# Phase-Coded and Noise-Based Brillouin Optical Correlation-Domain Analysis
## 2 Phase-Coded Brillouin Optical Correlation-Domain Analysis
### Principle of Operation
Given this modulation sequence, a correlation peak forms at the center of the fiber under test \(z=L/2\) with a fixed magnitude of the stimulated acoustic wave: \(Q(L/2,t)=j_{\Omega}A_{p0}A_{s0}^{*}/\Gamma_{A}(\Omega,z)\). The width of the correlation peak is \(\Delta z=\frac{1}{2}v_{g}T\). Periodic, higher-order correlation peaks appear at positions \(M\cdot N\cdot\Delta z\), where \(M\) is a positive or negative integer. Outside the correlation peaks, the magnitude of the stimulated acoustic waves is rapidly fluctuating, as the integrand in Equation (3) may assume positive or negative values. The expectation value of \(Q(z,t)\) equals zero for all times. Therefore, in principle, measurements of the SBS gain at the output end of the signal wave may retrieve the Brillouin gain spectrum at the position of the correlation peak only. Note, however, that even though off-peak Brillouin interactions are zero on average, their instantaneous magnitudes are non-zero with a finite variance. Off-peak Brillouin interactions may contribute significant noise to the measurements of output SBS gain, and they have to be taken into consideration. Phase-coded B-OCDA measurements of Brillouin gain spectra are unambiguous for fiber lengths \(L\) that are shorter than \(N\cdot\Delta z\). Since the repeating phase-modulation sequence can be chosen at any length, the measurement range of phase-coded B-OCDA may be arbitrarily long, with no effect on spatial resolution. In many realizations of the concept, the fiber paths leading the pump and signal waves into the measurement section of interest are deliberately imbalanced so that a high-order correlation peak (\(\left|M\right|\gg 1\)) is in overlap with the section of interest. With this choice, the position of the correlation peak can be conveniently scanned through small-scale variations in the bit duration \(T\). A map of Brillouin gain spectra as a function of position can be obtained by scanning the location of the correlation peak position, one resolution point at a time, and scanning the frequency \(\Omega\) at each position. Environmental drifts in the group delays along the pump and signal paths may be compensated for and stabilized based on chromatic dispersion [37]. A schematic illustration of a phase-coded B-OCDA measurement setup is provided in Figure 1. A single laser diode is typically used as the source of both pump and signal waves. The laser light passes through an electro-optic phase modulator, which is driven by the output voltage of an arbitrary waveform generator. The instrument is programmed to repeatedly generate the bit sequence of choice. The modulated waveform is then split into pump and signal paths. Pump light is upshifted in frequency through a single-sideband electro-optic modulator. The modulator is driven by sine-wave voltage from a microwave generator of frequency \(\Omega\approx\Omega_{B}\). The pump wave is then amplified by an erbium-doped fiber amplifier and launched into one end of a fiber section under test through a circulator. The signal path typically includes long fiber-optic delay as discussed above. An isolator is used to restrict the interaction between pump and signal to the fiber segment of interest. The Brillouin-amplified signal wave is routed through the circulator towards a photo-receiver. Figure 2 presents the first data of a phase-coded B-OCDA, reported in 2012 [18]. The SBS gain is shown as a function of position \(z\) and frequency detuning \(\Omega\)[18]. The measurement range was 40 m, and the measurement resolution was 1 cm. All 4000 resolution points were addressed in the analysis. A 1-cm-long hot-spot towards the end of the fiber under test was properly identified  
in the measurements [18]. The experimental uncertainty in the estimates of local BFS values was \(\pm 0.5\) MHz [18].