**3.3. Frequency domain analysis with carrier fringe method**

The consistency of the spectrum peak observed in **Figure 8** indicates that it is possible to evaluate the oscillation amplitude quantitatively. Consider the relation between the input voltage to the acoustic transducer and the oscillation amplitude of the film surface. At a given oscillation frequency, the electric power of the transducer is proportional to the square of the applied (input) voltage, and the mechanical power associated with the elastic motion of the film is proportional to the square of the oscillation amplitude. Thus, the input voltage and the film surface oscillation amplitude are proportional to each other. Under the condition that the oscillation frequency is orders of magnitude higher than the CCD's frame rate, an increase in the oscillation amplitude reduces the fringe contrast (because the CCD cannot follow the fast shift of the fringes). Therefore, it is expected that the blurriness of the fringe pattern increases with the input voltage to the transducer. With the reduction of fringe contrast, the peak value of the Fourier spectrum decreases. Further, by fitting the reduction in the fringe contrast with the use of Eq. (6), it is possible to estimate the oscillation amplitude accurately.

**Figure 10** compares fringe contrasts as a function of the input voltage. As expected, the fringe becomes blurrier with the increase in the input voltage. **Figure 11** plots the Fourier spectra obtained at the six voltages (including 0 V) indicated in **Figure 10**. With the increase in the input voltage, the spectrum peak decreases.

**Figure 10.** Fringe contrast at various input voltage to transducer.

function of time. Thus, it can be said that the peak height difference comes from the oscillation amplitude *δ* in Eq. (6). Here, the change in the total intensity depends on the initial phase *δ*0 ; when *δ*0 is closer to *π*/2 its change is greater as the cosine function has the greatest slope around π/2 (**Figure 3**). Since the initial phase changes randomly, we have no control over the

**Figure 9** illustrates the advantage of the carrier fringe method more explicitly. This figure depicts two representative cases observed at driving frequency of 8.5 and 14.0 KHz. For each frequency, the first and last frames of 100 consecutive frames are shown. **Figure 9(a)** and **(b)** shows the fringe images, the upper graphs of **Figure 9(c)** and **(d)** are the spatial intensity profiles over 640 horizontal pixels, and the lower graphs of **Figure 9(c)** and **(d)** are the corresponding Fourier spectra. In the case of 8.5 KHz driving, the fringe pattern shifts approximately by 10 pxl over the 100 frames, but its profile is unchanged. In this case, the peak

In the case of 14.0 KHz driving, the fringe shift is similar or less than the 8.5 KHz case but the intensity profiles are changed; the fringe image for the first frame (the left image of **Figure 9(b)**) shows that the right bright fringe is stronger than the left bright fringe in intensity whereas the image of the last frame (the right image of **Figure 9(b)**) shows that the left bright fringe is stronger in intensity. The top graph of **Figure 9(d)** indicates the difference in the intensity patterns between the first and last frames more explicitly. In this case,

As indicated above, the fringe shifts are due to the random change in the initial phase. The change in the intensity profile is most likely caused by angular misalignment of the reference and signal beams. These observations indicate that while the carrier fringe method is not affected by fluctuation of the initial phase, the angular misalignment must be suppressed as

The consistency of the spectrum peak observed in **Figure 8** indicates that it is possible to evaluate the oscillation amplitude quantitatively. Consider the relation between the input voltage to the acoustic transducer and the oscillation amplitude of the film surface. At a given oscillation frequency, the electric power of the transducer is proportional to the square of the applied (input) voltage, and the mechanical power associated with the elastic motion of the film is proportional to the square of the oscillation amplitude. Thus, the input voltage and the film surface oscillation amplitude are proportional to each other. Under the condition that the oscillation frequency is orders of magnitude higher than the CCD's frame rate, an increase in the oscillation amplitude reduces the fringe contrast (because the CCD cannot follow the fast shift of the fringes). Therefore, it is expected that the blurriness of the fringe pattern increases with the input voltage to the transducer. With the reduction of fringe contrast, the peak value of the Fourier spectrum decreases. Further, by fitting the reduction in the fringe contrast with the use of Eq. (6), it is possible to estimate

the peak values of the Fourier spectrum for the first and last frames are different.

value of the Fourier spectrum is the same for the first and last frames.

**3.3. Frequency domain analysis with carrier fringe method**

total intensity.

74 Optical Interferometry

much as possible to reduce errors.

the oscillation amplitude accurately.

**Figure 11.** Fourier spectra for six input voltages shown in **Figure 10**.

Based on the proportionality between the input voltage and the oscillation amplitude, it is possible to estimate the oscillation amplitude by fitting the experimental relation between the input voltage and the spectrum peak value to *J*0(*δ*) (Eq. (5)). When the input voltage is zero hence *δ* = 0, *J*<sup>0</sup> takes the maximum value of unity. As the oscillation amplitude increases, *J*0(*δ*) decreases to the first root at *δ* = 2.4048. Thus, by evaluating the peak value relative to the case when the transducer is turned off, it is possible to estimate the oscillation amplitude *d* = *δ*/*k*. **Figure 12** plots the peak values shown in **Figure 11** relative to the highest value obtained with the null input voltage using a factor *a* in *J*0(*aV*) as the fitting parameter. Here *V* denotes the input voltage. The measured spectrum peak values fit to the Bessel function reasonably well.

**Figure 12.** Spectrum peak value as a function of input voltage to transducer.

**Figure 13.** Oscillation amplitude estimated for each voltage input.

Now that the Bessel function-like behavior of the spectrum peaks is confirmed, the value of *δ* can be estimated from the spectrum peak value obtained for each input voltage relative to the peak value obtained with the transducer turned off (zero input voltage). Subsequently, the oscillation amplitude can be found from *d* = *δ/k*. **Figure 13** plots the value of *d* found in this fashion.
