# Phase-Coded and Noise-Based Brillouin Optical Correlation-Domain Analysis
## 2 Phase-Coded Brillouin Optical Correlation-Domain Analysis
### Optical Signal-to-Noise Ratio and the Choice of Codes  
When the frequency detuning \(\Omega\) matches the BFS at the correlation peak position, the fixed magnitude of the stimulated acoustic wave is given by \(|Q(L/2,t)|=2g_{1}\big{|}A_{p0}\big{|}|A_{s0}|\tau\)[21]. If a high-rate PRBS is used in phase-coded B-OCDA measurements, the variance of the acoustic wave magnitude at off-peak locations for the same \(\Omega\) may be estimated as \(\text{var}[Q(2\
eq L/2,t)]\approx g_{1}^{2}\big{|}A_{p0}\big{|}^{2}|A_{s0} \big{|}^{2}\tau T\)[21]. Off-peak SBS interactions accumulate over the entire length of the fiber under test and lead to undesirable fluctuations in the power of the output Brillouin signal wave. These variations can become the dominant noise mechanism in phase-coded B-OCDA and might mask out the Brillouin gain within the short correlation peak. The corresponding limitation on the optical signal-to-noise ratio (OSNR) is defined as the ratio between the coupling of power inside the correlation peak and the variance in output signal power due to the combined effect of all off-peak interactions. For PRBS phase modulation, the OSNR is estimated by a simple relation [21]:  
\[\text{OSNR}\approx\frac{2v_{g}\tau}{L}. \tag{5}\]  
Figure 1: Schematic illustration of a phase-coded Brillouin optical correlation domain setup. EDFA, erbium-doped fiber amplifier; Phase Mod., electro-optic phase modulator; SSB Mod., electro-optic single-sideband modulator.  
Figure 2: Phase-coded Brillouin optical correlation-domain analysis (B-OCDA) measurements [18]. (**a**) Brillouin gain mapping of a 40-m-long fiber with 1-cm resolution, corresponding to 4000 resolved points. A 1-cm-long section of the fiber was locally heated. (**b**) Magnified view of the Brillouin gain map in the vicinity of the heated section. Reproduced with permission from [18], [PERSON], 2012.  
The OSNR is inversely proportional to the length of the fiber under test, and it is independent of the analysis resolution. The OSNR is on the order of unity for a 2-m-long fiber. Consequently, the initial demonstrations of phase-coded B-OCDA over tens of meters of fiber required the averaging of collected traces over many repeating acquisitions and/or long integration durations at the detector [18].  
The OSNR can be improved if carefully synthesized phase codes are used instead of a PRBS. For example, Prof. [PERSON] of the University of Southern California (USC) developed a class of phase codes with cyclic auto-correlation values of exactly zero for all off-peak delays [38]. The employment of such perfect Golomb codes does not entirely eliminate off-peak SBS interactions, since the exponential windowing involved in Brillouin stimulation does not perfectly replicate the cyclic correlation operation [21; 39]. Nevertheless, both analysis and experiments show that the OSNR in Brillouin gain measurements may be improved by a factor of 3-4 when Golomb codes are used instead of a PRBS, depending on the sequence period \(NT\)[21; 39; 40]. Figure 3 shows an example of calculated \(|Q(z,t)|\) maps for a Golomb code and a PRBS of equal rates and periods [21]. Off-peak stimulation of the acoustic waves is significantly suppressed when the Golomb code is used. Perfect Golomb codes were also employed in demonstrations of all-optical variable delay of communication data and in microwave photonic filtering using Brillouin dynamic gratings [40; 41].