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

Optics Communications 281 (2008) 4291–4296

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Simultaneous measurement with one-capture of the two in-plane components of displacement by electronic speckle pattern interferometry Amalia Martínez a,*, J.A. Rayas a, Cruz Meneses-Fabián b, Marcelino Anguiano-Morales a a b

Centro de Investigaciones en Óptica, A.C. Apartado Postal 1-948, C.P. 37000 León, Gto., Mexico 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 2007 Received in revised form 29 April 2008 Accepted 14 May 2008

PACS: 06.20.f 62.20.x 62.20.Fe 42.30.M

a b s t r a c t We present the simultaneous measurement of the two in-plane displacement components by electronic speckle pattern interferometry with three object beams and without an in-line reference beam. Three interference fringe patterns, corresponding to three different sensitivity vectors, are recorded in a single interferogram and separated by means of the Fourier transform method. Then, two interference fringe patterns are selected to obtain the in-plane displacement components. Ó 2008 Elsevier B.V. All rights reserved.

Keywords: Speckle interferometry Displacement measurements Sensitivity matrix

1. Introduction Electronic speckle pattern interferometry (ESPI) is a well established technique that is especially useful for the static and dynamic measurement of deformation fields. ESPI utilizes digitally recorded speckle intensity patterns formed through the illumination of an optically rough surface with coherent light. The subtraction of speckle images taken before and after a deformation produces an interferogram containing a fringe pattern, where the fringes are contours of constant phase from which displacement can be obtained. Speckle interferometry systems can measure out-of-plane [1] and in-plane [2] displacements; in the case of simultaneous measurement of two in-plane components different methods have been suggested. A system with a 1  4 single-mode optical fiber beam-splitter to split the laser beam into four beams of equal intensity has been presented [3,4]. One pair of fibers is used to illuminate a diffuse target at equal angles in the horizontal plane; another pair of optical fibers is set to be sensitive only to vertical in-plane displace-

* Corresponding author. Tel.: +52 477 4414200; fax: +52 477 4414209. E-mail address: [email protected] (A. Martínez). 0030-4018/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2008.05.015

ment. The polarization directions of light emitted by the fibers are 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 not equal between the two pairs. In Ref. [5], an ESPI system was used to study resonant in-plane vibrations. An in-plane sensitive arrangement was used with dualbeam illumination for horizontal sensitivity. To complete the inplane study, the illumination beams were rotated through 90° about the viewing axis to make the system sensitive to vertical in-plane displacements. Another dual in-plane system that uses an electronic-optical switch to change between the illumination directions for x and y sensitivity has been proposed [6]. Finally some authors [7] describe an interferometer devised to measure two in-plane interferograms at the same time. In this work, we propose a system that uses three object illumination beams, and which has the interesting attribute of being sensitive only to two in-plane displacement components simultaneously. The technique allows simultaneous measurement of whole in-plane field from one image when using two pairs of components that are spatially separated in the Fourier domain. The phase distribution for each term is calculated by performing the Fourier transform method [4,8].

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2. Theory The phase difference is related to the deformation along each sensitivity vector as the vector scalar product [9]

D/k ¼ ~ ek  ~ d;

ð1Þ

where ~ ek is the sensitivity vector, and ~ d is the displacement vector with components u, v, and w. Since we have three components of the displacement vector, it is necessary to write three equations to 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 non coplanar. If of the three vectors, two are coplanar, this means that one has only one additional equation that is independent. In other words, two equations are linear combinations of each other, and hence, they are not independent. The sensitivity vectors are defined as functions of the illumination and observation directions [9]. It is only when one takes the phase difference corresponding to two illuminations vectors with a common observation vector, that the observation vector is removed. Hence only for this particular case the optical path difference depends on the illumination vectors. The sensitivity vectors for each of pair of illumination beam directions are defined as [10]

~ e1 ¼

  2 p x  xi x  xj ;  k ri rj   2p y  yi y  yj eky ¼ ;  k ri rj   2p z  zi z  zj ekz ¼ ;  k ri rj

ekx ¼

2p ^2 Þ; ^1  n  ðn k

~ e2 ¼

2p ^ 3 Þ; ^2  n  ðn k

~ e3 ¼

2p ^1 Þ ^3  n  ðn k

ð3Þ

^1 , n ^ 2 , and n ^ 3 are unit vectors along the illumination beam to where n each one of the sources. Notice that the illumination directions change for each point on the inspected area of the target object. We evaluate the displacement vector, ~ d, by decomposing the sensitivity vectors into their orthogonal components x, y, and z [i.e. ~ ek ¼ ðekx ; eky ; ekz Þ]. For the experiments presented here, the origin of a Cartesian coordinate system is placed at the object centre. This coordinate system is used to measure the location of the object illuminating beams and a CCD sensor (see Fig. 1). In order to solve for the all three components, three independent equations are necessary, namely three geometries. We analyze the case of multiple illumination beams that without the need of a cumbersome in-line reference beam, and when properly chosen and taken two by two in a sequential recording order, can have the interesting property of being sensitive only to the in-plane displacement components. With this in mind, the sensitivity vectors components due to the illumination sources Si(xi, yi, zi) and Sj(xj, yj, zj) can be written for the case of three illuminations sources, as

ð4Þ

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

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðx  xi Þ2 þ ðy  yi Þ2 þ ðz  zi Þ2 ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rj ¼ ðx  xj Þ2 þ ðy  yj Þ2 þ ðz  zj Þ2 :

ri ¼

ð5Þ

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

0

1 0 e1x u B C B ¼ v @ A @ e2x e3x w

e1y

e1z

11 0

C e2z A e3z

e2y e3y

1 D/1 B C  @ D/2 A D/3

ð6Þ

It should be noted that it must possible to invert the sensitivity matrix. Therefore, the experimental setup should be chosen in such a way that the three sensitivity vectors are linearly independent. The solution Eq. (6) is valid only if the determinant D of the sensitivity matrix is different from zero

D ¼ e1x ðe2y e3z  e2z e3y Þ  e1y ðe2x e3z  e2z e3x Þ þ e1z ðe2x e3y  e2y e3x Þ: ð7Þ By substituting of Eq. (4) in Eq. (7), D = 0 is obtained. Then, this proposed optical configuration with three object illumination beams does not allow separating the three displacement vector components because two vectors are coplanar. However, it is possible to obtain the two in-plane displacement components if the contributions of e1z, e2z, and e3z are very small compared to the other sensitivity vector components (e1z compared to e1x and e1y, e2z compared to 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 sources are located far enough from the object’s plane (or if collimated illumination is used). In this case, component w of displacement d cannot 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 e1x e2y  e1y e2x

ð9Þ

The above equation is true because two of three sensitivity vectors are linearly independent. S1 S2

r1 r2

S3

y

n1

S1

n2

S2

.P n3 r3

x

O

r3

y

n3

r1 r2

n1

.P

n2

z

S3 Fig. 1. Case A: representation of the illumination sources positions to analyze the sensitivity matrix.

O

z

x Fig. 2. Case B: representation of the illumination sources positions for the experimental case.

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

The geometry with three illumination sources allows the obtaining of the two in-plane components. Fig. 2 presents the studied experimental case and Fig. 3 shows the percentage of each one of the sensitivity vector components to each one of dual illumination 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 vector component can be negligible. The maximum value is actually nearly 2%. The optical system was designed in such a way that the in-plane sensitivity vector component to y-direction is greater than the in-plane sensitivity vector component to x-direction. The

3. Experimental part An He–Cd laser with a wavelength of 440 nm was used as a light source, which was split into three object beams, Fig. 4. Fig. 5 shows a photography of the arrangement that was used. The three object beams were conveyed through single-mode optical fibers to illuminate a test object from different directions

e1x (%)

e1y (%)

e1z (%)

20.4 20.3 20.1 24

78.7 78.1 77.6 24

1.9 1.5 1.1

12

0

y (mm)

-12

-24 -32

-16

0

32

16

12

0

y (mm)

x (mm)

-12

-16

-24 -32

0

32

16

24

e2y (%)

e2z (%)

21.3 21.1 21 24

77.9 77.4 76.9 24

1.9 1.5 1.1

0

y (mm)

-12

-24 -32

-16

0

16

32

12

0

y (mm)

x (mm)

-12

-24 -32

-16

0

16

32

100 99.9 24

12

y (mm)

0

-12

-24 -32

-16

0

16

32

12

0

y (mm)

-12

-12

-24 -32

-24 -32

-16

-16

0

0

16

32

x (mm)

16

32

x (mm)

e3z (%)

9x10-5 4x10-5 2x10-10 24

12

0

y (mm)

x (mm)

24

x (mm)

e3y (%)

e3x (%)

0

y (mm)

x (mm)

e2x (%)

12

12

-12

-24 -32

-16

0

16

32

0.034 0.017 3x10-6 24

12

0

y (mm)

x (mm)

-12

-24 -32

-16

0

16

32

x (mm)

Fig. 3. Percentage of each of the sensitivity vector components to each pair of sources for the case of Fig. 2.

He-Cd Laser M BS L L

BS L S3 S1 OF

OF S2 OF

n3 CCD

n1

y O

n2

z x

^1 , n ^ 2 and n ^ 3 ; M, mirror; Fig. 4. Experimental setup for simultaneous two-dimensional measurement with recording of one image corresponding to illumination directions n BS, beam-splitter; L, lens; OF, single-mode optical fibers; O object.

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

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

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

simultaneously. The incidence angles were 11° for S1 and S2 and 21° for S3, and the scattered light was imaged onto the CCD array. A total of three two-beam interferometers (constructed by the combination of sources S1–S2, S2–S3 and S3–S1) resulted from the three-beam configuration shown in Fig. 4. The experiment was conducted on an elastic surface of width w = 6.47 cm, height h = 4.74 cm and thickness t = .14 mm, which can be considered as a thin plate. One of the edges was clamped rigidly, while the other side of the plate was submitted to the action of tensile force in the x-direction, and which was uniformly distributed along the longitudinal side of the plate.

Speckle images were taken with an interline CCD monochromatic camera (Cohu, model 2122), and then captured and digitalized to 640  480 pixels on 8-bit gray scale with a framegrabber (National Instruments, model 1409). The first speckle image of the undeformed test object was captured and stored. The test object 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 generated, and so a total of six peaks appear in the Fourier transform plane (Fig. 7). A peak corresponding to each one of interferometers was extracted through appropriate masks, and an inverse Fourier transform was then applied to each of the masked images. The ratio between the imaginary and the real part at each point represents the phase of the deformed fringe pattern. Fig. 8 shows the fringe patterns obtained from each selected peak in the Fourier plane when the inversed Fourier transform was applied. In this case, the fringe pattern gradient coincides with the direction of the sensitivity vector, which happens to be true in the particular case for which the experiment was done, but it is not true in general [12]. The fringe patterns shown in Fig. 8 are so smooth due the filter used, which eliminated the frequency corresponding to speckle and the frequency associated to other the two interferometers. It is well known that the Fourier transform method (FTM) can be used to extract the fringe phase with certain conditions [8]. If the phase variation is slow, band-pass filtering in the spatial frequency domain isolates one of these terms. The isolated term is transferred to the origin to remove the carrier, and then the inverse Fourier transform is calculated again to obtain a complex function with real and imaginary non-zero parts. The main advantage of the FTM is that it only requires capturing a single fringe pattern. However, the type of fringes that it can demodulate should be open and of high frequency, and the FTM application to closed fringes yields biased results. Yet, even if the fringes have the proper frequency and are adequately open, the FTM outcomes depend on both the width and the location of the filter mask used to isolate the term carrying the phase information. Errors in the determination of these factors affect the FTM performance by changing the phase values retrieved by means of this technique. The influences of these error sources on the phase uncertainty have been characterized and compared by carrying out an uncertainty analysis on the data retrieved by using the FTM from an ESPI fringe pattern [13]. In a general case, carrier fringes are introduced to apply the Fourier method. In the experimental case presented, the fringes obtained are considered carrier fringes. Fig. 9 shows the wrapped phase associated to the fringe patterns shown in Fig. 8.

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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.

15 10 u 5 (μ m) 0 24 12

32 16

0 y (mm)

0

-12

-16 -24

x (mm)

-32

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

2 v 0.6 (μm) -0.6 -2 24 12

32 16

0 y (mm)

0

-12

-16 -24

x (mm)

-32

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

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24

y (mm)

12

0

-12

-24 -32

-12.6

7.2 x (mm)

27

46.7

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

Fig. 10 shows the 3D plot for the deformation that depicts the udirection. Fig. 11 shows 3D plot for the deformation that depicts the v-direction. The surface target topography is flat, which is considered when the sensitivity vector components are evaluated from Eqs. (4) and (5). With these data, the deformation in the Cartesian u, v, directions is finally mapped from Eq. (8). The result of the 2D deformation measurement (Figs. 10 and 11), combined with the shape data, is shown in Fig. 12. The figure shows a solid surface that represents the target before being loaded. The crossed grating surface, in the same figure, represents the target after being loaded. To appreciate the surface deformation, the graphic has been scaled with a factor of 1000 units for u and v, respectively. 4. Conclusion We have proposed an ESPI system that, in combination with the Fourier transform, is capable of simultaneously measuring the two in-plane components from a single interference pattern. The optical setup is built up with three illuminating sources. The advantage of this method is that it could be used for analysis in real time of inplane vibration. One more of the advantages of the presented technique is that uses a smaller number of illumination sources to get two in-plane displacement components. There is not redirection of the light like in some of the other reported cases.

Acknowledgments Authors wish to thank partial economical support from ‘‘Consejo de Ciencia y Tecnología del Estado de Guanajuato”, as well as to ‘‘Consejo Nacional de Ciencia y Tecnología”, and Mario Alberto Ruiz for his technical support. References [1] [2] [3] [4]

[5] [6] [7] [8] [9] [10] [11] [12] [13]

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