(b) Analysis with carrier fringe method

The fluctuation of the initial phase *δ*<sup>0</sup> can be evaluated with the carrier fringe method as well. In this case, a change in *δ*<sup>0</sup> causes a shift in the carrier fringe location. The top row of **Figure 8** shows the spatial shift of carrier fringes at three different driving frequencies. In conducting the measurement at the three driving frequencies, the voltage input to the transducer is adjusted so that the oscillation amplitude of the transducer surface is the same for all the frequencies. It is seen that the shift, i.e., the change in the reference phase *δ*0 is rather random, supporting the above speculation that convection of air in the interferometric arm causes the initial phase fluctuation.

The lower row of **Figure 8** is the Fourier spectrum of the spatial intensity variation of the fringe pattern observed at the respective driving frequencies. The first and main spectral peak observed at the spatial frequency of 0.02 (1/pxl) represents the carrier fringe periodicity of the light intensity; cos *αx* in Eq. (6) or the fringe pattern (see top of **Figure 9**). The Fourier spectrum is computed from the intensity profile over the horizontal span of 600 pxl (across a horizontal line near row 250 in the fringe image in **Figure 9**). Thus, the minimum frequency is 1/600 (1/ pxl). This means that the frequency of 0.02 (1/pxl) corresponds to the 0.02/(1/600) =12th harmonics, or the periodicity of 600/12=50 (pxl). The fringe patterns in **Figure 9** indicate this periodicity.

Application of Optical Interferometry for Characterization of Thin-Film Adhesion http://dx.doi.org/10.5772/66205 73

temperature easily rises over 0.1°C within 1 s after the transducer is turned on, and that the temperature fluctuates by ±0.1°C approximately every few minutes. In one set of measurement in which the transducer is turned on and off every 3 min, a total temperature rise of 0.4°C is recorded over a period of 30 min. It is suspected that air convection causes the temperature

It is informative to make a rough estimate of the phase change due to the above temperature change. The optical phase change due to the temperature dependence of the refractive index

Here *λ* is the wavelength, *l* is the path length, *n* is the refractive index of air and *dT* is the temperature change. The temperature coefficient ∂*n*/∂*T* of air is −0.87×10−6 (1/°C) [8]. The arm length of the interferometer used in this experiment is 10 (cm). The wavelength of the laser used in this study is 632.8 nm. So, the phase change due to a temperature change of ±0.1°C over the round trip in the interferometric arm is 20 (cm)/632.8 (nm) ×0.87×10−6×0.1=2.75% (of the period 2*π*). Accordingly, the phase error due to the air temperature change of 0.4°C

The issues of the deformed phase front and the initial phase fluctuation observed in Figures 6 and 7 make it difficult to use Eq. (5) with the total intensity method. In the next section, the carrier fringe method that greatly reduces the influence of the initial phase fluctuation is

The fluctuation of the initial phase *δ*<sup>0</sup> can be evaluated with the carrier fringe method as well. In this case, a change in *δ*<sup>0</sup> causes a shift in the carrier fringe location. The top row of **Figure 8** shows the spatial shift of carrier fringes at three different driving frequencies. In conducting the measurement at the three driving frequencies, the voltage input to the transducer is adjusted so that the oscillation amplitude of the transducer surface is the same for all the frequencies. It is seen that the shift, i.e., the change in the reference phase *δ*0 is rather random, supporting the above speculation that convection of air in the interferometric arm causes the

The lower row of **Figure 8** is the Fourier spectrum of the spatial intensity variation of the fringe pattern observed at the respective driving frequencies. The first and main spectral peak observed at the spatial frequency of 0.02 (1/pxl) represents the carrier fringe periodicity of the light intensity; cos *αx* in Eq. (6) or the fringe pattern (see top of **Figure 9**). The Fourier spectrum is computed from the intensity profile over the horizontal span of 600 pxl (across a horizontal line near row 250 in the fringe image in **Figure 9**). Thus, the minimum frequency is 1/600 (1/ pxl). This means that the frequency of 0.02 (1/pxl) corresponds to the 0.02/(1/600) =12th harmonics, or the periodicity of 600/12=50 (pxl). The fringe patterns in **Figure 9** indicate this

¶ <sup>=</sup> ¶ (8)

<sup>2</sup> *l n <sup>d</sup> dT T*

f p l

observed over 30 min is 2.75×4 = 11.0% of the period.

(b) Analysis with carrier fringe method

initial phase fluctuation.

periodicity.

fluctuation.

72 Optical Interferometry

discussed.

of air can be expressed as follows.

**Figure 8.** Fluctuation of the initial phase and corresponding change in Fourier spectrum observed with carrier fringe method.

**Figure 9.** Effect of fringe shift and intensity profile change on Fourier spectrum.

The lower row of **Figure 8** clearly illustrates the advantage of the carrier fringe method. The spectrum plotted for each driving frequency is superposition of 100 frames. Since the CCD frame rate is 30 fps, 100 frames correspond to approximately 3 s in time. Notice that the spectra observed at driving frequency 14.0 KHz show that the spectrum data are scattered in the frequency range left of the peak. This observation indicates that the random variation of the reference phase change is reflected in the low spatial frequency region of the spectrum; as cos*δ*0 changes, the total intensity detected by the CCD fluctuates, and that changes the low spatial frequency component. However, the peak height of the FFT spectrum is unaffected. This is because the peak value corresponds to the spatial dependence cos *αx*, not cos*δ*0 as a 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 total intensity.

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

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

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 much as possible to reduce errors.
