# Recent Advances in Brillouin Optical Correlation-Domain Reflectometry
## 2 Basic Principle of BOCDR
Note that a delay line is often inserted in the reference path so that the order of the correlation peak in the FUT may not be zero; the 0 th-order correlation peak cannot be scanned along the FUT by controlling\(f_{\mathrm{m}}\). Here, we give another explanation of the operating principle of BOCDR using Figure 2. When the laser frequency is sinusoidally modulated, the frequencies of the reference light and the Stokes light at the PD are expressed by the equations shown in Figure 2, where \(\varphi\) and \(\psi\) are phase factors. While \(\varphi\) is a constant, \(\psi\) changes according to where, in the FUT, the pump light is reflected. For example, when the pump light is reflected at Position A and \(\varphi=\psi+2\,\pi\,n\) (\(n\): integer), the frequency difference of the reference light and the Stokes light is always the same regardless of time, and so the beat signal can be detected. In contrast, when the pump light is reflected at Position B and \(\varphi\
eq\psi+2\,\pi\,n\), the frequency difference of the two light beams changes according to time, and so the beat signal cannot be observed. From the viewpoint of time averaging, we interpret that a correlation peak is standing at Position A. Now the modulation of the laser frequency is a periodical sinusoidal waveform, these correlation peaks appear periodically along the FUT, so only one correlation peak should be located within the range of the FUT to resolve the position properly, resulting in the limited measurement range. The position of the correlation peak can be scanned along the fiber by adjusting the modulation frequency \(f_{\mathrm{m}}\). Figure 1: Conceptual schematic of BOCDR. AC, alternating current; DC, direct current; ESA, electrical spectrum analyzer; FUT, fiber under test; PD, photodiode. The spatial resolution \(\Delta z\) and the measurement range \(d_{\rm m}\) (distance between the neighboring correlation peaks) of BOCDR are given by the same equations as those for BOCDA [25; 37]:  
\[\Delta z=\frac{v_{\rm g}\Delta v_{\rm B}}{2\pi f_{\rm m}\Delta f}, \tag{2}\]  
\[d_{\rm m}=\frac{v_{\rm g}}{2f_{\rm m}}, \tag{3}\]  
where \(v_{\rm g}\) is the group velocity of light, \(\Delta v_{\rm B}\) is the Brillouin gain bandwidth (\(\sim\)30 MHz) in optical fibers, and \(\Delta f\) is the modulation amplitude of the light source. Equation (2) was originally derived for BOCDA, so it is not evident that the spatial resolution of BOCDR is also given by this expression; refer to Reference [37] for the derivation of Equation (2) in BOCDR. According to Equations (2) and (3), it is evident that \(\Delta z\) and \(d_{\rm m}\) are in a trade-off relationship, and thus their ratio (=\(d_{\rm m}/\Delta z\)) is often used to evaluate the performance of BOCDR. Considering that \(\Delta f\) is limited to a half of the BFS, the resolution-to-range ratio of silica-fiber-based BOCDR is limited to 570, which can be extended by some special configurations (refer to the next section). Note that the SNR of BOCDR cannot be simply given as a function of system parameters, such as \(\Delta f,f_{\rm m}\), and fiber length. Finally, we compare the operating principle between BOCDA and BOCDR as shown in Figure 3. In both systems, the pump light is reflected everywhere in the FUT due to spontaneous Brillouin scattering. In order to resolve the position, in BOCDA, stimulated Brillouin scattering is strongly induced at one specific position in the FUT by injecting the frequency-modulated pump and probe light beams into each end of the FUT. In contrast, in BOCDR, the Stokes light due to spontaneous Brillouin scattering from one specific position in the FUT is selectively observed by the heterodyne detection with the frequency-modulated reference light. As a consequence, the Stokes light from the FUT in one-end-access BOCDR becomes much smaller than that in two-end-access BOCDA. Hence, another point of BOCDR is to utilize the heterodyne detection not only to resolve the position but also to enhance the beat signal. Figure 2: Operating principle of basic BOCDR.