# A High-Resolution Ultrasonic Ranging System Using Laser Sensing and a Cross-Correlation Method
## 2 Theory  
### Acousto-Optic Diffraction  
In this section, we describe the relationship between the continuum mechanics of ultrasound and the optical diffraction problem. Ultrasonic waves can be seen as a kind of mechanical wave. In an isotropic medium, such as water, the index of refraction changes with pressure, which is induced by the ultrasound. The relationship between the refractive index and density of the medium can be written as follows:  
\[n(\textbf{r},t)=\sqrt{1+\chi\rho(\textbf{r},t)/\rho_{0}}, \tag{1}\]  
where \(n(\textbf{r},t)\) is the refractive index of the medium at location **r** at time \(t\), \(\rho(\textbf{r},t)\) is the corresponding material density, \(\chi\) is the intrinsic susceptibility of the medium, and \(\rho_{0}\) is the ambient density of the medium. Using Taylor series expansion algebraic transformation, Equation (1) can be written as follows:  
\[n(\textbf{r},t)=n_{0}+\frac{n_{0}^{2}-1}{2\rho_{0}n_{0}}[\rho(\textbf{r},t)- \rho_{0}], \tag{2}\]  
where \(n_{0}\) represents the ambient refractive index. We assume that the sound pressure is small and that the density variation can be related to the pressure variation using the piezo-optic coefficient. As such, the relationship between the index of refraction and the sound pressure is directly proportional, as follows:  
\[n(\textbf{r},t)=n_{0}+\frac{\partial n}{\partial p}p(\textbf{r},t). \tag{3}\]  
The \(\partial n/\partial p\) is the piezo-optic coefficient, which can be expressed as follows:  
\[\frac{\partial n}{\partial p}=\frac{n_{0}^{2}-1}{2n_{0}\rho_{0}c_{0}^{2}}, \tag{4}\]  
where \(c_{0}\) is the sound velocity. Here, we establish the relationship between the sound pressure and the refractive index. The ultrasound can be a continuous wave or pulsing wave.  
When a laser beam intersects the ultrasound field, it is diffracted and split into several beams that propagate in different directions. [PERSON] and [PERSON] [31] presented the analytical solution of acousto-optic diffraction, which was known as Raman-Nath diffraction. [PERSON] [36] explained the weak scattering model for the interaction of the optical and acoustic pulses in the Raman-Nath regime. The criterion for the Raman-Nath diffraction regime is defined by the Klein-Cook parameter \(Q\)[32] as follows:  
\[Q=\frac{2\pi\lambda_{L}L}{n_{0}\lambda_{S}^{2}}, \tag{5}\]  
where \(\lambda_{L}\) is the wavelength of light, \(\lambda_{S}\) is the wavelength of ultrasound, and \(L\) represents the length of the interaction volume. When \(Q<1\), Raman-Nath diffraction occurs.  
When an ultrasonic pulse propagates through the medium, it modulates the amplitude and frequency of the laser. The \(m\)th order of diffracted light can be expressed as follows:  
\[E_{m}=A\cdot J_{m}(\varphi(t))e^{i(\omega+m\Omega)t}, \tag{6}\]  
where \(A\) is a constant, \(\omega\) is the angular frequency of light, and \(\Omega\) is the angular frequency of ultrasound. \(J_{m}\) is the Bessel function of the \(m\)th order. The frequency of the diffracted light corresponds to the frequency of the ultrasound and the diffracted order. The phase delay induced by the ultrasound \(\varphi(t)\) can be expressed as follows:  
\[\varphi(t)=k_{L}\frac{\partial n}{\partial p}\int_{0}^{L}p(y,t)dy, \tag{7}\]  
where \(k_{\mathrm{L}}\) is the wavenumber of the light, \(\partial n/\partial p\) is the piezo-optic coefficient of the medium, and \(p(y,t)\) is the pressure of ultrasound along the direction of light. The direction of light propagation is seen as the \(y\)-axis, which is the path of integration. The principle of acousto-optic diffraction is shown in Figure 1. At this point, we present the relationship between diffracted light and ultrasound. We can calculate an ultrasonic pulse through the diffracted light. As the width of laser beam is only 0.8 mm, it provides a noncontact and high spatial resolution means for ultrasound detection.