## 2 Theory
### Opto-Acoustic Method
#### 2.2.3 Proof of Principle  
Figure 3 (1) shows examples [17; 18] of fringe patterns and the Fourier spectrum of the intensity profile along a horizontal line around the vertical center. The left fringe pattern (a) is the case where the film was coated without a surface treatment; the middle (b) is where the film was coated after the silicon substrate was treated with the oxygen-plasma bombardment technique; and the right (c) is where the transducer was turned off. Careful observation will indicate that the fringe contrast increases from (a) to (c). The driving frequency was 11.5 kHz for all three cases. The observed fringe contrasts indicate that at this driving frequency, the surface treatment made the amplitude of the film-surface oscillation smaller than the untreated case. Naturally, the contrast is highest when the transducer was turned off. The left plots shown under the fringe images, Figure 3 (2), are the Fourier spectra of the optical intensity profile along the horizontal line at the vertical center as indicated in the leftmost fringe image. The unit of the horizontal axis of the spectrum is the inverse of the pixel numbers. The left group of three plots whose first spectrum peaks appear approximately at the spatial frequency of 0.0125 1/pxl result from the three fringe images (a) to (c). The peak values of the three spectra indicate the fringe contrast quantitatively; the spectrum peak decreases as the fringe contrast decreases, being consistent with Equation (5). The three plots shown on the right part of Figure 3 (2) are the Fourier spectra for the non-coated specimen (the bare silicon specimen) obtained in the same fashion as the left three plots. The three plots are the cases where the transducer oscillated one end-mirror only, oscillated the other end-mirror only and oscillated neither end-mirror, respectively (the use of the same plot marker as the left group indicates that the same mirror was oscillated). The peak values for the three cases are practically the same, indicating that the silicon surface oscillation amplitude was below the sensitivity of the interferometer. Figure 3: (1) Examples of fringe patterns; (2) Fourier spectra of intensity along the horizontal line indicated in Fringe Pattern (a); and (3) driving-voltage dependence of spectral peak value relative to the transducer-off (null driving voltage) case. In the frequency domain, the amplitude of the film-surface displacement is the product of the transfer functions of the layers between the transducer surface and the film surface and the Fourier transform of the transducer-surface displacement. The overall transfer function can be expressed as the product of the transfer functions of (a) the transducer surface to the rear surface of the silicon substrate, (b) the rear to the front surface of the silicon substrate, (c) the silicon to the film surface, (d) the rear to the front surface of the Ti film and (e) the transfer functions of the interface between the first and second metal layers (for the Ti-Au and Ti-Pt specimens only). Here: (a) is the transfer function of the glue (epoxy) used to bond the specimen to the transducer; (b) is the silicon substrate itself; (c) is the substrate-film interface; (d) is the film itself; and (e) is the product of the transfer functions of the metal-metal interface and the second metal film itself. Of these transfer functions, those of the silicon substrate and the metal film itself are considered unity due to the high stiffness as mentioned above. The metal-metal interface in (e) is also considered unity, as it is metallic bonding. As for the epoxy, a theoretical estimation indicates that the resonant frequency of the epoxy is over 30 MHz under the present condition and thereby can be assumed unity in the frequency range used in this experiment. Thus, of (a) through (e), all of the transfer functions, except (c), are considered unity under the driving frequency ranging from 5 kHz to 30 kHz.