**5.2.3 Spherical wave system**

46 Advanced Holography – Metrology and Imaging

This system has two sets of two-beam illuminations. One set, that is, an optical setup for displacement measurement using two-beam illuminations is shown in Figure 21. A collimated light beam from a laser is divided into an object wave and a reference wave by beam splitter 1. Additionally, the object wave is divided into object wave 1 and object wave 2 by beam splitter 2. Each object wave has a shutter on the light path. From the two holograms, the *x*-directional in-plane and the *z*-directional out-of-plane displacements are analyzed. Another set of two-beam illuminations provides the *y*-directional in-plane

(a) x (b) z (c) y (d) <sup>z</sup>

(e) *u* x (f) *u*y (g) ex (h) ey

Fig. 22. Phase-difference displacement and strain distributions measured by collimated

**5.2.2 Collimated wave system** 

Fig. 21. Optical using two beam illuminations

displacement.

beam system

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 at each pixel of the holograms.

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 obtained separately.

Fig. 23. Optical setup using three spherical waves

Three-Dimensional Displacement and Strain Measurements

The strain distribution in the x-direction is shown in Figure 27.

Fig. 26. Phase-difference distributions by each light sources

Fig. 27. Strain distribution in *x*-direction

**6. Conclusions** 

cantilever.

by Windowed Phase-Shifting Digital Holographic Interferometry 49

A strain distribution measurement of a T-shaped cantilever with a fixed line is performed by the system shown in Figure 23. The size of the cantilever is 5 mm width, 15 mm length, and 5 mm thickness. A load is applied on the cantilever at the point 12.5 mm from the base of the

Seven phase-shifted holograms are captured before and after deformation. The Fourier components corresponding to each light source are separated by the phase-shifting method using Fourier transformation and each phase value is obtained. The phase-difference distributions of the light sources 1, 2 and 3 are shown in Figs. 28 (a), (b) and (c), respectively.

In order to reduce the effect of speckle noise, the windowed digital holographic interferometry was proposed. In the method, the effect of number of windows, or window size on accuracy, spatial resolution and calculation time were examined. The results are that the number of windows increases, the accuracy becomes better at first because of speckle noise reduction and after that it becomes worse because of larger speckle size. The best accuracy is sub-nanometer when the number of windows is 1024 in our experiment.

Fig. 24. Brightness change of composite wave

Fig. 25. Fourier spectrum of composite wave

Fig. 24. Brightness change of composite wave

Fig. 25. Fourier spectrum of composite wave

A strain distribution measurement of a T-shaped cantilever with a fixed line is performed by the system shown in Figure 23. The size of the cantilever is 5 mm width, 15 mm length, and 5 mm thickness. A load is applied on the cantilever at the point 12.5 mm from the base of the cantilever.

Seven phase-shifted holograms are captured before and after deformation. The Fourier components corresponding to each light source are separated by the phase-shifting method using Fourier transformation and each phase value is obtained. The phase-difference distributions of the light sources 1, 2 and 3 are shown in Figs. 28 (a), (b) and (c), respectively. The strain distribution in the x-direction is shown in Figure 27.

Fig. 26. Phase-difference distributions by each light sources

Fig. 27. Strain distribution in *x*-direction
