## 2 Principles of DAS Fiber Optic
### Methodology  
Strain can significantly affect the optical properties of optical fibers [29]. The application of strain to optical fiber can induce changes in the refractive index profile and the mode field diameter, which can alter the optical properties of the fiber. This is due to the fact that strain can modify the distribution of dopants in the fiber core and cladding, which in turn can affect the refractive index profile.  
The measurement of the optical phase change that occurs as a result of the axial strain fluctuation in the optical fiber is how DAS is accomplished. The axial strain that the fiber experiences as a result of the action of the sound wave on the fiber causes a phase shift in the Rayleigh backscattering signal that is carried by the fiber. The photo-elastic effect [30] states that there is a linear relationship between the axial strain and the change in optical phase:  
\[\Delta\varphi=\beta\left[1-\frac{n^{2}}{2}(P_{12}+2P_{11})\right]\Delta L \tag{1}\]  
where \(\beta\) is the propagation constant of light, \(n\) is the refractive index of the core of optical fiber, \(P_{12}\) and \(P_{11}\) are tensor coefficients of the optical fiber, \(\Delta L\) is the change in fiber length from equation \(\Delta L=\varepsilon\kappa L\), \(\varepsilon\)s is the axial strain of the fiber, and \(L\) is the length of the fiber. Therefore, as long as the modifications in the phase difference between the points along the fiber of the probe light can be recovered, the modifications in axial strain can be known, and the quantitative perception of acoustic waves or vibrations can finally be detected [31]. In order for DAS to function properly, it is essential to establish distortion-free phase demodulation along the whole fiber.  
DAS is achieved by measuring the optical phase change caused by the axial strain variation of the optical fiber. When the sound wave acts on the fiber, the fiber will produce axial strain, which changes the phase of the Rayleigh backscattering signal in the fiber. According to the photo-elastic effect, there is a linear relationship between the axial strain and the optical phase change.  
DAS uses the backscattered light from an optical fiber to detect and analyze acoustic signals. The backscattered light is caused by changes in the refractive index of the fiber caused by acoustic waves. The strength of the backscattered light is related to the amplitude of the acoustic wave, and the phase of the backscattered light is related to the distance to the reflection.  
One of the key equations used in DAS is the equation for the backscattered light intensity [32], which is given by:  
\[I(z)=I(0)*(1-10^{(-\alpha*z)}) \tag{2}\]  
where \(I(z)\) is the backscattered light intensity at a distance \(z\), \(I(0)\) is the input light intensity, and \(\alpha\) is the fiber's attenuation coefficient.  
Another important equation in DAS is the phase shift equation, which describes the phase shift [33] of the backscattered light caused by a reflection. The phase shift is given by:  
\[\Delta\phi=\frac{2\pi nL}{\lambda} \tag{3}\]  
where \(\Delta\phi\) is the phase shift, \(n\) is the refractive index of the fiber, \(L\) is the distance to the reflection, and \(\lambda\) is the wavelength of the light.