**2.1 Principle of phase-shifting digital holography (Yamaguchi et al., 1997; Zhang et al., 1998)**

As an example of phase-shifting digital holography, a Twyman-Green type interferometer shown in Figure 1 is used. A collimated light from a laser is divided into an object wave and a reference wave by a beam splitter. The phase of the reference wave is shifted by with a PZT stage. The value is set as 0, /2, and 3/2 in this phase-shifting digital holography. The four phase-shifted digital holograms are recorded on the CCD plane in a CCD camera without any focusing lens. The intensity of the hologram with a phase-shift value at the pixel coordinates (*X, Y*) on the CCD plane is expressed as *I*(*X, Y,* ). The amplitude *a*0(*X, Y*), the phase 0(*X, Y*) and the complex amplitude *g*(*X, Y*) of the object wave are expressed at the pixel coordinates (*X, Y*) on the CCD plane as follows, respectively.

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,

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

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

**2. Principle of phase-shifting digital holographic interferometry** 

pixel coordinates (*X, Y*) on the CCD plane is expressed as *I*(*X, Y,* 

the pixel coordinates (*X, Y*) on the CCD plane as follows, respectively.

**2.1 Principle of phase-shifting digital holography (Yamaguchi et al., 1997; Zhang et al.,** 

As an example of phase-shifting digital holography, a Twyman-Green type interferometer shown in Figure 1 is used. A collimated light from a laser is divided into an object wave and a reference wave by a beam splitter. The phase of the reference wave is shifted by with a PZT stage. The value is set as 0, /2, and 3/2 in this phase-shifting digital holography. The four phase-shifted digital holograms are recorded on the CCD plane in a CCD camera without any focusing lens. The intensity of the hologram with a phase-shift value at the

0(*X, Y*) and the complex amplitude *g*(*X, Y*) of the object wave are expressed at

). The amplitude *a*0(*X, Y*),

2005c, 2007).

introduced.

**1998)** 

the phase

accuracy are also studied.

$$a\_o(X,Y) = \frac{1}{4}\sqrt{\left[I(X,Y,\frac{3\pi}{2}) - I(X,Y,\frac{\pi}{2})\right]^2 + \left\{I(X,Y,0) - I(X,Y,\pi)\right\}^2} \tag{1}$$

$$\tan \Phi\_o(X, Y) = \frac{I(X, Y, \frac{3\pi}{2}) - I(X, Y, \frac{\pi}{2})}{I(X, Y, 0) - I(X, Y, \pi)}\tag{2}$$

$$\log(X,Y) = a\_o(X,Y)\exp\left\{i\Phi\_o(X,Y)\right\} \tag{3}$$

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 as follows.

$$u(\mathbf{x}, y) = \exp\left[\frac{ik(\mathbf{x}^2 + y^2)}{2R}\right] F\left\{\exp\left[\frac{ik(\mathbf{X}^2 + Y^2)}{2R}\right] \mathbf{g}(\mathbf{X}, Y)\right\} \tag{4}$$

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 object surface, a holographically reconstructed image is obtained.

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

### **2.2 Principle of holographic interferometry**

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 before and after deformation are almost the same and only the phase changes by (*x, y*) i.e. the phase-difference before and after the deformation. The relationship between the out-ofplane displacement *w*(*x, y*) and the phase-difference (*x, y*) is expressed as follows.

$$
\Delta w(\mathbf{x}\_\prime \, y) = \frac{\lambda}{4\pi} \Delta \phi(\mathbf{x}\_\prime \, y) \tag{5}
$$

where is the wavelength of the light source.

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

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

Three-Dimensional Displacement and Strain Measurements

**3.1 Windowed hologram** 

by Windowed Phase-Shifting Digital Holographic Interferometry 33

**3. Reduction method of speckle noise error by averaging phase-differences** 

As mentioned above, the phase-difference is obtained from the complex amplitudes of the reconstructed holographic object before and after deformation. However, because coherent light is reflected from a rough surface, it provides random speckle patterns in the reconstructed image. And also, any measurement system has measurement error basically. Therefore, the obtained displacement distribution has also noise and the results are not so accurate. At the point where the intensity of the speckle is weak, the accuracy of the phase value of the light at the point becomes low. In holography, any part of a hologram has the optical information of the whole reconstructed image. By using this feature of holograms, the hologram is divided into many parts. The phase-difference at the same reconstructed point obtained from any part of the hologram should be the same if there is no error. The phase-difference obtained from the complex amplitude with high intensity is more reliable than the phase-difference obtained from the complex amplitude with low intensity. If there is speckle noise, among the phase-differences obtained from each divided hologram, the phase-difference with higher intensity at a reconstructed point is more reliable. Therefore, in our previous papers, the phase-difference was obtained by selecting the phase-difference with the maximum intensity at the same point (Morimoto et al. 2004, 2005a), or the phasedifference at the same point was obtained by averaging the phase-differences obtained from all the divided holograms by considering the weight of the intensity (Morimoto et al. 2005b,

**obtained by different windowed holograms (Morimoto et al., 2007)** 

2005c, 2007). It provided the displacement distribution with high-resolution.

In this section, the divided holograms are considered as windowed holograms using some window functions. A window function with value 1 in a small part of the whole hologram area and value 0 in the remaining area is superposed on an original hologram. By multiplying the window function with values 1 and 0 by the complex amplitude of the original hologram, the windowed hologram is obtained. By changing the position of the area with value 1 in the window function, many windowed holograms are formed. The reconstructed object image is calculated from each windowed hologram using Eq. (4). A point of the reconstructed object image has a speckle pattern. The speckle patterns obtained by different windowed holograms are all different from each other. However, the speckle pattern does not move as a result of small deformation but the phase is changed by the deformation. The intensity distributions of the reconstructed object and the phase-difference distribution before and after deformation are obtained from the windowed holograms with the same window function before and after deformation. After calculating the average intensity before and after deformation from *n* sets of the windowed holograms obtained from the different window functions, the average value of the *n* phase-difference values weighted proportional to the average intensity before and after deformation is calculated at each reconstructed point. The resultant average phase-difference value is highly reliable. In this study, especially, the effect of the size or the number *n* of the windows is examined. At first, for an example, let us consider the case of *n*=16. The hologram is divided into 16 square areas. That is, a window with value 1 in a square of 240 x 240 pixels and value 0 in the other area in the 960 x 960 pixels is superposed on the original hologram with 960 x 960 pixels. By moving the area with value 1 in the window, 16 windowed holograms are obtained and they are numbered as shown in Figure 4(a). The reconstructed object image is calculated from each windowed hologram using Eq. (4). The 16 intensity distributions of the

nm. The pixel size of the CCD sensor is 4.65 µm by 4.65 µm. A captured image is sampled by 1280(V) by 960(H) pixels and the image is digitized with 8 bits. The image of 960(V) by 960(H) pixels near the center of the recorded image is used for the analysis in this study.

As a specimen, a cantilever shown in Figure 2 is analyzed. (In this experiment, there is not a reference plane shown in the figure. The reference plane is used in Section 4). The cantilever is cut out from a thick stainless steel plate. The cantilever size is 10 mm wide, 25 mm long and 1mm thick. The loading point is 20 mm from the fixed end. The displacement at the loading point is given by a micrometer with a wedge. To improve the reflection from the specimen, lusterless white lacquer is sprayed on the surface of the cantilever. The distance R from the CCD to the cantilever is 280 mm. The phase of the reference wave is shifted by every /2 using a mirror controlled with a PZT stage. Then four phase-shifted digital holograms for one cycle of the phase-shifting are recorded in the memory of a personal computer. The complex amplitude of the brightness at each pixel on the hologram is calculated using the phase-shifting method expressed in Equations (1) to (3). The reconstructed complex amplitude of the object wave at a point of the reconstructed object surface is calculated from the complex amplitudes of the holograms using the Fresnel diffraction integral expressed in Eq. (4).

After the cantilever is deformed, the reconstructed complex amplitude of the object waves at the same point of the reconstructed object is obtained similarly. The reconstructed images and the phase distributions are obtained from the holograms with 960 x 960 pixels before and after deformation. The phase-difference distribution before and after deformation, that is, the out-of-plane displacement distribution is shown in Figure 3(a).

Fig. 3. Phase-difference distributions obtained by digital holographic interferometry

Fig. 2. Specimen (Cantilever)
