Optics Communications 281 (2008) 4291–4296

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Optics Communications

journal homepage: www.elsevier .com/locate /optcom

Simultaneous measurement with one-capture of the two in-plane componentsof displacement by electronic speckle pattern interferometry

Amalia Martínez a,*, J.A. Rayas a, Cruz Meneses-Fabián b, Marcelino Anguiano-Morales a

a Centro de Investigaciones en Óptica, A.C. Apartado Postal 1-948, C.P. 37000 León, Gto., Mexicob Benemérita Universidad Autónoma de Puebla, Facultad de Ciencias Físico-Matemáticas, Apartado Postal 1152, Puebla PUE 72000, Mexico

a r t i c l e i n f o

Article history:Received 24 September 2007Received in revised form 29 April 2008Accepted 14 May 2008

PACS:06.20.�f62.20.�x62.20.Fe42.30.M

Keywords:Speckle interferometryDisplacement measurementsSensitivity matrix

0030-4018/$ - see front matter � 2008 Elsevier B.V. Adoi:10.1016/j.optcom.2008.05.015

* Corresponding author. Tel.: +52 477 4414200; faxE-mail address: amalia@cio.mx (A. Martínez).

a b s t r a c t

We present the simultaneous measurement of the two in-plane displacement components by electronicspeckle pattern interferometry with three object beams and without an in-line reference beam. Threeinterference fringe patterns, corresponding to three different sensitivity vectors, are recorded in a singleinterferogram and separated by means of the Fourier transform method. Then, two interference fringepatterns are selected to obtain the in-plane displacement components.

� 2008 Elsevier B.V. All rights reserved.

1. Introduction

Electronic speckle pattern interferometry (ESPI) is a well estab-lished technique that is especially useful for the static and dynamicmeasurement of deformation fields. ESPI utilizes digitally recordedspeckle intensity patterns formed through the illumination of anoptically rough surface with coherent light. The subtraction ofspeckle images taken before and after a deformation produces aninterferogram containing a fringe pattern, where the fringes arecontours of constant phase from which displacement can be ob-tained. Speckle interferometry systems can measure out-of-plane[1] and in-plane [2] displacements; in the case of simultaneousmeasurement of two in-plane components different methods havebeen suggested.

A system with a 1 � 4 single-mode optical fiber beam-splitter tosplit the laser beam into four beams of equal intensity has beenpresented [3,4]. One pair of fibers is used to illuminate a diffusetarget at equal angles in the horizontal plane; another pair of opti-cal fibers is set to be sensitive only to vertical in-plane displace-

ll rights reserved.

: +52 477 4414209.

ment. The polarization directions of light emitted by the fibersare the same for each pair, but are at a right angle between pairs.The optical fibers are equal in length for each pair, but are notequal between the two pairs.

In Ref. [5], an ESPI system was used to study resonant in-planevibrations. An in-plane sensitive arrangement was used with dual-beam illumination for horizontal sensitivity. To complete the in-plane study, the illumination beams were rotated through 90�about the viewing axis to make the system sensitive to verticalin-plane displacements.

Another dual in-plane system that uses an electronic-opticalswitch to change between the illumination directions for x and ysensitivity has been proposed [6]. Finally some authors [7] describean interferometer devised to measure two in-plane interferogramsat the same time.

In this work, we propose a system that uses three object illu-mination beams, and which has the interesting attribute of beingsensitive only to two in-plane displacement components simulta-neously. The technique allows simultaneous measurement ofwhole in-plane field from one image when using two pairs ofcomponents that are spatially separated in the Fourier domain.The phase distribution for each term is calculated by performingthe Fourier transform method [4,8].

4292 A. Martínez et al. / Optics Communications 281 (2008) 4291–4296

2. Theory

The phase difference is related to the deformation along eachsensitivity vector as the vector scalar product [9]

D/k ¼~ek �~d; ð1Þ

where~ek is the sensitivity vector, and ~d is the displacement vectorwith components u, v, and w. Since we have three components ofthe displacement vector, it is necessary to write three equationsto get all the components,

D/1 ¼~e1 �~d; D/2 ¼~e2 �~d; D/3 ¼~e3 �~d: ð2Þ

The requirement is that the three sensitivity vectors must be noncoplanar. If of the three vectors, two are coplanar, this means thatone has only one additional equation that is independent. In otherwords, two equations are linear combinations of each other, andhence, they are not independent.

The sensitivity vectors are defined as functions of the illumina-tion and observation directions [9]. It is only when one takes thephase difference corresponding to two illuminations vectors witha common observation vector, that the observation vector is re-moved. Hence only for this particular case the optical path differ-ence depends on the illumination vectors.

The sensitivity vectors for each of pair of illumination beamdirections are defined as [10]

~e1 ¼2pk� ðn̂1 � n̂2Þ; ~e2 ¼

2pk� ðn̂2 � n̂3Þ; ~e3 ¼

2pk� ðn̂3 � n̂1Þ ð3Þ

where n̂1, n̂2, and n̂3 are unit vectors along the illumination beam toeach one of the sources. Notice that the illumination directionschange for each point on the inspected area of the target object.We evaluate the displacement vector,~d, by decomposing the sensi-tivity vectors into their orthogonal components x, y, and z [i.e.~ek ¼ ðekx; eky; ekzÞ]. For the experiments presented here, the originof a Cartesian coordinate system is placed at the object centre. Thiscoordinate system is used to measure the location of the object illu-minating beams and a CCD sensor (see Fig. 1). In order to solve forthe all three components, three independent equations are neces-sary, namely three geometries. We analyze the case of multiple illu-mination beams that without the need of a cumbersome in-linereference beam, and when properly chosen and taken two by twoin a sequential recording order, can have the interesting propertyof being sensitive only to the in-plane displacement components.With this in mind, the sensitivity vectors components due to theillumination sources Si(xi,yi,zi) and Sj(xj,yj,zj) can be written forthe case of three illuminations sources, as

S1

S2

S3

x

y

z

P.

O

r1

r2

r3

n1

n2

n3

Fig. 1. Case A: representation of the illumination sources positions to analyze thesensitivity matrix.

ekx ¼2pk

x� xi

ri� x� xj

rj

� �;

eky ¼2pk

y� yi

ri�

y� yj

rj

� �;

ekz ¼2pk

z� zi

ri� z� zj

rj

� �; ð4Þ

where ri and rj are the distances between the illumination sourcesand a point on the target. ri and rj can be written as

ri ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xiÞ2 þ ðy� yiÞ

2 þ ðz� ziÞ2q

;

rj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðx� xjÞ2 þ ðy� yjÞ

2 þ ðz� zjÞ2q

: ð5Þ

From Eq. (1) the deformation components d = (u,v,w) at each pointof the object are obtained by

u

vw

0B@

1CA ¼

e1x e1y e1z

e2x e2y e2z

e3x e3y e3z

0B@

1CA�1

�D/1

D/2

D/3

0B@

1CA ð6Þ

It should be noted that it must possible to invert the sensitivity ma-trix. Therefore, the experimental setup should be chosen in such away that the three sensitivity vectors are linearly independent.

The solution Eq. (6) is valid only if the determinant D of the sen-sitivity matrix is different from zero

D ¼ e1xðe2ye3z � e2ze3yÞ � e1yðe2xe3z � e2ze3xÞ þ e1zðe2xe3y � e2ye3xÞ:ð7Þ

By substituting of Eq. (4) in Eq. (7), D = 0 is obtained. Then, this pro-posed optical configuration with three object illumination beamsdoes not allow separating the three displacement vector compo-nents because two vectors are coplanar. However, it is possible toobtain the two in-plane displacement components if the contribu-tions of e1z, e2z, and e3z are very small compared to the other sensi-tivity vector components (e1z compared to e1x and e1y, e2z comparedto e2x and e2y and finally e3z compared to e3x and e3y). Then e1z, e2z,and e3z can be approximated to zero, which is possible if the sourcesare located far enough from the object’s plane (or if collimated illu-mination is used). In this case, component w of displacement d can-not be measured, and Eq. (6) can be reduced to

u

v

� �¼

e1x e1y

e2x e2y

� ��1 D/1

D/2

� �: ð8Þ

The determinant D is different from zero according to the equation

D ¼ e1xe2y � e1ye2x ð9Þ

The above equation is true because two of three sensitivity vectorsare linearly independent.

x

S1

S3

y

z

P.

O

r3

r1

r2

S2

n1

n2

n3

Fig. 2. Case B: representation of the illumination sources positions for the experi-mental case.

A. Martínez et al. / Optics Communications 281 (2008) 4291–4296 4293

The geometry with three illumination sources allows theobtaining of the two in-plane components. Fig. 2 presents the stud-ied experimental case and Fig. 3 shows the percentage of each oneof the sensitivity vector components to each one of dual illumina-tion systems [11]. The sources positions are S1 (�370 mm, 0 mm,1925 mm), S2 (370 mm, 0 mm, 1975 mm) and S3 (0 mm, 730 mm,1885 mm). It is observed that the out-of-plane sensitivity vectorcomponent can be negligible. The maximum value is actuallynearly 2%. The optical system was designed in such a way thatthe in-plane sensitivity vector component to y-direction is greaterthan the in-plane sensitivity vector component to x-direction. The

-24 0

1632

-12

12

-16-32

0

242121.121.3

x (mm)y (mm)

e2x

(%)

-24-12

12

-32

0

2476.977.477.9

y (mm)

e2y

(%)

-240

1632

-12

12

-16-32

0

2499.9

100

x (mm)y (mm)

e3x

(%)

-24-12

12

-32

0

242x10-104x10-59x10-5

y (mm)

e3y

(%)

-240

1632

-12

12

-16-32

0

2420.120.320.4

x (mm)y (mm)

e1x

(%)

-24-12

12

-3

0

2477.678.178.7

y (mm)

e1y

(%)

Fig. 3. Percentage of each of the sensitivity vector comp

OF

OF

OF

He-Cd Laser

M

BSBS

C

LL

L

S2

S3

Fig. 4. Experimental setup for simultaneous two-dimensional measurement with recordBS, beam-splitter; L, lens; OF, single-mode optical fibers; O object.

reason is that the load is applied in x-direction, which induces au-displacement field with higher values than the v-displacementfield.

3. Experimental part

An He–Cd laser with a wavelength of 440 nm was used as a lightsource, which was split into three object beams, Fig. 4. Fig. 5 showsa photography of the arrangement that was used.

The three object beams were conveyed through single-modeoptical fibers to illuminate a test object from different directions

016

32

-16 x (mm)

016

32

-16 x (mm)

-240

1632

-12

12

-16-32

0

241.11.51.9

x (mm)y (mm)

e1z

(%)

-240

1632

-12

12

-16-32

0

241.11.51.9

x (mm)y (mm)

e2z

(%)

-240

1632

-12

12

-16-32

0

243x10-60.0170.034

x (mm)y (mm)

e3z

(%)

016

32

-162 x (mm)

onents to each pair of sources for the case of Fig. 2.

n1

n3

x

y

zO

n2

CD

S1

ing of one image corresponding to illumination directions n̂1, n̂2 and n̂3; M, mirror;

Fig. 5. Photography of experimental set up shown in Fig. 4.

Fig. 6. Interference fringes obtained from the correlation of images taken beforeand after test object deformation.

Fig. 7. Fourier transform image of the interferogram shown in Fig. 6.

4294 A. Martínez et al. / Optics Communications 281 (2008) 4291–4296

simultaneously. The incidence angles were 11� for S1 and S2 and21� for S3, and the scattered light was imaged onto the CCD array.A total of three two-beam interferometers (constructed by thecombination of sources S1–S2, S2–S3 and S3–S1) resulted from thethree-beam configuration shown in Fig. 4.

The experiment was conducted on an elastic surface of widthw = 6.47 cm, height h = 4.74 cm and thickness t = .14 mm, whichcan be considered as a thin plate. One of the edges was clampedrigidly, while the other side of the plate was submitted to the ac-tion of tensile force in the x-direction, and which was uniformly dis-tributed along the longitudinal side of the plate.

Speckle images were taken with an interline CCD monochro-matic camera (Cohu, model 2122), and then captured and digital-ized to 640 � 480 pixels on 8-bit gray scale with a framegrabber(National Instruments, model 1409). The first speckle image ofthe undeformed test object was captured and stored. The test ob-ject was then deformed and a second speckle image was taken.Fig. 6 shows the fringe patterns obtained.

In our experiment, three sets of interference fringes were gener-ated, and so a total of six peaks appear in the Fourier transformplane (Fig. 7). A peak corresponding to each one of interferometerswas extracted through appropriate masks, and an inverse Fouriertransform was then applied to each of the masked images. The ra-tio between the imaginary and the real part at each point repre-sents the phase of the deformed fringe pattern.

Fig. 8 shows the fringe patterns obtained from each selectedpeak in the Fourier plane when the inversed Fourier transformwas applied. In this case, the fringe pattern gradient coincides withthe direction of the sensitivity vector, which happens to be true inthe particular case for which the experiment was done, but it is nottrue in general [12]. The fringe patterns shown in Fig. 8 are sosmooth due the filter used, which eliminated the frequency corre-sponding to speckle and the frequency associated to other the twointerferometers.

It is well known that the Fourier transform method (FTM) canbe used to extract the fringe phase with certain conditions [8]. Ifthe phase variation is slow, band-pass filtering in the spatial fre-quency domain isolates one of these terms. The isolated term istransferred to the origin to remove the carrier, and then the inverseFourier transform is calculated again to obtain a complex functionwith real and imaginary non-zero parts. The main advantage of theFTM is that it only requires capturing a single fringe pattern. How-ever, the type of fringes that it can demodulate should be open andof high frequency, and the FTM application to closed fringes yieldsbiased results. Yet, even if the fringes have the proper frequencyand are adequately open, the FTM outcomes depend on both thewidth and the location of the filter mask used to isolate the termcarrying the phase information. Errors in the determination ofthese factors affect the FTM performance by changing the phasevalues retrieved by means of this technique. The influences ofthese error sources on the phase uncertainty have been character-ized and compared by carrying out an uncertainty analysis on thedata retrieved by using the FTM from an ESPI fringe pattern [13].

In a general case, carrier fringes are introduced to apply theFourier method. In the experimental case presented, the fringes ob-tained are considered carrier fringes. Fig. 9 shows the wrappedphase associated to the fringe patterns shown in Fig. 8.

Fig. 8. Fringe patterns obtained from each selected peak in the Fourier plane and for application of the inversed Fourier transform.

Fig. 9. Wrapped phase associated to fringe patterns shown in Fig. 8.

016

32

-12

12

-16-32

0

24

-24

05

10

15

x (mm)y (mm)

u(μm)

Fig. 10. 3D Plot for the deformation that depicts the u-direction.

016

32

-12

12

-16-32

0

24

-24

-2-0.60.6

2

x (mm)y (mm)

v(μm)

Fig. 11. 3D plot for the deformation that depicts the v-direction.

A. Martínez et al. / Optics Communications 281 (2008) 4291–4296 4295

7.2 27 46.7

-12

12

-12.6-32

0

24

-24

x (mm)

y(m

m)

Fig. 12. Plot of combined shape and deformation data.

4296 A. Martínez et al. / Optics Communications 281 (2008) 4291–4296

Fig. 10 shows the 3D plot for the deformation that depicts the u-direction. Fig. 11 shows 3D plot for the deformation that depicts thev-direction.

The surface target topography is flat, which is considered whenthe sensitivity vector components are evaluated from Eqs. (4) and(5). With these data, the deformation in the Cartesian u, v, direc-tions is finally mapped from Eq. (8). The result of the 2D deforma-tion measurement (Figs. 10 and 11), combined with the shape data,is shown in Fig. 12. The figure shows a solid surface that representsthe target before being loaded. The crossed grating surface, in thesame figure, represents the target after being loaded. To appreciatethe surface deformation, the graphic has been scaled with a factorof 1000 units for u and v, respectively.

4. Conclusion

We have proposed an ESPI system that, in combination with theFourier transform, is capable of simultaneously measuring the twoin-plane components from a single interference pattern. The opticalsetup is built up with three illuminating sources. The advantage ofthis method is that it could be used for analysis in real time of in-plane vibration. One more of the advantages of the presented tech-nique is that uses a smaller number of illumination sources to gettwo in-plane displacement components. There is not redirection ofthe light like in some of the other reported cases.

Acknowledgments

Authors wish to thank partial economical support from ‘‘Cons-ejo de Ciencia y Tecnología del Estado de Guanajuato”, as well asto ‘‘Consejo Nacional de Ciencia y Tecnología”, and Mario AlbertoRuiz for his technical support.

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