Going beyond 1 000 000 resolved points in a Brillouin distributed fiber sensor: theoretical analysis and experimental demonstration Supporting material

where g1 is the electrostrictive coupling constant, τ is the acoustic phonon lifetime, and Ap, As are the complex amplitudes of counterpropagating pump and signal waves respectively. It can be observed that this equation represents convolution of the counterpropagating optical waves. For pump and probe modulated with a pseudo-random bit sequence (PRBS), this convolution leads to a triangular shape, with the bottomwidth given by the bit duration – for instance, 28mm for the 140 ps bit duration used in our experiment. This provides a full-width at half-maximum (FWHM) of 14mm, which is typically considered to be the spatial resolution for this type of sensors. However, it is important to consider that in real systems the transition between two phase values is far from being instantaneous; for instance, the manufacturer of the PRBS generator used in our experiment gives a typical 20%–80% transition time of 23 ps which is not negligible for a bit duration of 140 ps. Figure S1a shows how the non-instantaneous phase transition affects the correlation peak shape (solid blue line) in comparison to the ideal case (red dashed line). Interestingly, the FWHM decreases from 14mm down to 9mm, while the bottom width broadens from 28mm up to 40mm. Another effect given by the non-instantaneous phase transition is the generation of stable weak gratings along the fiber. Figure S1b shows three consecutive gratings separated by 14mm being generated outside the correlation peak (note that the vertical axis in Figure S1b is normalized to the maximum amplitude of the correlation peak shown in Figure S1a). The change in the correlation peak shape and the presence of weak permanent gratings can be also explained by examining the spectrumof the phase-modulated light. Figure S2 shows the predicted spectrum for both perfect and imperfect modulation of phases. First, it can be seen that non-instantaneous transition between two phase values suppresses higher-order components of the spectrum, effectively making it narrower. This explains the broadening of the correlation peak. Additionally, strong lines appear at frequencies equal to the integer number of the bitrate – where the spectrum of the perfectly-modulated light