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The measurement of slope using shearography
Optics and Lasers in Engineering 14 (1991) 13-24
The Measurement C. J. Tay, F. S. Chau,
of Slope Using Shearography
H. M. Shang,
V. P. W. Shim & S. L. Toh
Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 (Received 9 June 1989; revised version received and accepted 16 November 1989)
ABSTRACT This paper presents an optical method which is based on speckleshearing interferometry for the measurement of the slope at any point of an object. The object under study is illuminated by an expanded laser beam and its image is recorded on a photographic plate placed at the image plane of an image-shearing camera. A second exposure on the photographic film is made after the light source is moved slightly. The resulting photograph yields a fringe pattern which represents lines of constant displacement gradients. Theory of the method as well as some experimental results are presented.
1 INTRODUCTION The advent of laser, a highly coherent light source, has led to the development of new interferometers for studying surface deformations in diffused objects. This development includes holography,‘,* speckle interferometry,s5 and speckle-shearing interferometry.6*7 While holography measures surface displacements, speckle and speckle-shearing interferometry, including shearography,8T9 yields the derivatives of surface displacement. These laser-based techniques have the advantages of being extremely fast in response, noncontacting and generally full-field. This paper describes the use of a coherent optical method, namely shearography, for the identification of the slopes on the surface of simple geometrical objects. In most previous investigations using both speckle and speckle13
Optics and Lasers in Engineering 0143-8166/90/%03~50 @ 1990 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
C. J. Tay et al.
14
shearing interferometry,3-9 the slope contour fringes on an object surface are generated by the two interfering speckle patterns with a surface displacement between the patterns. In this study a method is proposed whereby the interference patterns are produced with a moving light source. Some experimental results are also included.
2 THE METHOD
An object is illuminated by a coherent light source through a system of beam expander and optical mirrors as shown in Fig. 1. The image of the object is captured by an image shearing camera which produces a pair of sheared images on the image plane. As the object is illuminated by coherent light, the two sheared images interfere with each other producing a random interference pattern. When the light source is moved to another position, the interference pattern is slightly modified. Superposition of the two interference patterns by double exposure results in a fringe pattern on the processed photograph. As shown in the next section, the fringes represent lines of constant displacement gradients of the object surface.
Image-shearing camera Fig.
1. Schematic diagram of shearography.
The measurement
of slope using shearography
3 THEORETICAL
15
DERIVATION
Figure 2 is a schematic diagram of an object illuminated by a laser light source and S and S’ represent two respective positions of the light source which are a small distance apart. The image of the object is captured on an image shearing camera. Consider a point P on the surface of the object and a neighbouring point P’ which is a small distance away from P. For simplicity the two points are assumed to lie on the X-Z plane. 0 represents the focal point of P and P’ on the image plane of the camera. Using the reference axis system as shown, the co-ordinates of the various points in Fig. 2 are denoted by P=(X,
Y, 2)
P’=(X+dX,
Y,Z+dZ)
S = (XW r,, ZJ S’=(X,+DX,
yS+DY, Z,+DZ)
0 = (X”, Y,, Z,) where dX, dZ, DX, DY and DZ denote small distances on the respective X, Y or Z directions. The optical path difference dl, between the light beams travelling from the light source S and S’ to the focal point 0 through the point P 2
s Y
0
Image-shearing camera
Fig. 2.
Rays from two neighbouring
points
brought
to meet in image plane.
16
C. J. Tay et al.
is given by dl, = (S’P + PO) - (SP + PO) = S’P - SP
(1)
= (S’P)2 - (SP)2 S’P + SP
where (SIP)* = (X - X, - DX)’ + (Y - y, - DY)” + (2 - 2, - DZ)’ (sP)2 = (X - XJ2 + (Y - ys), + (2 - Zs)2
Similarly, the optical path difference dl, between the light beams travelling from S and S’ to point 0 through P’ is given by dl, = (S’P’ + P’O) - (Sp’ + P’O) =S’P’-Sp’
(2)
= (S’P’)” - (SP’)2 S’P’ + SP’
where (S’P’)2=(X+dX-X,-DX)2+(Y-Y,-DY)2+(Z (SP’)2=(X+dX-X,)2+(Y-YJ2+(Z+dZ-2,)2 Since the relative distance between S and S’ is small S’P’=SP=S’P=SP’=R where R represents the distance between the light source and the object. This distance is dependent on the geometry of the experimental set-up. Hence, the change in optical path length (Yrecorded on the image plane due to the shifting of the light source from S to S’ is given by a=dZ,-dl,=;[DXdX+DZdZ] or
(3) It has been shown’ that a dark fringe is formed when the optical path difference A is given by A cos - = 0 2
0
The measurement
of slope using shearography
17
that is A = (2n + 1)~ where fn = 0, 1, 2, 3, 4, . . . represents
(4)
the fringe order number.
Also
(5) where A is the wavelength and (5) gives dx
dZ=-DX
of the light source. Combining
E
[
eqns (3), (4)
1+$(2n+1)&]
where
c, = If DZ/DX*O becomes
RA 2DX dZ
(See eqn (12)), C, can be neglected, dX E = C,(2n + 1)
hence,
eqn (6)
(7)
Hence, eqns (6) and (7) show that fringes are whole-field representations of the loci of dX/dZ. The slope at any point on the object surface can thus be readily determined.
4 EXPERIMENTAL
METHOD
4.1 Optical arrangement A 35 mW He-Ne laser is used and the light is expanded onto a mirror which can be rotated (Fig. 1). The rotation of the mirror serves to shift the virtual light source between the exposures. This technique differs from that used in Ref. 9 in that it does not require the application of load on the test object. The reflected light from the test specimen is recorded on the image shearing camera. The positions of the virtual light source are determined from the mirror rotation through the following formulation. Let L (Fig. 3) be the distance between the expander E and the centre
C. J. Tay et al.
18
-------_,
X
I
Fig. 3.
Positions of virtual light source.
of the mirror M and let 8, and & denote the positions of the mirror before and after rotation, respectively, i.e. 8, = 8, + d0. S and S’ are the positions of the virtual light source before and after rotation, respectively, and line MS makes an angle of /I with the X-axis. From simple geometry it can be shown that p=9V-20, Also, the vector
(8)
MS is given by MS = -i lMSl cos p -j
jMSl sin /3
(9)
where IMS( = IEtiI = L Hence MS = -iL cos (90” - 20,) - jL sin (90” - 2f3,) Replacing
13, by &, a similar expression
for MS’ can be obtained.
(10)
The measurement
of slope using shearography
19
Let DX and DZ represent the movement in the X and 2 directions when the light source is moved from S to S’, respectively. Hence iDX+jDZ=MS’-MS = il(sin 28, - sin 20,) + jl(cos
28, - cos 20,)
(11)
i.e. DZ -= DX
cos 28, - cos 20, sin 28, - sin 28,
(12)
The use of eqn (7) requires the condition that DZ/DX-, 0, which implies that the light source is shifted mainly in the x direction only. To achieve this condition 8, and de must be small. 4.2 Experimental
set-up
The experimental set-up is shown in Fig. 4. The beam from a He-Ne laser is expanded by a microscope objective onto a mirror which reflects the beam onto the areas of interest on the test object. The mirror is rotated by a small amount (0.07”) inbetween exposures of the photographic plate. White emulsion paint is applied on the test object to provide better and even reflection distribution. The image-shearing
Fig. 4.
The experimental set-up.
C. J. Tay et al.
20
Fig. 5.
Fig. 7.
Fig.
Fringe pattern on a hemispherical object.
Fringe
pattern
on an ellipsoidal
object.
6.
Fringe
Fig. 8.
pattern object.
on a cylindrical
Fringe pattern cal object.
on a coni-
The measurement of slope using shearography
21
camera used is a Mamiya RB67 Pro-5 with a specially fabricated glass wedge of 1” wedge angle placed in the iris plane of the lens. Kodak high resolution holographic plates are used for recording the fringes. 4.3 Test results A series of tests were performed on objects with simple geometrical shapes. The objects contain cylindrical, hemispherical, ellipsoidal and conical sections. Figures 5 to 8 show typical fringe patterns obtained from a hemispherical, cylindrical, ellipsoidal and conical section, respectively. By evaluating the fringes and applying eqn (7), the slope dX/dZ at various locations on the object surface was determined.
5 DISCUSSION
OF RESULTS
Figure 9 shows a plot of the slope dX/dZ versus the distance Z on a hemispherical surface of radius R = 60 mm. The theoretical values of dX/dZ are obtained from the equation of a circle. A comparison with the experimental results shows excellent agreement. Figure 10 shows a plot of the slope dX/dZ versus Z on an elliptical surface of 65 mm major axis and 40 mm minor axis. The theoretical values of dX/dZ on an elliptical surface can be readily obtained from the equation of an ellipse. Comparison of the experimental and theoretical results again shows good agreement. 1-o0 ExperImental
09.
0
Fig. 9.
A comparison
6
-
12
of experimental
TheoretIcal
16
24
and theoretical surface.
30
36
results
4:
for a hemispherical
C. J. Tay et al.
22 _I
0 Experimental
0
5
-Theoretical
10
15
20
25
30
Z(mm)
Fig. 10.
A comparison of experimental and theoretical results for an elliptical surface.
A similar comparison of theoretical dX/dZ values is made with the experimental for the case of a conical surface of 100 mm base diameter and 150 mm height. As can be seen (Fig. ll), the agreement is generally good except at both ends of the curve. The discrepancy is due to inaccuracy incurred during the fringe interpretation process caused by poor fringe quality. A plot of the slope dX/dZ versus Z on a cylindrical surface of 60 mm radius shows (Fig. 12) that there is a noticeable difference between the experimental and theoretical results. The discrepancy is due to the optical arrangement and the size of the object used which resulted in a significant variation of the value of the ‘constant’ C,.
Z(mm)
Fig. 11.
A comparison of experimental and theoretical results for a conical surface.
23
The measurement of slope using shearography o-9 -
l Expenmental
TheoretIcal
0.8‘
0
0.7 l
0.6
0
4
8
12
16
20
24
26
32
Z(mm)
Fig. 12.
A comparison
of experimental
and theoretical
CONCLUDING
results for a cylindrical surface.
REMARKS
In this paper a method for surface gradient measurements using shearographic technique has been presented. Unlike previous methods, the technique does not require the application of load on the test object. Tests were conducted on objects of simple geometrical shapes and the experimental results compared well with those from theoretical calculations. The results illustrated the potential of the technique being developed into a practical tool for surface gradient measurements.
ACKNOWLEDGEMENTS The authors are grateful to Professor Y. Y. Hung of Oakland University, Rochester, Michigan, USA for his expert and valuable advice on the work and to the National University of Singapore for providing the financial support. REFERENCES Vest, C. M., Holographic fnterferometry. John Wiley and Sons, New York, 1979. Erf, R. J. (ed.) Holographic Nondestructive Testing. Academic Press, New York, 1974. Erf, R. J. (ed.) Speckle Metrology. Academic Press, New York, 1978. Chiang, F. P. & Juang, R. M., Laser speckle interferometry for plate bending problems. Applied Optics, 15 (1976) 2199-204.
24
C. J. Tay et al.
5. Hung, Y. Y., Daniel, I. M. & Rowlands, R. E., Full-field optical strain measurement having postrecording sensitivity and direction selectivity. Experimental Mechanics, 18 (1976) 56-60. 6. Leendertz, J. A. & Butters, J. N., An image-shearing speckle patterns interferometer for measuring bending moment. J. Physics E. Scientific Zrzstruments, 6 (1973) 1107-10. 7. Boon, D. M., Determination of slope and strain contours by doubleexposure shearing interferometry. Experimental Mechanics, 15 (1975) 295-302. 8. Hung, Y. Y. & Liang, C. Y., Image-shearing camera for direct measurement of surface-strains. Applied Optics, 10,(1979) 1046-50. 9. Hung, Y. Y., Shearography: A new optical method for strain measurement and nondestructive testing. Optical Engineering, 21 (1982) 391-5.
The Measurement C. J. Tay, F. S. Chau,
of Slope Using Shearography
H. M. Shang,
V. P. W. Shim & S. L. Toh
Department of Mechanical and Production Engineering, National University of Singapore, 10 Kent Ridge Crescent, Singapore 0511 (Received 9 June 1989; revised version received and accepted 16 November 1989)
ABSTRACT This paper presents an optical method which is based on speckleshearing interferometry for the measurement of the slope at any point of an object. The object under study is illuminated by an expanded laser beam and its image is recorded on a photographic plate placed at the image plane of an image-shearing camera. A second exposure on the photographic film is made after the light source is moved slightly. The resulting photograph yields a fringe pattern which represents lines of constant displacement gradients. Theory of the method as well as some experimental results are presented.
1 INTRODUCTION The advent of laser, a highly coherent light source, has led to the development of new interferometers for studying surface deformations in diffused objects. This development includes holography,‘,* speckle interferometry,s5 and speckle-shearing interferometry.6*7 While holography measures surface displacements, speckle and speckle-shearing interferometry, including shearography,8T9 yields the derivatives of surface displacement. These laser-based techniques have the advantages of being extremely fast in response, noncontacting and generally full-field. This paper describes the use of a coherent optical method, namely shearography, for the identification of the slopes on the surface of simple geometrical objects. In most previous investigations using both speckle and speckle13
Optics and Lasers in Engineering 0143-8166/90/%03~50 @ 1990 Elsevier Science Publishers Ltd, England. Printed in Northern Ireland
C. J. Tay et al.
14
shearing interferometry,3-9 the slope contour fringes on an object surface are generated by the two interfering speckle patterns with a surface displacement between the patterns. In this study a method is proposed whereby the interference patterns are produced with a moving light source. Some experimental results are also included.
2 THE METHOD
An object is illuminated by a coherent light source through a system of beam expander and optical mirrors as shown in Fig. 1. The image of the object is captured by an image shearing camera which produces a pair of sheared images on the image plane. As the object is illuminated by coherent light, the two sheared images interfere with each other producing a random interference pattern. When the light source is moved to another position, the interference pattern is slightly modified. Superposition of the two interference patterns by double exposure results in a fringe pattern on the processed photograph. As shown in the next section, the fringes represent lines of constant displacement gradients of the object surface.
Image-shearing camera Fig.
1. Schematic diagram of shearography.
The measurement
of slope using shearography
3 THEORETICAL
15
DERIVATION
Figure 2 is a schematic diagram of an object illuminated by a laser light source and S and S’ represent two respective positions of the light source which are a small distance apart. The image of the object is captured on an image shearing camera. Consider a point P on the surface of the object and a neighbouring point P’ which is a small distance away from P. For simplicity the two points are assumed to lie on the X-Z plane. 0 represents the focal point of P and P’ on the image plane of the camera. Using the reference axis system as shown, the co-ordinates of the various points in Fig. 2 are denoted by P=(X,
Y, 2)
P’=(X+dX,
Y,Z+dZ)
S = (XW r,, ZJ S’=(X,+DX,
yS+DY, Z,+DZ)
0 = (X”, Y,, Z,) where dX, dZ, DX, DY and DZ denote small distances on the respective X, Y or Z directions. The optical path difference dl, between the light beams travelling from the light source S and S’ to the focal point 0 through the point P 2
s Y
0
Image-shearing camera
Fig. 2.
Rays from two neighbouring
points
brought
to meet in image plane.
16
C. J. Tay et al.
is given by dl, = (S’P + PO) - (SP + PO) = S’P - SP
(1)
= (S’P)2 - (SP)2 S’P + SP
where (SIP)* = (X - X, - DX)’ + (Y - y, - DY)” + (2 - 2, - DZ)’ (sP)2 = (X - XJ2 + (Y - ys), + (2 - Zs)2
Similarly, the optical path difference dl, between the light beams travelling from S and S’ to point 0 through P’ is given by dl, = (S’P’ + P’O) - (Sp’ + P’O) =S’P’-Sp’
(2)
= (S’P’)” - (SP’)2 S’P’ + SP’
where (S’P’)2=(X+dX-X,-DX)2+(Y-Y,-DY)2+(Z (SP’)2=(X+dX-X,)2+(Y-YJ2+(Z+dZ-2,)2 Since the relative distance between S and S’ is small S’P’=SP=S’P=SP’=R where R represents the distance between the light source and the object. This distance is dependent on the geometry of the experimental set-up. Hence, the change in optical path length (Yrecorded on the image plane due to the shifting of the light source from S to S’ is given by a=dZ,-dl,=;[DXdX+DZdZ] or
(3) It has been shown’ that a dark fringe is formed when the optical path difference A is given by A cos - = 0 2
0
The measurement
of slope using shearography
17
that is A = (2n + 1)~ where fn = 0, 1, 2, 3, 4, . . . represents
(4)
the fringe order number.
Also
(5) where A is the wavelength and (5) gives dx
dZ=-DX
of the light source. Combining
E
[
eqns (3), (4)
1+$(2n+1)&]
where
c, = If DZ/DX*O becomes
RA 2DX dZ
(See eqn (12)), C, can be neglected, dX E = C,(2n + 1)
hence,
eqn (6)
(7)
Hence, eqns (6) and (7) show that fringes are whole-field representations of the loci of dX/dZ. The slope at any point on the object surface can thus be readily determined.
4 EXPERIMENTAL
METHOD
4.1 Optical arrangement A 35 mW He-Ne laser is used and the light is expanded onto a mirror which can be rotated (Fig. 1). The rotation of the mirror serves to shift the virtual light source between the exposures. This technique differs from that used in Ref. 9 in that it does not require the application of load on the test object. The reflected light from the test specimen is recorded on the image shearing camera. The positions of the virtual light source are determined from the mirror rotation through the following formulation. Let L (Fig. 3) be the distance between the expander E and the centre
C. J. Tay et al.
18
-------_,
X
I
Fig. 3.
Positions of virtual light source.
of the mirror M and let 8, and & denote the positions of the mirror before and after rotation, respectively, i.e. 8, = 8, + d0. S and S’ are the positions of the virtual light source before and after rotation, respectively, and line MS makes an angle of /I with the X-axis. From simple geometry it can be shown that p=9V-20, Also, the vector
(8)
MS is given by MS = -i lMSl cos p -j
jMSl sin /3
(9)
where IMS( = IEtiI = L Hence MS = -iL cos (90” - 20,) - jL sin (90” - 2f3,) Replacing
13, by &, a similar expression
for MS’ can be obtained.
(10)
The measurement
of slope using shearography
19
Let DX and DZ represent the movement in the X and 2 directions when the light source is moved from S to S’, respectively. Hence iDX+jDZ=MS’-MS = il(sin 28, - sin 20,) + jl(cos
28, - cos 20,)
(11)
i.e. DZ -= DX
cos 28, - cos 20, sin 28, - sin 28,
(12)
The use of eqn (7) requires the condition that DZ/DX-, 0, which implies that the light source is shifted mainly in the x direction only. To achieve this condition 8, and de must be small. 4.2 Experimental
set-up
The experimental set-up is shown in Fig. 4. The beam from a He-Ne laser is expanded by a microscope objective onto a mirror which reflects the beam onto the areas of interest on the test object. The mirror is rotated by a small amount (0.07”) inbetween exposures of the photographic plate. White emulsion paint is applied on the test object to provide better and even reflection distribution. The image-shearing
Fig. 4.
The experimental set-up.
C. J. Tay et al.
20
Fig. 5.
Fig. 7.
Fig.
Fringe pattern on a hemispherical object.
Fringe
pattern
on an ellipsoidal
object.
6.
Fringe
Fig. 8.
pattern object.
on a cylindrical
Fringe pattern cal object.
on a coni-
The measurement of slope using shearography
21
camera used is a Mamiya RB67 Pro-5 with a specially fabricated glass wedge of 1” wedge angle placed in the iris plane of the lens. Kodak high resolution holographic plates are used for recording the fringes. 4.3 Test results A series of tests were performed on objects with simple geometrical shapes. The objects contain cylindrical, hemispherical, ellipsoidal and conical sections. Figures 5 to 8 show typical fringe patterns obtained from a hemispherical, cylindrical, ellipsoidal and conical section, respectively. By evaluating the fringes and applying eqn (7), the slope dX/dZ at various locations on the object surface was determined.
5 DISCUSSION
OF RESULTS
Figure 9 shows a plot of the slope dX/dZ versus the distance Z on a hemispherical surface of radius R = 60 mm. The theoretical values of dX/dZ are obtained from the equation of a circle. A comparison with the experimental results shows excellent agreement. Figure 10 shows a plot of the slope dX/dZ versus Z on an elliptical surface of 65 mm major axis and 40 mm minor axis. The theoretical values of dX/dZ on an elliptical surface can be readily obtained from the equation of an ellipse. Comparison of the experimental and theoretical results again shows good agreement. 1-o0 ExperImental
09.
0
Fig. 9.
A comparison
6
-
12
of experimental
TheoretIcal
16
24
and theoretical surface.
30
36
results
4:
for a hemispherical
C. J. Tay et al.
22 _I
0 Experimental
0
5
-Theoretical
10
15
20
25
30
Z(mm)
Fig. 10.
A comparison of experimental and theoretical results for an elliptical surface.
A similar comparison of theoretical dX/dZ values is made with the experimental for the case of a conical surface of 100 mm base diameter and 150 mm height. As can be seen (Fig. ll), the agreement is generally good except at both ends of the curve. The discrepancy is due to inaccuracy incurred during the fringe interpretation process caused by poor fringe quality. A plot of the slope dX/dZ versus Z on a cylindrical surface of 60 mm radius shows (Fig. 12) that there is a noticeable difference between the experimental and theoretical results. The discrepancy is due to the optical arrangement and the size of the object used which resulted in a significant variation of the value of the ‘constant’ C,.
Z(mm)
Fig. 11.
A comparison of experimental and theoretical results for a conical surface.
23
The measurement of slope using shearography o-9 -
l Expenmental
TheoretIcal
0.8‘
0
0.7 l
0.6
0
4
8
12
16
20
24
26
32
Z(mm)
Fig. 12.
A comparison
of experimental
and theoretical
CONCLUDING
results for a cylindrical surface.
REMARKS
In this paper a method for surface gradient measurements using shearographic technique has been presented. Unlike previous methods, the technique does not require the application of load on the test object. Tests were conducted on objects of simple geometrical shapes and the experimental results compared well with those from theoretical calculations. The results illustrated the potential of the technique being developed into a practical tool for surface gradient measurements.
ACKNOWLEDGEMENTS The authors are grateful to Professor Y. Y. Hung of Oakland University, Rochester, Michigan, USA for his expert and valuable advice on the work and to the National University of Singapore for providing the financial support. REFERENCES Vest, C. M., Holographic fnterferometry. John Wiley and Sons, New York, 1979. Erf, R. J. (ed.) Holographic Nondestructive Testing. Academic Press, New York, 1974. Erf, R. J. (ed.) Speckle Metrology. Academic Press, New York, 1978. Chiang, F. P. & Juang, R. M., Laser speckle interferometry for plate bending problems. Applied Optics, 15 (1976) 2199-204.
24
C. J. Tay et al.
5. Hung, Y. Y., Daniel, I. M. & Rowlands, R. E., Full-field optical strain measurement having postrecording sensitivity and direction selectivity. Experimental Mechanics, 18 (1976) 56-60. 6. Leendertz, J. A. & Butters, J. N., An image-shearing speckle patterns interferometer for measuring bending moment. J. Physics E. Scientific Zrzstruments, 6 (1973) 1107-10. 7. Boon, D. M., Determination of slope and strain contours by doubleexposure shearing interferometry. Experimental Mechanics, 15 (1975) 295-302. 8. Hung, Y. Y. & Liang, C. Y., Image-shearing camera for direct measurement of surface-strains. Applied Optics, 10,(1979) 1046-50. 9. Hung, Y. Y., Shearography: A new optical method for strain measurement and nondestructive testing. Optical Engineering, 21 (1982) 391-5.