# Phase-Coded and Noise-Based Brillouin Optical Correlation-Domain Analysis
## 2 Phase-Coded Brillouin Optical Correlation-Domain Analysis
### Transient Analysis of the Brillouin Signal Wave  
The construction of local Brillouin gain spectra typically requires the measurement of the output signal wave at several tens of different values of the frequency detuning \(\Omega\). The output signal power is recorded when it approaches a steady state after the SBS interaction has already taken place over several acoustic lifetimes. Alternatively, in 2016 [PERSON] and co-workers were able to show that the BFS can also be extracted from transient analysis of the instantaneous signal power during the stimulation process [24]. Figure 7: Double pulse-pair, phase-coded B-OCDA map of Brillouin gain as a function of position along a fiber under test and the frequency offset between the pump and signal waves [23]. A 43-m-long fiber was monitored with 2.7 cm resolution. All 1600 resolution points were addressed in only 11 scans of correlation peak positions [23]. Three 8-cm-wide heated regions are well identified in the measurements. Reproduced with permission from [23], Optical Society of America, 2016. Consider a single pump pulse that is launched at \(t=0\) into a uniform fiber under test of short length \(L\) and a uniform BFS \(\Omega_{B}\), and a continuous counter-propagating signal wave with a central optical frequency that is lower than that of the pump by a difference \(\Omega\). The temporally varying output power of the Brillouin-amplified signal wave may be expressed in terms of an instantaneous gain coefficient \(g(t)\): \(|A_{s}(z=0,t)|^{2}=|A_{s0}|^{2}\exp\Bigl{[}g(t,\Omega)\bigl{|}A_{p0}\bigr{|}^{2 }L\Bigr{]}\approx|A_{s0}|^{2}\Bigl{[}1+g(t,\Omega)\bigl{|}A_{p0}\bigr{|}^{2}L \Bigr{]}\)[24]. The functional form of \(g(t)\) is uniquely determined by the difference in frequencies \(|\Delta\Omega|\equiv|\Omega-\Omega_{B}|\)[24]:  
\[g\bigl{(}t^{\prime},\Omega\ \bigr{)}=g_{0}\frac{\Gamma_{B}^{2}}{\left(2\cdot \Delta\Omega\right)^{2}+\Gamma_{B}^{2}}\biggl{\{}1-\exp(-\Gamma_{B}t)\biggl{[} \cos(\Delta\Omega t\prime)-\frac{2\cdot\Delta\Omega}{\Gamma_{B}}\sin(\Delta \Omega t\prime)\biggr{]}\biggr{\}}. \tag{8}\]  
Here, \(g_{0}\) is the steady-state SBS gain coefficient for \(\Omega=\Omega_{B}\), \(t^{\prime}\equiv t-2L/v_{g}\), and the expression is valid for \(0\leq t\leq T_{p}\). The instantaneous gain coefficient approaches its known steady-state, frequency-dependent value when \(t\prime\gg\tau\). In the proposed transient analysis protocol, an arbitrary value of \(\Omega\) is chosen, and the instantaneous output power of the signal wave is recorded. The measured trace of \(g(t\prime)\) is then compared against a set of reference functions calculated through Equation (8) to identify the value of \(\Delta\Omega\) that best fits the experimental results. The procedure can therefore recover an unknown BFS with no frequency scanning. Sign ambiguity may be removed by repeating the experiment with two different random choices of \(\Omega\). Transient analysis of SBS requires that the interaction is confined to short fiber sections with a uniform BFS. These conditions are automatically met in correlation-domain analysis. Figure 8 presents an example of transient, phase-coded B-OCDA [24]. The output signal power is shown as a function of time in nine different correlation peaks, each 4 cm wide. One of the positions was in overlap with a local hot-spot. A different functional form of the transient output signal power is clearly observed at the heated segment. A standard phase-coded B-OCDA map of the SBS gain as a function of position and frequency is presented as well for comparison [24]. The BFS estimates of transient analysis are in quantitative agreement with those of traditional Brillouin sensing. An average error of 2.9 MHz between transient and steady-state analyses was found in a set of calibration B-OCDA experiments with 4 cm spatial resolution [24].