Simultaneous measurement of deformation and the first derivative with spatial phase-shift digital shearography

Optics Communications 286 (2013) 277–281

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Discussion

Simultaneous measurement of deformation and the first derivative with spatial phase-shift digital shearography Xin Xie a, Nan Xu a, Jianfei Sun a,b, Yonghong Wang b, Lianxiang Yang a,n a b

Optical Laboratory, Department of Mechanical Engineering, Oakland University, Rochester, MI 48309, USA School of Instrument Science and Opto-electronics Engineering, Hefei University of Technology, Hefei, Anhui, 230009, China

a r t i c l e i n f o

abstract

Article history: Received 12 July 2012 Received in revised form 20 August 2012 Accepted 22 August 2012 Available online 8 September 2012

This paper presents a spatial phase-shift digital shearography system based on a modified Michelson Interferometer, which is capable of measuring out-of-plane deformation and its first derivative simultaneously. A reference beam is introduced into the Michelson Interferometer using an optical fiber. Two sheared images from the Michelson Interferometer are combined with the reference beam to generate two holograms and one shearogram. The Fourier Transform method is utilized to calculate the phase difference. To separate the spectrum in the Fourier Domain, the reference beam is tilted a small angle to create a spatial phase shift. An out-of-plane deformation and its first derivative can then be determined simultaneously from a single speckle interferogram by properly selecting filters for the spectrum image. The theory and the experimental results are presented. & 2012 Elsevier B.V. All rights reserved.

Keywords: Shearography Spatial phase shift Fourier transform Simultaneous measurement of deformation and its first derivative

1. Introduction Digital shearography is a non-contact, whole field, coherent optical measurement method [1–3]. The use of the phase shift technique improves the sensitivity of the measurement [4]. Most shearography systems use the temporal phase shift technique, which requires a stable speckle pattern distribution between the phase shifting steps and, thus, limits its usage to the static situations only [5,6]. A spatial phase shift technique should be introduced into the digital shearography setup to improve its applicability for such dynamic measurements as fast moving online testing and biomedical measurement. Basanta Bhaduri and N. K. Mohan applied the spatial phase technique on the digital shearography system in 2006. This shearography system utilizes a multiple-aperture mask diffraction to introduce a spatial phase shift and has firstly enabled evaluation of deformation and the first derivative of deformation in one measurement with the spatial phase shift technique [6,7]. However, the field of view of this system is greatly limited by the size of the beamsplitter so that the target size is for sure greatly limited. In addition the multiple-aperture mask diffraction structure is too complicated which has to be specially designed for every single test object and it is too sensitive to the environment turbulence as well which may reduce the system stability. In 1996, G Pedrini introduced the Fourier transform method into the Mach–Zehnder based digital

n

Corresponding author. Fax: þ1 248 370 4416. E-mail address: [email protected] (L. Yang).

0030-4018/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.optcom.2012.08.072

shearography system [8]. He uses Fourier transform to extract the spectrum in frequency domain and calculates the phase of the shearogram. It is a simple and efficient way to introduce and evaluate the spatial frequency shift in an optical system. But with a simple Mach–Zehnder interferometer based shearography system, the deformation and first derivative of deformation cannot be measured simultaneously. Traditional digital shearography system measures only the first derivative of deformation, i.e. slop, and has been known as a practical NDT(Non-Destructive Testing) method for more than twenty years. However, it has its limitation for certain kinds of defects which has a relative larger deformation but smaller slop, while this kind of defects can be easily identified by the Holography technique which measures the deformation directly. The newly developed digital shearography system measures the deformation and its first derivative simultaneously, which can provide multiple choice for the detection of different types of defects. This paper introduces an innovative approach to apply the spatial phase shift technique to the modified Michelson Interferometerbased shearography system for simultaneous measurement of deformation and its first derivative. Theoretical derivation and experiment results are also presented.

2. Theory and experiment The optical setup is illustrated in Fig. 1. This setup uses a modified Michelson Interferometer as the shearing device [9].

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following equations: u1 ðx,yÞ ¼ 9u1 ðx,yÞ9expð2pif 0x x2pif 0y yÞ

ðReference Beam 1Þ ð1aÞ

u2 ðx,yÞ ¼ 9u2 ðxþ Dx,y þ DyÞ9expð2pif 0x x2pif 0y y2pif 1 xÞ

ðReference Beam 2Þ

ð1bÞ u3 ðx,yÞ ¼ 9u3 ðx,yÞ9exp½ijðx,yÞ

ðShearing Beam 1Þ

  u4 ðx,yÞ ¼ 9u4 ðx þ Dx,y þ DyÞ9exp ijðx þ Dx,y þ DyÞ2pif 1 x

ð1cÞ

ðShearing Beam 2Þ

ð1dÞ where u1, u2 are the two components of the reference beam and u3,u4 are the object beam. The values f0x, f0y, and f1 represent the components of the spatial frequency. These components can be represented by the following equations: f 0x ¼ ðsiny0x =lÞ

ð1eÞ

f 0y ¼ ðsiny0y =lÞ

ð1fÞ

f 1 ¼ ðsiny1 =lÞ

ð1gÞ

y0x and y0y represent the x and y components of the angle between the reference beam and the normal of CCD sensor plane, respectively. y1 represents the shearing angle of the shearing device. Dx and Dy are the shearing distance in the x and y directions, respectively, and l is the laser wavelength. The intensity recorded on the CCD camera can be expressed as: Fig. 1. Experimental setup for out-of-plane deformation and its first derivative measurement.

I ¼ ðu1 þu2 þ u3 þ u4 Þðun1 þ un2 þ u3 n þ un4 Þ ¼ u1 un1 þ u2 un2 þ u3 un3 þu4 un4 þ u1 un2 þu1 un3 þ u1 un4 þ u2 un1 þ u2 un3 þ u2 un4 þu3 un1 þ u3 un2 þu3 un4 þ u4 un1 þ u4 un2 þu4 un3

ð2Þ

n

Laser light from a Helium-Neon laser, with a wavelength of 632.8 nm, is separated into two beams using a beam splitter (BS): an object beam and a reference beam. The object beam is expanded by a beam expander (BE), and used to illuminate the object. Diffused light reflected from the object is then received by the CCD camera. The aperture (AP), located just before the image lens (L1), is used to control the speckle size and the spectrum pattern size on the Fourier domain. The reference beam goes through a convex lens (L2). This beam then converges into a single-mode optical fiber, propagates through the optical fiber, and exits behind the lens (L1). The reference beam is tilted at an angle of approximately 21 from the normal of the CCD plane to create a frequency shift on the Fourier plane. The shearogram is generated by the object beam, through the use of the Michelson interferometer. A hologram is also gained through interferometry between the object beam and reference beam. The shearogram and hologram are simultaneously recorded by the CCD camera in the form of an intensity difference. An edge-clamped and center loaded metal plate was tested using this shearography system. The size of the plate is 115 mm  115 mm. A 1392  1024 CCD camera with a pixel size (dx ndx) of 4.65  4.65 mm is used for recording pictures. Pictures before and after loading are captured for evaluation. The original speckle image captured by the CCD camera contains the information from both the hologram and shearogram. A Fourier Transform (FT) was applied to this image to separate the hologram and shearogram information. The CCD camera captures two shared wave front from the reference beam and two sheared wave fronts from the object beam. For this experiment, the object beam wave fronts were sheared in the x-direction. These wave fronts can be represented by the

The denotes the complex conjugate of u. A Fourier Transform is used to transmit the image from the spatial domain to the Fourier domain. Using the Fourier Transform, Eq. (2) transforms into the following form:     FTðIÞ ¼ U 1 f 0x ,f 0y  U n1 f 0x ,f 0y þU 2 ðf 0x þ f 1 ,f 0y Þ  U n2 ðf 0x þ f 1 ,f 0y Þ þ U 3 ðf x ,f y Þ  U n3 ðf x ,f y Þ þ U 4 ðf 0x þ f 1 ,f y Þ  U n4 ðf 0x þ f 1 ,f y Þ þ U 1 ðf 0x ,f 0y Þ  U n2 ðf 0x þ f 1 ,f 0y Þ þ U 1 ðf 0x ,f 0y Þ  U n3 ðf x ,f y Þ þ U 1 ðf 0x ,f 0y Þ  U n4 ðf 0x þ f 1 ,f y Þ þ U 2 ðf 0x þ f 1 ,f 0y Þ  U n1 ðf 0x ,f 0y Þ þ U 2 ðf 0x þf 1 ,f 0y Þ  U n3 ðf x ,f y Þ þ U 2 ðf 0x þ f 1 ,f 0y Þ  U n4 ðf 0x þ f 1 ,f y Þ þU 3 ðf x ,f y Þ  U n1 ðf 0x ,f 0y Þ þ U 3 ðf x ,f y Þ  U n2 ðf 0x þf 1 ,f 0y Þ þ U 3 ðf x ,f y Þ  U n4 ðf 0x þ f 1 ,f y Þ þ U 4 ðf 0x þ f 1 ,f y Þ  U n1 ðf 0x ,f 0y Þ þU 4 ðf 0x þ f 1 ,f y Þ  U n2 ðf 0x þ f 1 ,f 0y Þ þ U 4 ðf 0x þ f 1 ,f y Þ  U n3 ðf x ,f y Þ

ð3Þ

Here  is the convolution operation, U1(f0x, f0y)¼FT(u1), U2(f0x þ f1, f0y)¼FT(u2), U3(fx, fy)¼FT(u3), U4(f0x þf1, fy)¼FT(u4). Fig. 2(a) shows the Fourier spectrum of the images. On the Fourier domain, there are 11 spectrums that can clearly be observed, corresponding to the sixteen terms in Eq. (3). Only sections A, C, D, D’, F, and F’ are useful for evaluation of the deformation and the first derivative. Sections A and C represent the U4Un3 and U3Un4 parts, respectively. These two sections contain the information of the shearogram. Sections D, D’, F, and F’ represent the U3Un2, U2Un3 parts, U4Un1, and U1Un4 parts, respectively. These four parts contain the information of the holograms. The other spectrums correspond to the remaining ten terms. Spectrum B corresponds to the U1Un1 þ U2Un2 þU3Un3 þ U4Un4 terms. These low frequency components represent the background in the spatial domain. Spectrum E and E’ correspond to U1Un3 þ U2Un4 and U3Un1 þU4Un2 terms, respectively. These terms are transformed from the phase combination of two

X. Xie et al. / Optics Communications 286 (2013) 277–281

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Fig. 2. (a) Spectrum after Fourier Transform (FT); (b) Phase map of a hologram due to a deformation after Windowed Inverse Fourier Transform (WIFT) operation from Section F0 ; (c) Phase map of a shearogram due to a deformation after WIFT operation from Section C. (d) 7  7 Smoothed Phase map from (b); and (e) 7  7 Smoothed Phase map from (c).

holograms. Part G and G’ correspond to the U1 Un2 and U2 Un1 terms respectively, which result the interference of two sheared reference beams. The phase distributions are calculated, using the complex amplitudes, by employing an Inverse Fourier Transform. This results in the following equations:

hologram and shearogram, due to the deformation calculated from Sections D and A, respectively. The relation between the phase difference and deformation is given by

h

Djh ¼

jðx,yÞ þ 2pxf 0x þ 2pyf 0y

i

Im½u3 un1  ¼ arctan Re½u3 un1 





jðx,yÞjðx þ Dx,y þ DyÞ þ 2pxf 1 ¼ arctan

ðhologramÞ

ð4aÞ

Im½u4 un3  Re½u4 un3 

ðshearogramÞ

For Eq. (4a) and (4b), Im and Re denote the imaginary and real part of the complex numbers. In the same sense, after the object deforms, two additional phase distributions can be obtained by the same method using the recorded images i

j0 ðx,yÞ þ2pxf 0x þ2pyf 0y ¼ arctan





Im½u3 un1  Re½u3 un1 

j0 ðx,yÞj0 ðx þ Dx,y þ DyÞ þ 2pxf 1 ¼ arctan

ðhologramÞ

Im½u4 un3  Re½u4 un3 

ð5aÞ

ðshearogramÞ

ð5bÞ The phase difference due to a deformation can then be calculated as

Djh ¼ j0 ðx,yÞjðx,yÞ 0

l

ds

ð7aÞ

And the derivative of deformation:

ð4bÞ

h

2p

Djs ¼

2pDx

l

Ds

ð7bÞ

For Eq. 7a and 7b, d denotes the deformation vector (u, v, w) and D represents the vector for the first derivative of deformation @v @w ð@u @x , @x , @x Þ. The sensitivity vector s is expressed by the formulas ¼ ki ko , where ki is the unit vector of the illumination direction and ko is the unit vector of the observation direction. For this case, s ¼ ðsin a,0,1 þ cos aÞ, the shearing amount in the x direction is denoted by Dx. Using the method specified in this section, the deformation and the first derivative of the deformation can be measured quantitatively and simultaneously. While the illuminating angle is close to perpendicular to the object surface, the system measures simultaneously an out-of-plane deformation and its first derivative in the shearing direction [10].

3. Discussion and application

ð6aÞ 0

Djs ¼ ½j ðx,yÞjðx,yÞ½j ðx þ Dx,y þ DyÞjðx þ Dx,y þ DyÞ

ð6bÞ

Fig. 2(b) and (c) shows the hologram and the shearogram depicting the phase differences, also called phase maps of

The key of the spatial phase shift technique presented in this paper is to separate the spectrums in the Fourier domain [11]. There are three parameters in the setup that affect the ability of the system to separate the spectrums: the tilting angle of the optical fiber, the aperture size and the shearing amount.

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The tilting angle introduces a spatial frequency (f0x, f0y), defined by Eq. 1e and 1f. This establishes the position of E and E’ in the Fourier domain at (f0x  fx, f0y fy). The maximum tilting angle ymax is limited by [12] f max ¼

2

l

sin



ymax 2

ð8Þ

Where fmax is the maximum spatial frequency that can be captured by the CCD camera. For our case, fmax ¼1/(2dx), and dx is the pixel size. The aperture works as a spatial frequency filter, which limits the maximum spatial frequency that can be captured. The aperture size should be carefully chosen to use the full spatial resolution, gain more useful spectrum information, and obtain a high S/N (signal/noise) ratio. The aperture size is expected to be as big as possible to satisfy these three needs. However, as the aperture increases, so do the sizes of the three spectrums (E, E’ and B) which do not contribute to the evaluation. If the aperture size becomes too big, frequency aliasing will occur. Once the useless and useful spectrums overlay each other, extraction of the frequency in the interested area becomes difficult, and the phase map will contain a lot of noise. The shearing amount results in a frequency shift f1 in the Fourier domain. This represents the distances between spectrums D and F from E, spectrums D’ and F’ from E’, and spectrums G and G’ from B. Therefore, the shearing amount must be large enough to avoid aliasing. In the spectrum shown in Fig. 2(a), the sample size is 115  115 mm2 and the shearing amount of 15 mm which is 12% of object size. With this shearing amount, portions of the useful spectrum are almost separated each other. A smaller shearing amount will result in a larger aliasing area. The aliasing part of the spectrum has to be windowed out when applying the Windowed Inverse Fourier Transform (WIFT), otherwise it generates noise in the phase map. A precisely chosen WIFT window can reduce the noise. However, the edge of the aliasing

area is practically difficult to identify. Therefore, an additional aperture can be introduced when the shearing amount is relatively small. This added aperture works as a spatial frequency filter to block the aliasing part of the spectrum. An improved spectrum without aliasing area by adding an additional aperture for a shearing amount of 8 mm which about 7% of the object size is shown in Fig. 3(a), and the phase maps of shearogram and hologram due to a deformation are presented in Fig. 3(b) and (c), respectively. Adding the additional aperture will reduce the spectrum area, however, it can greatly suppress the aliasing phenomenon. Although the decrease of the useful spectrum area may lower down the S/N ratio in the phase map a little, the resulting phase map is still acceptable, and also this can be improved by using a higher spatial resolution camera and a higher power laser. Fig. 4 shows the results of an application in NDT area. A honeycomb structure plate (100 mm  100 mm) which has a small disbond under the skin is tested. The sample is thermal loaded with a heat gun, a series of images is recorded by the CCD camera with an acquisition rate of 10 Hz. The phase maps indicate the location of the defect clearly. In this application, the titling angle of optical fiber is pre-set to gain a properly distributed spectrum. The shearing amount could be decided by the required measuring sensitivity. As the measuring sensitivity of shearography is increasing with the shearing amount, a relative big shearing amount is selected (15 mm). As long as there is no frequency aliasing, the aperture size should be set as big as possible to make the full use of spatial resolution.

4. Conclusion A spatial phase-shift digital shearography system has been developed to measure the out-of-plane deformation and the first

Fig. 3. Spectrum and phase maps in a small shearing amount.

Fig. 4. Non-destructive testing results for a honeycomb structure under dynamic loading.

X. Xie et al. / Optics Communications 286 (2013) 277–281

derivative of deformation simultaneously. This system has the following advantages: (1) Extremely high measuring speed and being capable of dynamic applications. (2) Simultaneous measurement capability for both deformation and its derivative, which can provide multiple choices for different NDT situations. (3) Simpler structure and higher stability: the hologram and shearogram share a single light path, and all the components can be fixed on one rigid platform. The reference beam is inducted by a fiber and can be fixed on the rigid platform too. As a result, the system is relatively insensitive to the rigid body movement. (4) Adjustable field of view which is no longer limited by the size of beamspliter, by using different imaging lens (L1 in Fig. 1), this system can have an adjustable field of view, so that it can provide an adjustable measurable size.

The spatial resolution of this technique is, however, a little bit lower than the traditional temporal phase shift technique due to a relative lower S/N ratio. This problem can be solved by using higher special frequency CCD camera, making full use of spatial resolution, using higher power laser and better frequency filter.

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Acknowledgments The authors would like to express their sincere thanks to Mr. Bernard Sia of the Optical Lab of Oakland University who carefully and thoroughly read the manuscript and provide valuable criticisms. References [1] Lianxiang Yang, W. Steinchen, G. Kupfer, P. Maeckel, F. Voessing, Optics and Lasers in Engineering 30 (1998) 199. [2] Y.Y. Hung, Optical Engineering 21 (1982) 391. [3] Wolfgang Steinchen, Lianxiang Yang, Digital Shearography: Theroy and Application of Digital Speckle Pattern Shearing Interferometry, SPIE Press, Bellingham, 2003. (Chapter 1). [4] W. Steinchen, L.X. Yang, G. Kupfer, P. Maeckel, Journal of Aerospace Engineering 212 (1998) 21. [5] M. Schuth, F. Voessing, L.X. Yang, Journal of Holography and Speckle 1 (1) (2004) 46. [6] Basanta Bhaduri, N.K. Mohan, M.P. Kothiyal, R.S. Sirohi, Optics Express 14 (24) (2006) 11598. [7] B. Bhaduri, N.K. Mohan, M.P. Kothiyal, Applied Optics 46 (2007) 5680. [8] G. Pedrini, Y.-L. Zou, H.J. Tiziani, Pure and Applied Optics 5 (1996) 313. [9] Y.Y. Hung, H.M. Shang, L.X. Yang, Optical Engineering 42 (5) (2003) 1197. [10] L.X. Yang, F. Chen, W. Steinchen, Y.Y. Hung, Journal of Holography and Speckle 1 (2) (2004) 69. [11] Giancarlo Pedrini, Wolfgang Osten, Mikhail E. Gusev, Applied Optics 45 (15) (2006) 3456. [12] Yu Fu, Hongjian Shi, Hong Miao, Applied Optics 48 (11) (2009) 1990.