# Phase-Coded and Noise-Based Brillouin Optical Correlation-Domain Analysis
## 3 Brillouin Optical Correlation-Domain Analysis Based on Amplified Spontaneous Emission Noise  
### Principle of Operation  
In nearly all Brillouin analysis schemes, the pump and signal waves are drawn from a common, narrowband light source. To begin with, the magnitude of the cross-correlation function between the complex envelopes of the pump and signal waves is appreciable over a long range of relative delays. Correlation-domain analysis protocols deliberately degrade the initial coherence of the light source through broad-bandwidth frequency or phase modulation. However, the modulation bandwidth is restricted to several tens of GHz by technological limitations. The scaling of B-OCDA resolution below 1 mm of silica fiber requires bandwidths of 100 GHz and higher. A different approach was taken for the first time by [PERSON] and co-workers in 2014 [PERSON] and co-workers in 2014 [PERSON] _et al._ (2014). Rather than reduce the coherence of laser light, they used the polarized ASE of an erbium-doped fiber amplifier, which is highly incoherent to begin with, as the source of the Brillouin pump and signal [PERSON] _et al._ (2014). The modulation function \(u(t)\) in this case is no longer deterministic. It is fluctuating in both amplitude and phase, and may be described in terms of a stationary circular-Gaussian random process [PERSON] _et al._ (2014); [PERSON] _et al._ (2014). The magnitude of the stimulated acoustic wave, at a certain position and a given time, can be predicted only in the statistical sense. Following an initial buildup (\(t\gg\tau\)), the expectation value of \(Q(z,t)\) becomes independent of time [PERSON] _et al._ (2014):  
\[ \begin{split}\overline{Q(z,t)}=j_{1}A_{\varphi 0}A_{\mathrm{s}0}^{*}\int\limits_{- \infty}^{t}&\exp[\Gamma_{A}(\Omega,z)\cdot(t-t^{\prime})] \overline{u\left(t^{\prime}-\frac{z}{v_{x}}\right)u^{*}\left[t^{\prime}- \frac{z}{v_{x}}+\theta(z)\right]}dt^{\prime}\\ &=j\frac{\delta_{1}A_{\varphi 0}A_{\mathrm{s}0}^{*}}{\Gamma_{A}( \Omega z)}C_{\mathrm{s}}[\theta(z)].\end{split}  \tag{9}\]  
The SBS interaction is therefore effectively confined to a single correlation peak, with a spatial extent \(\Delta z\) that equals half the coherence length of the ASE source [PERSON] _et al._ (2014). That length can be controlled by  
Figure 10: Phase-coded B-OCDA spectra along a silica micro-wire as a function of position for multiple frequency ranges [PERSON] _et al._ (2014): (**a**) 5.5–5.7 GHz, (**b**) 8.5–9.0 GHz, (**c**) 9.5–10.0 GHz, (**d**) 10.0–10.5 GHz, (**e**) 10.5–11.0 GHz, and (**f**) 11.0–11.5 GHz. Position 0 cm is the center of the tapered fiber waist [PERSON] _et al._ (2014). Reproduced with permission from [PERSON] _et al._ (2014), Optical Society of America, 2018.  
spectral filtering of the ASE source. In principle, it might be as short as tens of microns. The spatial resolution of B-OCDA based on an ASE source can be scaled beyond those of systems that employ frequency modulation or phase coding. The spatial resolution of ASE-based B-OCDA is restricted in practice by poor SNRs. Two factors limit the SNR of the measurements. First, the relative power gain of the signal wave over ultra-short resolution points is very weak: on the order of 1 \(\times\) 10\({}^{-4}\) for 1 mm of standard fiber with 1 W of pump power. The weak gain would be the same for ASE-based and phase-coded setups of equal resolution. The second restriction, however, is specific to ASE-based realizations: the power of the output signal wave \(\left|A_{s}(z=0,t)\right|^{2}\) is inherently stochastic, due to amplitude fluctuations of the optical source, regardless of SBS [31; 32]. In contrast, the power of the signal wave in all other Brillouin sensors can be regarded as deterministic, and intensity noise of the laser source has little effect on the measurements. This noise mechanism sets a fundamental limitation on the measurement SNR, which cannot be overcome by simply raising the average signal power.