**5. Three-dimensional displacement analysis by digital holographic interferometry**

In order to measure three-dimensional displacement components, a three-directionalillumination method or three-directional-observation method is usually employed (Zhang et al., 1998; Kolenovic et al. 2003). However, in order to analyze strain distributions, the displacement is spatially differentiated which requires accurate measurement of displacement distributions. Windowed PSDHI is useful for measuring quantitatively nanometer displacement of objects (Morimoto et al., 2007; 2008b). The authors previously proposed a method of simultaneous measurement of in-plane and out-of-plane displacements using two beam illuminations. (Okazawa et al. 2005) A three-directional illumination method was also proposed (Fujigaki et al. 2005; Morimoto et al. 2008a, 2008c). To miniaturize the equipment for practical use, laser beams with spherical wavefronts were used. However, if spherical waves are used, the incident angles are different for each point and then each point has different sensitivity vectors. The incident angle for each point on an object is determined by the three-dimensional position of the point and the point source of the laser beam. It is, however, difficult to measure the incident angle accurately. The authors proposed a calibration method with a reference flat plane. The reference plane was installed on an XYZ three-axis piezo stage which was movable in the XYZ directions by a very small amount (Morimoto et al., 2008a).

By calculating the each phase-difference between before and after deformation using digital holography, the parameter for the relationship between the displacement and the phasedifference can be obtained. Tabulation of parameters for each point helps to measure the displacement in high speed from the phase-difference of a specimen. Displacement measurement using spherical waves can be realized with this calibration method. The theoretical treatment and experimental results of some three-dimensional displacement measurements using this method are shown in this study.

### **5.1 Relationship between phase-differences and displacement components**

The schematic positions of an object and an observation direction when the object is illuminated at a particular incident angle are shown in Figure 18. The phase-difference for a unit displacement depends on the position of the object (Morimoto et. al., 2008; Fujigaki et al., 2006). When the positional relationship is as expressed in Figure 18, the equation at a point P before deformation on an object is expressed as shown in Equation (6).

$$
\Delta \phi = \mathbf{e} \cdot d \tag{6}
$$

where, *e* is the sensitivity vector which depends on the half of the angle between the incident angle i and the observation angle of the point P, and *d* is the displacement vector for the point P before deformation which moves to a point P' after deformation. As the displacement is very small compared with the distance between the light source and the object, it is assumed that the incident angle 1 and the observation angle 0 do not change. And is the phase-difference resulted from the displacement at the point P.

The displacement vector *d* and the sensitivity vector *e* each have components in the *x*, *y*, and *z* directions. Then Eq. 6 is written as

windows is 16, 64 or 256, as in our experiment by considering the balance of the

In order to measure three-dimensional displacement components, a three-directionalillumination method or three-directional-observation method is usually employed (Zhang et al., 1998; Kolenovic et al. 2003). However, in order to analyze strain distributions, the displacement is spatially differentiated which requires accurate measurement of displacement distributions. Windowed PSDHI is useful for measuring quantitatively nanometer displacement of objects (Morimoto et al., 2007; 2008b). The authors previously proposed a method of simultaneous measurement of in-plane and out-of-plane displacements using two beam illuminations. (Okazawa et al. 2005) A three-directional illumination method was also proposed (Fujigaki et al. 2005; Morimoto et al. 2008a, 2008c). To miniaturize the equipment for practical use, laser beams with spherical wavefronts were used. However, if spherical waves are used, the incident angles are different for each point and then each point has different sensitivity vectors. The incident angle for each point on an object is determined by the three-dimensional position of the point and the point source of the laser beam. It is, however, difficult to measure the incident angle accurately. The authors proposed a calibration method with a reference flat plane. The reference plane was installed on an XYZ three-axis piezo stage which was movable in the XYZ directions by a very small

By calculating the each phase-difference between before and after deformation using digital holography, the parameter for the relationship between the displacement and the phasedifference can be obtained. Tabulation of parameters for each point helps to measure the displacement in high speed from the phase-difference of a specimen. Displacement measurement using spherical waves can be realized with this calibration method. The theoretical treatment and experimental results of some three-dimensional displacement

The schematic positions of an object and an observation direction when the object is illuminated at a particular incident angle are shown in Figure 18. The phase-difference for a unit displacement depends on the position of the object (Morimoto et. al., 2008; Fujigaki et al., 2006). When the positional relationship is as expressed in Figure 18, the equation at a

where, *e* is the sensitivity vector which depends on the half of the angle between the

vector for the point P before deformation which moves to a point P' after deformation. As the displacement is very small compared with the distance between the light source and the

The displacement vector *d* and the sensitivity vector *e* each have components in the *x*, *y*, and

*e d* (6)

1 and the observation angle

of the point P, and *d* is the displacement

0 do not change.

**5.1 Relationship between phase-differences and displacement components** 

point P before deformation on an object is expressed as shown in Equation (6).

is the phase-difference resulted from the displacement at the point P.

**5. Three-dimensional displacement analysis by digital holographic** 

computation time and accuracy.

amount (Morimoto et al., 2008a).

incident angle

And  *z* directions. Then Eq. 6 is written as

object, it is assumed that the incident angle

measurements using this method are shown in this study.

i and the observation angle

**interferometry** 

$$
\Delta\phi = \begin{bmatrix} e\_{\mathcal{X}} & e\_{\mathcal{Y}} & e\_{\mathcal{Z}} \end{bmatrix} \begin{bmatrix} d\_{\mathcal{X}} \\ d\_{\mathcal{Y}} \\ \vdots \\ d\_{\mathcal{Z}} \end{bmatrix} = e\_{\mathcal{X}}d\_{\mathcal{X}} + e\_{\mathcal{Y}}d\_{\mathcal{Y}} + e\_{\mathcal{Z}}d\_{\mathcal{Z}} \tag{7}
$$

Fig. 18. Relationship between displacement of object, observation direction and incident angle at Point P

When an object is illuminated from three different directions, the number of parameters of the sensitivity vector components increases, and Equation (7) can be extended as Equation (8);

$$
\begin{bmatrix}
\Delta\phi\_1\\\Delta\phi\_2\\\Delta\phi\_3
\end{bmatrix} = \begin{bmatrix}
e\_{1x} & e\_{1y} & e\_{1z} \\
e\_{2x} & e\_{2y} & e\_{2z} \\
e\_{3x} & e\_{3y} & e\_{3z}
\end{bmatrix} \begin{bmatrix} d\_x\\d\_y\\d\_z \end{bmatrix} \tag{8}
$$

$$S = \begin{bmatrix} e\_{1x} & e\_{1y} & e\_{1z} \\ e\_{2x} & e\_{2y} & e\_{2z} \\ e\_{3x} & e\_{3y} & e\_{3z} \end{bmatrix} \tag{9}$$

where the suffixes 1, 2 and 3 show the corresponding illumination directions.

Each component of the sensitivity vector matrix *S* is obtained by the geometric parameters of the optical system. When each component of the matrix *S* of Equation (7) is specified, the displacement components *dx dy* and *dz* can be obtained from the phase-difference and for each incident light, respectively, using the inverse matrix *S*-1 of the sensitivity vector matrix *S* as follows;

Three-Dimensional Displacement and Strain Measurements

effect of speckle noise.

Figs. 20 (c), (d) and (e).

(c)

1 (d)

Fig. 20. Displacement measurement of L-shaped cantilever

(f) Displacement components along *<sup>x</sup>*0*x*1 (g) Displacement components along *x*1*x*<sup>2</sup>

experimental setup shown in Figure 19.

by Windowed Phase-Shifting Digital Holographic Interferometry 45

The pixel size of the used CCD (SONY XCD-900) is 4.65 m×4.65 m. The analyzed image size is 960×960 pixels. A stereoscopic microscope (NIKON) using a CCD camera adapter is used. In the analysis, windowed PSDHI with 64 windows is used in order to decrease the

Displacement measurement of an L-shaped cantilever is performed by using the

An L-shaped cantilever with a fixed end, 2 mm width, 8 mm length, and 1 mm thickness shown in Figure 20(a) is measured. Figure 20(b) shows the reconstructed image. Figures 20(c), (d) and (e) shows the phase-difference distribution respectively obtained by a laser beams ch1, ch2 and ch3 in Figure 19. Figures 20 (f) and (g) show the resultant threedimensional displacement component distributions along the lines *x*0*x*1 and *x*1*x*2 shown in

(a) Specimen (b) reconstructed image

2 (e)

3

$$
\begin{bmatrix} d\_x \\ d\_y \\ d\_z \end{bmatrix} = \begin{bmatrix} f\_{x1} & f\_{x2} & f\_{x3} \\ f\_{y1} & f\_{y2} & f\_{y3} \\ f\_{z1} & f\_{z2} & f\_{z3} \end{bmatrix} \begin{bmatrix} \Delta \phi\_1 \\ \Delta \phi\_2 \\ \Delta \phi\_3 \end{bmatrix} \tag{10}
$$

$$S^{-1} = \begin{bmatrix} f\_{x1} & f\_{x2} & f\_{x3} \\ f\_{y1} & f\_{y2} & f\_{y3} \\ f\_{z1} & f\_{z2} & f\_{z3} \end{bmatrix} \tag{11}$$

where *f ij* is the (*i, j*) component of *S*-1.

### **5.2 Displacement and strain measurement of L-shaped cantilever 5.2.1 Laboratory system for microscope**

Figure 19 shows a schematic view of an optical system for digital holographic interferometry with a microscope. A parallel collimated laser beam illuminates an object. The light source is a He-Ne laser (Output: 8mW, Wavelength: 632.8mm). The laser beam from the light source is separated into three object beams ch.1, ch.2 and ch.3 and one reference beam by using three beam splitters, one half mirror and several mirrors. The reflected object beam arrives in a CCD plane through the microscope. A parallel reference beam is incident from a beam splitter between the object and the microscope. The reference beam also arrives in the same CCD plane. Each object beam is interfered with the reference beam by cutting off the other object beams. The phase of the reference beam is changed by using the PZT stage.

Fig. 19. Optical setup for experiment

*x xxx y yyy z zzz*

*d fff d fff d fff*

1

**5.2 Displacement and strain measurement of L-shaped cantilever** 

where *f ij* is the (*i, j*) component of *S*-1.

using the PZT stage.

Fig. 19. Optical setup for experiment

**5.2.1 Laboratory system for microscope** 

123 1 123 2 3 123

> 123 123 123

*xxx yyy zzz*

*fff*

*fff*

Figure 19 shows a schematic view of an optical system for digital holographic interferometry with a microscope. A parallel collimated laser beam illuminates an object. The light source is a He-Ne laser (Output: 8mW, Wavelength: 632.8mm). The laser beam from the light source is separated into three object beams ch.1, ch.2 and ch.3 and one reference beam by using three beam splitters, one half mirror and several mirrors. The reflected object beam arrives in a CCD plane through the microscope. A parallel reference beam is incident from a beam splitter between the object and the microscope. The reference beam also arrives in the same CCD plane. Each object beam is interfered with the reference beam by cutting off the other object beams. The phase of the reference beam is changed by

*S fff*

(10)

(11)

The pixel size of the used CCD (SONY XCD-900) is 4.65 m×4.65 m. The analyzed image size is 960×960 pixels. A stereoscopic microscope (NIKON) using a CCD camera adapter is used. In the analysis, windowed PSDHI with 64 windows is used in order to decrease the effect of speckle noise.

Displacement measurement of an L-shaped cantilever is performed by using the experimental setup shown in Figure 19.

An L-shaped cantilever with a fixed end, 2 mm width, 8 mm length, and 1 mm thickness shown in Figure 20(a) is measured. Figure 20(b) shows the reconstructed image. Figures 20(c), (d) and (e) shows the phase-difference distribution respectively obtained by a laser beams ch1, ch2 and ch3 in Figure 19. Figures 20 (f) and (g) show the resultant threedimensional displacement component distributions along the lines *x*0*x*1 and *x*1*x*2 shown in Figs. 20 (c), (d) and (e).

Fig. 20. Displacement measurement of L-shaped cantilever

Three-Dimensional Displacement and Strain Measurements

distributions, respectively.

**5.2.3 Spherical wave system** 

at each pixel of the holograms.

Fig. 23. Optical setup using three spherical waves

obtained separately.

by Windowed Phase-Shifting Digital Holographic Interferometry 47

An L-shaped cantilever with a fixed line is measured by the system shown in Figure 21. Figures 22(a) and (b) show the phase-difference distributions along the *x*- and *z*-directions, respectively, obtained by the *x*-directional set. Figures 22(c) and (d) show the phasedifference distributions along the *y*- and *z*-directions, respectively, obtained by the *y*directional set. Figures 22(e) and (f) show the *x*- and *y*-directional displacement distributions, respectively. Figures 22(g) and (h) show the *x*- and *y*- directional strain

In this experiment, three holograms using light sources 1, 2 and 3 are recorded on a CCD simultaneously as shown in Figure 23. Although the phases of the three reference waves are shifted simultaneously seven times, the phase-shift amounts during the seven times are different for each light source. The total phase-shifts are 2, 4 and 6for the light sources 1, 2 and 3, respectively. The each fringe pattern by the light sources 1, 2 and 3 is extracted from the continuous seven holograms using the Fourier transformation of the brightness change

Let us explain the procedures. Figure 24 illustrates the captured brightness changes at a pixel and the brightness change corresponding to each of the three light sources. The discrete Fourier transformation of the captured brightness along the time axis provides the Fourier spectrum shown in Figure 25. It has seven frequency components from -3 to 3. Here, the components of the frequencies 1, 2 and 3 arise from the light sources 1, 2 and 3, respectively. By extracting these components and calculating the amplitudes and the phases of the components, the complex amplitudes of the three holographic fringe patterns are
