## 2 Theory
### Opto-Acoustic Method
#### 2.2.2 Principle of Operation  
The reduction in the fringe contrast in a given image frame due to the dithering motion of the fringes can be argued quantitatively as follows. The optical intensity on the screen placed behind the beam splitter is given by the following expression.  
\[I(t)=2I_{0}+2I_{0}\cos[k(l_{10}-l_{20})+kd\sin\omega t] \tag{1}\]  
Here, \(I_{0}\) is the intensity of the individual arm; \(k\) is the wave number of the laser beam; \(d\) is the amplitude of the film surface oscillation; and \(\omega\) is the driving angular frequency provided by the transducer. Since one of the end-mirrors is tilted, the relative phase difference on the screen can be resolved as a linear function of \(x\) with a constant \(\alpha\) as \(k(l_{10}-l_{20})=\alpha x\). Here, the \(x\)-axis is set perpendicular to the dark fringes. Thus, Equation (1) can be rewritten in the following form.  
\[I(t)=2I_{0}+2I_{0}[\cos(\alpha x)\cos(\delta\sin\omega t)-\sin(\alpha x)\sin( \delta\sin\omega t)] \tag{2}\]  
where \(\delta=kd\). Using Bessel functions of the first kind, Equation (2) can further be rewritten as follows.  
\[I(t)=2I_{0}[1+\cos(\alpha x)\{I_{0}(\delta)+2J_{2}(\delta)\cos(2\omega t)+ \cdots\}-\sin(\alpha x)\{2J_{1}(\delta)\sin(\omega t)+2J_{3}\sin(3\omega t) \cdots] \tag{3}\]  
Therefore, the signal of the digital camera is:  
\[S(\tau) \propto \int_{0}^{\tau}\{1+\cos(\alpha x)\{J_{0}(\delta)\}dt \tag{4}\] \[+ 2\cos(\alpha x)\sum_{m=1}^{N}J_{2m}(\delta)\int_{0}^{\tau}\cos( 2m\omega t)dt\] \[- 2\sin(\alpha x)\sum_{m=1}^{N}J_{2m-1}(\delta)\int_{0}^{\tau} \sin\{(2m-1)\omega t\}dt\]  
Since the frame rate of the digital camera is much lower than the driving frequency, the exposure time \(\tau\) is much longer than the period of the film surface oscillation. Consequently, the values of the oscillatory terms in Equation (4) \(\int_{0}^{\tau}\cos(2m\omega t)dt\) and \(\int_{0}^{\tau}\sin\{(2m-1)\omega t\}dt\) sweep through a number of periods as \(t\) is varied from zero to \(\tau\) in the integration. On the other hand, the values of the first two terms that do not contain an oscillatory term increase in proportion to \(\tau\). Therefore, the signal from the digital camera is dominated by the first non-oscillatory terms of Equation (4). Thus, the signal \(S(\tau)\) can be approximately expressed as follows (The concept is similar to the technique known as the amplitude-fluctuation electronic speckle pattern interferometry (AF-ESPI) [15; 16]. The difference is that AF-ESPI involves image subtraction from a reference image, and the present method extracts the oscillation amplitude from a given image frame of the interferometric fringe pattern without subtracting a reference image.):  
\[S(\tau)\propto\tau+J_{0}(\delta)\cos(\alpha x)a\tau \tag{5}\]  
Equation (5) indicates that the peak value of the spatial Fourier spectrum of \(S(\tau)\) (the value of the peak at the lowest spatial frequency) is proportional to \(J_{0}(\delta)\). When the film surface is still or the acoustic transducer is turned off, \(d=0\), hence \(\delta=0\). Since \(J_{0}(0)=1\) and \(J_{0}(\delta)\) monotonically decreases to zero in the range of \(0<\delta<2.4\), the peak value of the Fourier spectrum decreases as the oscillation amplitude increases in the corresponding range. The Fourier spectrum peak value of a given signal obtained with the film surface oscillation amplitude can be evaluated relative to the still film surface case (called the relative peak value) as \(J_{0}(\delta)/J_{0}(0)=J_{0}(\delta)\). By obtaining the relative peak value experimentally, _i.e._, by comparing the Fourier spectrum peak value of a given signal with that obtained with no film surface oscillation, it is possible to estimate \(\delta=kd\) and, hence, \(d\).