## 2 The Waveguide
In the case of acoustically guiding fibers, this additional attenuation term is far less in magnitude than the intrinsic (viscoelastic) material damping loss and can be neglected. Interestingly, should the density and shear velocity values be the same in the core and cladding, the waveguide loss term vanishes [39]. Making this approximation can then simplify the mathematical analysis to some extent. In the case of acoustically anti-guiding fibers, the longitudinal wave also becomes leaky (again assuming an infinite cladding) and can contribute waveguide loss comparable to, or even much larger than [47] that of the material [44]. As discussed below, this acts to broaden the Brillouin gain spectrum (BGS) and reduce the peak gain coefficient; potentially undesirable characteristics for some distributed sensor applications. Ultimately, the determination (or design) of the BGS is the desired outcome. Therefore, once the components of the displacement vector are known, the scattering amplitude can be determined. Using similar modal symmetry arguments as above, the analyses in [38; 40; 42] show that the dominant scattering term arises from the \(p_{12}S_{zz}\) contribution. Therefore, the scattering integral in Equation (3) can be replaced by a simplified (and normalized) overlap integral taking the form [48]  
\[\Gamma_{m}^{1/2}=\frac{\int\limits_{0}^{\infty}\overset{\rightarrow}{E}^{*}(r) u_{z,0m}(r)\overset{\rightarrow}{E}(r)rdr}{\sqrt{\left(\int\limits_{0}^{ \infty}|u_{z,0m}(r)|^{2}rdr\right)\left(\int\limits_{0}^{\infty}\left|\overset{ \rightarrow}{E}(r)\right|^{4}rdr\right)}}. \tag{8}\]  
Owing to the simple exponential decay of the acoustic wave, a BGS possesses the natural (or Lorentzian) lineshape [49], with the possibility of multiple overlapping peaks given the presence of multiple acoustic modes possessing non-zero overlap with the optical field [42]. The scattered intensity is proportional to \(\Gamma_{m}\)[38], and, therefore, the BGS can be calculated by the following  
\[BGS=\sum_{m}g_{B}(\
u_{a_{0m}})\frac{\left(\Delta\
u_{B_{0m}}/2\right)^{2}}{ \left(\
u-\
u_{a_{0m}}\right)^{2}+\left(\Delta\
u_{B_{0m}}/2\right)^{2}}\Gamma _{m} \tag{9}\]  
where \(\Delta\
u_{B}\) is the spectral width of the mode, which possesses contributions from both material damping and acoustic waveguide-based losses. As previously stated, the waveguide part can be neglected for typical acoustically-guiding fiber. \(g_{B}\) is the Brillouin gain coefficient, which possesses a number of material parameters, including the photoelastic constant, \(p_{12}\), which is dropped from the normalized overlap integral, and given by [49]  
\[A_{eff}=\frac{\left(\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}rE(r,\theta )E^{*}(r,\theta)drd\theta\right)^{2}}{\int\limits_{0}^{2\pi}\int\limits_{0}^{ \infty}r(E(r,\theta)E^{*}(r,\theta))^{2}drd\theta} \tag{1}\]0with all the material properties previously defined, except for the mass density, \(\rho\). In general, this equation should not have a wavelength dependence. \(\Delta v_{B}\), if only resulting from material damping, typically is proportional to \({\
u_{a}}^{2}\), which, in turn, is proportional to \({\lambda_{o}}^{-2}\) thereby canceling the wavelength dependence in the denominator. This is discussed in the next section. However, there may be chromatic dispersion in the index [50] or photoelastic constant [51; 52; 53], or it is possible that each acoustic mode possesses a unique \(g_{B}\) (more on this later). Now, a very simple example calculation of the BGS is presented. Figure 2 shows the RIP and longitudinal acoustic velocity profile for a fictitious fiber wherein all acoustic modes have the same \(\Delta v_{B}\) = 35 MHz and the same \(g_{B}\). How these profiles came to be, with regard to the dopants in Table 1, is not important for this example, although this is briefly revisited in the next section. Following the procedure outlined above, \(n_{eff}\) is calculated first, establishing the acoustic wavelength and eventually the real part of the acoustic propagation constant.