- Email: [email protected]
Evaluation of error in the measurement of displacement vector components by using electronic speckle pattern interferometry
Optics Communications 271 (2007) 445–450 www.elsevier.com/locate/optcom
Evaluation of error in the measurement of displacement vector components by using electronic speckle pattern interferometry Amalia Martı´nez *, J.A. Rayas ´ ptica, A. C. Apartado Postal 1-948, C. P. 37000, Leo´n, Guanajuato, Mexico Centro de Investigaciones en O Received 16 March 2006; received in revised form 11 October 2006; accepted 23 October 2006
Abstract We report on the errors obtained by comparing in- and out-of-plane displacements calculated from the sensitivity matrix with all its components, and when only the component from the largest contributing of each one of the three interferometers is considered. Divergent illumination is considered in the evaluation of sensitivity vector to measure displacement vector components. This analysis is performed for a flat elastic target which is loaded in the x-direction and after in the z-direction. The technique applied is electronic speckle pattern interferometry. 2006 Elsevier B.V. All rights reserved. PACS: 06.20.f; 07.60.Ly Keywords: Optical metrology; In-plane and out-of-plane displacements; ESPI; Sensitivity matrix
1. Introduction Electronic speckle pattern interferometry (ESPI) [1,2] is an effective technique for the non-destructive and non-contact measurement of displacement or deformation. The distribution of displacement along a sensitivity vector, defined by the configuration of the interferometer, can be observed as a two-dimensional contour or fringe map. Both in-plane and out-of-plane displacements can be detected with ESPI. To retrieve the phase of the displacement fringes, one can use either phase shifting [3] or the Fourier transform method [4]. However, the phase gives only a first impression of the deformation of the surface. In practice, the three-dimensional displacement components are required if the mechanical behaviour of the object under load is to be investigated. To calculate displacement components, some other variables are necessary: the three coordinates of the object points and the evaluation of sensitivity vector *
Corresponding author. Tel.: +52 477 4414200; fax: +52 477 4414209. E-mail address: [email protected] (A. Martı´nez).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.046
components. In general, the evaluation of the displacement is carried out without considering the spatial variation of the sensitivity vector when divergent illumination is used [5,6]. Only few works regarding the error introduced by the use of a divergent illumination wavefront to calculate displacement vector components have been reported in the literature [7–11]. In our experiment, a combination of double dual and simple illumination interferometers [12] is used to measure 3D deformation. The optical system can be switched to get simple and double dual illumination. The optical system uses divergent illumination for both cases: in-plane and out-of-plane sensitivity. Due to this kind of illumination, the experimental system sensitivity has spatial variation of the sensitivity vector, which is taken into account to calculate the displacements. The error of evaluation of the in-plane and out-of-plane displacement components is computed considering all the contributions from the sensitivity vector components in the sensitivity matrix [3] and those obtained after ignoring the contribution from the y and
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
446
z-components of the sensitivity vector to measure u-displacement, ignoring the contribution of x and z-components of the sensitivity vector to measure v-displacement, and ignoring the contribution of x and y-sensitivity components to measure w-displacement. ESPI results for an elastic surface under mechanical load are shown. Those results are important because they show that each component of the displacement vector can be obtained by ignoring the two components of the sensitivity vector which make very small contributions. It is shown that for a plane sample of size 5.6 · 5.6 cm, the maximum error is of approximately 0.018 lm when measuring u-displacement, 0.016 lm for v-displacement and 0.12 lm for w-displacement. However, the effect of spatial variation of the sensitivity vector for object divergent illumination must be not ignored [9].
consisting of a subtraction of two speckle images recorded before and after deformation of a test object shows fringes that contour the object deformation. The phase change arising from the object deformation, D/, is related to the three-dimensional displacement vector, ~ d, through a sensitivity vector ~ e as
laser M1 switcher S2
y
CCD θ1
BS
x
M2 S1
2. Principles behind of the measurement When the propagation vectors of two laser beams forming a speckle interferometer are denoted as ^e1 and ^e2 , the sensitivity vector, ~ e, of the speckle interferometer is represented as ~ e¼
2p ð^e1 ^e2 Þ: k
z Object Plane
a M3
ð1Þ
The sensitivity vector depends on the geometry of the arrangement and on the wavelength of the laser source. In the double-illumination method, both ^e1 and ^e2 correspond to unit vectors that describe the vectorial characteristics of illuminating beams emerging from S1 and S2, respectively. In this case, the system sensitivity does not depend on the observation direction. In the reference beam method, one vector corresponds to the vectorial characteristics of illuminating vector and the other corresponds to the observation vector. In this case, the system sensitivity depends on the observation direction. An interferogram
S3
laser switcher
y
CCD θ2 x
M4
S4
z Object Plane
b
laser switcher M5 CCD BS
y
S5 θ3 x
S6 M6 Object Plane
z
c Fig. 1. Photograph of ESPI system which is formed by three optical circuits showed in Fig. 2. The optical components are indicated: B is the laser illumination; M1. . .M6 are mirrors; SS is a switched system; CCD is the camera; O is the object; S1. . .S6 are spatial filters.
Fig. 2. (a) Dual-beam optical setup with sensitivity mostly along x, (b) dual illumination interferometer with sensitivity mostly along y and (c) interferometer with one beam to sensitivity mostly along z.
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
D/ðP Þ ¼ ~ dðP Þ ~ eðP Þ;
ð2Þ
where the point denotes a scalar dot product. To extract the phase from an interferogram, we use the phase shifting method. The use of multiple interferometers whose sensitivity vectors are different permits the measurement of the three displacement components. A sensitivity matrix, consisting of three sensitivity vectors that relate the three measured phase maps to the three components of displacement, can be derived. Because the sensitivity matrix is not singular, the inverse sensitivity matrix exists, and therefore, displacement maps can be calculated from the phase maps. Let us consider the three sensitivity vectors e1(P), e2(P) and e3(P), in which the upper index denotes correspondence to each one of the interferometers. In this case, e1(P) is associated to dual illumination with x-sensitivity, e2(P) to dual illumination with y-sensitivity, and e3(P) associated to z-sensitivity. Then, at each point P, we have to solve the system of linear equations: 0 1 0 1 1 1 0 1 ex ey e1z uðP Þ D/1 ðP Þ B C B C B C ð3Þ @ D/2 ðP Þ A ¼ @ e2x e2y e2z A @ vðP Þ A D/3 ðP Þ
e3x
e3y
wðP Þ
e3z
to obtain d(u, v, w). The solution is dðu; v; wÞ ¼ E1 ðP Þ D/ðP Þ;
ð4Þ
where E(P) is the sensitivity matrix. The sensitivity vector components for each case are given by [12]. In the first case, we have calculated the sensitivity matrix with all the sensitivity vector components. In the second case, the sensitivity matrix was reduced to 1 0 1 0 1 0 1 ex 0 0 uðP Þ D/1 ðP Þ C B C B C B ð5Þ @ D/2 ðP Þ A ¼ @ 0 e2y 0 A @ vðP Þ A; 3 3 wðP Þ D/ ðP Þ 0 0 ez where it has been supposed that some sensitivity vector components can be ignored because each interferometer
447
presents the advantage of a larger sensitivity in each of one directions u, v and w, respectively. 3. Evaluation of error in the measurement Eq. (2) shows that the measured phase difference D/(P) is equivalent to the component ds of the displacement vector ~ d in the direction of the sensitivity vector ~ eðP Þ. However, this direction depends on the location P(x, y, z) where the phase D/(P) is measured, since the observation and illumination directions vary with the object points. Often, this dependence is depreciated by assuming a constant sensitivity across the surface of the object. In this way, the measurement is simplified considerably, because the geometry of the object is without influence. Consequently, the shape of the object does not have to be measured, and the displacement measurement is reduced to the reconstruction of the interference phase for every object point. This simplified technique may cause serious errors [9]. If the vector sensitivity is calculated considering its variation for each surface point, but some vector sensitivity components are ignored, so such that the sensitivity matrix is reduced to the expression given by Eq. (5), then error associated to measurement when the displacement is calculated from Eqs. (3) and (5) can be written as Eu ¼ juf ud j;
ð6Þ
where uf is the displacement measured by the sensitivity matrix given by 3 and ud is the displacement measured by the using sensitivity matrix given by Eq. (5). By using similar equations, the error associated to measurement of v and w was calculated. 4. Phase calculation by phase-stepping method The method utilized to obtain the wrapped phase from a fringe pattern is phase-stepping method. One of the advantages of this technique, using n-steps, is that laser speckle noise is reduced. The wrapped phase is given by [13]
y
x
Work area
→
Fx →
Fz
a
→
F
b
Fig. 3. Representation of the object target and the loads applied.
z
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
448
Ið0ÞIðN Þ ctg 2p 2 N
tanðD/Þ ¼ Ið0ÞþIðN Þ 2
þ
NP 1
IðnÞ sin
2pn
n¼1 NP 1 n¼1
IðnÞ cos
2pn
N
;
ð7Þ
N
where I(n) is the nth image and N it is the number of steps. 5. Experimental setup Fig. 1 illustrates the optical system used for speckle correlation. An He-CD laser is utilized with a power of 100 mW and k = .440 lm. For improved clarity of Fig. 1, the three interferometers are shown separately in Fig. 2. Fig. 2a shows an in-plane interferometer which has the largest sensitivity component in the x-direction. The sources coordinates were: Ps1 = (46.5 cm, 0 cm, 46 cm) and Ps2 = (46.5 cm, 0 cm, 46 cm). Afterwards, the dual illumination is redirected to the object surface to obtain the largest sensitivity component in the y-direction, Fig. 2b, where the sources coordinates were: Ps3 = (0 cm, 47 cm, 46 cm) and Ps4 = (0 cm, 47 cm, 46 cm). In the case of both dual illuminations, the incidence angle was h = 45. In each one of dual illumination, one of the illumination beams was reflected from a piezoelectric mirror to implement the phase stepping technique. Finally, the illumination is redirected but using only one object beam of illumination to get an interferometer with the largest sensitivity component in the z-direction, Fig. 2c. The following coordinates were considered for observation and illumination respectively: P0 = (0 cm, 0 cm, 85 cm) and P5 = (5 cm, 0 cm, 66 cm). The incidence angle of object illumination to the out-of-plane system was h = 4. The target object consisted in an elastic surface with width l = 5.73 cm and thickness h = .33 mm which can be considered 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, was uniformly distributed along the longitudinal side of the plate. The plate was then deflected in such a way that the points of the plate initially lying on a normal to the middle plane of the plate remain on the normal to the middle surface of the plate after bending, as shown in Fig. 3. PZT1 and PZT2 are shown which induce displacements on the target. Fringe patterns were captured by means of a CCD camera of 640 · 480 pixels, and 255 levels of gray. It used a phase stepping technique for 15-steps to get the optical phase [13]. The camera observing the object along its normal records two optical fields, one before and another second after final deformation. Three recordings, one for each interferometer are made. The recordings made before the deformation are subtracted one-by-one from the corresponding set of recordings taken after the deformation. As a result, we end up with three interferograms that represent the displacement components in three sensitivity directions (u, v, w). Fig. 4 shows the wrapped phase associated to displacements u, v, and w, respectively. Fig. 5 shows
Fig. 4. Wrapped phase associated to (a) u, (b) v and (c) w.
the 3D graph associated to the deformation on the x-direction, which was obtained by using Eqs. (3) and (5), respectively, which corresponds to a and b, and c the error associated to displacement. Figs. 6 and 7, in the same fashion, show the deformation on v and w directions, respectively. The obtained maximum error is of approximately
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
449
Fig. 5. u-Displacement field calculated (a) with all the elements of the sensitivity matrix, and (b) when only sensitivity vector component ex was considered and (c) plotting of error among results shown in (a) and (b).
Fig. 6. v-Displacement field calculated (a) with all the elements of the sensitivity matrix, and (b) when only sensitivity vector component ey was considered and (c) plotting of error among results shown in (a) and (b).
0.018 lm when measuring u-displacement, 0.016 lm for v-displacement and 0.12 lm for w-displacement. Displacement v is a consequence of the Poisson effect and hence, smaller than u. The surface target topography
is plane, which is considered when the sensitivity matrix is evaluated from Eqs. (3) and (5). With these data, the displacements in the Cartesian u, v, w directions are finally mapped. In this case, all the components of the sensitivity
450
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
matrix (Eq. (3)) were calculated. The sensitivity matrix represented by Eq. (5) was calculated considering the same coordinates, but taking only the component from largest contribution of each interferometer. 6. Conclusions We calculated the associated error to each one of the displacement vector components when these were obtained with all the elements of sensitivity matrix, and when were taken only the major contributions of each sensitivity vector associated to each interferometer and considering divergent illumination. The results show the possibility to simplify the calculation when only the largest component in each interferometer is considered. In this case, the displacement vector components can be obtained in an independent manner. Also, it is observed that in the consideration of collimated illumination when divergent illumination is used, in the computation of displacement vector components, the introduced errors must not be ignored. Acknowledgements The authors wish to thank Consejo Nacional de Ciencia y Tecnologı´a (CONACYT) and Consejo de Ciencia y Tecnologı´a del Estado de Guanajuato (CONCYTEG) for their partial economical support. References
Fig. 7. w-Displacement field calculated (a) with all the elements of the sensitivity matrix, and (b) when only sensitivity vector component ez was considered and (c) plotting of error among results shown in (a) and (b).
[1] J.N. Butters, J.A. Leendertz, J. Meas. Control 4 (1971) 349. [2] R.S. Sirohi, in: R.S. Sirohi (Ed.), Optical Methods of Measurement, Marcel Dekker Inc., New York, 1999. [3] T. Kreis, Holographic Interferometry, Quantitative Evaluation of the Interference Phase, in Holographic Interferometry, Akademie Verlag, New York, 1996. [4] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1982) 156. [5] S. Yoshida, Suprapedi, R. Widiastuti, Marincan, Septriyanti, Julinda, A. Faisal, A. Kusnowo, Appl. Opt. 36 (1997) 266. [6] J. Zhang, T. Chong Chong, Appl. Opt. 37 (1998) 6707. [7] D. Albrecht, Opt. Lasers Eng. 31 (1999) 63. [8] H.J. Puga, R. Rodrı´guez-Vera, Amalia Martı´nez, Opt. Laser Tech. 34 (2002) 81. [9] Amalia Martı´nez, R. Rodrı´guez-Vera, J.A. Rayas, H.J. Puga, Opt. Comm. 223 (2003) 239. [10] R. Cordero, Amalia Martı´nez, R. Rodrı´guez-Vera, P. Roth, Opt. Comm. 241 (2004) 279. [11] N. Sujatha, V.M. Murukeshan, A.R. Ganesan, J. Hologr. Speckle 2 (2005) 20. [12] Amalia Martı´nez, J.A. Rayas, R. Rodrı´guez-Vera, H.J. Puga, Appl. Opt. 43 (2004) 4652. [13] Y. Surrel, Appl. Opt. 32 (1993) 3598.
Evaluation of error in the measurement of displacement vector components by using electronic speckle pattern interferometry Amalia Martı´nez *, J.A. Rayas ´ ptica, A. C. Apartado Postal 1-948, C. P. 37000, Leo´n, Guanajuato, Mexico Centro de Investigaciones en O Received 16 March 2006; received in revised form 11 October 2006; accepted 23 October 2006
Abstract We report on the errors obtained by comparing in- and out-of-plane displacements calculated from the sensitivity matrix with all its components, and when only the component from the largest contributing of each one of the three interferometers is considered. Divergent illumination is considered in the evaluation of sensitivity vector to measure displacement vector components. This analysis is performed for a flat elastic target which is loaded in the x-direction and after in the z-direction. The technique applied is electronic speckle pattern interferometry. 2006 Elsevier B.V. All rights reserved. PACS: 06.20.f; 07.60.Ly Keywords: Optical metrology; In-plane and out-of-plane displacements; ESPI; Sensitivity matrix
1. Introduction Electronic speckle pattern interferometry (ESPI) [1,2] is an effective technique for the non-destructive and non-contact measurement of displacement or deformation. The distribution of displacement along a sensitivity vector, defined by the configuration of the interferometer, can be observed as a two-dimensional contour or fringe map. Both in-plane and out-of-plane displacements can be detected with ESPI. To retrieve the phase of the displacement fringes, one can use either phase shifting [3] or the Fourier transform method [4]. However, the phase gives only a first impression of the deformation of the surface. In practice, the three-dimensional displacement components are required if the mechanical behaviour of the object under load is to be investigated. To calculate displacement components, some other variables are necessary: the three coordinates of the object points and the evaluation of sensitivity vector *
Corresponding author. Tel.: +52 477 4414200; fax: +52 477 4414209. E-mail address: [email protected] (A. Martı´nez).
0030-4018/$ - see front matter 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.optcom.2006.10.046
components. In general, the evaluation of the displacement is carried out without considering the spatial variation of the sensitivity vector when divergent illumination is used [5,6]. Only few works regarding the error introduced by the use of a divergent illumination wavefront to calculate displacement vector components have been reported in the literature [7–11]. In our experiment, a combination of double dual and simple illumination interferometers [12] is used to measure 3D deformation. The optical system can be switched to get simple and double dual illumination. The optical system uses divergent illumination for both cases: in-plane and out-of-plane sensitivity. Due to this kind of illumination, the experimental system sensitivity has spatial variation of the sensitivity vector, which is taken into account to calculate the displacements. The error of evaluation of the in-plane and out-of-plane displacement components is computed considering all the contributions from the sensitivity vector components in the sensitivity matrix [3] and those obtained after ignoring the contribution from the y and
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
446
z-components of the sensitivity vector to measure u-displacement, ignoring the contribution of x and z-components of the sensitivity vector to measure v-displacement, and ignoring the contribution of x and y-sensitivity components to measure w-displacement. ESPI results for an elastic surface under mechanical load are shown. Those results are important because they show that each component of the displacement vector can be obtained by ignoring the two components of the sensitivity vector which make very small contributions. It is shown that for a plane sample of size 5.6 · 5.6 cm, the maximum error is of approximately 0.018 lm when measuring u-displacement, 0.016 lm for v-displacement and 0.12 lm for w-displacement. However, the effect of spatial variation of the sensitivity vector for object divergent illumination must be not ignored [9].
consisting of a subtraction of two speckle images recorded before and after deformation of a test object shows fringes that contour the object deformation. The phase change arising from the object deformation, D/, is related to the three-dimensional displacement vector, ~ d, through a sensitivity vector ~ e as
laser M1 switcher S2
y
CCD θ1
BS
x
M2 S1
2. Principles behind of the measurement When the propagation vectors of two laser beams forming a speckle interferometer are denoted as ^e1 and ^e2 , the sensitivity vector, ~ e, of the speckle interferometer is represented as ~ e¼
2p ð^e1 ^e2 Þ: k
z Object Plane
a M3
ð1Þ
The sensitivity vector depends on the geometry of the arrangement and on the wavelength of the laser source. In the double-illumination method, both ^e1 and ^e2 correspond to unit vectors that describe the vectorial characteristics of illuminating beams emerging from S1 and S2, respectively. In this case, the system sensitivity does not depend on the observation direction. In the reference beam method, one vector corresponds to the vectorial characteristics of illuminating vector and the other corresponds to the observation vector. In this case, the system sensitivity depends on the observation direction. An interferogram
S3
laser switcher
y
CCD θ2 x
M4
S4
z Object Plane
b
laser switcher M5 CCD BS
y
S5 θ3 x
S6 M6 Object Plane
z
c Fig. 1. Photograph of ESPI system which is formed by three optical circuits showed in Fig. 2. The optical components are indicated: B is the laser illumination; M1. . .M6 are mirrors; SS is a switched system; CCD is the camera; O is the object; S1. . .S6 are spatial filters.
Fig. 2. (a) Dual-beam optical setup with sensitivity mostly along x, (b) dual illumination interferometer with sensitivity mostly along y and (c) interferometer with one beam to sensitivity mostly along z.
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
D/ðP Þ ¼ ~ dðP Þ ~ eðP Þ;
ð2Þ
where the point denotes a scalar dot product. To extract the phase from an interferogram, we use the phase shifting method. The use of multiple interferometers whose sensitivity vectors are different permits the measurement of the three displacement components. A sensitivity matrix, consisting of three sensitivity vectors that relate the three measured phase maps to the three components of displacement, can be derived. Because the sensitivity matrix is not singular, the inverse sensitivity matrix exists, and therefore, displacement maps can be calculated from the phase maps. Let us consider the three sensitivity vectors e1(P), e2(P) and e3(P), in which the upper index denotes correspondence to each one of the interferometers. In this case, e1(P) is associated to dual illumination with x-sensitivity, e2(P) to dual illumination with y-sensitivity, and e3(P) associated to z-sensitivity. Then, at each point P, we have to solve the system of linear equations: 0 1 0 1 1 1 0 1 ex ey e1z uðP Þ D/1 ðP Þ B C B C B C ð3Þ @ D/2 ðP Þ A ¼ @ e2x e2y e2z A @ vðP Þ A D/3 ðP Þ
e3x
e3y
wðP Þ
e3z
to obtain d(u, v, w). The solution is dðu; v; wÞ ¼ E1 ðP Þ D/ðP Þ;
ð4Þ
where E(P) is the sensitivity matrix. The sensitivity vector components for each case are given by [12]. In the first case, we have calculated the sensitivity matrix with all the sensitivity vector components. In the second case, the sensitivity matrix was reduced to 1 0 1 0 1 0 1 ex 0 0 uðP Þ D/1 ðP Þ C B C B C B ð5Þ @ D/2 ðP Þ A ¼ @ 0 e2y 0 A @ vðP Þ A; 3 3 wðP Þ D/ ðP Þ 0 0 ez where it has been supposed that some sensitivity vector components can be ignored because each interferometer
447
presents the advantage of a larger sensitivity in each of one directions u, v and w, respectively. 3. Evaluation of error in the measurement Eq. (2) shows that the measured phase difference D/(P) is equivalent to the component ds of the displacement vector ~ d in the direction of the sensitivity vector ~ eðP Þ. However, this direction depends on the location P(x, y, z) where the phase D/(P) is measured, since the observation and illumination directions vary with the object points. Often, this dependence is depreciated by assuming a constant sensitivity across the surface of the object. In this way, the measurement is simplified considerably, because the geometry of the object is without influence. Consequently, the shape of the object does not have to be measured, and the displacement measurement is reduced to the reconstruction of the interference phase for every object point. This simplified technique may cause serious errors [9]. If the vector sensitivity is calculated considering its variation for each surface point, but some vector sensitivity components are ignored, so such that the sensitivity matrix is reduced to the expression given by Eq. (5), then error associated to measurement when the displacement is calculated from Eqs. (3) and (5) can be written as Eu ¼ juf ud j;
ð6Þ
where uf is the displacement measured by the sensitivity matrix given by 3 and ud is the displacement measured by the using sensitivity matrix given by Eq. (5). By using similar equations, the error associated to measurement of v and w was calculated. 4. Phase calculation by phase-stepping method The method utilized to obtain the wrapped phase from a fringe pattern is phase-stepping method. One of the advantages of this technique, using n-steps, is that laser speckle noise is reduced. The wrapped phase is given by [13]
y
x
Work area
→
Fx →
Fz
a
→
F
b
Fig. 3. Representation of the object target and the loads applied.
z
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
448
Ið0ÞIðN Þ ctg 2p 2 N
tanðD/Þ ¼ Ið0ÞþIðN Þ 2
þ
NP 1
IðnÞ sin
2pn
n¼1 NP 1 n¼1
IðnÞ cos
2pn
N
;
ð7Þ
N
where I(n) is the nth image and N it is the number of steps. 5. Experimental setup Fig. 1 illustrates the optical system used for speckle correlation. An He-CD laser is utilized with a power of 100 mW and k = .440 lm. For improved clarity of Fig. 1, the three interferometers are shown separately in Fig. 2. Fig. 2a shows an in-plane interferometer which has the largest sensitivity component in the x-direction. The sources coordinates were: Ps1 = (46.5 cm, 0 cm, 46 cm) and Ps2 = (46.5 cm, 0 cm, 46 cm). Afterwards, the dual illumination is redirected to the object surface to obtain the largest sensitivity component in the y-direction, Fig. 2b, where the sources coordinates were: Ps3 = (0 cm, 47 cm, 46 cm) and Ps4 = (0 cm, 47 cm, 46 cm). In the case of both dual illuminations, the incidence angle was h = 45. In each one of dual illumination, one of the illumination beams was reflected from a piezoelectric mirror to implement the phase stepping technique. Finally, the illumination is redirected but using only one object beam of illumination to get an interferometer with the largest sensitivity component in the z-direction, Fig. 2c. The following coordinates were considered for observation and illumination respectively: P0 = (0 cm, 0 cm, 85 cm) and P5 = (5 cm, 0 cm, 66 cm). The incidence angle of object illumination to the out-of-plane system was h = 4. The target object consisted in an elastic surface with width l = 5.73 cm and thickness h = .33 mm which can be considered 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, was uniformly distributed along the longitudinal side of the plate. The plate was then deflected in such a way that the points of the plate initially lying on a normal to the middle plane of the plate remain on the normal to the middle surface of the plate after bending, as shown in Fig. 3. PZT1 and PZT2 are shown which induce displacements on the target. Fringe patterns were captured by means of a CCD camera of 640 · 480 pixels, and 255 levels of gray. It used a phase stepping technique for 15-steps to get the optical phase [13]. The camera observing the object along its normal records two optical fields, one before and another second after final deformation. Three recordings, one for each interferometer are made. The recordings made before the deformation are subtracted one-by-one from the corresponding set of recordings taken after the deformation. As a result, we end up with three interferograms that represent the displacement components in three sensitivity directions (u, v, w). Fig. 4 shows the wrapped phase associated to displacements u, v, and w, respectively. Fig. 5 shows
Fig. 4. Wrapped phase associated to (a) u, (b) v and (c) w.
the 3D graph associated to the deformation on the x-direction, which was obtained by using Eqs. (3) and (5), respectively, which corresponds to a and b, and c the error associated to displacement. Figs. 6 and 7, in the same fashion, show the deformation on v and w directions, respectively. The obtained maximum error is of approximately
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
449
Fig. 5. u-Displacement field calculated (a) with all the elements of the sensitivity matrix, and (b) when only sensitivity vector component ex was considered and (c) plotting of error among results shown in (a) and (b).
Fig. 6. v-Displacement field calculated (a) with all the elements of the sensitivity matrix, and (b) when only sensitivity vector component ey was considered and (c) plotting of error among results shown in (a) and (b).
0.018 lm when measuring u-displacement, 0.016 lm for v-displacement and 0.12 lm for w-displacement. Displacement v is a consequence of the Poisson effect and hence, smaller than u. The surface target topography
is plane, which is considered when the sensitivity matrix is evaluated from Eqs. (3) and (5). With these data, the displacements in the Cartesian u, v, w directions are finally mapped. In this case, all the components of the sensitivity
450
A. Martı´nez, J.A. Rayas / Optics Communications 271 (2007) 445–450
matrix (Eq. (3)) were calculated. The sensitivity matrix represented by Eq. (5) was calculated considering the same coordinates, but taking only the component from largest contribution of each interferometer. 6. Conclusions We calculated the associated error to each one of the displacement vector components when these were obtained with all the elements of sensitivity matrix, and when were taken only the major contributions of each sensitivity vector associated to each interferometer and considering divergent illumination. The results show the possibility to simplify the calculation when only the largest component in each interferometer is considered. In this case, the displacement vector components can be obtained in an independent manner. Also, it is observed that in the consideration of collimated illumination when divergent illumination is used, in the computation of displacement vector components, the introduced errors must not be ignored. Acknowledgements The authors wish to thank Consejo Nacional de Ciencia y Tecnologı´a (CONACYT) and Consejo de Ciencia y Tecnologı´a del Estado de Guanajuato (CONCYTEG) for their partial economical support. References
Fig. 7. w-Displacement field calculated (a) with all the elements of the sensitivity matrix, and (b) when only sensitivity vector component ez was considered and (c) plotting of error among results shown in (a) and (b).
[1] J.N. Butters, J.A. Leendertz, J. Meas. Control 4 (1971) 349. [2] R.S. Sirohi, in: R.S. Sirohi (Ed.), Optical Methods of Measurement, Marcel Dekker Inc., New York, 1999. [3] T. Kreis, Holographic Interferometry, Quantitative Evaluation of the Interference Phase, in Holographic Interferometry, Akademie Verlag, New York, 1996. [4] M. Takeda, H. Ina, S. Kobayashi, J. Opt. Soc. Am. 72 (1982) 156. [5] S. Yoshida, Suprapedi, R. Widiastuti, Marincan, Septriyanti, Julinda, A. Faisal, A. Kusnowo, Appl. Opt. 36 (1997) 266. [6] J. Zhang, T. Chong Chong, Appl. Opt. 37 (1998) 6707. [7] D. Albrecht, Opt. Lasers Eng. 31 (1999) 63. [8] H.J. Puga, R. Rodrı´guez-Vera, Amalia Martı´nez, Opt. Laser Tech. 34 (2002) 81. [9] Amalia Martı´nez, R. Rodrı´guez-Vera, J.A. Rayas, H.J. Puga, Opt. Comm. 223 (2003) 239. [10] R. Cordero, Amalia Martı´nez, R. Rodrı´guez-Vera, P. Roth, Opt. Comm. 241 (2004) 279. [11] N. Sujatha, V.M. Murukeshan, A.R. Ganesan, J. Hologr. Speckle 2 (2005) 20. [12] Amalia Martı´nez, J.A. Rayas, R. Rodrı´guez-Vera, H.J. Puga, Appl. Opt. 43 (2004) 4652. [13] Y. Surrel, Appl. Opt. 32 (1993) 3598.