to Eq. (7) H p
H − jRj2 ≅ R O  ε;

(7)

recording regime is equivalent to a freezing of the object at the instant at which the recording is performed (“impulse regime”) [44–52]. When, on the contrary, α ≫ 1, the regime is said to be “time-average.” The object reconstructed from the digital hologram is then amplitude-modulated by a Bessel function [25]. In experiments for which 0 ≪ α < 1, the cyclic ratio is too high to be classified as “impulse” and too short to be considered as “time-average” and this intermediary regime is called “quasi-time-averaging” [28]. It follows that considering the cyclic ratio α precisely defines the recording regime. For a transient regime, the cyclic ratio can not be defined since the movement/ vibration includes a large number of harmonic signals. Then the influence of the exposure time on the dynamic phase measurement must be evaluated in a way which completely differs from the classical analysis. In this paper, we propose to analyze the effect of the time exposure by taking into account a numerical simulation of the full holographic process, from the dynamic object producing the phase variation to the final quantitative phase evaluation after recording and processing the hologram. In order to simulate a realistic transient phase variation, we consider the case of an object submitted to a local impact. This object is supposed to be illuminated by a laser, and diffracts a wave onto the sensor to produce a digital hologram. Section 4.B discusses the full numerical simulation. B. Full Simulation of the Holographic Process

The numerical simulation was developed by taking into account the full acquisition and reconstruction process, including the object excitation, optical wave propagation, interference pattern, recording and processing, and the final quantitative phase evaluation. Figure 1 shows the scheme of the full simulation. The dynamic object was chosen to be a simply supported aluminum beam submitted to a local impact at its center. This choice of structure and boundary conditions was motivated by the fact that beams are structures having vibration modes with analytic solutions [74,75]. The impact generates a transient flexural vibration of the beam. This transient vibration

and used for the computation with Eq. (4). The variable ε in Eq. (7) includes the noise produced by the weak terms jOj2 and RO . 4. Case of Transient Dynamic Vibration A.

Time Evolution

Section 3 discussed the specific characteristic of highspeed sensors. Section 4 aims at discussing about specificities related to the time evolution of the transient phase change that has to be recovered. Classically, one distinguishes several recording regimes for stationary sinusoidal movements/vibrations. The parameter of interest is the cyclic ratio α
T∕T 0 defined by the ratio between the exposure time T and the vibration period T 0 . Typically, if α ≪ 1, the 3188

APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015

Fig. 1. Scheme for simulating the full holographic process.

includes vibration of a lot of modes, giving a temporal bandwidth to the dynamic phase that is encoded in the digital hologram. The physical parameters included in the model of the transient vibration are the Young’s modulus E, the moment of inertia of the cross section of the beam I, the Poisson ratio ν, the thickness h, the dimensions Lx × Ly (with Lx ≪ Ly ), and the mass density ρ. Equation (8) gives the solution, expressed in the modal basis, of the out-of-plane displacement field due to the flexural motion [74,75] wx; y; t


X F 0 ϕn x0 ϕn x p 2 2 n M n ωn 1 − ξn  q × sin ωn 1 − ξ2n t exp−ξn ωn t:

(8)

In Eq. (8), we have ϕn x
sin

  nπx ; Lx

(9)

being the expression of eigenshapes of the simply supported beam, x0 ; y0  being the coordinates of the excitation point M n
ρh

Lx Ly ; 2

is the generalized modal mass, and  2 s nπ EI ; ωn
Lx ρhLy

(10)

(11)

2π 1  cosθwx; y; t; λ

Effect of Exposure Time

The effect of the exposure time can be gauged for the first instants of the transient phase, for which slopes in the signal are relatively strong. When the temporal slope is very strong, the phase variation is integrated during the exposure time, and this creates a distortion in the recorded phase. Consider the temporal integration of H during T, and focus on the complex term related to the transient phase given in Eq. (13). Then, we have Z t T 0 expiψx;y;tdt
T expiψx;y;t0  t0

(14) (12)

corresponds to the modal damping factor. The term η is the structural damping and F 0 is the modulus of the applied force. The dynamic optical phase generated by the vibration of the structure is related to the displacement field by ψx; y; t


C.

 Tqψ;t0 ;TexpiΘψ;t0 ;T:

is the modal angular frequency, and ξn
η∕2;

1024 pixels × 1024 pixels. Physical parameters are F 0
1 N, Lx × Ly
20 mm × 500 mm, h
1.5 mm, Young’s modulus E
70 GPa, Poisson ratio ν
0.33, and a damping ratio ξn
0.5 × 10−3 . The excitation is performed at the location (x0
10 mm, y0
250 mm) and a bandwidth (1,5,10,20) kHz. The displacement field is calculated by using a truncated modal basis of the beam with the 50 first modes and is considered to be the exact displacement in the following (high-order modes having smaller amplitudes and higher attenuations). Figure 2 shows the time evolution of the displacement [Eq. (8)] at the impact point x0 ; y0  for different modal bandwidths. It can be seen that it exhibits large amplitude at the first instants, especially for bandwidth larger than 10 kHz. After the first few 0.01 s, the time evolution is smoother and exhibits smaller slopes.

(13)

where θ is the illumination angle and λ the laser wavelength. The second part of the full simulation deals with the optical wave propagation (diffraction and interferences), and the recording process (sensor with spatial and temporal rates, digitization with 8 bits). After recovering the phases from the digital holograms, the error between the initial input phase [Eq. (13] and the one obtained with the simulation can be evaluated. This error is the key to highlight the critical parameters of the experimental setup. For the simulation, these recording parameters were chosen: exposure time T
0.1; 1; 5; 10 μs, frame rate f s
1; 5; 10; 50; 100 kHz, and M × N


The complex term q expiΘ depends on the dynamic phase ψ, the instant at which the recording is started, t0 , and the exposure time T. From Eq. (14) and Ref. [48] the estimated phase from the reconstructed hologram can be written as  q sinψ − Θ : (15) ψˆ
ψ − arctan 1  q cosψ − Θ The last term of Eq. (15) is the error generated by the temporal integration. This is a nontrivial error

Fig. 2. Displacement amplitude at the impact point versus temporal evolution for different modal truncatures (1,5,10,20) kHz. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS

3189

that cannot be modeled by a simple expression. Reference [48] discussed the modeling of such a distortion in the case of a sinusoidal phase variation. Unfortunately, this model cannot be applied to the case discussed in this paper. However, the full numerical simulation may help to better understand the limitations due to the temporal integration, especially at the first instants of the transient phase (refer to the first instants in Fig. 2). As examples, for a frame rate at 100 kHz, Fig. 3 shows the modulo 2π phase map evaluated between t
0 and 10 μs with different values of the exposure time, for a measured area including the center of the beam (impact point at the center of the field of view). The impact force is assimilated to a Dirac-shaped pulse which generates a transient flexural vibration of the beam, with a 20 kHz bandwidth. Note that the observation area is 170 mm × 20 mm (beam 500 mm × 20 mm) centered at the center of beam, corresponding to the area reconstructed with the holographic process in the numerical simulation. Figure 3 shows that the noise level is increasing when increasing the exposure time, since the phase difference for T
10 μs is more noisy than that for T
0.1 μs. Figure 4 shows the phase error obtained by computing the difference between the evaluated phase and the initial input phase. A distortion exists at the impact point since Fig. 4 exhibits localized errors on the beam. In addition, the noise level is increased, as was observed in Fig. 3. The error is larger for T
10 μs than for T
0.1 μs. This distortion in the phase map is related to the high slope at these instants. The slope is related to the velocity of the structure between the two instants. It depends on the modal density of the structure, the damping coefficient, and the nature of the impact. So, for a few specific practical cases, the post-correction of this distortion might be required to get an error-free measurement. Figure 5 shows the error map on the phase, few instants later, between t
100 and 110 μs. Figure 5

Fig. 3. Evaluated modulo 2π phase between t
0 and 10 μs with T
0.1; 1; 5; 10 μs. 3190

APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015

Fig. 4. Error on the phase for phase difference between t
0 and 10 μs with T
0.1; 1; 5; 10 μs.

shows that a distortion on the measurement is always observable but that the distortion is less compared to Fig. 4. In addition, the noise is less than for the first instants of the impact, for T
10 μs. This is due to the “moderate” slope at this moment of the temporal evolution of the phase. Figure 4 shows that the phase distortion occurs where phase jumps exists. This is also the case for Fig. 5 (evaluated phase not shown). The simulation permits to study the distortion on phase at the border of the beam, e.g., far from the impact point which is localized at the center of the beam. Figure 6 presents the phase error for the five exposure times, a few instants later after the shock, for time between t
100 and 100.01 ms. The phase maps do not exhibit any distortion and noise. This numerical analysis exhibits the effect of the temporal integration in the phase measurement. When the optical phase includes rapid time evolution,

Fig. 5. Error on the phase for phase difference between t
100 and 110 μs with T
0.1; 1; 5; 10 μs (excitation at center of the beam).

Fig. 6. Error on the phase for phase difference between t
100 and 100.01 ms with T
0.1; 1; 5; 10 μs.

an error-free measurement might be not obtained, especially near the excitation point. The error is localized at the phase jumps. Noise increases with the increase of the exposure time. Then, for these few first instants, compensation algorithms have to be developed to compensate for the phase distortion. Section 5 presents experimental results obtained for transient vibration by using the in-line holographic method. Comparison with a classical pointwise vibrometer is provided. 5. Experimental Results A.

Experimental Setup

The experimental setup is described in Fig. 7. The light source is from a diode pumped solid state (DPSS) laser at λ
532 nm with maximum power at 6 W. The half-wave plate at the output of the laser is used to adjust the power in the object and reference paths. The laser is separated into a reference wave and an object wave by use of a polarizing beam splitter (PBS). The polarization of the object wave is then rotated 90° to be parallel with that of the reference

Fig. 7. Experimental setup for multipoint vibrometer based on digital holography. PBS, polarizing beam splitter.

wave, so that interferences can occur. The object and reference waves are combined by the 50% beam splitter cube placed just in front of the high-speed sensor. The reference wave is expanded, spatially filtered using a spatial filter (microscope objective and microscopic pinhole), and collimated to produce a smooth plane reference wave impacting the sensor at normal incidence. So the carrier spatial frequencies along the reference beam are u0 ; v0  ≈ 0; 0, giving an in-line configuration. The object wave is spatially expanded to illuminate the structure, using a lens assembly and mirrors (not detailed in Fig. 7). A set of negative lenses are inserted in the object path, in front of the cube, between the object and the sensor and permit increasing the studied area [72,73]. This negative lens induces a change in the objectto-sensor distance, which becomes now d00, instead of d0 initially. By this way, the optical magnification between the initial object plane and the virtual image plane is about 0.0233 and the distance between the virtual object and the sensor is 151 mm. The sensor is a high-speed camera (Photron), with pixel pitch at 20 μm and a maximum spatial resolution including M × N
1024 pixels × 1024 pixels. At the full spatial resolution, the maximum frame rate is 13,500 Hz. When increasing the frame rate, the spatial resolution is degraded. The exposure time can be set from 380 ns to a few milliseconds (ms). The object under interest is an aluminum beam 89.5 cm × 2 cm × 0.2 cm, suspended at one of its end, and excited by a mechanical shaker clamped at its center. In the following, the excitation signal is transient having a bandwidth from 20 Hz to 10 kHz. The object wave is shaped to produce a quasi-uniform rectangular illumination sized 10 cm × 2 cm from the bottom of the beam. Since the surface of the aluminum beam is not cooperative, the ratio between the reference wave and the object wave was measured at jRj2 ∕jOj2 ≈ 1000. Considering Sections 3.B and 3.C, the experimental conditions are quite adapted to the in-line digital holography. B. Reconstructed Images

Holograms are recorded at 50 kHz and the spatial resolution is M × N
352 pixels × 256 pixels, with an exposure time T
1 μs. The spatial resolutions in the reconstructed plane are about 48.9 and 67.3 μm for the vertical and horizontal directions, respectively. It follows that the reconstructed useful area includes 204 × 29 independent data points. Preliminarily to the recording of a hologram sequence, the reference wave is recorded separately by masking the object beam, and is then systematically subtracted to each hologram of the sequence. The complex object field is reconstructed with the discrete Fresnel transform. Figure 8 shows the reconstructed amplitude from digitally recorded holograms. Figure 8(a) shows the reconstructed amplitude without subtracting the reference wave, and Fig. 8(b) shows the reconstructed amplitude after having subtracted the reference wave. As can be seen, the image of the beam emerges 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS

3191

Fig. 8. Reconstructed amplitudes: (a) amplitude reconstructed without subtracting the reference wave; (b) amplitude reconstructed after subtracting the reference wave.

from the background. Note that the conclusion in Section 3.B about the average-illumination ratio between the focused and the unfocused reconstructed images was established for the case of single-pixel objects. For multi-pixel objects, the unfocused images of neighboring pixels should merge behind each of the pixels of the focused image, thus reducing the average illumination ratio between both images. However, Fig. 8(b) shows that the image of the focused object is clearly observable and merges from the “noise” induced by the out-of-focus twin image; as a consequence, the power-to-area ratios (average illuminations) of the focused and unfocused images are demonstrated to be quite different, as expected by the analysis in Section 3.B. C.

pointwise vibrometer. Figure 9 shows the experimental configuration, including the localization of three particular points, P1, P2, and P3, at which a pointwise measurement is performed using a classical vibrometer. The force applied by the shaker, is measured by a force sensor placed at Point S (see Fig. 9). The acquisition is performed over a temporal window of 0.5 s and a frequency width of 12.8 kHz to a sampling frequency of 32.7 kHz. The force signal has 0.4 s duration. For each measurement at P1, P2, and P3 using the pointwise vibrometer, the applied force is recorded. The three power spectrum densities recorded for the three points P1, P2, and P3 are shown in Fig. 10. The cut-off frequency of the excitation signal is about 10 kHz, as expected. After processing the hologram sequence, Fig. 11 shows a set of nine modulo 2π phase maps versus the time evolution, obtained after calculating the temporal phase difference between two consecutive recorded holograms. The start time is 60 μs and the time delay between each sub-image is 20 μs. A mechanical wave-front emitted at the excitation point, and propagating along the beam can be clearly observed in the measured area. Figure 12 shows the unwrapped phase maps, demonstrating that the digital holographic in-line method is quite able to provide full-field contactless quantitative dynamic phase measurements. This

Transient Vibration

Signals are generated and analyzed via the acquisition system PULSE (Brüel & Kjær). The excitation signal delivered to the shaker is a linear sweep whose frequency varies from 20 Hz to 10 kHz over a period of 0.4 s. A hologram sequence is recorded with 25,000 frames, corresponding of a temporal window of 0.5 s. The use of such window is useful to provide a comparison with the experimental results given by a

Fig. 10. Power spectrum density of the force signal recorded at the three excitation points P1, P2, and P3.

Fig. 9. Experimental configuration with excitation shaker, sensor force, and points P1, P2, and P3. 3192

APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015

Fig. 11. Transient phase variation at 50 kHz, sequence of nine consecutive phase variations having a 20 μs delay in between, between 0.06 and 0.22 ms.

vz x; y; t


λf s ψx; y; t  1∕f s  2π1  cosθ − ψx; y; t:

(16)

Figure 13(b) shows the temporal evolution of the measured velocity, after scaling the unwrapped phase with Eq. (16). D.

Fig. 12. Unwrapped transient phase variation at 50 kHz, sequence of nine consecutive phase variations having a 20 μs delay in between, between 0.06 and 0.22 ms.

unwrapped phase change is proportional to the relative displacement between consecutive instants. Each phase map includes 204 × 29
5916 independent data points. Note that there exists a global phase jump between t
0.16 and 0.22 ms. Such phase jump will be removed by a temporal phase unwrapping. Figure 13(a) shows the temporal evolution of the measured phase at Point P1 (see Fig. 9). The data appears to be wrapped between −π; π (see also Fig. 12). Then a temporal unwrapping has to be performed to recover a continuous signal. From this “continuous” temporal phase evolution, the velocity of the structure can be calculated. The temporal phase was evaluated from phase differences between two consecutive instants in the hologram sequence. Then, from Eq. (13), the velocity is estimated with Eq. (16), where f s is the frame rate of the hologram sequence

Fig. 13. Temporal evolution of the measured phase at Point P1: (a) raw data; (b) measured velocity after temporal unwrapping and scaling.

Comparison with Pointwise Vibrometer

A laser vibrometer was used to measure the vibration of the structure at different points at the surface of the beam. Especially, Points P1, P2, and P3 were chosen (see Fig. 9). The vibrometer is able to directly yield the velocity at the considered point. Figure 14 shows the experimentally measured velocity, from the vibrometer, at the surface of the structure at Point P1, and the comparison with that obtained from the holographic setup at the same location. As can be seen, there exists a very good agreement between the measurement provided by the vibrometer and that given by the proposed method. With the velocity measurements and force signals from the force sensor, the power spectrum densities and frequency response function (FRF) can be calculated. The FRF is obtained by calculating the cross spectral density between the force and the velocity over the power spectrum of the force. Figure 15 shows both the FRF obtained with the pointwise vibrometer and from the holographic setup, for measurements at Points P1, P2, and P3. Results correspond to the bandwidth 20–12,000 Hz. The comparison shows a quite good agreement in particular at eigenfrequencies of the beam. Both measurements exhibit same quality factor. Note that the FRFs from the vibrometer were obtained after averaging four measurements whereas the holographic measurement was extracted from a single hologram sequence. That is why the pointwise measurements include less random fluctuations than the holographic one. In addition, in the upper part of the bandwidth the results are not coinciding; especially the vibrometer measurement includes resonance frequencies in the range 10–12 kHz, whereas no excitation is provided by the shaker in this frequency range (see Fig. 10).

Fig. 14. Comparison of signals from the vibrometer and from the holographic approach, at Point P1, continuous line; velocity measured at Point P1 with digital holography, dashed line; and velocity measured at Point P1 with the pointwise vibrometer. 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS

3193

simulation of the full acquisition process, including physical behavior of the structure, propagation and diffraction of wave, recording, and digital reconstruction of holograms. The simulation shows that the first instants of the transient vibration are critical, especially at the excitation point. Indeed, distortion due to the finite value of the exposure time might occur, thus requiring the development of new correction strategies. Note that one main limiting factor is related to the close relation between the acquisition frame rate and the spatial resolution of the sensor. The current state-of-the-art of high-speed sensor technology induces a decrease of the spatial resolution when increasing the frame rate. When increasing the frame rate and decreasing the spatial resolution, noise artifacts will appear in the reconstructed phase differences. This point was studied with the full numerical simulation, although no results were presented in this paper. In addition we have investigated the minimal detectable displacement by acquiring a hologram sequence at 72 kHz and by measuring the noise in the phase map. This noise was found to be at 0.26 rad rms, and considering the experimental parameters in Section 5, the minimum detectable displacement between two exposures is estimated at about λ∕25 ≈ 20 nm. Application of such experimental methodology, with frame rates up to 100 kHz, will give opportunities to analyze vibration and vibroacoustics problems related to friction-induced vibrations or structure deformation due to impacts, for example. It is also particularly interesting for the analysis of displacements due to noncoherent excitations like aero or hydro-acoustic sources. Industrial applications for which a full-field measurement is required are also expected. Fig. 15. Comparison of signals from the vibrometer and from the holographic approach: (a) FRF measured at Point P1; (b) FRF measured at Point P2; (c) FRF measured at Point P3.

6. Conclusion

This paper demonstrates the possibility of multipoint vibrometry based on very high-speed digital in-line holography. At a frame rate at 50 kHz, the measurement includes 5916 independent data points and is the best performance yet obtained until now for multipoint vibrometer measurements. The in-line configuration is made possible by the specific properties of the high-speed sensor (large pixels and very restrictive Shannon conditions). The proposed setup is quite adapted to the recording of the spatial and temporal evolution of a dynamic object producing a transient phase modulation. The experimental setup was tested with a transient excitation and the experimental results were compared to pointwise measurement obtained with a classical vibrometer. The results confirm the ability of the proposed method to provide full-field contactless measurements of a transient vibration variation, at its time scale. The main parameters governing the performances of the setup were investigated using a numerical 3194

APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015

This work is part of the Chair program VIBROLEG (Vibroacoustics of Lightweight structures) supported by IRT Jules Verne (French Institute in Research and Technology in Advanced Manufacturing Technologies for Composite, Metallic, and Hybrid Structures). The authors wish to acknowledge the industrial and academic partners of this project: Airbus, Alstom Power, Bureau Veritas, CETIM, Daher, DCNS Research, STX, and University of Maine in France. References 1. L. E. Drain, The Laser Doppler Technique (Wiley, 1980). 2. C. B. Scruby and L. E. Drain, Laser-Ultrasonics: Techniques and Applications (Adam Hilger, 1990). 3. J.-P. Monchalin, “Progress towards the application of laserultrasonics in industry,” in Review of Progress in Quantitative Nondestructive Evaluation, D. O. Thompson and D. E. Chimenti, eds. (Plenum, 1993), Vol. 12, p. 495. 4. P. Castellini, G. M. Revel, and E. P. Tomasini, “Laser Doppler vibrometry: a review of advances and applications,” Shock Vib. Dig. 30, 443–456 (1998). 5. S. Donati, M. Norgia, and G. Giuliani, “Self-mixing differential vibrometer based on electronic channel subtraction,” Appl. Opt. 45, 7264–7268 (2006). 6. A. Chebbour, T. Gharbi, and G. Tribillon, “Laser-diode interferometric heterodyne vibrometer: application to linear motor control,” Appl. Opt. 40, 5610–5614 (2001).

7. A. Chijioke and J. Lawall, “Laser Doppler vibrometer employing active frequency feedback,” Appl. Opt. 47, 4952–4958 (2008). 8. J. Shang, Y. He, D. Liu, H. Zang, and W. Chen, “Laser Doppler vibrometer for real-time speech-signal acquirement,” Chin. Opt. Lett. 7, 732–733 (2009). 9. K. Bertling, J. Perchoux, T. Taimre, R. Malkin, D. Robert, A. D. Rakić, and T. Bosch, “Imaging of acoustic fields using optical feedback interferometry,” Opt. Express 22, 30346–30356 (2014). 10. U. Zabit, R. Atashkhooei, T. Bosch, S. Royo, F. Bony, and A. Rakic, “Adaptive self-mixing vibrometer based on a liquid lens,” Opt. Lett. 35, 1278–1280 (2010). 11. U. Zabit, O. D. Bernal, T. Bosch, and F. Bony, “MEMS accelerometer embedded in a self-mixing displacement sensor for parasitic vibration compensation,” Opt. Lett. 36, 612–614 (2011). 12. Y. Li, S. Meersman, and R. Baets, “Realization of fiber-based laser Doppler vibrometer with serrodyne frequency shifting,” Appl. Opt. 50, 2809–2814 (2011). 13. A. Magnani, A. Pesatori, and M. Norgia, “Self-mixing vibrometer with real-time digital signal elaboration,” Appl. Opt. 51, 5318–5325 (2012). 14. P. Roos, M. Stephens, and C. Wieman, “Laser vibrometer based on optical-feedback-induced frequency modulation of a single-mode laser diode,” Appl. Opt. 35, 6754–6761 (1996). 15. S. Vanlanduit, P. Guillaume, B. Cauberghe, and P. Verboven, “An automatic position calibration method for the scanning laser Doppler vibrometer,” Meas. Sci. Technol. 14, 1469–1476 (2003). 16. P. O’Malley, T. Woods, J. Judge, and J. Vignola, “Five-axis scanning laser vibrometry for three-dimensional measurements of non-planar surfaces,” Meas. Sci. Technol. 20, 115901 (2009). 17. B. J. Halkon and S. J. Rothberg, “Vibration measurements using continuous scanning laser Doppler vibrometry: theoretical velocity sensitivity analysis with applications,” Meas. Sci. Technol. 14, 382–393 (2003). 18. W. MacPherson, M. Reeves, D. Towers, A. Moore, J. Jones, M. Dale, and C. Edwards, “Multipoint laser vibrometer for modal analysis,” Appl. Opt. 46, 3126–3132 (2007). 19. K. Sun, L. Yuan, Z. Shen, Z. Xu, Q. Zhu, X. Ni, and J. Lu, “Scanning laser-line source technique for nondestructive evaluation of cracks in human teeth,” Appl. Opt. 53, 2366– 2374 (2014). 20. M. Connelly, P. Szecówka, R. Jallapuram, S. Martin, V. Toal, and M. Whelan, “Multipoint laser Doppler vibrometry using holographic optical elements and a CMOS digital camera,” Opt. Lett. 33, 330–332 (2008). 21. Y. Fu, M. Guo, and P. Phua, “Multipoint laser Doppler vibrometry with single detector: principles, implementations, and signal analyses,” Appl. Opt. 50, 1280–1288 (2011). 22. Y. Fu, M. Guo, and P. Phua, “Spatially encoded multibeam laser Doppler vibrometry using a single photodetector,” Opt. Lett. 35, 1356–1358 (2010). 23. Th. Kreis, Holographic Interferometry—Principles and Methods, Vol. 1 of Akademie Series in Optical Metrology (Akademie, 1996). 24. R. L. Powell and K. A. Stetson, “Interferometric analysis by wavefront reconstruction,” J. Opt. Soc. Am. 55, 1593–1598 (1965). 25. K. A. Stetson and R. L. Powell, “Interferometric hologram evaluation and real-time vibration analysis of diffuse objects,” J. Opt. Soc. Am. 55, 1694–1695 (1965). 26. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003). 27. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Some opportunities for vibration analysis with time-averaging in digital Fresnel holography,” Appl. Opt. 44, 337–343 (2005). 28. M. Leclercq, M. Karray, V. Isnard, F. Gautier, and P. Picart, “Evaluation of surface acoustic waves on the human skin using quasi-time-averaged digital Fresnel holograms,” Appl. Opt. 52, A136–A146 (2013). 29. O. J. Lokberg and K. Hogmoen, “Use of modulated reference wave in electronic speckle pattern interferometry,” J. Phys. E 9, 847–851 (1976).

30. T. R. Moore, J. D. Kaplon, G. D. McDowall, and K. A. Martin, “Vibrational modes of trumpet bells,” J. Sound Vib. 254, 777–786 (2002). 31. F. Pinard, B. Laine, and H. Vach, “Musical quality assessment of clarinets reeds using optical holography,” J. Acoust. Soc. Am. 113, 1736–1742 (2003). 32. N. Demoli and D. Vukicevic, “Detection of hidden stationary deformations of vibrating surfaces by use of time-averaged digital holographic interferometry,” Opt. Lett. 29, 2423–2425 (2004). 33. C. W. Sim, F. S. Chau, and S. L. Toh, “Vibration analysis and non-destructive testing with real-time shearography,” Opt. Laser Technol. 27, 45–49 (1995). 34. F. Zhang, J. D. R. Valera, I. Yamaguchi, M. Yokota, and G. Mills, “Vibration analysis by phase-shifting digital holography,” Opt. Rev. 11, 297–299 (2004). 35. D. N. Borza, “High-resolution time average electronic holography for vibration measurement,” Opt. Lasers Eng. 41, 515–527 (2004). 36. M. Karray, P. Slangen, and P. Picart, “Comparison between digital Fresnel holography and digital image-plane holography: the role of the imaging aperture,” Exp. Mech. 52, 1275–1286 (2012). 37. S. Ellingsrud and O. J. Lokberg, “Full field amplitude and phase measurement of loudspeakers by using TV-holography and digital image processing,” J. Sound Vib. 168, 193–208 (1993). 38. A. F. Doval, C. Trillo, D. Cernadas, B. V. Dorrio, C. Lopez, J. L. Fernandez, and M. Perez-Amor, “Measuring amplitude and phase of vibration with double exposure stroboscopic TV holography,” in Interferometry in Speckle Light—Theory and Applications, Lausanne, Switzerland, 25–28 September 2000, pp. 281–288. 39. D. J. Anderson, J. D. R. Valera, and J. D. C. Jones, “Electronic speckle pattern interferometry using diode laser stroboscopic illumination,” Meas. Sci. Technol. 4, 982–987 (1993). 40. G. Pedrini, S. Schedin, and H. J. Tiziani, “Pulsed digital holography combined with laser vibrometry for 3D measurements of vibrating objects,” Opt. Lasers Eng. 38, 117–129 (2002). 41. C. R. Hazell and S. D. Liem, “Vibration analysis of plates by real time stroboscopic holography,” Exp. Mech. 13, 339–344 (1973). 42. I. Alexeenko, M. Gusev, and V. Gurevich, “Separate recording of rationally related vibration frequencies using digital stroboscopic holographic interferometry,” Appl. Opt. 48, 3475–3480 (2009). 43. D. De Greef, J. Soons, and J. J. J. Dirckx, “Digital stroboscopic holography setup for deformation measurement at both quasistatic and acoustic frequencies,” Int. J. Optomechatron. 8, 275–291 (2014). 44. G. Pedrini and H. J. Tiziani, “Digital double pulse holographic interferometry using Fresnel and image plane holograms,” Measurement 15, 251–260 (1995). 45. G. Pedrini, Y. L. Zou, and H. J. Tiziani, “Digital double pulse holographic interferometry for vibration analysis,” J. Mod. Opt. 42, 367–374 (1995). 46. G. Pedrini, H. Tiziani, and Y. Zou, “Digital double pulse-TV holography,” Opt. Lasers Eng. 26, 199–219 (1997). 47. G. Pedrini, Ph. Froening, H. Fessler, and H. Tiziani, “Transient vibration measurements using multipulse digital holography,” Opt. Laser Technol. 29, 505–511 (1998). 48. J. Leval, P. Picart, J.-P. Boileau, and J.-C. Pascal, “Full field vibrometry with digital Fresnel holography,” Appl. Opt. 44, 5763–5771 (2005). 49. J. C. Pascal, X. Carniel, V. Chalvidan, and P. Smigielski, “Determination of phase and magnitude of vibration for energy flow measurements in a plate using holographic interferometry,” Opt. Lasers Eng. 25, 343–360 (1996). 50. J. P. Chambard, V. Chalvidan, X. Carniel, and J. C. Pascal, “Pulsed TV-holography recording for vibration analysis applications,” Opt. Lasers Eng. 38, 131–143 (2002). 51. C. Trillo, A. F. Doval, D. Cernadas, O. Lopez, B. V. Dorrio, J. L. Fernandez, and M. Pérez-Amor, “Measurement of the complex amplitude of transient surface acoustic waves using 10 April 2015 / Vol. 54, No. 11 / APPLIED OPTICS

3195

52.

53.

54. 55. 56. 57. 58.

59.

60. 61. 62.

double-pulsed TV holography and a two-stage spatial Fourier transform method,” Meas. Sci. Technol. 14, 2127–2134 (2003). P. Picart, J. Leval, F. Piquet, J.-P. Boileau, Th. Guimezanes, and J.-P. Dalmont, “Tracking high amplitude auto-oscillations with digital Fresnel holograms,” Opt. Express 15, 8263–8274 (2007). S. S. Munoz, F. Mendoza Santoyo, and M. del Socorro Hernandez-Montes, “3D displacement measurements of the tympanic membrane with digital holographic interferometry,” Opt. Express 20, 5613–5621 (2012). F. Joud, F. Laloe, M. Atlan, J. Hare, and M. Gross, “Imaging a vibrating object by sideband digital holography,” Opt. Express 17, 2774–2779 (2009). F. Joud, F. Verpillat, F. Laloë, M. Atlan, J. Hare, and M. Gross, “Fringe-free holographic measurements of large-amplitude vibrations,” Opt. Lett. 34, 3698–3700 (2009). B. Samson, F. Verpillat, M. Gross, and M. Atlan, “Video-rate laser Doppler vibrometry by heterodyne holography,” Opt. Lett. 36, 1449–1451 (2011). J. M. Huntley, G. H. Kaufmann, and D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999). J. M. Kilpatrick, A. J. Moore, J. S. Barton, J. D. C. Jones, M. Reeves, and C. Buckberry, “Measurement of complex surface deformation by high-speed dynamic phase-stepped digital, speckle pattern interferometry,” Opt. Lett. 25, 1068–1070 (2000). A. J. Moore, D. P. Hand, J. S. Barton, and J. D. C. Jones, “Transient deformation measurement with electronic speckle pattern interferometry and a high-speed camera,” Appl. Opt. 38, 1159–1162 (1999). G. Pedrini, W. Osten, and M. Gusev, “High-speed digital holographic interferometry for vibration measurement,” Appl. Opt. 45, 3456–3462 (2006). C. Pérez-López, M. De la Torre-Ibarra, and F. Mendoza Santoyo, “Very high speed cw digital holographic interferometry,” Opt. Express 14, 9709–9715 (2006). Y. Fu, G. Pedrini, and W. Osten, “Vibration measurement by temporal Fourier analysis of a digital hologram sequence,” Appl. Opt. 46, 5719–5727 (2007).

3196

APPLIED OPTICS / Vol. 54, No. 11 / 10 April 2015

63. C. Trillo, A. F. Doval, F. Mendoza-Santoyo, C. Pérez-López, M. de la Torre-Ibarra, and J. L. Deán, “Multimode vibration analysis with high-speed TV holography and a spatiotemporal 3D Fourier transform method,” Opt. Express 17, 18014–18025 (2009). 64. D. Aguayo, F. M. Mendoza Santoyo, M. de la Torre-Ibarra, M. D. Salas-Araiza, C. Caloca-Mendez, and D. A. Gutierrez Hernandez, “Insect wing deformation measurements using high speed digital holographic interferometry,” Opt. Express 18, 5661–5667 (2010). 65. S. Li, K. D. Mohan, W. W. Sanders, and A. L. Oldenburg, “Toward soft-tissue elastography using digital holography to monitor surface acoustic waves,” J. Biomed. Opt. 16, 116005 (2011). 66. K. Mohan, W. Sanders, and A. Oldenburg, “Elastic depth profiling of soft tissue by holographic imaging of surface acoustic waves,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2012), paper DSu4C.5. 67. U. Schnars and W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994). 68. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1996). 69. I. Yamaguchi and T. Zhang, “Phase shifting digital holography,” Opt. Lett. 22, 1268–1270 (1997). 70. P. Picart and J. Leval, “General theoretical formulation of image formation in digital Fresnel holography,” J. Opt. Soc. Am. A 25, 1744–1761 (2008). 71. P. Picart and P. Tankam, “Analysis and adaptation of convolution algorithms to reconstruct extended objects in digital holography,” Appl. Opt. 52, A240–A253 (2013). 72. U. Schnars, T. M. Kreis, and W. Jüptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996). 73. J. Mundt and T. Kreis, “Digital holographic recording and reconstruction of large scale objects for metrology and display,” Opt. Eng. 49, 125801 (2010). 74. K. F. Graff, Wave Motion in Elastic Solids (Dover, 1975). 75. J.-L. Guyader, Vibration in Continuous Media (ISTE–Wiley, 2006).