# Phase-Coded and Noise-Based Brillouin Optical Correlation-Domain Analysis
## 2 Phase-Coded Brillouin Optical Correlation-Domain Analysis
### Combination of Amplitude and Phase Sequence Coding  
The acquisition duration of a phase-coded B-OCDA map is determined by the number of repeating acquisitions that must be averaged in the collection of each raw trace, by the number of position scans that are necessary to address the entire fiber under test, and by the number of frequency offsets used between pump and signal. As noted above, the overlay of an amplitude pulse on the continuously phase-coded pump wave reduces the first two of these values drastically. Yet, additional attempts were made to further reduce the necessary number of repeating averages, position scans, and frequency scans. The approaches taken towards these objectives are presented in the next three sub-sections. The number of repeating acquisitions is discussed first. When noise due to off-peak Brillouin interactions is sufficiently suppressed through amplitude pulse modulation and/or the application of Golomb codes, the OSNR is typically limited by additive noise sources associated with the measurement setup. Repeating pulses that are used for the averaging of collected traces must be separated in time by the two-way time-of-flight along the fiber under test to remove ambiguity. The OSNR can be improved, at least in principle, by using stronger pump pulses. However, the pump power is ultimately limited by the onset of competing nonlinear effects and depletion [45; 46]. Alternatively, the pump power may be spread in time among a large number \(k\) of weaker pulses, representing an amplitude coding sequence [19; 25]:  
\[A_{p}(z,t)=A_{p0}\sum_{k=1}^{K}b_{k}\text{rect}\bigg{(}\frac{t-kT_{p}-z/v_{g}}{ T_{p}}\bigg{)}u\bigg{(}t-\frac{z}{v_{g}}\bigg{)}. \tag{7}\]  
The value \(b_{K}\) for each symbol in the amplitude sequence may be 1 or 0 according to design. The acoustic wave at the location of each peak position is repeatedly stimulated by each pump  
Figure 5: Measured normalized Brillouin gain as a function of correlation peak position and frequency offset between the pump and signal waves [43]. A magnified view of a 130-mm-long section of fiber is presented. The section is located at the far end of a 17.5-km-long fiber under test. A 14-mm-wide hot-spot is clearly identified in the measurements. Reproduced with permission from [43], [PERSON], EPFL, 2015.  
pulse, and may decay between successive pulses. Each stimulated event is associated with local SBS amplification of the signal wave. Therefore, the Brillouin interaction at each peak location imprints a delayed replica of the pulse amplitude sequence on the instantaneous power of the signal wave [19; 25]. Since the pump pulses are closely spaced, and many correlation peaks take place along the fiber, a large number of such replicas are superimposed on the signal wave power, with large overlap in time. Nevertheless, the Brillouin amplification events at individual peak locations may be resolved through proper post-processing of the collected data [19; 25]. The protocol effectively compresses each replica of the extended code into a short and intense virtual pulse, with weak residual side-lobes. The analysis of compressed traces is therefore analogous to a hypothetic measurement with a single intense pulse, which cannot be transmitted in practice. The amplitude codes are designed to achieve a non-cyclic auto-correlation function with minimal off-peak values [47]. Figure 6a shows a Brillouin gain map of 8.8 km of fiber obtained with a spatial resolution of 2 cm [19]. Both pump and signal waves were phase-modulated by a perfect Golomb code with a period of 211 bits and symbol duration of 200 ps. In addition, the amplitude of the pump wave was modulated by a 9785-bits-long sequence with a pulse duration of 20 ns. All 440,000 resolution points are addressed in the analysis. This number remains the largest set of data points measured by a correlation-domain Brillouin sensor [19]. As many as 2000 correlation peaks are simultaneously and unambiguously monitored in each B-OCDA trace.