# Recent Progress in Fast Distributed Brillouin Optical Fiber Sensing
## 4 Developments of the Fast BOTDA Schemes
### Slope-Assisted Fast BOTDA
#### 4.3.2 Enlarging the Monotonous Slope of the Sensing Curve  
Instead of the slope of the BGS, the slopes of the BPSS [43; 44] and RF phase-shift profile [45] are selected as the sensing curve resulting in the dynamic ranges of 40 MHz and 128 MHz, respectively. Recently, a dimensionless coefficient \(K\)[11] is defined as the expression below. \[K(\
u_{S},z)=\frac{\varphi_{SBS}(\
u_{S},z)}{g_{SBS}(\
u_{S},z)} \tag{7}\]  
where the coefficient \(K\) spectrum (KS) is a function of the frequency detuning \(\Delta\
u\). The KS is insensitive to the power fluctuation of the pump pulse so as to improve the SNR. Especially for the situation of a continuous pump wave, Equation (7) can be simplified as the formula below. \[K(\
u_{S},z)=\frac{2\Delta\
u}{\Delta\
u_{B}} \tag{8}\]  
where the KS is a straight line with the slope value of \(2/(\Delta\
u_{B})\). The measured spectra of the BGS, the BPSS, and the KS are obtained as shown in Figure 11 for a pump pulse of 20 ns. By Lorentzian-fitting (blue line) the BGS, its BFS and linewidth are computed to be 10.694 GHz and 48 MHz, respectively. The available frequency range of the slopes of the BGS and BPSS for dynamic sensing is narrower than the linewidth of the BGS while that of the monotonous slope of the KS is beyond 200 MHz, which is wider than fourfold of the linewidth of the BGS. Moreover, the SNR of the KS is better than that of the BGS and the BPSS. Figure 10: The dynamic measurement of the multi-slope-assisted BOTDA [42] for the frequency intervals of **(a)** 20 MHz, **(b)** 40 MHz, **(c)** 60 MHz, and **(d)** 80 MHz. A single-slope-assisted BOTDA scheme is proposed using KS as the sensing curve, which is shown in Figure 12a. The probe wave and a reference wave (Ref) are injected into the sensing fiber while the pump pulse is injected in the opposite direction. The beating Brillouin signal between the probe wave and the Ref wave can be detected via the heterodyne detection and demodulated based on the IQ demodulation algorithm [46], which results in the BGS value and the BPSS value. Then, the K value can be calculated based on Equation (7). Generally, working at the K = 0 point, the dynamic BFS is computed by substituting the K value into the expression of the KS sensing curve. Then, to further enlarge the dynamic range, a multi-slope-assisted BOTDA scheme is also assembled through the combination of the KS and the OFA techniques, which is shown in Figure 12b,c. In our scheme, the pump pulses are frequency-modulated using the OFA technique so that each pump pulse corresponds to one K value. Subsequently, the K value closest to zero is selected and substituted into the expression of the KS sensing curve to calculate the BFS of the FUT. The results of dynamic strain measurements using the multi-slope-assisted BOTDA are shown in Figure 13, which are demodulated based on two methods i.e., the traditional curve-fitting method and the hybrid KS + OFA method, respectively. The reference time traces are demodulated through curve-fitting the BGS and the KS to calculate the BFS with a frequency number of \(N_{f}=101\) and a frequency interval of 4 MHz. As shown in Figure 13a-d, the time traces (red line) are demodulated  
Figure 11: The experimental results for a 20-ns pump pulse [11]. BGS: blue line and dot. BPSS: magenta line and dot. KS: red line and dot. The dots are the experimental data and the lines are their fitting curves. Figure 12: The operation principle of **(a)** the single-slope and **(b)** the multi-slope assisted BOTDA. **(c)** The time relationship between the pump pulse and the probe wave [11]. by searching the minimum \(\mid\)K\(\mid\) to compute the BFS for the frequency interval of 80 MHz, 120 MHz, 160 MHz, and 180 MHz, and for the frequency number \(N_{f}\) of 6, 4, 3, and 3, respectively.