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Curvature measurement using three-aperture digital shearography and fast Fourier transform
ARTICLE IN PRESS
Optics and Lasers in Engineering 45 (2007) 1001–1004 www.elsevier.com/locate/optlaseng
Curvature measurement using three-aperture digital shearography and fast Fourier transform Basanta Bhaduri, M.P. Kothiyal, N. Krishna Mohan Applied Optics Laboratory, Indian Institute of Technology Madras, Chennai, TN 600 036, India Received 15 March 2007; received in revised form 17 April 2007; accepted 18 April 2007 Available online 20 June 2007
Abstract Curvature measurement using a three-aperture digital shearography (DS) system is reported in this paper. The outer apertures are covered with wedge plates for introducing shear. Four images by sequentially blocking the outer apertures are used for quantitative measurement. Fourier transform technique is used to determine two sheared slope phase maps from two images at a time representing initial and deformed states. Subtraction of these two-phase maps yields the curvature phase map. Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center. r 2007 Elsevier Ltd. All rights reserved. Keywords: Digital shearography; Slope; Curvature; Fast Fourier transform
1. Introduction For plate bending problems and flexural analysis, the second order derivatives (curvature) of displacement components are needed for calculating the stress components. Hence it is necessary to perform the optical differentiation of the slope data to obtain the curvature information. Shearography is a powerful optical technique for extracting the first- and second-order derivatives of displacement components [1]. Various methods have been reported for the measurement of curvature using photographic processing and Fourier filtering [1–4]. In all these methods, the curvature information is coded in the two sheared slope contours as a Moire´ pattern. The Moire´ pattern being additive in nature, inherently has a low contrast. A method using a Michelson interferometer setup in tandem was first reported by Rastogi [5] by storing the two slope phase maps with a shift between the two for the measurement of curvature. The method is based on digital shearography [6,7] and eliminates the intermediate photographic processing. This method provides high-quality curvature phase maps and provides flexibility in handling Corresponding author. Tel.: +91 44 22574892; fax: +91 44 22570509.
E-mail address: [email protected] (N. Krishna Mohan). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.04.005
the data for processing. However, the analysis uses the temporal phase shifting and requires a large number of frames for quantitative evaluation [1]. Method that employs digital multiple-exposure technique is also proposed in which the object is loaded, shifted in the x-direction and again unloaded between the exposures [8]. It is always not possible to unload the object perfectly because of the loading system. Recently, a method for retrieving the transient curvature and twist of a continuously deforming object has been reported by using a combination of digital shearography and the continuous wavelet transform [9]. In this paper, a three-aperture digital shearographic arrangement is demonstrated for quantitative determination of curvature using fast Fourier transform (FFT). Multi-aperture mask placed in front of the imaging lens is useful in generating spatial carrier fringes inside the speckle [10]. The outer two apertures of the three-aperture arrangement carry two identical wedge plates, while a suitable glass plated is mounted in the central aperture to compensate for the optical path. The method needs only four images by sequentially blocking the outer apertures; twice before and twice after the object deformation. The proposed method allows only two sheared object waves to interfere at the CCD plane for each recording. A 2D FFT
ARTICLE IN PRESS 1002
B. Bhaduri et al. / Optics and Lasers in Engineering 45 (2007) 1001–1004
is performed on the images to obtain three diffraction halos including the central halo at the Fourier spectrum. A 2D inverse FFT (IFFT) is performed using one of the firstorder halos [11]. The analysis yields two slope phase maps that are sheared with respect to each other. Subtraction of these two phase maps directly generates the desired curvature phase map. Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center.
carrier fringe width can be given by D ¼ lV =D, where l is the wavelength of the light used, V is the image distance and D is the aperture separation [12]. As we block one of the apertures A1 or A3 as shown in Fig. 1 for sequential recordings, there will be interference between the beams u1 and u2 or u2 and u3 for each recording. Thus there will be a contribution either of u1 or u3 with respect to u2 at the image plane. We can express the amplitudes u1 and u3 in general as u1;3 ðx; yÞ ¼ juðx Dx; yÞj exp½iffðx Dx; yÞ 2pf 0x xg.
2. Optical arrangement and theory
(2)
Fig. 1 represents the schematic arrangement. It is similar to the arrangement reported earlier by Sharma et al. [2] except that a CCD is used at the image plane instead of a photographic plate. The imaging lens carries a mask containing three equi-spaced apertures along x-direction. The outer apertures A1 and A3 carry two identical wedge plates, while a suitable flat glass plate is mounted in front of the central aperture A2 to compensate for the optical path. The object is illuminated normally by a collimated laser beam. The stops S1 and S2 are used to block the apertures A1 and A3 for sequential recording of images. When all the apertures are open, three adjacent points on the object will be imaged as one point in the image plane. Each of the two scattered fields (from the outer apertures) is sheared with respect to the field from the central aperture by an amount Dx. The shear Dx in the object plane can be given by Dx ¼ Uðn 1Þa, where U is the object distance from the imaging lens, n is the refractive index of the wedge material and a is the wedge angle [1]. Let us consider the complex wave-amplitude of respective beams coming from the apertures A1, A2 and A3 as u1 ðx; yÞ ¼ juðx þ Dx; yÞj exp½iffðx þ Dx; yÞ þ 2pf 0 xg, u2 ðx; yÞ ¼ juðx; yÞj exp½iffðx; yÞg, u3 ðx; yÞ ¼ juðx Dx; yÞj exp½iffðx Dx; yÞ 2pf 0 xg,
ð1Þ
where |u|’s are the modulus of the amplitudes, f’s are the phases, f 0 is the phase bias and is given by, f 0 ¼ 1=D, where D is the carrier fringe width inside the speckle. The
Considering the interference of u1,3 with respect to u2 at the image plane, one can express the intensity I(x,y) as Iðx; yÞ ¼ ju2 ðx; yÞ þ u1;3 ðx; yÞj2 ¼ u2 u2 þ u1;3 u1;3 þ u2 u1;3 þ u1;3 u2 .
ð3Þ
Using Eqs. (1) and (2), the Eq. (3) can now be written as Iðx; yÞ ¼ juðx; yÞj2 þ juðx Dx; yÞj2 þ juðx; yÞj:juðx Dx; yÞj exp½iffðx; yÞ fðx Dx; yÞ 2pf 0 xg þ juðx; yÞj:juðx Dx; yÞj exp½iffðx; yÞ fðx Dx; yÞ 2pf 0 xg.
ð4Þ
The Fourier transform of the Eq. (3) can be obtained as i ¼ FT½I ¼ U 2 U 2 þ U 1;3 U 1;3 þ U 2 U 1;3 þ U 1;3 U 2 ,
ð5Þ
where U j ¼ FT½uj ; j ¼ 1; 2; 3 and denotes the convolution operation. The first and second terms are the halos centered at the origin, corresponding to the spectrum of the image speckle patterns due to two apertures A1 or A3 and A2 whereas, the third and fourth terms of the above equation are the diffraction halos due to the cross terms and they appear shifted in the horizontal direction due to the angular offset between the object beams. Thus there will be three distinct diffraction halos, two first order halos and the central halo in the spectrum. These first order halos contain the phase information. If we select either of them and perform inverse Fourier transform, we can retrieve the phase. For example, if we select the fourth term, U 1;3 U 2 and perform the inverse Fourier transform, we will obtain u1;3 u2 . The phase term ½fðx; yÞ fðx Dx; yÞ 2pf 0 x is then calculated by using the relation 1 Imðu1;3 u2 Þ ½fðx; yÞ fðx Dx; yÞ 2pf 0 x ¼ tan , (6) Reðu1;3 u2 Þ where Im and Re represent the imaginary and real part respectively. The object wave-fields after the object deformation can be given by
Fig. 1. Schematic of the multi-aperture digital shearography setup for quantitative curvature measurement: A1, A3, wedge plates, A2, glass plate, S1, S2, aperture stops.
u01;3 ðx; yÞ ¼ juðx Dx; yÞj exp½iff0 ðx Dx; yÞ 2pf 0 xg u02 ðx; yÞ ¼ juðx; yÞj exp½iff0 ðx; yÞg.
ARTICLE IN PRESS B. Bhaduri et al. / Optics and Lasers in Engineering 45 (2007) 1001–1004
Now using the same procedure we can calculate the phase term ½f0 ðx; yÞ f0 ðx Dx; yÞ 2pf 0 x after object deformation as " # Imðu01;3 u0 2 Þ 0 0 1 ½f ðx; yÞ f ðx Dx; yÞ 2pf 0 x ¼ tan . Reðu01;3 u0 2 Þ (7) The phase difference Df due to 1st-order derivative of the out-of-plane displacement (slope) can then be given by [12] " # Imðu01;3 u0 2 Þ 1 1 Imðu1;3 u2 Þ Df ¼ tan tan . (8) Reðu1;3 u2 Þ Reðu01;3 u0 2 Þ
1003
As explained earlier, if we block the aperture A3, then there will be contributions of u01 and u02 via the apertures A1 and A2 respectively at the CCD plane. The images stored before and after deformation with this condition are used for analysis. Under this condition, Eq. (8) yields the slope phase map Df12 as 0 0 1 Imðu1 u 2 Þ 1 Imðu1 u2 Þ Df12 ¼ tan tan . (9) Reðu01 u0 2 Þ Reðu1 u2 Þ Similarly, if we block the aperture A1, then there will be contributions of u02 and u03 via the apertures A2 and A3 respectively at the CCD plane. Following the similar analysis as explained above, we obtain the slope phase map
Fig. 2. Quantitative measurement of curvature: (a) phase map as obtained by subtracting the slope phase maps, (b) unwrapped 2D plot and (c) 3D plot.
ARTICLE IN PRESS B. Bhaduri et al. / Optics and Lasers in Engineering 45 (2007) 1001–1004
1004
Df23 as Df23 ¼
Imðu03 u0 2 Þ tan1 Reðu03 u0 2 Þ
Imðu3 u2 Þ tan1 Reðu3 u2 Þ
.
(10)
The second order derivative or curvature information can be obtained as [2] 4p q2 w 2 Dx . (11) l qx2 The resulting phase map obtained in this method is noisy, thus filtering is necessary in order to improve it. For the present case, we have filtered using 5 5 window sinecosine phase filtering [7]. The filtered phase distributions are the wrapped or modulo-2p phase map, which range from p to p and require unwrapping [13]. Df12 Df23 ¼
3. Results and discussion The three-aperture mask has three circular holes of diameter 2.7 mm and they are separated each other along x by an amount 4.3 mm. The experiments were conducted on a white matt paint coated 60 mm diameter circular diaphragm with its edge rigidly clamped and loaded at the center. The object is illuminated by a laser beam from a He–Ne laser (20 mW) after collimation. Two wedge plates having identical wedge angles ( 25 arc min) are attached to the outer apertures A1 and A3. The shear elements provide an object plane shear around 4 mm along the x-direction with respect to the object wave passing through the central aperture A2. The three-aperture mask is placed close to an imaging lens of focal length 89 mm. We have chosen the lens in order to image full object on to the CCD sensor. The waves enter the apertures and combine coherently at the CCD plane (image plane). The stops S1 and S2 are used in order to block the waves passing through A1 and A3 respectively for sequential recording of images. We have used a Jai CV-A1 CCD camera, with matrix 1376 1035 and 4.65 4.65 mm2 pixel size. The average size of the speckle on the CCD for each image is around six pixels of the CCD and the spatial carrier fringe has a width of three pixels in the x direction. For analysis we have adopted the following sequential procedure for storing the images. First we have blocked the aperture A3 and then stored the first images. The second image is the then stored after opening the aperture A3 and blocking aperture A1. Both these images represent the initial state of the object. Now the diaphragm is given a small deflection at the center and the third frame is stored. The fourth and the final image is then stored after opening the aperture A1 and blocking aperture A3. As explained earlier, we have performed FFT and IFFT on first and fourth image to get the slope phase map corresponding to the Eq. (9). Similarly the slope phase map that corresponding to the Eq. (10) is obtained from the
second and third images. Subtraction of these slope phase maps as described in Eq. (11), generates the desired curvature phase map. As the resultant phase map is noisy, it has been filtered with 5 5 window sine-cosine phase filter [7]. We have used the overlapped region for analysis and the evaluated curvature phase map as obtained is shown in Fig. 2(a) whereas Fig. 2(b)–(c) show the corresponding unwrapped 2D and 3D plots respectively. It is interesting to note that the addition of these slope phase patterns yields slope with doubled the sensitivity [2]. 4. Conclusions We have presented a method for curvature measurement which requires only four images to be recorded. Further, the proposed technique is simple and cost effective as it requires only a three-aperture mask in front of the imaging lens instead of a conventional PZT-driven phase shifting unit. References [1] Sirohi RS, editor. Speckle metrology. New York: Marcel Dekker; 1993. [2] Sharma DK, Sirohi RS, Kothiyal MP. Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer. Appl Opt 1984;23:1542–6. [3] Tay CJ, Toh SL, Shang HM, Lin QY. Direct determination of second order derivatives in plate bending using multiple-exposure shearography. Opt Laser Technol 1994;26:91–8. [4] Krishna Mohan N. The influence of multiple-exposure recording on curvature pattern using multi-aperture speckle shear Interferometry. Opt Commun 2000;186:253–9. [5] Rastogi PK. Measurement of curvature and twist of a deformed object by electronic speckle-shearing pattern Interferometry. Opt Lett 1996;21:905–7. [6] Rastogi PK, editor. Digital speckle pattern interferometry and related techniques. England: Wiley; 2001. [7] Steinchen W, Yang L. Digital shearography: theory and application of digital speckle pattern shearing interferometry. Washington, DC: PM100, Optical Engineering Press, SPIE; 2003. [8] Chau FS, Zhou J. Direct measurement of curvature and twist of plates using digital shearography. Opt Lasers Eng 2003;3:431–9. [9] Tay CJ, Fu Y. Determination of curvature and twist by digital shearography and wavelet transforms. Opt Lett 2005;30:2873–5. [10] Bhaduri B, Krishna Mohan N, Kothiyal MP, Sirohi RS. Use of spatial phase shifting technique in digital speckle pattern interferometry (DSPI) and digital shearography (DS). Opt Exp 2006;14: 11598–607. [11] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringepattern analysis for computer-based topography and Interferometry. J Opt Soc Am 1982;72:156–60. [12] Pedrini G, Zou YL, Tiziani H. Quantitative evaluation of digital shearing interferogram using the spatial carrier method. Pure Appl Opt 1996;5:313–21. [13] Ghiglia DC, Romero LA. Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods. J Opt Soc Am A 1994;11:107–17.
Optics and Lasers in Engineering 45 (2007) 1001–1004 www.elsevier.com/locate/optlaseng
Curvature measurement using three-aperture digital shearography and fast Fourier transform Basanta Bhaduri, M.P. Kothiyal, N. Krishna Mohan Applied Optics Laboratory, Indian Institute of Technology Madras, Chennai, TN 600 036, India Received 15 March 2007; received in revised form 17 April 2007; accepted 18 April 2007 Available online 20 June 2007
Abstract Curvature measurement using a three-aperture digital shearography (DS) system is reported in this paper. The outer apertures are covered with wedge plates for introducing shear. Four images by sequentially blocking the outer apertures are used for quantitative measurement. Fourier transform technique is used to determine two sheared slope phase maps from two images at a time representing initial and deformed states. Subtraction of these two-phase maps yields the curvature phase map. Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center. r 2007 Elsevier Ltd. All rights reserved. Keywords: Digital shearography; Slope; Curvature; Fast Fourier transform
1. Introduction For plate bending problems and flexural analysis, the second order derivatives (curvature) of displacement components are needed for calculating the stress components. Hence it is necessary to perform the optical differentiation of the slope data to obtain the curvature information. Shearography is a powerful optical technique for extracting the first- and second-order derivatives of displacement components [1]. Various methods have been reported for the measurement of curvature using photographic processing and Fourier filtering [1–4]. In all these methods, the curvature information is coded in the two sheared slope contours as a Moire´ pattern. The Moire´ pattern being additive in nature, inherently has a low contrast. A method using a Michelson interferometer setup in tandem was first reported by Rastogi [5] by storing the two slope phase maps with a shift between the two for the measurement of curvature. The method is based on digital shearography [6,7] and eliminates the intermediate photographic processing. This method provides high-quality curvature phase maps and provides flexibility in handling Corresponding author. Tel.: +91 44 22574892; fax: +91 44 22570509.
E-mail address: [email protected] (N. Krishna Mohan). 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.04.005
the data for processing. However, the analysis uses the temporal phase shifting and requires a large number of frames for quantitative evaluation [1]. Method that employs digital multiple-exposure technique is also proposed in which the object is loaded, shifted in the x-direction and again unloaded between the exposures [8]. It is always not possible to unload the object perfectly because of the loading system. Recently, a method for retrieving the transient curvature and twist of a continuously deforming object has been reported by using a combination of digital shearography and the continuous wavelet transform [9]. In this paper, a three-aperture digital shearographic arrangement is demonstrated for quantitative determination of curvature using fast Fourier transform (FFT). Multi-aperture mask placed in front of the imaging lens is useful in generating spatial carrier fringes inside the speckle [10]. The outer two apertures of the three-aperture arrangement carry two identical wedge plates, while a suitable glass plated is mounted in the central aperture to compensate for the optical path. The method needs only four images by sequentially blocking the outer apertures; twice before and twice after the object deformation. The proposed method allows only two sheared object waves to interfere at the CCD plane for each recording. A 2D FFT
ARTICLE IN PRESS 1002
B. Bhaduri et al. / Optics and Lasers in Engineering 45 (2007) 1001–1004
is performed on the images to obtain three diffraction halos including the central halo at the Fourier spectrum. A 2D inverse FFT (IFFT) is performed using one of the firstorder halos [11]. The analysis yields two slope phase maps that are sheared with respect to each other. Subtraction of these two phase maps directly generates the desired curvature phase map. Experimental results are presented for a circular diaphragm clamped along the edges and loaded at the center.
carrier fringe width can be given by D ¼ lV =D, where l is the wavelength of the light used, V is the image distance and D is the aperture separation [12]. As we block one of the apertures A1 or A3 as shown in Fig. 1 for sequential recordings, there will be interference between the beams u1 and u2 or u2 and u3 for each recording. Thus there will be a contribution either of u1 or u3 with respect to u2 at the image plane. We can express the amplitudes u1 and u3 in general as u1;3 ðx; yÞ ¼ juðx Dx; yÞj exp½iffðx Dx; yÞ 2pf 0x xg.
2. Optical arrangement and theory
(2)
Fig. 1 represents the schematic arrangement. It is similar to the arrangement reported earlier by Sharma et al. [2] except that a CCD is used at the image plane instead of a photographic plate. The imaging lens carries a mask containing three equi-spaced apertures along x-direction. The outer apertures A1 and A3 carry two identical wedge plates, while a suitable flat glass plate is mounted in front of the central aperture A2 to compensate for the optical path. The object is illuminated normally by a collimated laser beam. The stops S1 and S2 are used to block the apertures A1 and A3 for sequential recording of images. When all the apertures are open, three adjacent points on the object will be imaged as one point in the image plane. Each of the two scattered fields (from the outer apertures) is sheared with respect to the field from the central aperture by an amount Dx. The shear Dx in the object plane can be given by Dx ¼ Uðn 1Þa, where U is the object distance from the imaging lens, n is the refractive index of the wedge material and a is the wedge angle [1]. Let us consider the complex wave-amplitude of respective beams coming from the apertures A1, A2 and A3 as u1 ðx; yÞ ¼ juðx þ Dx; yÞj exp½iffðx þ Dx; yÞ þ 2pf 0 xg, u2 ðx; yÞ ¼ juðx; yÞj exp½iffðx; yÞg, u3 ðx; yÞ ¼ juðx Dx; yÞj exp½iffðx Dx; yÞ 2pf 0 xg,
ð1Þ
where |u|’s are the modulus of the amplitudes, f’s are the phases, f 0 is the phase bias and is given by, f 0 ¼ 1=D, where D is the carrier fringe width inside the speckle. The
Considering the interference of u1,3 with respect to u2 at the image plane, one can express the intensity I(x,y) as Iðx; yÞ ¼ ju2 ðx; yÞ þ u1;3 ðx; yÞj2 ¼ u2 u2 þ u1;3 u1;3 þ u2 u1;3 þ u1;3 u2 .
ð3Þ
Using Eqs. (1) and (2), the Eq. (3) can now be written as Iðx; yÞ ¼ juðx; yÞj2 þ juðx Dx; yÞj2 þ juðx; yÞj:juðx Dx; yÞj exp½iffðx; yÞ fðx Dx; yÞ 2pf 0 xg þ juðx; yÞj:juðx Dx; yÞj exp½iffðx; yÞ fðx Dx; yÞ 2pf 0 xg.
ð4Þ
The Fourier transform of the Eq. (3) can be obtained as i ¼ FT½I ¼ U 2 U 2 þ U 1;3 U 1;3 þ U 2 U 1;3 þ U 1;3 U 2 ,
ð5Þ
where U j ¼ FT½uj ; j ¼ 1; 2; 3 and denotes the convolution operation. The first and second terms are the halos centered at the origin, corresponding to the spectrum of the image speckle patterns due to two apertures A1 or A3 and A2 whereas, the third and fourth terms of the above equation are the diffraction halos due to the cross terms and they appear shifted in the horizontal direction due to the angular offset between the object beams. Thus there will be three distinct diffraction halos, two first order halos and the central halo in the spectrum. These first order halos contain the phase information. If we select either of them and perform inverse Fourier transform, we can retrieve the phase. For example, if we select the fourth term, U 1;3 U 2 and perform the inverse Fourier transform, we will obtain u1;3 u2 . The phase term ½fðx; yÞ fðx Dx; yÞ 2pf 0 x is then calculated by using the relation 1 Imðu1;3 u2 Þ ½fðx; yÞ fðx Dx; yÞ 2pf 0 x ¼ tan , (6) Reðu1;3 u2 Þ where Im and Re represent the imaginary and real part respectively. The object wave-fields after the object deformation can be given by
Fig. 1. Schematic of the multi-aperture digital shearography setup for quantitative curvature measurement: A1, A3, wedge plates, A2, glass plate, S1, S2, aperture stops.
u01;3 ðx; yÞ ¼ juðx Dx; yÞj exp½iff0 ðx Dx; yÞ 2pf 0 xg u02 ðx; yÞ ¼ juðx; yÞj exp½iff0 ðx; yÞg.
ARTICLE IN PRESS B. Bhaduri et al. / Optics and Lasers in Engineering 45 (2007) 1001–1004
Now using the same procedure we can calculate the phase term ½f0 ðx; yÞ f0 ðx Dx; yÞ 2pf 0 x after object deformation as " # Imðu01;3 u0 2 Þ 0 0 1 ½f ðx; yÞ f ðx Dx; yÞ 2pf 0 x ¼ tan . Reðu01;3 u0 2 Þ (7) The phase difference Df due to 1st-order derivative of the out-of-plane displacement (slope) can then be given by [12] " # Imðu01;3 u0 2 Þ 1 1 Imðu1;3 u2 Þ Df ¼ tan tan . (8) Reðu1;3 u2 Þ Reðu01;3 u0 2 Þ
1003
As explained earlier, if we block the aperture A3, then there will be contributions of u01 and u02 via the apertures A1 and A2 respectively at the CCD plane. The images stored before and after deformation with this condition are used for analysis. Under this condition, Eq. (8) yields the slope phase map Df12 as 0 0 1 Imðu1 u 2 Þ 1 Imðu1 u2 Þ Df12 ¼ tan tan . (9) Reðu01 u0 2 Þ Reðu1 u2 Þ Similarly, if we block the aperture A1, then there will be contributions of u02 and u03 via the apertures A2 and A3 respectively at the CCD plane. Following the similar analysis as explained above, we obtain the slope phase map
Fig. 2. Quantitative measurement of curvature: (a) phase map as obtained by subtracting the slope phase maps, (b) unwrapped 2D plot and (c) 3D plot.
ARTICLE IN PRESS B. Bhaduri et al. / Optics and Lasers in Engineering 45 (2007) 1001–1004
1004
Df23 as Df23 ¼
Imðu03 u0 2 Þ tan1 Reðu03 u0 2 Þ
Imðu3 u2 Þ tan1 Reðu3 u2 Þ
.
(10)
The second order derivative or curvature information can be obtained as [2] 4p q2 w 2 Dx . (11) l qx2 The resulting phase map obtained in this method is noisy, thus filtering is necessary in order to improve it. For the present case, we have filtered using 5 5 window sinecosine phase filtering [7]. The filtered phase distributions are the wrapped or modulo-2p phase map, which range from p to p and require unwrapping [13]. Df12 Df23 ¼
3. Results and discussion The three-aperture mask has three circular holes of diameter 2.7 mm and they are separated each other along x by an amount 4.3 mm. The experiments were conducted on a white matt paint coated 60 mm diameter circular diaphragm with its edge rigidly clamped and loaded at the center. The object is illuminated by a laser beam from a He–Ne laser (20 mW) after collimation. Two wedge plates having identical wedge angles ( 25 arc min) are attached to the outer apertures A1 and A3. The shear elements provide an object plane shear around 4 mm along the x-direction with respect to the object wave passing through the central aperture A2. The three-aperture mask is placed close to an imaging lens of focal length 89 mm. We have chosen the lens in order to image full object on to the CCD sensor. The waves enter the apertures and combine coherently at the CCD plane (image plane). The stops S1 and S2 are used in order to block the waves passing through A1 and A3 respectively for sequential recording of images. We have used a Jai CV-A1 CCD camera, with matrix 1376 1035 and 4.65 4.65 mm2 pixel size. The average size of the speckle on the CCD for each image is around six pixels of the CCD and the spatial carrier fringe has a width of three pixels in the x direction. For analysis we have adopted the following sequential procedure for storing the images. First we have blocked the aperture A3 and then stored the first images. The second image is the then stored after opening the aperture A3 and blocking aperture A1. Both these images represent the initial state of the object. Now the diaphragm is given a small deflection at the center and the third frame is stored. The fourth and the final image is then stored after opening the aperture A1 and blocking aperture A3. As explained earlier, we have performed FFT and IFFT on first and fourth image to get the slope phase map corresponding to the Eq. (9). Similarly the slope phase map that corresponding to the Eq. (10) is obtained from the
second and third images. Subtraction of these slope phase maps as described in Eq. (11), generates the desired curvature phase map. As the resultant phase map is noisy, it has been filtered with 5 5 window sine-cosine phase filter [7]. We have used the overlapped region for analysis and the evaluated curvature phase map as obtained is shown in Fig. 2(a) whereas Fig. 2(b)–(c) show the corresponding unwrapped 2D and 3D plots respectively. It is interesting to note that the addition of these slope phase patterns yields slope with doubled the sensitivity [2]. 4. Conclusions We have presented a method for curvature measurement which requires only four images to be recorded. Further, the proposed technique is simple and cost effective as it requires only a three-aperture mask in front of the imaging lens instead of a conventional PZT-driven phase shifting unit. References [1] Sirohi RS, editor. Speckle metrology. New York: Marcel Dekker; 1993. [2] Sharma DK, Sirohi RS, Kothiyal MP. Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer. Appl Opt 1984;23:1542–6. [3] Tay CJ, Toh SL, Shang HM, Lin QY. Direct determination of second order derivatives in plate bending using multiple-exposure shearography. Opt Laser Technol 1994;26:91–8. [4] Krishna Mohan N. The influence of multiple-exposure recording on curvature pattern using multi-aperture speckle shear Interferometry. Opt Commun 2000;186:253–9. [5] Rastogi PK. Measurement of curvature and twist of a deformed object by electronic speckle-shearing pattern Interferometry. Opt Lett 1996;21:905–7. [6] Rastogi PK, editor. Digital speckle pattern interferometry and related techniques. England: Wiley; 2001. [7] Steinchen W, Yang L. Digital shearography: theory and application of digital speckle pattern shearing interferometry. Washington, DC: PM100, Optical Engineering Press, SPIE; 2003. [8] Chau FS, Zhou J. Direct measurement of curvature and twist of plates using digital shearography. Opt Lasers Eng 2003;3:431–9. [9] Tay CJ, Fu Y. Determination of curvature and twist by digital shearography and wavelet transforms. Opt Lett 2005;30:2873–5. [10] Bhaduri B, Krishna Mohan N, Kothiyal MP, Sirohi RS. Use of spatial phase shifting technique in digital speckle pattern interferometry (DSPI) and digital shearography (DS). Opt Exp 2006;14: 11598–607. [11] Takeda M, Ina H, Kobayashi S. Fourier-transform method of fringepattern analysis for computer-based topography and Interferometry. J Opt Soc Am 1982;72:156–60. [12] Pedrini G, Zou YL, Tiziani H. Quantitative evaluation of digital shearing interferogram using the spatial carrier method. Pure Appl Opt 1996;5:313–21. [13] Ghiglia DC, Romero LA. Robust two-dimensional weighted and unweighted phase unwrapping that uses fast transforms and iterative methods. J Opt Soc Am A 1994;11:107–17.