# The Influence of Laser Linewidth on the Brillouin Shift Frequency Accuracy of BOTDR
## 2 Numerical Simulation
The peak power is normalized by that of the spectrum profile obtained when the pulse width is 12 ns. The spectrum width and relative power versus the probe pulse width are plotted in Figure 3b. From Figure 3a, it is obvious that the BGS spectrum becomes narrower and higher as the pulse width increases. The spectrum width of \(H_{\text{B}}\)(_f_, _N_), denoted as \(W_{\text{BGS}}\), decreases from 107.9 MHz to 87.15 MHz. The relative peak power of \(H_{\text{B}}\)(_f_, _N_), denoted as \(P_{\text{rp}}\), increases as the pulse width is increased from 12 ns to 52 ns. Figure 2: Numerical simulation of the broadening effect of Brillouin gain spectrum (BGS) in FS-BOTDR: \(G_{\text{B}}\)(_f_) is the original BGS for the continuous light; \(Q_{\text{B}}\)(_f_) is the BGS for the pulse light with a width of 22 ns; \(H_{\text{B}}\)(_f_, _N_) is the final measured BGS through the frequency-scanning process when the width of the probe pulse is 22 ns. Following simulation analysis of the BGS broadening, the relationship between the system SNR and the laser linewidth should be emphatically discussed. The photocurrent in coherent detection can be expressed as Equation (8) neglecting the polarization mismatch [28]:  
\[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]0  
where \(G\) and \(R\) are, respectively, the gain and responsivity of the detector, \(P_{\text{SpB}}(t)\) and \(P_{\text{Rf}}\) are the power of the spontaneous Brillouin backscattering and the reference light, respectively, \(f_{\text{B}}\) is the BFS, and \(\Delta\varphi\) is the phase difference. Because the coherent detection is highly phase sensitive, the phase fluctuation caused by \(\Delta\varphi\) will greatly decrease the SNR and even lead to the central frequency jitter [25]. It has been proven that the phase fluctuation in the coherent detection can be viewed as a nonstationary random process and follows normal distribution with the mean value of \(\mu\) and the variance of \(\sigma^{2}\), given by Equation (9) [29]:  
\[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]1  
where \(n\) is the refractive index of optical fiber, \(c\) is the light speed in vacuum, \(\Delta f\) is the laser linewidth, and \(\Delta L\) is the optical path difference between the probe beam and the reference beam. Figure 4 gives a histogram of \(\Delta\varphi\) utilized in the simulation when \(\Delta f=1.05\) MHz, \(\Delta L=20\) km. \[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]2 Histogram of \(\Delta\varphi\) utilized in the simulation when \(\Delta f=1.05\) MHz and \(\Delta L=20\) km. \[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]3 Numerical simulation results: (**a**) Profiles of \(H_{\text{B}}(f,N)\) under different probe pulse widths; (**b**) spectrum width and relative power versus probe pulse width. To analyze the relationship between SNR and \(\Delta\varphi\) succinctly, we treated phase fluctuation as a major noise contribution, regardless of the thermal noise and shot noise. The power of useful signal can be expressed as:  
\[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]4  
The power of noise can be expressed as:  
\[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]5  
where \(\Delta t\) is the sampling interval of \(i(t,\Delta\varphi)\), and \(N_{\text{t}}\)=\(\tau/\Delta t\) is the number of samples. Hence, the SNR can be given by Equation (12):  
\[G_{\text{B}}(f)=g_{0}\frac{(w/2)^{2}}{\left(f-f_{\text{B}}\right)^{2}+\left(w /2\right)^{2}} \tag{2}\]6  
where \(P_{\text{rp}}(\tau)\) is the relative peak power of the measured BGS closely related with the width of the probe pulse. The relative SNR versus laser linewidth under different pulse widths was numerically simulated and is plotted in Figure 5a according to Equation (12).