# Phase-Coded and Noise-Based Brillouin Optical Correlation-Domain Analysis
## 2 Phase-Coded Brillouin Optical Correlation-Domain Analysis  
### Principle of Operation  
Consider a Brillouin pump wave that is propagating in the positive \(\hat{z}\) direction along an optical fiber under test, and a counter-propagating Brillouin signal wave in the negative \(\hat{z}\) direction. The central optical frequency of the pump wave is higher than that of the signal by an offset \(\Omega\), which is close to the Brillouin frequency shift (BFS) of the fiber under test \(\Omega_{B}\). The two waves are drawn from a common optical source. The complex envelope of the pump wave at position \(z\) and time \(t\) may be expressed as follows:  
\[A_{p}(z,t\ )=A_{p0}u\bigg{(}t-\frac{z}{v_{g}}\bigg{)}. \tag{1}\]  
Here, \(A_{p0}\) is a constant amplitude, \(v_{g}\) is the group velocity of light in the fiber, and \(u(t)\) denotes a modulation function of the optical source with an average norm \(|u|^{2}\) of unity. The corresponding complex envelope of the counter-propagating signal wave is given by:  
\[A_{s}(z,t\ )=A_{s0}u\bigg{(}t-\frac{L-z}{v_{g}}\bigg{)}. \tag{2}\]  
In Equation (2), \(A_{s0}\) is a constant amplitude which may be different from that of the pump, and \(L\) is the length of the fiber under test. In most cases of interest for phase-coded and ASE-based B-OCDA, the Brillouin interactions are confined to short sections of fiber. As a first-order approximation, the changes in the magnitudes of the pump and signal waves due to the SBS interactions are neglected below. Linear losses along the fiber are omitted as well for simplicity. Subject to these conditions, the magnitude of the stimulated acoustic wave is given by the following expression [36]:  
\[Q(z,t\ )=jg_{1}A_{p0}A_{s0}^{*}\int\limits_{-\infty}^{t}\exp[\Gamma_{A}( \Omega,z)\cdot(t-t^{\prime})]u\bigg{(}t-\frac{z}{v_{g}}\bigg{)}u^{*}\bigg{[}t^ {\prime}-\frac{z}{v_{g}}+\theta(z)\bigg{]}\mathrm{d}t\prime. \tag{3}\]  
In Equation (3), \(g_{1}\) is an electro-strictive parameter of the fiber, \(\theta(z)\equiv\big{[}(2z-L)/v_{g}\big{]}\) is a position-dependent time lag, and \(\Gamma_{A}(\Omega,z)\equiv j\Big{\{}\Big{[}\Omega_{B}^{2}(z)-\Omega^{2}-j \Omega\Gamma_{B}\Big{]}/2\Omega\Big{\}}\) denotes a complex linewidth that is position-dependent through possible local changes in \(\Omega_{B}\). The complex linewidth reduces to half the inherent Brillouin linewidth: \(\Gamma_{A}(\Omega,z)=\frac{1}{2}\Gamma_{B}\), when \(\Omega=\Omega_{B}(z)\). The inherent linewidth \(\Gamma_{B}\) is on the order of \(2\pi\times 30\) MHz. Let us denote the auto-correlation function of the modulation waveform \(u(t)\) as \(C_{u}(\xi)\), where \(\xi\) is a delay variable. Equation (3) suggests that the magnitude of the stimulated acoustic wave at \(z\), and hence the strength of the local SBS coupling between pump and signal, is closely related with \(C_{u}[\theta(z)]\). This characteristic is at the basis of all variants of B-OCDA [13; 14; 18; 22; 31; 36]. Note, however,that the acoustic wave magnitude does not perfectly follow the auto-correlation function due to the exponential weighing window \(\exp[\Gamma_{A}(\Omega,z)\cdot(t-t)]\) that is associated with Brillouin stimulation. In phase-coded B-OCDA, the modulation function \(u(t)\) is chosen as a binary phase sequence, with symbol duration \(T\) and a period of \(N\) bits:  
\[u(t)\ =\sum_{n}a_{n}\text{rect}\bigg{(}\frac{t-nT}{T}\bigg{)}. \tag{4}\]  
Here, \(\text{rect}(\xi)=1\) if \(\left|\xi\right|\leq 0.5\) and equals zero otherwise, and \(a_{n}\) is the value of bit \(n\) in the sequence. A magnitude of unity \(\left|u(t)\right|^{2}=1\) is retained at all times. The bit duration \(T\) is taken to be considerably shorter than the Brillouin lifetime \(\tau\equiv 1/\Gamma_{B}\). In the first proposition of this concept, the values of \(a_{n}\) were those of a pseudo-random bit sequence (PRBS), in which each symbol assumes a value of either \(+1\) or \(-1\) with equal probabilities [18].