# Phase-Coded and Noise-Based Brillouin Optical Correlation-Domain Analysis
## 2 Phase-Coded Brillouin Optical Correlation-Domain Analysis
### The Overlay of Pump Pulses  
The initial phase-coded B-OCDA scheme suffers from two main drawbacks that are common to many correlation-based sensing protocols. The OSNR is degraded due to the accumulated contributions of multiple off-peak interactions, and the mapping of the entire fiber one point at a time may require prohibitively long experimental durations. While the former concern is partially addressed with the application of carefully synthesized phase codes, the OSNR over long fibers under test remains insufficient. A breakthrough in addressing both issues has been achieved with the overlaying of a single amplitude pulse on the phase-modulated pump wave. The input waveform of the pump wave in this case is given by [20; 22]:  
\[A_{p}(z,t\ )=A_{p0}\text{rect}\bigg{(}\frac{t-z/v_{g}}{T_{p}}\bigg{)}u\bigg{(}t -\frac{z}{v_{g}}\bigg{)}, \tag{6}\]  
Figure 3: Calculated magnitude of the stimulated acoustic wave density fluctuations (in units of kg/mm\({}^{3}\), see color bar) as a function of time and position within 1 m of fiber [21]. The pump and signal waves were jointly modulated by a high-rate binary phase sequence. A pseudo-random bit sequence (PRBS) was used in the left panel, and a perfect Golomb code was used in the right panel. Both codes were 63 bits long with bit duration of 100 ps. Stationary stimulation of the acoustic wave may be observed at a narrow correlation peak at the center of the fiber. The magnitude of residual off-peak acoustic waves is considerably reduced with the employment of the Golomb code. Reproduced with permission from [21], Optical Society of America, 2012.  
where \(T_{p}\) is the duration of a pulse centered at \(t=z/v_{g}\) and the modulation function \(u(t)\) is otherwise unchanged. The complex envelope of the Brillouin signal wave remains that of Equation (2). The pulse duration is chosen to be shorter than the phase code period: \(T_{p}<NT\), but sufficiently longer than the Brillouin lifetime \(\tau\)[22]. The length of the fiber under test \(L\) may be much longer than the separation \(N\cdot\Delta z\) between adjacent correlation peak locations. With these choices of parameters, acoustic waves are stimulated in multiple positions along the fiber. However, the SBS interaction in each peak is restricted to the duration of the pump pulse. The amplification of the signal wave at any given moment takes place at a single location at most (see Figure 4a, [22]). Local Brillouin gain spectra may be unambiguously resolved through the analysis of the output signal power in the time domain, similar to B-OTDA. The spatial resolution of the analysis, however, is determined by the short phase-code symbol duration \(T\) rather than by the longer pump pulse duration \(T_{p}\). Pulse amplitude modulation of the pump wave brings about two significant advantages. First, the Brillouin gain spectra along the entire fiber may be mapped with only \(N\) scans of correlation peaks positions, regardless of the fiber length. Figure 4b shows one of the earlier implementations of this principle [42]. B-OCDA was performed over a 1600-m-long fiber with a spatial resolution of 2 cm. All 80,000 resolution points were addressed using only 127 position scans. The figure shows a magnified view of the SBS gain map in the final 3 m of the fiber. A few-centimeters-long heated region is clearly identified in the measurements. Compared with the first proposition of phase-coded B-OCDA (Figure 1), the number of resolution points was already increased 20-fold. No less important is the effect of pulse modulation on the off-peak SBS interactions. Rather than accumulate over the entire length of the fiber under test, these interactions are confined to the spatial extent of the pump pulse only, which is typically on the order of few meters (see for example the calculations in Figure 4a). The OSNR of the output signal power measurements may be estimated as  
Figure 4: Phase-coded B-OCDA with overlaid pulse modulation of the pump wave.