## 2 The Waveguide
The scattering integral that ultimately provides the strength of the Brillouin interaction is [38]  
\[D_{p,q}=\int\limits_{Vol}E_{p}^{*}\delta\varepsilon_{p,q}E_{q}dV \tag{3}\]  
where _p,q_ = _r,q_, \(z\) are the cylindrical coordinates, \(E\) is the electric field, and \(\delta\epsilon\) are the dielectric perturbations resulting from the pressure wave (or strain field). The relevant dielectric perturbations (\(\delta\varepsilon_{p,q}\)) are functions of the acoustic strain fields (\(S_{p,q}\)) and are given by [38; 43]  
\[\delta\varepsilon_{rr}=-\varepsilon_{0}n_{o}^{4}\big{(}p_{11}S_{rr}+p_{12}S_{ \phi\phi}+p_{12}S_{zz}\big{)} \tag{4}\]  
\[\delta\varepsilon_{\phi\phi}=-\varepsilon_{0}n_{o}^{4}\big{(}p_{11}S_{\phi \phi}+p_{12}S_{rr}+p_{12}S_{zz}\big{)} \tag{5}\]  
\[\delta\varepsilon_{\phi r}=-\varepsilon_{0}n_{o}^{4}2p_{44}S_{\phi r} \tag{6}\]  
where \(\varepsilon_{0}\) is the permittivity of free space and the _p_\(\bar{\eta}\)s are the usual photoelastic coefficients of the material. The generalized form of the acoustic strain field can be written as a function of the displacement vector [36; 37]  
\[S_{pq}=\frac{1}{2}\bigg{(}\frac{\partial u_{q}}{\partial p}+\frac{\partial u_ {p}}{\partial q}\bigg{)}. \tag{7}\]  
The displacement vector now remains to be found. In the case of the fiber structure of Figure 1, the cladding is taken to be infinite and so there is one boundary at \(r\) = \(a\). To determine the acoustic propagation constant(s), the boundary conditions are applied to assumed solutions (_u_(_r,q_,_z_)) of the wave equations (Bessel functions in this case). In the case of the acoustic waveguide, there are four boundary conditions. First, the displacement vector must be continuous at any interface. Second, the two shear stresses at the interface and the normal compressional stress must be continuous at any boundary. These can be found mathematically listed in [44]. Once the propagation constants are determined by solving the characteristic matrix equation, they can be utilized to determine the spatial  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & **GeO\({}_{2}\)** & **P\({}_{2}\)O\({}_{5}\)** & **TiO\({}_{2}\)** & **B\({}_{2}\)O\({}_{3}\)** & **F** & **Al\({}_{2}\)O\({}_{3}\)** & **Y\({}_{2}\)O\({}_{3}\)** & **RE\({}_{2}\)O\({}_{3}\)** \\ \hline \(n_{o}\) & \(\uparrow\) & \(\uparrow\) & \(\uparrow\) & \(\downarrow\) & \(\downarrow\) & \(\uparrow\) & \(\uparrow\) & \(\uparrow\) \\ \(n_{a}\) & \(\uparrow\) & \(\uparrow\) & \(\uparrow\) & \(\uparrow\) & \(\uparrow\) & \(\uparrow\) & \(\downarrow\) & \(\downarrow\) & \(\uparrow\) \\ \hline \end{tabulardistribution of the displacement vector for the subsequent analysis. Similar, more recent approaches to calculating the longitudinal acoustic modes of an optical fiber can be found in several places [44; 45; 46]. The solutions to the wave equation are acoustic modes that have radial, longitudinal, and azimuthal components [38]. The fundamental optical mode in the fiber is azimuthally-symmetric, and therefore will have greatest overlap with azimuthally-symmetric acoustic modes. [PERSON] et al. [43] identified these as axial-radial modes R\({}_{0m}\), but which are dominated by the longitudinal component (L\({}_{0m}\)). Nevertheless, the presence of these radial wave components causes the propagation constants, even in acoustically guiding fibers, to be complex-valued (unlike with the optical waves). Simply put, the wave phase velocity exceeds the shear velocity in both the core and cladding; the Bessel functions become complex (Hankel) functions in the cladding (by assuming that it is infinite in extent), giving rise to radiation, or leaky, modes [39]. The complex propagation constant can be expressed as \(\beta=k_{a}-j\gamma_{mg}\) and, therefore, a z-propagating displacement field will take the form \(\widetilde{\mathrm{u}}\sim e^{-j\beta z}=e^{-\gamma_{mg}z}e^{-j\delta_{az}}\), possessing an attenuation term associated with the radiation of acoustic energy.