**1. Introduction**

28 Advanced Holography – Metrology and Imaging

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It is important to measure displacement and strain distributions to prevent failures of structures. However, one point measurement method such as strain gage is widely used in industrial practice. It is time–consuming and difficult for one-point method to find the point with the maximum strain on the structures. Whole-field optical methods such as moire method, digital image correlation, speckle method and holography are effective to find the point with the maximum strain (Sharpe, 2008). Especially, holography provides three dimensional displacement information and high resolution.

In conventional holography, an object beam and a reference beam interfere each other on a high-resolution photographic plate. It is necessary to develop the photographic plate (hologram) and it is time-consuming and bothersome work. In conventional holographic interferometry, holograms recorded before and after deformation are superposed on a photographic plate and the displacement is obtained from the interference fringe pattern (Valery et al., 1996; Ranson et al., 1993; Hayashi et al. 1986). In digital holography, a hologram is usually recorded on a CCD or C-MOS sensor instead of a photographic plate (Yaroslavskii et al., 1980; Pedriniet al., 1998; Schnars, et al. 2005). In phase-shifting digital holographic interferometry (PSDHI), the complex amplitude of the object is analyzed from the phase-shifted holograms obtained by shifting the phase of the reference beam. The reconstructed image can be calculated from the complex amplitudes of the hologram using the Fresnel diffraction integral with a computer (Yamaguchi et al., 1997; Zhang et al., 1998). The displacement at each point of an object is obtained from the phase-difference between the reconstructed images analyzed from the digital holograms recorded before and after deformation. The equipment is simple and the analysis is fast. It is useful for practical field measurement of displacement and strain distributions.

Holograms and reconstructed images have speckle noise and they provide large error in the calculation of displacement and strain analysis. In order to reduce the effect of speckle noise, the authors developed a novel method, i.e., the windowed phase-shifting digital holographic interferometry (Windowed PSDHI). In holography, any part of a hologram has the optical information about the whole reconstructed image. By using this feature of

Three-Dimensional Displacement and Strain Measurements

as follows.

where 

by Windowed Phase-Shifting Digital Holographic Interferometry 31

<sup>2</sup> 1 3 <sup>2</sup> ( , ) ( , , ) ( , , ) ( , ,0) ( , , ) <sup>422</sup> *<sup>o</sup> a XY IXY IXY IXY IXY*

<sup>3</sup> (,, ) (,,) 2 2 tan , ( , ,0) ( , , ) *<sup>o</sup>*

By calculating the Fresnel diffraction integral from the complex amplitudes *g*(*X, Y*) on the CCD plane, the complex amplitude *u*(*x, y*) of the reconstructed image at the position (*x, y*) on the reconstructed object surface being at the distance *R* from the CCD plane is expressed

> 2 2 2 2 ( ) ( ) ( , ) exp exp (,) 2 2 *ik x y ik X Y u x y F g X Y R R*

where *k* and *F* denote the wave number of the light and the operator of Fourier transform, respectively. The optical axis is normal to the CCD plane and the origin is at the center of the CCD plane. By calculating the intensities of these complex amplitudes on the reconstructed

 

*IXY IXY*

*IXY IXY*

*X Y*

object surface, a holographically reconstructed image is obtained.

**2.2 Principle of holographic interferometry** 

plane displacement *w*(*x, y*) and the phase-difference

is the wavelength of the light source.

Fig. 1. Twyman-Green type interferometer for phase-shifting digital holography

before and after deformation are almost the same and only the phase changes by

**2.3 Experiment of displacement measurement (Morimoto et al., 2007)** 

Let us consider the deformation of an object (cantilever) shown in Figure 1. If the out-ofplane displacement *w*(*x, y*) of the object is small, the amplitudes of the reconstructed object

the phase-difference before and after the deformation. The relationship between the out-of-

(,) (,) <sup>4</sup> *wxy xy* 

As an experiment, the deflection of a cantilever is measured using the optical system shown in Figure 1. The light source is a He-Ne laser. The power is 8 mW and the wavelength is 632.8

*g*( , ) ( , )exp ( , ) *XY a XY i XY o o* (3)

(2)

(1)

(4)

(*x, y*) is expressed as follows.

(5)

(*x, y*) i.e.

holograms, a hologram is divided into several parts by superposing many different windows. The phase-difference values at the same reconstructed point obtained from any other different part of the hologram should be the same. If there is speckle noise, the phase-difference with higher intensity at a reconstructed point is more reliable than that with lower intensity. Therefore, the phase-difference at each point is selected when the intensity is the largest at the same point (Morimoto et al. 2004, 2005a), or the phasedifference is calculated by averaging the phase-differences obtained from all the windowed holograms by considering the weight of the intensity (Morimoto et al. 2005b, 2005c, 2007).

In this study, in order to check the effect of the number *n* of the windowed holograms on accuracy, the number *n* is changed. The weight for averaging is also changed. When the weight is the *m*-th power of the absolute amplitude of the complex amplitude of the reconstructed object, the accuracy, that is, the standard deviation of the analyzed displacement error changes according to the power *m*. The optimal weight is studied experimentally for out-of-plane movement of a flat plate. The resolution becomes better when the number of windows becomes larger. However, the spatial resolution may be lower when the number of windows becomes larger. Therefore, the effects of the number of windows, the window size, the displacement resolution and the spatial resolution on the accuracy are also studied.

Holography has basically three-dimensional information. It is useful to analyze threedimensional displacement and strain distributions of an object. (Zhang et al., 1998; Kolenovic et. al., 2003) In this study, the Windowed PSDHI is extended to analyze threedimensional displacement components. The authors developed three systems for digital holographic interferometry using three-directional illuminations. The first one was for a laboratory bench system using a microscope (Morimoto et al., 2008a). The second one used collimated light beams from one laser source and 3 shutters (Fujigaki et al., 2005). It provided a stable system for static measurement (Shiotani et al., 2008). The third one used three spherical waves from three laser sources. It provided a compact system for static and dynamic measurement. In this study, after discussing the relationship between phase differences and displacement components, examples of three measurements using microscope, collimated light beams, and three spherical waves from three laser sources, are introduced.
