# A Fading Suppression Method for \(\Phi\)-OTDR Systems Based on Multi-Domain Multiplexing
## 2 Multi-Domain Multiplexing Principle  
In a heterodyne \(\Phi\)-OTDR system, fluctuations of the RBS are randomly variable, and the electric field of the RBS is as follows:  
\[E_{RBS}(t,z)=\sqrt{2a_{s}(t,z)SWe^{-2 az}}E_{s}\exp\left[\omega_{s}\cdot\frac{2 z}{v_{g}}+\varphi_{s}(t,z)\right] \tag{1}\]  
where \(W\) is the fiber length corresponding to the pulse width \(\tau\) of the probe pulse light, \(a\) is the fiber attenuation coefficient, \(a_{s}(t,z)\) is the RBS scattering coefficient whose value varies with position, \(S\) is the capture factor of the RBS, and \(\omega_{s}=\omega_{0}+\Delta\omega\), where \(\omega_{0}\) is the frequency of the laser, \(\Delta\omega\) is the frequency shift, \(\varphi_{s}(t,z)\) is the phase, and \(v_{g}\) is the speed of light in the optical fiber. The above model is based on the complete matching of the polarization state of the RBS signal and that of the local light. In fact, due to the fiber birefringence effect, the polarization state of light evolves continuously along the length of the fiber, showing complex variation characteristics. To analyze the influence of the polarization state, the optical signal is decomposed into two mutually orthogonal components, the P-state and the S-state, through the Jones matrix:  
\[ \begin{pmatrix}E_{s1}(t)\\ E_{p1}(t)\end{pmatrix} =E_{s}\exp[j(\omega_{s}t+\varphi_{s})] \begin{pmatrix} \cos\theta_{s}\cos\varepsilon_{s}-j\sin\theta_{s}\sin\varepsilon_{s}\\ \sin\theta_{s}\cos\varepsilon_{s}+j\cos\theta_{s}\sin\varepsilon_{s}\end{pmatrix}  \tag{2}\]  
\[ \begin{pmatrix}E_{s2}(t)\\ E_{p2}(t)\end{pmatrix} =E_{l}\exp[j(\omega_{l}t+\varphi_{l})] \begin{pmatrix} \cos\theta_{l}\cos\varepsilon_{l}-j\sin\theta_{l}\sin\varepsilon_{l}\\ \sin\theta_{l}\cos\varepsilon_{l}+j\cos\theta_{l}\sin\varepsilon_{l}\end{pmatrix}  \tag{3}\]where \(\omega_{l}=\Delta\omega\), \(\varphi_{l}\) is the phase of local light, \(\theta_{s}\) and \(\theta_{l}\) are the azimuth angles of the RBS signal and the local light, respectively, and \(\varepsilon_{s}\) and \(\varepsilon_{l}\) are the ellipticity angles of the RBS signal and the local light, respectively. When the probe light and the local light pass through the analyzer with an angle of \(\theta\) to the \(x\)-axis, the output light intensity is as follows:  
\[I(t)=I_{0}(t)+E_{s}(t)E_{l}(t)\sqrt{a_{\theta}^{2}+b_{\theta}^{2}}\cos\left[ \varphi(t)+\arctan\left(\frac{b_{\theta}}{a_{\theta}}\right)\right] \tag{4}\]  
where \(\varphi(t)\) is \(\omega_{s}-\omega_{l}\), \(a_{\theta}\) and \(b_{\theta}\) can be written as follows:  
\[a_{\theta}=\cos(\theta_{s}-\theta_{l})\cos(\varepsilon_{s}-\varepsilon_{l})+ \cos(\theta_{s}+\theta_{l}-2\theta)\cos(\varepsilon_{s}+\varepsilon_{l}) \tag{5}\]  
\[b_{\theta}=\sin(\theta_{s}-\theta_{l})\sin(\varepsilon_{s}-\varepsilon_{l})+ \sin(\theta_{s}+\theta_{l}-2\theta)\sin(\varepsilon_{s}+\varepsilon_{l}) \tag{6}\]  
The visibility of the detection signal is as follows:  
\[V=\frac{\sqrt{a_{\theta}^{2}+b_{\theta}^{2}}}{2} \tag{7}\]  
When the probe pulse light propagates in the optical fiber, it is affected by the random changes in fiber birefringence. As a result, the polarization state of the RBS signal cannot maintain an optimal match with the local light. That is, \(\theta\) is different along the fiber link, which leads to different visibilities of the signals detected by the detector. In the worst scenario, when the polarization state of the RBS signal is completely orthogonal to that of the local light, the detector cannot output an effective signal. The fading noise can cause the IF signal to enter a low amplitude range, which causes fading and then results in low accuracy in the phase demodulation result.