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

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

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


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

Three-Dimensional Displacement and Strain Measurements

displacement distribution is shown in Figure 10(a).

Fig. 9. Reconstructed image obtained by Windowed PSDH

Fig. 10. Phase difference distribution obtained by Windowed PSDH

spatial resolution, as shown in Figure 1.

**interferometry** 

**processing time** 

by Windowed Phase-Shifting Digital Holographic Interferometry 37

As an experiment, the deflection of a cantilever is also measured using the optical system shown in Figure 1. As a specimen, a same size cantilever shown in Figure 2 is also analyzed. In this experiment, a fixed reference plate is set at 1.3 mm behind the cantilever to check

The reconstructed images (the intensity distribution, i.e., the amplitude squared) and the phase distributions are obtained from the holograms with 960 x 960 pixels before and after deformation. The reconstructed image before deformation is shown in Figure 9(a). The phase-difference distribution before and after deformation, that is, the out-of-plane

(a) *n* = 1 (b) *n* = 16 (c) *n* = 64 (d) *n* = 1024 (e) *n* = 57600

(a) *n* = 1 (b) *n* = 16 (c) *n* = 64 (d) *n* = 1024 (e) *n* = 57600

The effect on accuracy by changing the number *n* of windowed holograms into 1, 4, 16, 64, 256, 1024, 4096, 14400, 25600 and 57600 is examined. Some reconstructed images obtained from each one of the windowed holograms are shown in Figure 9. When the number of windows becomes larger, that is, the window size becomes smaller, the speckle size becomes larger. It is considered that the spatial resolution would be worse when the speckle size becomes larger. Figure 10 shows some of the results. The displacement distributions along the centerline of the beam are shown in Figure 11. The theoretical displacement distribution for a cantilever is obtained by fitting a cubic curve with the minimum error by means of 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 2 and Figure 11. The standard deviation changes according to the number *n*. Though the standard deviation when *n* =1 is 22 nm, it is 2.4 nm when *n* =16. The value becomes the minimum value of 680 pm when *n* =1024. It shows the proposed method

**4.2 Effect of number of windowed holograms on accuracy and computation** 

**4.1 Displacement measurement of cantilever by phase-shifting digital holographic** 

Fig. 6. Relationship between number of windows and standard deviation of errors

Fig. 7. Displacement distributions when flat plate is moved out of plane

Fig. 8. Standard deviations of errors when number of windows and power of the amplitude as weight of averaging are changed

### **4. Effect of window size on accuracy and spatial resolution in windowed phase-shifting digital holographic interferometry (Morimoto, 2008b)**

As shown in Figure 6, the accuracy becomes better according to the number of windows. However, the spatial resolution may be lower when the number of windows becomes larger. In this section, therefore, the effects of the number of windows, the window size, the displacement resolution and the spatial resolution on the accuracy are analyzed.

Fig. 6. Relationship between number of windows and standard deviation of errors

Fig. 7. Displacement distributions when flat plate is moved out of plane

as weight of averaging are changed

Fig. 8. Standard deviations of errors when number of windows and power of the amplitude

As shown in Figure 6, the accuracy becomes better according to the number of windows. However, the spatial resolution may be lower when the number of windows becomes larger. In this section, therefore, the effects of the number of windows, the window size, the

**4. Effect of window size on accuracy and spatial resolution in windowed** 

**phase-shifting digital holographic interferometry (Morimoto, 2008b)**

displacement resolution and the spatial resolution on the accuracy are analyzed.
