Optics and Lasers in Engineering 45 (2007) 1001–1004 Curvature measurement using three-aperture digital shearography and fast Fourier transform Basanta Bhaduri, M.P. Kothiyal, N. Krishna Mohan à 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 compo- nents. 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 photo- graphic 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 photo- graphic processing. This method provides high-quality curvature phase maps and provides flexibility in handling 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 pro- posed 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 continu- ously 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 determina- tion 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 www.elsevier.com/locate/optlaseng 0143-8166/$ - see front matter r 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.optlaseng.2007.04.005 à Corresponding author. Tel.: +91 44 22574892; fax: +91 44 22570509. E-mail address: nkmohan@iitm.ac.in (N. Krishna Mohan).