# Spontaneous Brillouin Scattering Spectrum and Coherent Brillouin Gain in Optical Fibers
## 3 Discussion  
The usual model of stimulated Brillouin scattering in optical fibers [10] was originally conceived as a set of coupled equations for slowly varying amplitudes of scalar plane waves, with spontaneous Brillouin scattering originating from noise fluctuations in time and along an axis \(z\)[13]. The coupled equations were later generalized to modal amplitudes for optical waves and for elastic waves, with transverse dimensions taken into account (see, e.g., Refs. [14; 18]). The experimental observation that the coherent Brillouin gain and spontaneous Brillouin scattering in optical fibers have similar spectral shapes is supported by the scalar model. Equations (12) and (13), however, describe different experimental situations. In the former case, typical of Brillouin sensing, a signal is coherently amplified or attenuated through the generation of elastic waves resulting of optical mixing. In the latter case, typical of stimulated Brillouin scattering at low pump powers, thermally activated acoustic phonons backscatter incident pump photons. Similar Lorentzian spectral shapes are found because of a property of the acoustic equation: the shape of the spectrum is decided by the left-hand side of (9), not by the deterministic or stochastic nature of the applied force.  
When the elastodynamic equation is introduced to describe the polarization of elastic waves and acoustic phonons, the picture remains similar if only isolated elastic modes are present. This picture applies typically to microstructured fibers, nanoscale waveguides, and microwires. We found that in this case it is only necessary to update the definition of the coefficients \(\omega_{0}\) and \(\Gamma\) in the spectral response of Equation (18). It remains that the \(z\)-axis evolution is not described in the elastodynamic Equation (14), since this is a 2D model. This is consistent, however, with a description of local small-signal amplification and spontaneous Brillouin scattering, in the non depleted pump approximation. Furthermore, as we have exemplified numerically, the stochastic excitation is spatially filtered by the elastodynamic equation to produce the isolated mode near its eigenfrequency.  
If there is no isolated mode but a continuum of bulk elastic waves, the above arguments do not apply to noise initiation of Brillouin scattering. This case corresponds for instance to usual optical fibers with a rather large cladding diameter and a central core ensuring smooth optical guidance but not necessarily elastic wave guidance by the fiber structure. The situation is intrinsically 3D and neither the scalar model nor the 2D elastodynamic equation describe it properly. We have suggested, based on a heuristic argument, that spatial filtering is provided by optical reading of the acoustic phonon density, resulting in the selection of only paraxial phonons. In this case, the canonical Lorentzian spectral shape is once again obtained. As a note, it has been predicted that the optoacoustic phonon spectrum for similar conditions could be non-symmetrically broadened towards higher frequencies, as a result of the selective excitation of the bulk elastic wave continuum [27; 30]. It remains that the same spatial filtering argument we used for spontaneous Brillouin scattering also applies to the coherent Brillouin gain for guided optical waves: not all generated phonons are collected by backward Brillouin light scattering, but only those that are compatible with phase matching.