# High Resolution Brillouin Sensing of Micro-Scale Structures
## 4 Correlation-Based Brillouin Sensing  
BOCDA, first introduced by [PERSON] et al. [43; 89], relies on the correlation between the counter-propagating optical pump and probe waves. The pump and probe waves are driven from a low coherence source so that their interference in time is only partially correlated and can be defined by the auto-correlation function between the two counter-propagating waves. The length of the correlation peak determines the spatial resolution in this technique. Low-coherence sources involved so far in BOCDA include frequency-modulated [43] or random-phase-modulated laser source [44], amplified spontaneous emission (ASE) source [47], and random lasers [90]. As illustrated in Figure 3, all these techniques involve a source whose spectral bandwidth is defined either by frequency and phase modulation or a random process. This source is then split into pump and probe signals, which are frequency shifted by the BFS. The interference of the pump and probe signals in the medium results in the generation of the correlation peak in a certain positions along the waveguide, as illustrated in Figure 3. Different BOCDA approaches are discussed in the following. Figure 2: Overview of the spatial resolution evolution the time domain and correlation domain techniques. The first demonstration of BOCDA was based on frequency-modulated pump and probe waves to create a localized correlation peak for the local excitation and detection of SBS response along an optical fibre with a spatial resolution of 40 cm, as shown in Figure 4a[43]. The spatial resolution (\(\Delta z\)) and the measurement range (\(d_{m}\)) of this technique are determined by the following equations [45]:  
\[d_{m}=\frac{v_{g}}{2f_{m}}, \tag{3}\]  
\[\Delta z=\frac{v_{g}\Delta v_{B}}{2\pi f_{m}\Delta F}, \tag{4}\]  
where \(f_{m}\) is the modulation frequency, \(v_{g}\) is the group velocity, \(\Delta v_{B}\) is the Brillouin gain bandwidth, and \(\Delta F\) is the modulation amplitude, as shown in Figure 3. From Equations (3) and (4), for a fixed measurement range, the spatial resolution is inversely related to the modulation amplitude. However, increasing the modulation amplitude beyond \(\frac{v_{B}}{2}\) raises the problem of spectral overlap between the residual pump back-reflection--due to Rayleigh scattering and, in photonic chip experiments, a strong reflection from the chip facets--and the amplified probe, which cannot be separated using an optical filter. This issue was addressed by introducing two lock-in amplifiers (LIA) in the detection stage to distinguish electronically between the slow intensity-modulated pump and the amplified probe [92]. A distributed strain measurement with a spatial resolution of 1.6 mm based on this technique was demonstrated by [PERSON] et al. [92]. In this technique, the frequency modulation of the pump and probe results in the generation of periodic correlation peaks, which in turn limit the measurement range,  
Figure 3: Schematic of various sources for BOCDA measurement from the top to the bottom: frequency-modulated laser source with a modulation amplitude of \(\Delta F\)[43], random phase coded laser source [44] (Copyright ©2012 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim), amplified spontaneous emission (ASE) source with spectral width of \(\Delta f\)[47; 48] (adapted from [48] under a Creative Commons Attribution 4.0 International Public License), and a random laser based on a distributed feedback laser diode in an external feedback cavity [91] with a spectral width of \(\Delta f\) (adapted with permission from [91], OSA.). as expressed in Equation (3). In addition, from Equations (3) and (4), it becomes clear that there is a trade-off between the measurement range and the spatial resolution. BOCDA measurement based on random phase-coded pump and probe waves introduced by [PERSON] et al. [44] significantly extended the measurement range while achieving high spatial resolution.