# Spontaneous Brillouin Scattering Spectrum and Coherent Brillouin Gain in Optical Fibers
## 2 Results
### Spatial Noise Filtering
#### 2.4.1 Discrete Spectrum of Guided Phonons  
In case the spectrum of the elastodynamic equation is composed of isolated frequencies, the solution of the elastodynamic equation produces a spatial noise filtering of the stochastic response that restores deterministic phase fronts at a particular position along the fiber axis. In order to illustrate this point we consider the following numerical experiment. We solve the elastodynamic Equation (14) for a silica microwire with a diameter of 800 nm (Figures 2 and 3). As a note, experimental results for sub-micron diameter microwires have become available recently [PERSON] et al. (2000); [PERSON] et al. (2000); [PERSON] et al. (2000). In Figure 2, we apply a Gaussian random force field to the whole microwire cross-section. In the numerical computation, only one noise realization is considered. The axial wavenumber \(q=4\pi n/\lambda\) is imposed for an optical wavelength in a vacuum \(\lambda=1550\) nm, with \(n=1.376\) the effective index for the fundamental optical mode. The elastodynamic response, as represented by the total energy of the solution, \(E(\omega)\), is plotted as a function of frequency. The response is composed of a superposition of Lorentzians, each centered on the frequency of a particular elastic mode of the microwire. At the maximum of each peak, the displacement field is proportional to the elastic mode distribution at that frequency. For comparison, we consider in Figure 3 the case of an incident Gaussian distributed optical wave. Given the circular symmetry of the source, only a few peaks appear in the elastodynamic response. Elastic modal shapes are shown for the three main peaks. These three modes were already present in the stochastic response of Figure 2, though with quite different relative levels of response. Significantly, it is seen that the random spatial fluctuations of the stochastic force are filtered in the elastodynamic response, since the displacement distributions are rather similar in Figures 2 and 3 at resonance frequencies. Figure 3: Electrostriction response of a silica microwire. **(a)** The cross-section of the microwire is circular and the diameter is \(d=800\) nm. The incident optical field is a Gaussian wave (normalized square modulus is shown). **(b)** The electrostriction response, computed as the total energy of the solution of the elastodynamic equation, is shown as a function of frequency. Examples of the solution field (norm of the total displacement) are shown for the three main peaks in the response. Figure 2: Stochastic response of a silica microwire. **(a)** The cross-section of the microwire is circular and the diameter is \(d=800\) nm. A Gaussian random Langevin force (the real part of the longitudinal component is shown) is applied. **(b)** The stochastic response, computed as the total energy of the solution of the elastodynamic equation, is shown as a function of frequency. Examples of the solution field (norm of the total displacement) are shown for three peaks in the response. The solution (18) gives a hint at the form of the stochastic response, if the power spectral density can be obtained. Let us now extend the Langevin noise model to include a spatial dependence. If we consider two different points in space, we can generalize the Gaussian noise properties in Equation (10) to  
\[\langle f_{i}(t,\mathbf{r})\rangle=0\,\ \ \langle f_{i}(t,\mathbf{r})f_{j}^{*}(t^{ \prime},\mathbf{r^{\prime}})\rangle=Q_{ij}\delta(t-t^{\prime})\delta(\mathbf{r }-\mathbf{r^{\prime}}), \tag{20}\]  
where the \(f_{i}\) are the components of the vector noise \(\mathbf{f}\). This amounts to considering that generating fluctuations are uncorrelated at different locations and times. Defining the spatially averaged noise \(g(t)=\int_{\Omega}\mathbf{f}\cdot\mathbf{u}_{0}=\int_{\Omega}f_{i}u_{0i}\), for a given mode \(\mathbf{u}_{0}\), we proceed to evaluate its statistical properties. By linearity, we have \(\langle g(t)\rangle=0\). Furthermore,  
\[