- Email: [email protected]
Digital speckle pattern shearing interferometry using diffraction gratings
__ __ RB
15 May 1996
2@3 ELSIZVIER
OPTICS
COMMUNICATIONS Optics Communications
126 ( 1996) 191- 196
Digital speckle pattern shearing interferometry using diffraction gratings Hector Rabal
* *2,
Rodrigo Henao lp2,Roberto Torroba 2
Ceniro de 1nvestigucione.s Opticas ICfOp).
Cusilla de Correo
Received 22 August 1995; revised version received 5 December
124, 1900 La Plats, Argentina 1995; accepted 7
December 1995
Abstract A digital speckle pattern shearing interferometer with the aid of a diffraction grating as shearing element, is presented. A brief theoretical approach for qualitative purposes is outlined. A comparison with Michelson shearing and phase stepping measurements are presented to show the validity of the proposal. Real time visualization, compactness of the setup and good quality fringes are main advantages.
1. Introduction Diffraction gratings were extensively studied for different purposes. Talbot [l] and Lau [2] effects are examples of their use in analog image processing and optical metrology. They were also used to provide several spatially disjoint viewpoints of an object, and the stereoscopic properties of the image replicas were described [3]. High frequency diffraction gratings provided viewpoints widely apart. It was found that in speckle interferometry a better signal-to-noise ratio can be obtained if the speckle
* Corresponding author. Fax: +54 21 712771; E-mail: [email protected]. ’ Permanent address: ANIRT, Apartado Aereo 2073, Medellin, Colombia. * All authors are also with the Facultad de Ingenieria, Universidad National de La Plats, Argentina. 0030~4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved PII SOO30-40 18(96)000 17-X
image is modulated by a grating [4]. A digital single-beam coherent contouring method was also reported [5]. It is based on the use of a diffraction grating in the illumination beam, the position of which is modified after the reference frame acquisition. All these techniques allow to compact the recording device into a single camera unit and an illumination system. In this last context of compact optical systems, a measurement technique called shearography [6] appears as a prospective non-destructive tool. It is more practical than holographic interferometry as it provides a fringe pattern depicting displacement derivatives. Then, it does not require differentiating the measured displacements as in holography and hence, yield strains. Advantages over holography include simplicity of the optical setup, relaxation of vibration isolation requirements and reduced recording resolution requirements. Shearography is accomplished by the collection of
192
H. Rabal et d/Optics
Communicutioru
two image-sheared image plane interferograms, one before and one after loading the object. Hence shearography is essentially a common path interferometric technique and should be well suited for investigation under turbulence, as its effects between the object and the recording medium tend to cancel out. Besides, coherence restrictions on the light source are alleviated. There are two ways to obtain the two interferograms, either by the double-exposure technique or individual grabbing of the interferograms, i.e. using commercially available PC-based image processing equipment. There exist two basic setups to produce the shear. One consists of an imaging lens together with a thin glass wedge located just in the front of the lens to cover one-half of its aperture [7]. In the other, two mirrors are arranged as in a Michelson interferometer but one of them is slightly tilted with respect to the other. For the light coming from the object entering this setup, the result is the superposition on the imaging device of two laterally displaced images of the object, with a shear proportional to the tilt of the mirror [8]. Frame grabbing offers the posibility of depicting the fringe information in real time [9,10]. Limitations and prospects of the individual frame grabbing used in correlation with digital speckle pattern interferometry (DSPI) based on the above mentioned setups has been described in the literature [ 11 I. In this work we propose the use of a coarse diffraction grating in front of the camera in a DSPI arrangement as shear device. It will produce the overlapping of the speckle patterns so that spatially neighbor speckles are added and the results compared for the before and the after states in comparison or deformation analysis. This shearing interferometer is governed by the same restrictions as the displacement-sensitive interferometers. We outline a theoretical analysis for a qualitative estimation of the expected results. Experimental results for mechanically and thermally induced strains example are given.
2. Theoretical
analysis
126 (1996) 191-196
object. The point spread function of such an image forming system consists of a zero order and several replicas. In the case of holographic gratings, the replicas are reduced to two. If the object is rough, the image will consist of the coherent superposition of the spectra of every single speckle. Some speckles will overlap with the repiclas produced by neighbor speckles in the shearing direction. This situation for the zero and + 1 order can be expressed in mathematical terms by A,( x,
Y>=A(x+Y)exp[icp( x3r>l
as the contribution
from the zero order, and
A’,(x, y)=A(x,
y)exp[icp(x+Ax,
Y)]
(2)
as that from the + 1 order. The shear is represented by AX in the x direction. We assume the same constant amplitude for each beam. The (x, y> denotes image coordinates. The intensity distribution taken by the camera for this primary interferogram is
1,(x,
y)=2
IA12[1 +cos
(P,~(x,Y)],
(3)
where (p,? = cp(x, y> - +4x + A X, y>. After loading the object a phase change 6(x, y) is introduced, then the zero and + 1 orders are modified as A,(x,
y)=A(x,
Y) exp{i[cp(x, Y) +a(~,
Y)]]? (4)
and A’&x,
Y) exp(i[cp(x+Ax7
y)=A(x,
+S(x+Ax,
respectively. 1,(x,
y)=2
(5)
intensity
IA12{1 +co+P,,(x, +6(x,
Y)
Y)]},
The corresponding
~)-6(.x+Ax,
is
Y) Y)]].
(6)
We assume that the object intensity is not alterated by deformation. The difference intensity displayed on the TV monitor, according to the normal DSPI procedure is
II, -1, I = 14 IA -co4
Consider the situation of a grating just in front of an optical system which forms the image of an
(1)
12{cos 4x, (P,2(
-6(x+Ax,
x9
Y)
Y) + 6(x, r)]}l
Y) (7)
H. Ruhal et d/Optics
Communicutions
126 (1996) 191-196
193
or alternatively A(x, (p12(x,y)-
Y) 2
(8) with A(x,
y)=a(x+Ax, =
@(x2
Y)-6(x>
Y) Ax
ax The phase difference S= F[cf
Y)
sin 8+w(l
(9)
. 6 can be expressed fcos
e)],
as
Fig. 2. Experimental setup used to produce the shear for the different test objects. A single laser beam is used to illuminate the specimen. The sinusoidal diffraction grating DG is placed in front of the CCD camera to produce the shear image.
(10)
while 0 is the angle of the object illumination, u and w are the components of the in-plane and out-of-plane deformation, respectively, and A is the wavelength of the light used. A comparison of Eqs. (8), (9) and (10) shows that the resulting correlation fringes depict information on the displacement derivatives, as expected in our proposal. When considering the interaction of the zero and - 1 orders, similar expressions are formed, so that the information is replicated. For the case of the + 1 and - 1 orders, the shear is 2Ax, thus giving the same fringe pattern but twice the frequency. In prac-
LX
Fig. 1. Schematic arrangement representing the grating position as shear device. The shear A x is a function of the z, positions along the z axis, measured with respect to an arbitrary coordinate system. A represents a limiting aperture to lens L. The object and image planes are denoted by 0 and I, respectively.
Fig. 3. (a) Shear fringes along the x (horizontal) axis for a clamped metal plate loaded with a point pressure applied in the center of the plate. The slope contours correspond to the derivatives of the out-of-plane displacements along the x axis. (b) Shows the comparison of the same example using a Michelson shearing interferometer. No significant differences in performance are observed.
194
H. Rub& et d/Optics
Communications
126 (1996) 191-196
tice the contrast of this last fringe system is very low in comparison with the previous one and constitutes no essential source of errors.
700
((3
--------__. ,A
__
---_
1 0 y’
4
Fig. 4. Sequence showing heating device applied in show the derivatives of lence due to heat transfer could be expected.
the effect at different instants of a point the center of the metal plate. The fringes the out-of-plane displacements. Turbudoes not produce decorrelation effect as
Fig. 5. (a) Shear fringes obtained with our proposed setup for a clamped membrane centrally loaded. (b) The phase map of the fringes shown in (a). (c) The phase distribution analyzed from the fringes pattern of (a).
H. Rubal et al./Optics
3. Experimental Referring arrangement grating is
Communications
results
to Fig. 1. the shear introduced by the of a lens and a sinusoidal diffraction
Ax = ADZ,,
;,*/%3 I1 + (
zddl
’
(11)
where f is the frequency of the grating, and z,, = zi - zj are distances defined in Fig. 1. In our experiment we study the case of out-ofplane displacements only (U = 0). Recalling Eqs. (9) and (lo), we found that bright fringes are formed whenever aM ?K=
(2n + 1) ~Ax( 1 + cos 0) ’
(12)
where n is an integer. The grating is a holographically phase grating of 8 l/mm registered such that the zero and + 1 orders are transmitted. This avoids the noise eventually generated by the combinations of the other normal diffraction orders. The employed setup is depicted in Fig. 2. In one experiment we used as test object a clamped metal plate mechanically loaded with a centered point charge from behind. Fig. 3a corresponds to a shear along the _Yaxis, and the fringes give slope contours of aw/ax. As an objective comparison to show that this technique is not outweighed by any significant disadvantage in its performance, we carried out the same experiment with a Michelson shearing interferometer. The results are shown in Fig. 3b. The sequence in Figs. 4a-4c shows the same plate, heated in the center with the point of a welder device, for different instants. In this case, expected decorrelation effects due to the turbulence of the heat transfer phenomenon are diminished. However, speckle decorrelation effects are inherent, as every frame in DSPI shows only the current state of an object compared to the first acquired image. Phase shifting can be used as an evaluation procedure to demonstrate that the technique is fully compatible with the conventional alternatives. It can be implemented by an in-plane translation of the grating. The grating, of spacing d is moved through a distance d/k (that is, the phase is moved through
126 (1996) 191-196
195
2r/k) and a second image is read into the frame grabber. This process is repeated until the grating has moved through (k - 1) equal steps and a total of the k images have been stored in the computer [ 121. The phase maps are calculated then in the usual manner. The sequence of Fig. 5 shows a pratical example with (a) the shear fringes, (b) phase stepping of (a), and (c) the corresponding phase map.
4. Conclusions We have proposed and implemented a digital speckle pattern shearing interferometer by using a diffraction grating as shearing element. This technique represents an attractive alternative to other shearographic setups as it provides real time fringe formation, compactness of the optical setup and the possibility of phase-stepping implementation. In shearography, there are constraints in the rather limited space-bandwidth product (contrast> but in our case, there is no such restriction, as seen from the experimental data. Although only out-of-plane derivatives were considered here, extensions of this method can be made to determine in-plane derivations. The magnitude of the shear, and hence the sensibility can be varied. Experimental demonstration performed so far has used only a planar surface, but the technique can be extended to three-dimensional surfaces.
Acknowledgements This work was supported by CONICET grant PID No. 3974/92 (Argentina), Fundaci6n Antorchas (Argentina), TWAS (The Third World Academy of Sciences) grant 93-389 and Alexander von Humboldt Foundation (Germany). We thank J. Pomarico and R. Arizaga for providing the software and an anonymous referee for helpful suggestions. H.R. performed part of this work at the Bremen Institute for Applied Optics (Germany) under the auspices of the German Academic Exchange Service (DAAD). R.H. acknowledges the support of the Mutis Program (Spain).
196
H. Rahal et d/Optics
Communications
References 111 A. Lohman and D. Silva, Optics Comm 2 (1971) 413. [2] I. Pomarico and R. Torroba, Eur. J. Phys. 14 (1993) 114. [3] R. Henao, F. Medina, H. Rabal and M. Trivi, Appl. Optics 32 (1993) 726. [4] A. Tai, Optics Lett. 5 (1980) 552. 151 N. Bolognini, H. Rabal and R. Torroba, Appl. Optics 31 (1992) 1009. [6] Y. Hung, Opt. Eng. 21 (1982) 391.
126 (1996) 191-196
[7] S. Togooka, H. Nishida and J. Takezaki, Opt. Eng. 28 (1989) 55. [8] J. Leedertz and J. Butters, J. Phys. E 6 (1973) 1107. [9] C. Joenathan and R. Torroba, Optics Lett. 15 (1990) 1159. [lOI A. Ganesan, D. Sharma and M. Kothiyal, Appl. Optics 27 (1988) 473 I. [l I] M. Owner-Petersen, Appl. Optics 30 (1991) 2730. 1121 G.T. Reid, R.C. Rixon and H.I. Messer, Optics Laser Technol. 16 (1984) 315.
15 May 1996
2@3 ELSIZVIER
OPTICS
COMMUNICATIONS Optics Communications
126 ( 1996) 191- 196
Digital speckle pattern shearing interferometry using diffraction gratings Hector Rabal
* *2,
Rodrigo Henao lp2,Roberto Torroba 2
Ceniro de 1nvestigucione.s Opticas ICfOp).
Cusilla de Correo
Received 22 August 1995; revised version received 5 December
124, 1900 La Plats, Argentina 1995; accepted 7
December 1995
Abstract A digital speckle pattern shearing interferometer with the aid of a diffraction grating as shearing element, is presented. A brief theoretical approach for qualitative purposes is outlined. A comparison with Michelson shearing and phase stepping measurements are presented to show the validity of the proposal. Real time visualization, compactness of the setup and good quality fringes are main advantages.
1. Introduction Diffraction gratings were extensively studied for different purposes. Talbot [l] and Lau [2] effects are examples of their use in analog image processing and optical metrology. They were also used to provide several spatially disjoint viewpoints of an object, and the stereoscopic properties of the image replicas were described [3]. High frequency diffraction gratings provided viewpoints widely apart. It was found that in speckle interferometry a better signal-to-noise ratio can be obtained if the speckle
* Corresponding author. Fax: +54 21 712771; E-mail: [email protected]. ’ Permanent address: ANIRT, Apartado Aereo 2073, Medellin, Colombia. * All authors are also with the Facultad de Ingenieria, Universidad National de La Plats, Argentina. 0030~4018/96/$12.00 0 1996 Elsevier Science B.V. All rights reserved PII SOO30-40 18(96)000 17-X
image is modulated by a grating [4]. A digital single-beam coherent contouring method was also reported [5]. It is based on the use of a diffraction grating in the illumination beam, the position of which is modified after the reference frame acquisition. All these techniques allow to compact the recording device into a single camera unit and an illumination system. In this last context of compact optical systems, a measurement technique called shearography [6] appears as a prospective non-destructive tool. It is more practical than holographic interferometry as it provides a fringe pattern depicting displacement derivatives. Then, it does not require differentiating the measured displacements as in holography and hence, yield strains. Advantages over holography include simplicity of the optical setup, relaxation of vibration isolation requirements and reduced recording resolution requirements. Shearography is accomplished by the collection of
192
H. Rabal et d/Optics
Communicutioru
two image-sheared image plane interferograms, one before and one after loading the object. Hence shearography is essentially a common path interferometric technique and should be well suited for investigation under turbulence, as its effects between the object and the recording medium tend to cancel out. Besides, coherence restrictions on the light source are alleviated. There are two ways to obtain the two interferograms, either by the double-exposure technique or individual grabbing of the interferograms, i.e. using commercially available PC-based image processing equipment. There exist two basic setups to produce the shear. One consists of an imaging lens together with a thin glass wedge located just in the front of the lens to cover one-half of its aperture [7]. In the other, two mirrors are arranged as in a Michelson interferometer but one of them is slightly tilted with respect to the other. For the light coming from the object entering this setup, the result is the superposition on the imaging device of two laterally displaced images of the object, with a shear proportional to the tilt of the mirror [8]. Frame grabbing offers the posibility of depicting the fringe information in real time [9,10]. Limitations and prospects of the individual frame grabbing used in correlation with digital speckle pattern interferometry (DSPI) based on the above mentioned setups has been described in the literature [ 11 I. In this work we propose the use of a coarse diffraction grating in front of the camera in a DSPI arrangement as shear device. It will produce the overlapping of the speckle patterns so that spatially neighbor speckles are added and the results compared for the before and the after states in comparison or deformation analysis. This shearing interferometer is governed by the same restrictions as the displacement-sensitive interferometers. We outline a theoretical analysis for a qualitative estimation of the expected results. Experimental results for mechanically and thermally induced strains example are given.
2. Theoretical
analysis
126 (1996) 191-196
object. The point spread function of such an image forming system consists of a zero order and several replicas. In the case of holographic gratings, the replicas are reduced to two. If the object is rough, the image will consist of the coherent superposition of the spectra of every single speckle. Some speckles will overlap with the repiclas produced by neighbor speckles in the shearing direction. This situation for the zero and + 1 order can be expressed in mathematical terms by A,( x,
Y>=A(x+Y)exp[icp( x3r>l
as the contribution
from the zero order, and
A’,(x, y)=A(x,
y)exp[icp(x+Ax,
Y)]
(2)
as that from the + 1 order. The shear is represented by AX in the x direction. We assume the same constant amplitude for each beam. The (x, y> denotes image coordinates. The intensity distribution taken by the camera for this primary interferogram is
1,(x,
y)=2
IA12[1 +cos
(P,~(x,Y)],
(3)
where (p,? = cp(x, y> - +4x + A X, y>. After loading the object a phase change 6(x, y) is introduced, then the zero and + 1 orders are modified as A,(x,
y)=A(x,
Y) exp{i[cp(x, Y) +a(~,
Y)]]? (4)
and A’&x,
Y) exp(i[cp(x+Ax7
y)=A(x,
+S(x+Ax,
respectively. 1,(x,
y)=2
(5)
intensity
IA12{1 +co+P,,(x, +6(x,
Y)
Y)]},
The corresponding
~)-6(.x+Ax,
is
Y) Y)]].
(6)
We assume that the object intensity is not alterated by deformation. The difference intensity displayed on the TV monitor, according to the normal DSPI procedure is
II, -1, I = 14 IA -co4
Consider the situation of a grating just in front of an optical system which forms the image of an
(1)
12{cos 4x, (P,2(
-6(x+Ax,
x9
Y)
Y) + 6(x, r)]}l
Y) (7)
H. Ruhal et d/Optics
Communicutions
126 (1996) 191-196
193
or alternatively A(x, (p12(x,y)-
Y) 2
(8) with A(x,
y)=a(x+Ax, =
@(x2
Y)-6(x>
Y) Ax
ax The phase difference S= F[cf
Y)
sin 8+w(l
(9)
. 6 can be expressed fcos
e)],
as
Fig. 2. Experimental setup used to produce the shear for the different test objects. A single laser beam is used to illuminate the specimen. The sinusoidal diffraction grating DG is placed in front of the CCD camera to produce the shear image.
(10)
while 0 is the angle of the object illumination, u and w are the components of the in-plane and out-of-plane deformation, respectively, and A is the wavelength of the light used. A comparison of Eqs. (8), (9) and (10) shows that the resulting correlation fringes depict information on the displacement derivatives, as expected in our proposal. When considering the interaction of the zero and - 1 orders, similar expressions are formed, so that the information is replicated. For the case of the + 1 and - 1 orders, the shear is 2Ax, thus giving the same fringe pattern but twice the frequency. In prac-
LX
Fig. 1. Schematic arrangement representing the grating position as shear device. The shear A x is a function of the z, positions along the z axis, measured with respect to an arbitrary coordinate system. A represents a limiting aperture to lens L. The object and image planes are denoted by 0 and I, respectively.
Fig. 3. (a) Shear fringes along the x (horizontal) axis for a clamped metal plate loaded with a point pressure applied in the center of the plate. The slope contours correspond to the derivatives of the out-of-plane displacements along the x axis. (b) Shows the comparison of the same example using a Michelson shearing interferometer. No significant differences in performance are observed.
194
H. Rub& et d/Optics
Communications
126 (1996) 191-196
tice the contrast of this last fringe system is very low in comparison with the previous one and constitutes no essential source of errors.
700
((3
--------__. ,A
__
---_
1 0 y’
4
Fig. 4. Sequence showing heating device applied in show the derivatives of lence due to heat transfer could be expected.
the effect at different instants of a point the center of the metal plate. The fringes the out-of-plane displacements. Turbudoes not produce decorrelation effect as
Fig. 5. (a) Shear fringes obtained with our proposed setup for a clamped membrane centrally loaded. (b) The phase map of the fringes shown in (a). (c) The phase distribution analyzed from the fringes pattern of (a).
H. Rubal et al./Optics
3. Experimental Referring arrangement grating is
Communications
results
to Fig. 1. the shear introduced by the of a lens and a sinusoidal diffraction
Ax = ADZ,,
;,*/%3 I1 + (
zddl
’
(11)
where f is the frequency of the grating, and z,, = zi - zj are distances defined in Fig. 1. In our experiment we study the case of out-ofplane displacements only (U = 0). Recalling Eqs. (9) and (lo), we found that bright fringes are formed whenever aM ?K=
(2n + 1) ~Ax( 1 + cos 0) ’
(12)
where n is an integer. The grating is a holographically phase grating of 8 l/mm registered such that the zero and + 1 orders are transmitted. This avoids the noise eventually generated by the combinations of the other normal diffraction orders. The employed setup is depicted in Fig. 2. In one experiment we used as test object a clamped metal plate mechanically loaded with a centered point charge from behind. Fig. 3a corresponds to a shear along the _Yaxis, and the fringes give slope contours of aw/ax. As an objective comparison to show that this technique is not outweighed by any significant disadvantage in its performance, we carried out the same experiment with a Michelson shearing interferometer. The results are shown in Fig. 3b. The sequence in Figs. 4a-4c shows the same plate, heated in the center with the point of a welder device, for different instants. In this case, expected decorrelation effects due to the turbulence of the heat transfer phenomenon are diminished. However, speckle decorrelation effects are inherent, as every frame in DSPI shows only the current state of an object compared to the first acquired image. Phase shifting can be used as an evaluation procedure to demonstrate that the technique is fully compatible with the conventional alternatives. It can be implemented by an in-plane translation of the grating. The grating, of spacing d is moved through a distance d/k (that is, the phase is moved through
126 (1996) 191-196
195
2r/k) and a second image is read into the frame grabber. This process is repeated until the grating has moved through (k - 1) equal steps and a total of the k images have been stored in the computer [ 121. The phase maps are calculated then in the usual manner. The sequence of Fig. 5 shows a pratical example with (a) the shear fringes, (b) phase stepping of (a), and (c) the corresponding phase map.
4. Conclusions We have proposed and implemented a digital speckle pattern shearing interferometer by using a diffraction grating as shearing element. This technique represents an attractive alternative to other shearographic setups as it provides real time fringe formation, compactness of the optical setup and the possibility of phase-stepping implementation. In shearography, there are constraints in the rather limited space-bandwidth product (contrast> but in our case, there is no such restriction, as seen from the experimental data. Although only out-of-plane derivatives were considered here, extensions of this method can be made to determine in-plane derivations. The magnitude of the shear, and hence the sensibility can be varied. Experimental demonstration performed so far has used only a planar surface, but the technique can be extended to three-dimensional surfaces.
Acknowledgements This work was supported by CONICET grant PID No. 3974/92 (Argentina), Fundaci6n Antorchas (Argentina), TWAS (The Third World Academy of Sciences) grant 93-389 and Alexander von Humboldt Foundation (Germany). We thank J. Pomarico and R. Arizaga for providing the software and an anonymous referee for helpful suggestions. H.R. performed part of this work at the Bremen Institute for Applied Optics (Germany) under the auspices of the German Academic Exchange Service (DAAD). R.H. acknowledges the support of the Mutis Program (Spain).
196
H. Rahal et d/Optics
Communications
References 111 A. Lohman and D. Silva, Optics Comm 2 (1971) 413. [2] I. Pomarico and R. Torroba, Eur. J. Phys. 14 (1993) 114. [3] R. Henao, F. Medina, H. Rabal and M. Trivi, Appl. Optics 32 (1993) 726. [4] A. Tai, Optics Lett. 5 (1980) 552. 151 N. Bolognini, H. Rabal and R. Torroba, Appl. Optics 31 (1992) 1009. [6] Y. Hung, Opt. Eng. 21 (1982) 391.
126 (1996) 191-196
[7] S. Togooka, H. Nishida and J. Takezaki, Opt. Eng. 28 (1989) 55. [8] J. Leedertz and J. Butters, J. Phys. E 6 (1973) 1107. [9] C. Joenathan and R. Torroba, Optics Lett. 15 (1990) 1159. [lOI A. Ganesan, D. Sharma and M. Kothiyal, Appl. Optics 27 (1988) 473 I. [l I] M. Owner-Petersen, Appl. Optics 30 (1991) 2730. 1121 G.T. Reid, R.C. Rixon and H.I. Messer, Optics Laser Technol. 16 (1984) 315.