## 2 State-of-the-Art
### Cliv  
When a laser beam passes through an acoustically active volume, the phase of the light wave changes as a function of the sound wave due to the sound-induced change of the refractive index. This is called the optoacoustic effect [9].  
Consider a simple LIV set-up, a plane wave sound source and a photo detector. The laser beam travels through a known constant distance \(l\). Let the plane sound wave excite the medium along \(l\). Over time the sound wave creates areas of higher density and areas of lower density in dependence of the current spatial and temporal sound pressure wave propagation. This oscillation is given by the frequency and amplitude of the sound wave. The changes in the light intensity are detected as a phase shift [9]. The intensity signal \(I(t)\) of the laser light can be described as:  
\[I(t)\sim\{1+V\cdot cos(\Delta\Theta(t))\}\quad\text{with }0\leq V\leq 1, \tag{1}\]  
depending on the modulation depth \(V\) and the phase shift \(\Delta\Theta\). The intensity signal oscillates with the instant frequency  
\[f_{l}(t)=\frac{1}{2\pi}\frac{\partial}{\partial t}\Delta\Theta=f_{B}+\Delta f (t), \tag{2}\]  
where \(f_{B}\) represents the carrier frequency and \(\Delta f(t)\) the frequency shift. A fluctuation of the sound field results in a fluctuation of the frequency shift \[\Delta f(t)=\frac{L(t)}{\lambda_{Laser}} \tag{3}\]  
in dependency of the laser wavelength \(\lambda_{Laser}\) and the time derivative \(\dot{L}=\mathrm{d}L/\mathrm{d}t\) of the optical path length  
\[L(t)=\int_{l}n(z,t)\mathrm{d}z, \tag{4}\]  
i.e., the line integral over the refractive index \(n\) along the laser beam [10]. The Gladstone-Dale theorem [30] provides the mathematical link between the refractive index \(n(z,t)\) and the density \(\rho\) of the medium:  
\[n-1=G\rho, \tag{5}\]  
with the material dependent Gladstone-Dale constant \(G\). Using Equations (2)-(5) and the assumption of an adiabatic change of state, the one-dimensional acousto-optical relationship between the instant frequency \(f_{l}(t)\) and the time derivative \(\dot{p}(z,t)=\mathrm{d}p(z,t)/\mathrm{d}t\) of the sound pressure along the optical path can be established:  
\[\Delta f_{l}(t)=\frac{1}{\lambda_{Laser}}\frac{n_{o}-1}{\kappa p_{0}n_{0}}\int _{l}\dot{p}(z,t)\mathrm{d}z, \tag{6}\]  
where \(\kappa\) is the adiabatic exponent, \(n_{0}\) and \(p_{0}\) are the refractive index and the pressure under reference conditions, respectively. This equation is valid for a projection line in the measurement volume. Thus, the integral sound pressure can be estimated from the detected intensity signals instant frequency \(f_{l}(t)\). In order to detect negative and positive frequency shifts, i.e., heterodyne detection, and to define a measurement range for \(f_{l}(t)\), the laser light has to be modulated with a carrier frequency \(f_{B}\)[7]. This can be achieved by using an acousto-optical modulator (AOM). Finally, the photo detector has to be fast enough to capture the high-frequency fluctuations \(f_{B}+\Delta f\).  
The extension of this technique is the matrix measurement of many projections with a camera. Figure 1 shows the camera-based laser interferometer. This enables a three-dimensional tomographic measurement of local sound pressure fields with a spatial resolution of 31.5 \(\upmu\)m at a sampling rate of 120 kHz.