# Non-Local Effects in Brillouin Optical Time-Domain Analysis Sensors
## 3 Mitigation of Non-Local Effects
### Mitigation of Second-Order Non-Local Effects  
One approach to overcome the distortion of the pump pulse spectrum is the use of an alternative scanning method in the dual-probe BOTDA sensor configuration [19]. The method is based on changing the conventional scanning of the Brillouin spectra to ensure the continuous overlapping of the gain and loss spectra induced by both probe waves on the pump. This is achieved by maintaining a fixed frequency difference between both probe wave sidebands equal to twice the BFS of the fiber. Recently, a more elaborate solution making use of this technique but with four probe waves was also presented [20]. These methods enable an increase of the probe power over the second-order non-local effects limit, at the cost of a somewhat increased complexity of the setup. Nevertheless, the maximum probe wave power injected into the fiber is still limited by its Brillouin threshold. We have also introduced a method to compensate second-order non-local effects that also enables dual-probe BOTDA setups to overcome the Brillouin threshold of the fiber. The basis of this technique is to introduce a modulation or \"dithering\" to the optical frequency of the probe waves, as it is schematically depicted in Figure 4, where the optical waves involved in the technique as well as the Brillouin interaction on the pump and probe waves are highlighted. As is depicted in Figure 4a, the optical frequency of the probe waves is modulated in the time domain following a saw-tooth shape [21]. Alternatively, sinusoidal or a triangular shape could be deployed, which have been shown to have a similar performance [14; 22]. Additionally, this frequency modulation (FM) is synchronized to the pump pulses so that a series of pulses experience the same optical frequency of the probe waves at any given location. Note that in this particular experiment both, the pump pulse wave and the probe wave, are generated with the same optical source, a 1550-nm distributed feedback laser. Altogether, this makes the effective optical frequency of the probe wave vary along the optical fiber following the applied FM modulation shape. Therefore, the pump pulse, as it travels along the fiber, experiences stimulated Brillouin scattering interaction with probe wavefronts that have a different frequency detuning. The net effect is completely analogous to having a virtual BFS distribution in the fiber, in the sense of having a fiber whose BFS profile has the same shape as the FM modulation imposed upon the probe [22]. As the pulse depletion is cumulative along the fiber, interacting with different frequencies within the Brillouin spectrum leads to a decreased depletion factor of the pulse. Furthermore, as highlighted in Figure 4a, the Brillouin interaction induced by both probe waves upon the optical pulse spreads over a large frequency range. Therefore, the net gain and loss spectra inducedby both probe waves broadens so that no distortion is introduced in the pump pulse spectrum and second-order non-local effects are suppressed [14]. In addition, the gain and loss spectra mutually compensate, so there is no depletion of the pulse. It is important to point out that, as it is depicted in Figure 4b, the FM of the probe wave is also used to perform the frequency scan of the Brillouin spectra, instead of doing it by a conventional frequency sweep. At each location of the fiber, the pump pulse is made to interact with different probe wave frequencies simply by changing the relative delay between the probe wave FM modulation and the pump pulse [14]. Note that the number of relative delay steps is given by the relation between the peak-to-peak frequency deviation of the FM, which defines the range of frequencies to scan, and the desired frequency step; hence, it is exactly the same as in a conventional BOTDA scheme. Thus, there is no penalty at all regarding measurement time in comparison with a conventional frequency sweep.