**3.1 Windowed hologram**

32 Advanced Holography – Metrology and Imaging

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)

(a) Front view (b) Top view (c) Photograph

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

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

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

is, the out-of-plane displacement distribution is shown in Figure 3(a).

pixels near the center of the recorded image is used for the analysis in this study.

Fig. 2. Specimen (Cantilever)

diffraction integral expressed in Eq. (4).

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, 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

Three-Dimensional Displacement and Strain Measurements

**3.3 Effect of weight of averaging on accuracy** 

errors is 88 pico-meters when *n*=1024 and *m*=2.

*n*= 1 and 1024 and *m*=2.

Fig. 5. Displacement distributions along lines A and B shown in Figure3

by Windowed Phase-Shifting Digital Holographic Interferometry 35

The effect on accuracy of changing the number of windowed holograms and the weight of the averaging is examined. When the weight is proportional to the *m*-th power of the amplitude of the complex amplitude, the standard deviations of errors are compared by changing the number *m*. In Section 3.2, the case of *m*=2 as the weight was examined by considering that reliability is larger according to the intensity, i.e., the 2nd power of the absolute amplitude of the complex amplitude. In this section, though the theoretical curve of the cantilever mentioned in the previous chapter is a cubic function, the experimental curve is not cubic in a precise sense because of the anisotropy of the materials, the actual boundary conditions, etc. In this section, parallel movement of a flat plate is adopted because the anisotropy of the materials, the actual boundary conditions, etc. are almost cut off. The parallel movement is a half wavelength, that is, about 316 nano- meters along the direction of the normal to the flat surface. The displacement distributions are shown in Figure 7 when

By changing 1, 4, 16, 64, 256 and 1024 as the number *n*, and 1/4, 1/2, 1, 2, 4 and 8 as *m* for the *m*-th power of the amplitude of the complex amplitude for the weight of the averaging of the phase-difference, i.e., the displacement distributions, the errors are examined. The displacement distributions along the centerline of the flat plate are analyzed by phaseshifting digital holographic interferometry. The theoretical displacement distribution for the centerline is obtained by fitting a linear expression with the minimum error by the least square method to each obtained distribution. The standard deviations of the errors from the theoretical linear expressions are shown in Figure 8. The standard deviation decreases according to the number *n*, as same as in the case of Section 3.2. By changing *n* and *m*, the standard deviation is examined. In the same *n*, the standard deviation is the minimum when *m* is 2, that is, when the weight is proportional to the intensity of the complex amplitude. It is appropriate to adopt *m*=2 as the weight of averaging. The minimum standard deviation of

Number of windows *n* 1 4 16 64 256 1024 Standard deviation [nm] 16.39 4.02 1.95 1.09 0.78 0.67

Table 1. Relationship between number of windows and standard deviation of errors

reconstructed object images before deformation are shown in Figure 4(b). At each point on the reconstructed object, 16 complex amplitudes are obtained. Sixteen phase-difference values before and after deformation at each point are obtained. The average phasedifference is calculated by considering the weight of the average intensity before and after deformation. The result is shown in Figure 3(b) though the details of the displacement are described later. It provides more accurate phase-difference distribution than that shown in Figure 3(a).

In order to check the effect of the number of the windows on accuracy, the hologram is divided into *n* square areas by changing *n* (*n*=1, 4, 16, 64, 256 and 1024). That is, the original hologram with 960 x 960 pixels is windowed with *n* window functions whose small square area has value 1 and the remaining area has value 0 in the each window as in Figure 4(a). The average values of the *n* phase-difference values obtained from the windowed holograms are calculated. The results are shown later.

Fig. 4. Divided holograms and reconstructed object images

### **3.2 Effect of number of windowed holograms on accuracy**

The effect on accuracy of changing the number of windowed holograms is examined. By changing the number *n* into 1, 4, 16, 64, 256 and 1024, the displacement distributions are obtained. The displacement distributions along the centerline of the beam, shown as lines A and B in Figure 3 are shown in Figures 5(a) to 5(f). The theoretical displacement distribution for a cantilever is obtained by fitting a cubic curve with the minimum error by the least square method to each obtained distribution from the fixed point to the loading point. The standard deviations of the errors from the theoretical cubic curves are shown in Table 1 and Figure 6. The standard deviation decreases according to the number *n*, and the value becomes 670 pm when *n*=1024. It shows the proposed method provides very high-resolution measurement.

reconstructed object images before deformation are shown in Figure 4(b). At each point on the reconstructed object, 16 complex amplitudes are obtained. Sixteen phase-difference values before and after deformation at each point are obtained. The average phasedifference is calculated by considering the weight of the average intensity before and after deformation. The result is shown in Figure 3(b) though the details of the displacement are described later. It provides more accurate phase-difference distribution than that shown in

In order to check the effect of the number of the windows on accuracy, the hologram is divided into *n* square areas by changing *n* (*n*=1, 4, 16, 64, 256 and 1024). That is, the original hologram with 960 x 960 pixels is windowed with *n* window functions whose small square area has value 1 and the remaining area has value 0 in the each window as in Figure 4(a). The average values of the *n* phase-difference values obtained from the windowed

holograms are calculated. The results are shown later.

Fig. 4. Divided holograms and reconstructed object images

**3.2 Effect of number of windowed holograms on accuracy** 

The effect on accuracy of changing the number of windowed holograms is examined. By changing the number *n* into 1, 4, 16, 64, 256 and 1024, the displacement distributions are obtained. The displacement distributions along the centerline of the beam, shown as lines A and B in Figure 3 are shown in Figures 5(a) to 5(f). The theoretical displacement distribution for a cantilever is obtained by fitting a cubic curve with the minimum error by the least square method to each obtained distribution from the fixed point to the loading point. The standard deviations of the errors from the theoretical cubic curves are shown in Table 1 and Figure 6. The standard deviation decreases according to the number *n*, and the value becomes 670 pm when *n*=1024. It shows the proposed method provides very high-resolution

Figure 3(a).

measurement.

Fig. 5. Displacement distributions along lines A and B shown in Figure3
