October 15, 2004 / Vol. 29, No. 20 / OPTICS LETTERS 2423

Detection of hidden stationary deformations of vibratingsurfaces by use of

time-averaged digital holographic interferometry

Nazif Demoli and Dalibor Vukicevic

Institute of Physics, Bijenicka c. 46, P.O. Box 304, Zagreb HR-10001, Croatia, andUniversity Louis Pasteur, Ecole Nationale Superieure de Physique de Strasbourg,

Boulevard Sébastien Brant, BP 10413, Illkirch F-67412, France

Received April 22, 2004

A method of detecting displacements of a surface from its steady-state position to its equilibrium positionwhile it is vibrating has been developed by use of time-average digital holographic interferometry. Thismethod permits extraction of such a hidden deformation by creating two separated systems of interferogramfringes: one corresponding to a time-varying resonantly oscillating optical phase, the other to the stationaryphase modification. A mathematical description of the method and illustrative results of experimental veri-fication are presented. © 2004 Optical Society of America

OCIS codes: 090.0090, 090.2880, 090.1760.

There has been increasing interest in digital hologra-phy for interferometric applications. Recent examplesinclude characterizing transparent material1 or studiesof semiconductor ablation.2 A well-established tech-nique for studying surface vibrations is time-averagedholographic interferometry,3 in which the fringe inten-sity is described by the Bessel functions. The idea ofexposing an electronic photosensor array with manyvibration cycles has extended the time-averaged holo-graphic interferometry concept in three areas, namely,in electronic speckle-pattern interferometry,4 in digi-tal speckle-pattern interferometry,5 and most recentlyin digital holography.6 In this Letter we propose anew method that exploits particular aspects of digitalholography for, in addition, detection of stationary de-formations of a vibrating surface. Namely, if an objectis subjected to a resonant vibration, then the pointson its surface oscillate around their equilibrium posi-tions, which form an equilibrium surface that can beeither identical to the steady surface or different fromit. If a difference occurs, we refer to it as to the sta-tionary object surface deformation, which is a conse-quence of many parameters, such as object structure,material, and geometrical composition as well as of thecharacter and intensity of the excitation mechanism.A schematic of the problem addressed here is shownin Fig. 1. The standard time-averaged holographic in-terferometry technique is not able to detect such ahidden deformation; i.e., the vibration fringe patternwill be the same whether the stationary deformationis present or not. Quantifying hidden deformations isimportant because its evaluation allows direct mea-surement of a stationary bias strain in the dynamicanalysis of the vibration object. To achieve this goalwe are proposing an additional numerical procedure tobe applied while phase evaluating from time-averageddigital holographic interferograms.

Consider a quasi-Fourier setup with an off-axisreference point source and an opaque object, bothlocated at the input plane and a CCD sensor located

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at the hologram plane. Digital holograms are storedin a computer memory by CCD recordings of theprimary fringe patterns. The reconstructions areobserved by first Fourier transforming the recordeddata and, second, calculating the modulus of theobtained Fourier transform. We describe the inputplane as U �x, y� � d�x 2 X, y 2 Y � 1 s�x, y�, wherethe point source is located at the �X,Y � position ands�x, y� � js�x, y�jexp�if�x, y�� is the object wave front.At a distance d from the input plane the diffracted f ieldis calculated according to the Fresnel approximation:U �u, v� ~ exp�ip�u2 1 v2��F �U �x, y�exp�ij�x, y���,where F denotes the Fourier-transform operator,j�x, y� � �p�ld� �x2 1 y2�, l is the wavelength, andwhere the constant terms are omitted. Thus, atime-integrating photodetector will record exposureE�u, v� �

Rt

0 jU �u, v, t�j2dt (t is the exposure time),where one of additive terms, namely,

E�u, v� ~ exp�iz �u, v��F �s�x, y�exp�ij�x, y��� , (1)

in which z �u, v� � 2p�uX 1 vY �; calculatingthe inverse Fourier transform and then takingthe squared modulus, yields the reconstruction:jUr�x, y�j2 ~ js�x 2 X, y 2 Y �j2. First we supposethat the object deformation consists of a staticdisplacement that inf luences the phase change

Fig. 1. Hidden deformation of a vibrating surface. A, B,fixed points.

© 2004 Optical Society of America

2424 OPTICS LETTERS / Vol. 29, No. 20 / October 15, 2004

Df�x, y� � Dw�x, y� 1 Dc�x, y�, where Dw�x, y� denotesits deterministic part and Dc�x, y� is its random partresulting from the surface roughness of the object.Note that the phase Dc�x, y� is a rapidly varying func-tion of position, unlike Dw�x, y�, which varies slowlyacross the object. The exposure term of interest is

E�u, v� ~ exp�iz �u, v��

3 F �s�x, y�exp�iDf�x,y� 1 ij�x, y��� . (2)

A standard procedure for observing interferometricfringes (corresponding to double-exposure holographicinterferometry) is the addition of the fringe patternsdescribed by relations (1) and (2):

E1�u, v� ~ exp�iz �u, v��F ���s�x, y� �1 1 exp�iDf�x,y���

3 exp�ij�x, y����� . (3)

Thus after calculating the inverse Fourier transformof the right-hand side of relation (3) and takingthe squared modulus, we obtain �jUr

1�x, y�j2 ~js�x 2 X, y 2 Y �j2�1 1 cos�Dw�x 2 X, y 2 Y ���, wherethe spatial averaging denoted by � is carried outover an area (for example, a pixel of the CCD sensor)smaller than the fringe spacing and is used to removethe random speckle phase Dc�x, y�.

Now we suppose that the object deformation iscomposed of a harmonic vibration with frequency v

and amplitude h�x, y� in addition to the previouslydescribed static deformation. If only out-of-plane vi-brations occur with the sensitivity vector normal to theobject surface, the additional phase term of the objectwave front can be described as �4p�l�h�x, y�sin vt.Then, if the integration time of the CCD sensorsatisfies the condition t .. 1�v, the exposure termbecomes

E�u, v� ~ exp�iz �u, v��F �s�x, y�J0��4p�l�h�x,y��

3 exp�iDf�x, y� 1 ij�x, y��� , (4)

where J0 is the zero-order Bessel function of the f irstkind. The reconstruction procedure (i.e., the inverseFourier transform, squared modulus, and averaging)yields

�jUr�x, y�j2 ~ js�x 2 X, y 2 Y �j2J02�l,h� , (5)

where the vibration fringes are described by thesquared Bessel function J0

2�l,h� J02��4p�l�h�x 2

X, y 2 Y �� and where the information about the staticdeformation is lost.

To retrieve the static displacement, we propose thefollowing procedure:

(i) Subtract relations (1) and (4), followed bythe reconstruction procedure, yielding

�jUr2�x, y�j2 ~ js�x 2 X, y 2 Y �j2j1 2 J0�l,h�

3 exp�iDw�x 2 X, y 2 Y ��j2. (6)

(ii) Add relations (1) and (4), followed by the re-construction procedure, yielding

�jUr1�x, y�j2 ~ js�x 2 X, y 2 Y �j2j1 1 J0�l,h�

3 exp�iDw�x 2 X, y 2 Y ��j2. (7)

(iii) Subtract relations (6) and (7), yielding

�jUr1�x, y�j2 2 �jDur

2�x, y�j2 ~ js�x 2 X, y 2 Y �j2

3 jJ0�l,h�j jcos�Dw�x 2 X, y 2 Y ��j . (8)

The right-hand side of relation (8) contains threemultiplicative terms: the reconstructed image ofthe object; the modulus of the Bessel function, whichdescribes the vibration fringes; and the modulus of thecosine function, which describes the stationarydeformation fringes. Compared with relation (5),relation (8) demonstrates better visibility of vibrationfringes (the modulus versus squaring operations).Furthermore, relation (8) shows enhanced sensitivityfor the stationary deformation because of the modulusof the cosine function (doubling of the fringe effect).

For experimental verification of our method weemployed a quasi-Fourier optical setup and thesubtraction digital holography technique.7 Thistechnique has been shown to be effective7,8 in re-moving the zero-order disturbance in off-axis digitalholography. We used a krypton-ion laser (647 nm) asa light source and an integration type of CCD sensor(Kodak Megaplus, 1008 3 1018 pixels, 9 mm 3 9 mmeach pixel) as an array photodetector. We adjustedthe intensity ratio between the reference and objectbeams to 3:1 and recorded holograms at a distanceof 1.5 m from the object, using an exposure timeof 64 ms. To apply the subtraction technique twoholograms were captured and stored in computermemory for each situation. As an example, we usedan oscillating membrane with a diameter of 32 mm.The membrane, attached to a massive carrier in foursymmetrical spots at its edge, was excited to vibratesinusoidally by a function generator. The offset volt-age (load) was used for simulating static deformation.First we selected the resonant frequency (1020 Hz)with a simple vibration mode. Then we recorded twotime-averaged digital holograms, f irst one without andsecond one with the offset voltage imposed. Both re-constructions resulted, as expected, in the same fringepattern shown in Fig. 2(a). To observe fringes corre-sponding to the offset deformation alone, we applieddouble-exposure digital holographic interferometry.Figure 2(b) shows these fringes. As expected, the off-set voltage caused maximum displacement in thecentral part of the membrane. By applying themethod of interferogram analysis introduced here,we obtain the fringe pattern shown in Fig. 3. Theresult indicates clearly that the vibration movementsare in fact accompanied by additional deformation.Comparing Figs. 2 and 3, it is possible to separate theadditional fringe structure numerically. Althoughit is self-evident, we nevertheless demonstrate thatthe same separation between fringes corresponding

October 15, 2004 / Vol. 29, No. 20 / OPTICS LETTERS 2425

Fig. 2. Fringe patterns obtained for the displacements:(a) vibration and (b) offset.

Fig. 3. Fringe pattern containing both the vibration andoffset displacements.

Fig. 4. Fringe patterns obtained for the displacements:(a) shift and (b) vibration 1 offset 1 shift.

to pure vibrations and pure deformations is possiblefor the most general case, when, while the vibration isaccompanied with some offset deformation, the objectis additionally displaced. The object is now mountedon a rotational stage. By choosing two static positionsof the membrane, initial and rotated, additional phase

shift is imposed. Three digital holograms of themembrane were recorded corresponding to (a) initialposition, (b) rotated position, and (c) rotated positionwith offset and the membrane vibrating. Figure 4(a)shows double-exposure interferometric fringes ob-tained by superimposing holograms of the initialand rotated object positions (the phase shift withoutthe offset and vibration displacements). Vibrationexcitation with offset is then applied, and through theproposed procedure the fringe pattern is obtained [seeFig. 4(b)]. Again, the vibration fringes are clearlyseparated from displacement and are stationary (i.e.,hidden deformation) fringes. Although they aredetected, the hidden deformation fringes contain thesign ambiguity problem that is typical of holographicinterferometry.9 In addition to the proposed proce-dure, possible solutions for obtaining full quantitativephase data are applying the phase-shifting or themultiwavelength approaches.

In conclusion, we have presented a method forstationary hidden deformation visualization and mea-surements through time-averaged digital holographicinterferometry. The proposed method is an originalway of numerically processing experimental data andis effective for possible analysis of strain distributionacross the vibrating surface. The method allows oneto detect the hidden deformations that often accom-pany the dynamic deformations of a vibrating surface.

This work was supported by the French–CroatianCogito project “Dynamic Multiwave Digital Holo-graphic Interferometry” and by the Croatian Ministryof Science and Technology (project 0035005). N.Demoli’s e-mail address is demoli@ifs.hr.

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