# Recent Progress in Fast Distributed Brillouin Optical Fiber Sensing
## 2 Operation Principle  
When a high power optical wave propagates along an ordinary standard single-mode fibers (SMF), a moving density grating (i.e., the acoustic wave) will be excited in the core of the SMF via an electrostriction effect through SBS effect, which will induce a Doppler shift (i.e., the BFS) for its backscattered Stokes wave and the BFS is given by References [8; 9; 27]. \[\
u_{B}=\frac{\Omega_{B}}{2\pi}\equiv\frac{2n_{\text{eff}}V_{A}}{\lambda_{p}} \tag{1}\]  
where \(n_{\text{eff}}\) is the effective core refractive index, \(V_{A}\) is the acoustic velocity, and \(\lambda_{p}\) is the wavelength of the pump wave in a vacuum. Generally, for a silica SMF, the BFS is \(\sim\)11 GHz [1; 5; 9] at the communication wavelength of 1550 nm. It is noted that the BFS has a good linear relationship [1; 11] with the strain and temperature of the fiber and is given by the equation below. \[\
u_{B}=\text{C}_{B}^{\epsilon}\Delta\epsilon+\text{C}_{B}^{T}\Delta T+\
u_{ B0} \tag{2}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Time-Domain** & **Frequency-Domain** & **Correlation-Domain** \\ \hline Reflectometer & Brillouin optical time domain reflectometer & Brillouin optical frequency domain reflectometry (BOTDR) [18; 19] & Brillouin optical correlation domain reflectometry (BOCDR) [6; 21] \\ \hline Analysis & Brillouin optical time domain analysis (BOTDA) [22; 23] & Brillouin optical frequency domain analysis (BOFDA) [24; 25] & Brillouin optical correlation domain analysis (BOCDA) [6; 26] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Several Brillouin-based optical fiber sensing schemes. where \(\
u_{B0}\) is the initial BFS of the fiber, \(\mathrm{C}_{B}^{\varepsilon}\) and \(\mathrm{C}_{B}^{\mathrm{T}}\) are the strain and temperature coefficients of the BFS, respectively, and \(\Delta\varepsilon\) and \(\Delta T\) are the variations of strain and temperature. Based on Equation (2), the distributed variations of the strain or temperature are demodulated by measuring the distributed BFS. To measure the distributed BFS, the typical BOTDA scheme is operated by the \"Pump-Probe\" method [5; 11; 28; 29], which is shown in Figure 1. A pulsed-pump is injected into the FUT at the position \(z=0\) while a CW-probe wave is injected into the FUT at the position \(z=L\), which is shown in Figure 1a. For a probe wave with different frequencies, their amplitudes will be amplified by obeying the BGS curve while their phases will be modulated obeying the Brillouin phase-shifted spectrum (BPSS), which is illustrated in Figure 1b. If the frequency of the pump pulse maintains constant, the distributed BGS will be constructed through segment-by-segment scanning of the frequency of the probe wave. A well-defined position distribution along the FUT corresponding to a round-trip time of the pump pulse can be achieved by identifying the start and end points of the amplified Brillouin signal. The spatial resolution is defined as \(c\tau/(2n_{\mathrm{eff}})\), which corresponds to the pump pulse's width \(\tau\). Then, the distributed BGS can be obtained through a frequency-sweeping process for the probe wave and is curve-fitted to get the distributed BFS. Based on the Maxwell equations representing the optical waves and the Navier-Stokes equation describing the acoustic variation in the FUT [30; 31], the BOTDA scheme can be depicted by the formula below. \[ \begin{array}{c}\frac{\partial E_{p}}{\partial z}+\frac{n}{c}\frac{\partial E _{p}}{\partial t}=ig_{o}\rho E_{S}-\frac{\alpha}{2}E_{P}\\ -\frac{\partial E_{S}}{\partial z}+\frac{\alpha}{c}\frac{\partial E_{S}}{ \partial t}=ig_{o}\rho^{*}E_{L}-\frac{\alpha}{2}E_{S}\\ \frac{\partial p}{\partial t}+\Gamma\rho=ig_{a}E_{P}E_{S}^{*}\end{array}  \tag{3}\]  
where \(E_{P}\) and \(E_{S}\) are the electric fields of the optical pump and probe fields, respectively. \(\rho\) is the material density along the FUT. \(*\) is the complex conjugate operator. \(i\) is the imaginary symbol.