# The Influence of Laser Linewidth on the Brillouin Shift Frequency Accuracy of BOTDR
## 2 Numerical Simulation  
Figure 1 gives a classical schematic diagram of BOTDR with coherent heterodyne detection [26]. One branch of the seed laser is modulated by an optical modulator to generate probe pulse light, which is injected into the tested fiber through an optical circulator. The other one branch passing a polarization controller is injected into a photodetector to beat with the Brillouin backscattering. The output electronic signal is orderly down-converted by mixing with a frequency scanner, filtered by a band-pass filter (BPF), and acquired by a data acquisition (DAQ) digitalizer. The measurement error of BFS in FS-BOTDR is related with both the width of BGS denoted as \(W_{\text{BGS}}\) and the system SNR, which is given by Equation (1) [1]:  
\[\Delta\alpha=\frac{W_{\text{BGS}}}{\sqrt{2}(SNR)^{1/4}} \tag{1}\]  
hence, the values of \(W_{\text{BGS}}\) and \(SNR\) need to be both analyzed for figuring out the BFS accuracy of \(\Delta\alpha\), as specified below. When the continuous lightwave is injected into the tested fiber, the obtained BGS presents a Lorentzian shape, given by [27]:  
\[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]  
where \(w\) is the full-width at half maximum (FWHM) of BGS, \(f_{\text{B}}\) is the Brillouin frequency shift, and g\({}_{0}\) is the Brillouin gain coefficient, which is the BGS peak value when \(f=f_{\text{B}}\). When the continuous lightwave is modulated into the pulse light with a peak power of \(P_{0}\) and pulse width of \(\tau\), the power spectrum of the pulse probe light can be expressed as:  
\[P_{\text{P}}(f,\ f_{0})=P_{0}[\frac{\sin\pi(f-f_{0})\tau}{\pi(f-f_{0})}]^{2}, \tag{3}\]  
where \(f_{0}\) is the optical frequency of the probe light. Hence, the Brillouin backscattered-light power spectrum for pulse light can be calculated by [24]:  
\[Q_{\text{B}}(f)=\int_{-\infty}^{+\infty}P_{\text{P}}(f,\ f_{0})G_{\text{B}}(f) =\frac{g_{0}P_{0}}{\eta^{2}+1}\Big{\{}1+\frac{\eta^{2}-1-e^{-\tau\eta\tau}[( \eta^{2}-1)\cos\tau\eta\eta\tau+2\sigma\sin\eta\tau\eta\tau]}{\tau\upsilon( \eta^{2}+1)\pi}\Big{\}} \tag{4}\]  
where:  
\[\eta=\frac{f-f_{\text{B}}}{w/2} \tag{5}\]  
In the FS-BOTDR, the power spectrum, \(Q_{\text{B}}(f)\), is always measured by the frequency-scanning method. When the frequency scanner is tuned at a certain frequency point, the integrated power of every frequency segment passing the BPF is acquired by DAQ. Hence, as the frequency scanner is tuned step by step, the final measured spectrum after frequency-scanning process, denoted as \(H_{\text{B}}(f,\ N)\), can be given by:  
\[H_{\text{B}}(f,\ N)=\Big{\{}\ Q_{\text{B}}^{1}(f),\ Q_{\text{B}}^{2}(f),\ \dots\,Q_{\text{B}}^{\text{N}}(f)\ \Big{\}} \tag{6}\]  
Figure 1: Schematic diagram of frequency-scanning BOTDR (FS-BOTDR) with coherent heterodyne detection: AWG = Arbitrary Waveform Generator; BPF = Band-pass Filter; DAQ = Data Acquisition. \[Q_{\text{B}}^{i}(f)=\int_{f_{\text{s}}+(i-1)f_{\text{step}}-B/2}^{f_{\text{s}}+(i- 1)f_{\text{step}}+B/2}Q_{\text{B}}(f)df,\;i=1,\;2,\;\ldots,\;N. \tag{7}\]  
where \(f_{\text{s}}\) and \(f_{\text{step}}\) are, respectively, the start frequency and frequency step during the frequency-scanning process, \(N\) is the number of frequency points, and \(B\) is the bandwidth of the BPF. Based on Equation (4)\(\sim\) Equation (7), the broadening effect of BGS in FS-BOTDR is numerically simulated. The model parameters utilized for simulations are listed in Table A1 of Appendix A. The simulation results are shown in Figure 2. It can been clearly seen that the finally measured BGS width in FS-BOTDR, denoted as \(W_{\text{BGS}}\), is larger than both the original BGS width for continuous light and the BGS width for pulse light. Hence, the broadening effect of BGS in FS-BOTDR must be taken into account when the BFS accuracy is evaluated according to Equation (2). Figure 3a gives profiles of \(H_{\text{B}}\)(_f_, _N_) under different probe pulse widths.