# Spontaneous Brillouin Scattering Spectrum and Coherent Brillouin Gain in Optical Fibers
## 2 Results  
### The Scalar Model of Stimulated Brillouin Scattering  
The usual model of stimulated Brillouin scattering is based on the works of [PERSON] [11], [PERSON] [12], and [PERSON] et al. [10; 13], among others. This theory is expressed for optical plane waves, i.e., the transverse variations of the optical fields are considered unimportant or known in the form of a guided modal shape [18]. Isotropic propagation and linear polarization are also implied. The total optical field is written  
\[E(z,t)=E_{P}(z,t)+E_{S}(z,t). \tag{1}\]  
with the pump (P) and the signal (S) components given by  
\[E_{P}(z,t) =A_{P}(z,t)\exp(i(\omega_{P}t-k_{P}z))+\text{c.c.}, \tag{2}\] \[E_{S}(z,t) =A_{S}(z,t)\exp(i(\omega_{S}t+k_{S}z))+\text{c.c.}, \tag{3}\]  
where \(A_{P}(z,t)\) and \(A_{S}(z,t)\) are slowly-varying amplitudes around carrier plane waves with given frequency and wavenumber. Referring to Figure 1, the pump wave travels to the right while the  
Figure 1: Brillouin light scattering (BLS) in an optical fiber or an optical waveguide, in the counter-propagating interaction setting. **(a)** The coherent mixing of two optical waves produces a dynamical acoustic wave grating satisfying the phase-matching conditions \(\omega=\omega_{P}-\omega_{S}\) and \(q=k_{P}-k_{S}\). The signal wave can experience a gain from BLS of the pump wave on the dynamic grating. **(b)** Spontaneous Brillouin scattering occurs for photons of an incident optical wave on thermally generated acoustic phonons. The random phonons are modeled as arising from a stochastic body force distributed in the whole solid. Some of them pick up the phase-matched wavevector **q**.  
signal wave travels to the left. The notations \(k_{P}=|\mathbf{k}_{P}|\) and \(k_{S}=|\mathbf{k}_{S}|\) are used. Coupled optical wave equations are obtained by neglecting second-order and higher terms and read  
\[\frac{\partial A_{P}}{\partial z}+\frac{n}{c}\frac{\partial A_{P}}{ \partial t} =\frac{-i\gamma\omega_{P}}{2\rho_{0}nc}A_{a}A_{S}, \tag{4}\] \[-\frac{\partial A_{S}}{\partial z}+\frac{n}{c}\frac{\partial A_{ S}}{\partial t} =\frac{-i\gamma\omega_{S}}{2\rho_{0}nc}A_{a}^{*}A_{P}, \tag{5}\]  
where \(A_{a}\) is a slowly-varying acoustic amplitude introduced as  
\[\rho(z,t)=A_{a}(z,t)\exp(i(\omega t-qz))+\text{c.c.} \tag{6}\]  
The fluctuations of mass density accompanying the acoustic wave or the acoustic phonons, \(\rho(z,t)\), are thus also modeled as plane waves with a slowly-varying amplitude. \(n\) is the index of refraction, \(c\) the speed of light in vacuum, \(\rho_{0}\) is the mass density, and \(\gamma\) is the electrostriction coefficient. Phase-matching is strictly imposed through relations \(q=k_{P}+k_{S}\) and \(\omega=\omega_{P}-\omega_{S}\).  
The density fluctuations also satisfy a scalar wave equation  
\[\frac{\partial^{2}\rho}{\partial t^{2}}-\Gamma^{\prime}\frac{\partial}{ \partial t}\frac{\partial^{2}\rho}{\partial z^{2}}-v^{2}\frac{\partial^{2} \rho}{\partial z^{2}}=-\frac{\gamma}{4\pi}\frac{\partial^{2}}{\partial z^{2} }E^{2}+f_{a}(z,t), \tag{7}\]  
where \(v\) is the longitudinal velocity, \(\Gamma^{\prime}\) is a viscosity constant, and \(f_{a}\) is a random force giving rise to thermal phonons. The acoustic amplitude follows the coupled-wave equation  
\[\frac{\partial A_{a}}{\partial t}+\frac{1}{2}\Gamma A_{a}+v\frac{\partial A_{ a}}{\partial z}=\frac{-i\gamma q^{2}}{8\pi\omega_{B}}A_{P}A_{S}^{*}+f(z,t). \tag{8}\]  
with the Brillouin frequency \(\omega_{B}=vq\), \(\Gamma=\Gamma^{\prime}q^{2}\), and \(f_{a}(z,t)=2i\omega f(z,t)\exp(i(\omega t-qz))+\text{c.c.}\) In what follows, we will implicitly make the undepleted pump assumption, i.e., \(A_{P}\) is a constant, since we are mostly interested in spontaneous scattering and small signal gain.