# Non-Local Effects in Brillouin Optical Time-Domain Analysis Sensors
## 2 Non-Local Effects Induced by Pump Pulse Depletion  
Two optical waves are used in a BOTDA sensor: a continuous wave (CW) probe and a counter-propagating pump pulse. When these two waves have an optical frequency detuning close to the BFS of the fiber, they interact exchanging energy via stimulated Brillouin scattering (SBS) [11]. In a gain-based BOTDA, a pump pulse at higher optical frequency transfers energy to the probe wave that undergoes amplification at the cost of the pump pulse being depleted. The probe wave only interacts with the pump pulse over a small length of the fiber, which is delimited by the pulse duration, which provides the spatial resolution of the sensor. Therefore, the probe gain, which is used for sensing, is rather small. On the other hand, the pump pulse is subjected to a continuous interaction with wavefronts of the probe wave as it crosses the fiber, losing a small percentage of energy at every step, so that the total accumulated depletion can be significant. The longer the fiber, the higher the pulse depletion. In a fiber with a constant BFS, the depletion of the pulse can be characterized by a dimensionless depletion factor which is given by [9]:  
\[d=1-exp\left(-\frac{g_{B}}{A_{eff}}P_{Si}L_{eff}\right) \tag{1}\]  
where \(g_{B}\) and \(A_{eff}\) are the Brillouin gain coefficient and the effective area of the fiber, respectively, \(P_{Si}\) is the probe wave power at the input of the fiber and \(L_{eff}=(1-exp(-\alpha L))/\alpha\), being \(\alpha\) the attenuation of the fiber and \(L\) its total length. This expression conveys the fact that the pulse depletion depends on the probe power injected into the fiber: the larger the probe wave, the larger the energy transferred to it to attain a given amplification and, hence, the larger the pump depletion. On the other hand, the depletion is shown not to depend on the power or duration of the pump pulse itself. Figure 1 schematically depicts the impairment brought by pulse depletion to the measurement. A worst-case scenario is assumed in which a uniform BFS fiber is followed by a small section with different BFS, where the gain spectrum is measured. The pump depletion depends on the frequency detuning between pump and probe optical waves along the fiber and this frequency dependence induces an additional transfer function being superimposed on the measured gain spectrum. This distorts the measured spectrum and introduces a bias in the obtained BFS that entails a systematic error in the measurement performed [9]. Notice that this measurement impairment is a non-local effect in the sense that the BFS measurement at a particular location in the fiber is affected by the Brillouin interaction between pump and probe at other locations. In fact, the shape of the transfer function associated with pump depletion at a particular location in the fiber depends on the BFS distribution along all previous locations. It has been found that, in the worst-case scenario, the BFS measurement error is maximum when the BFS difference between the long uniform BFS section and the short final section is around a quarter of the full width at half maximum of the gain spectrum [9]. Moreover, the error increases as the depletion factor increases. For a maximum 1-MHz error in the BFS measurement, the maximum tolerable depletion factor is \(d\leq 0.2\), which, for typical standard single-mode fiber, translates to an upper limit of \(-14\) dBm on the maximum probe power that can be injected into a long-range BOTDA sensing link [9]. One of the simplest ways to mitigate the appearance of pulse depletion is the use of two probe waves, equally separated in optical frequency from the pump [12]. This leads to two simultaneous Brillouin interactions (gain and loss) upon the pulse that compensate each other; hence, pulse depletion is suppressed. However, this is only true for first order approximation if the amplification and attenuation experienced by the lower and higher frequency probes, respectively, during their interaction with the pump, are neglected.