34 S. Yoshida et al. Fig. 5.8 Three patterns of fringe shift and Fourier spectrum of spatial intensity profile (c) is the greatest among the three, and the corresponding Fourier spectra exhibit greater feature of the low frequency side scattering. The Fourier spectral peak also shows some changes among different frames. The above-mentioned features in the three graphs lead to the following observations. From comparison of Figs. 5.8a, b, we can say that a low frequency (slow) fringe shift does not necessarily compromise the shape of the Fourier spectrum. The spectrum peak height and position on the frequency axis are unchanged. Comparison of Fig. 5.8c with the other two indicates that the fringe shift or some accompanying effect can cause the lower frequency side scattering in the Fourier spectrum. Since the comparison of (a) and (b) indicates that a slow fringe shift does not always change the Fourier spectrum shape, it is likely that the reason for the change in the Fourier spectral shape exhibited by Fig. 5.8c is due to some accompany effect than the fringe shift itself. The fact that a slow fringe shift does not compromise the Fourier spectral shape is favorable for our analysis of the thin-film specimen. As we reported previously [3, 4], we use the spectral peak height to evaluate the blurriness of the carrier fringes. By comparing the peak height with the case when the acoustic oscillation is removed we evaluate the value of J0(ı) in Eq. (5.8), and in turn, estimate the oscillation amplitude d from the value of ı that corresponds to the evaluated J0(ı). Hence, it is essential that the spectral peak height is not influenced by the low frequency phase fluctuation. However, the slight change in the spectral peak height and the noisy low frequency side of the spectrum observed in Fig. 5.8c are sources of concern. It is worth while identifying what causes the features observed in the Fourier spectra shown in Fig. 5.8. To clarify the causes of the spectral features observed in Fig. 5.8, we made a simple numerical model and conducted simulations. The numerical model allows us to shift the fringes on the x-axis as a sinusoidal function of time for a given amplitude and frequency (i.e., it can shift the fringes right and left at a given frequency and amplitude), and change the spatial periodicity. The former is to simulate the change in the operation-point phase ı0, and the latter to simulate a change in the angular alignment of the interferometric arms. The optical intensity profile is in the form of a product of a Gaussian profile and a cosine function. The Gaussian profile is to simulate the laser beam’s intensity profile and the cosine function is to simulate the cos˛x term in Eq. (5.7). The model also allows us to vary the spot size of the Gaussian profile. This is to simulate the effect that the signal beam can have a smaller spot size depending on the reflectivity of the specimen surface at the spot where the laser beam is reflected. Figure 5.9 shows some results of the numerical simulation. Here (a) is when both the fringe shift and the spatial periodicity change is negligible; (b) is when both the fringe shift and spatial periodicity fluctuate; (c) is when only the fringe shift fluctuates in the same fashion as (a); and (d) is when only the spatial periodicity fluctuates in the same fashion as (a). The top plot of each set shows the superposition of optical intensity for 100 frames and the bottom plot shows the corresponding spatial Fourier spectrum. The amplitude of the fringe shift is approximately 15% of the spatial periodicity as the top of Fig. 5.9c indicates. Similarly, the amount of the variation in the spatial periodicity is seen in the top of Fig. 5.9d where the left side of the leftmost peak and the right side of the rightmost peak are expanded to the left and right, respectively; as the spatial periodicity increases the beam profiles expands on the spatial axis. It is clear that the translational shift of the fringes barely affects the spectral shape. This is understandable because purely translational movement of the fringes changes neither the fringe spacing (the spatial periodicity) nor the intensity profile; thus in the frequency domain, the spectrum is unchanged. On the other hand, the angular fluctuation affects the spectrum shape to a certain extent. The bottom plots of Fig. 5.9d shows the spectral peak shows some fluctuation at the high and frequency corner of the peak. This is also understandable as the angular
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