5 Noise Reduction in Amplitude-Fluctuation Electronic Speckle-Pattern Interferometry 33 Fig. 5.7 (a) Temporal fluctuation of interference intensity peak location; (b) Frequency dependence of (a) trend from one another. The 11 kHz case shows almost sinusoidal fluctuation with a frequency of approximately 1.25 Hz (0.8 s in the period); the 11.2 kHz shows similar trend with a slightly lower frequency; and the 11.3 kHz show much slower change with approximately the same amplitude as the 11.0 kHz and 11.2 kHz cases. There is no correlation between these frequencies and the driving frequencies. It is interpreted that the oscillatory behavior of these time trends is due to a random fluctuation of the phase ıo. Figure 5.7b plots the frequency dependences of these three time trends. There is no peak in the driving frequency, supporting the interpretation that the oscillatory behaviors seen in Fig. 5.7a is due to a random fluctuation of the phase ıo. The actual mechanism of the random fluctuation including the reason why the fluctuation is greater when the transducer is turned on is unclear at this time. One possibility is that when turned on the acoustic transducer raises the air temperature and that changes the phase via Eq. (5.6). Our temperature measurement indicates that the air temperature near the specimen can rise of the order of 0.1ıC within 1 s. The corresponding phase change according to Eq. (5.6) is a few percent of the wavelength. 5.3.2 Carrier Fringe Configuration With the carrier fringe configuration, relative optical path length change causes the dark fringes to shift in a direction perpendicular to the fringes. If such shifts are caused by the acoustic transducer, the digital camera is unable to track the shift because the frame rate is significantly lower than the shift frequency. Consequently, the fringes become blurry, as Fig. 5.4b shows. Mathematically, the fast oscillation decreases the value of J0(ı) in Eq. (5.7) as the Bessel function decreases monotonically from 1 to zero as ı increases from zero towards the first root at 2.4. If the relative optical path length changes slowly, the digital camera can capture the corresponding shift of the carrier fringes without compromising the fringe contrast. By tracking the location of the carrier fringes, we can identify the slow change in the relative optical path length. Mathematically, we can track the shift of cos˛x function in Eq. (5.7) along the x-axis. The top three graphs in Fig. 5.8 plot the location of the carrier fringes as a function of frame number (time). The fringe locations are expressed as deviation from the first frame in the unit of the relative value to the fringe-to-fringe distance (the spatial period). For example, a fringe shift of 0.08 at a certain frame means the fringes of that frame are shifted from the first frame by 8% of the spatial period. These plots correspond to Fig. 5.7a that represents the trend of relative optical path length change observed with the normal incident configuration. As expected, the behavior of the fluctuation is similar to Fig. 5.7a. The bottom three graphs in Fig. 5.8 are the Fourier spectrum of the spatial intensity variation of the fringe pattern for each frame number; the Fourier spectra of multiple frames out of the 100 frames are superposed in the same graph for each driving frequency. The Fourier spectra are evaluated along a horizontal line crossing the fringes perpendicularly (along row 250 of the fringe image shown in Fig. 5.4). The higher the fringe contrast, the higher the main peak of the Fourier spectrum. Three sets of graphs are selected because of the following features they show. The top graph of (a) exhibits a slow shift of the fringes. In 100 frames (approximately in 3.3 s), the fringes keep shifting in the same direction as much as 10% of the spatial period. However, the lower graph indicates that the Fourier spectrum is practically unchanged. In (b) the fringes remain almost at the same location. For the 3.3 s period, the shift is within a few percent. Yet, the Fourier spectra exhibit some scattered feature at the low frequency side of the spectral peak (between 0.01 and 0.02 1/pixel). The fringe shift in
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