## 2 The Waveguide
As such, the optical fiber would naturally be expected to exhibit acoustic waveguide effects. This would then suggest that \(V_{a}\) should be considered to be the phase velocity of the acoustic mode. Thus, similar to the optical RIP, a radial profile can be identified that describes the acoustic waveguide. Usually, the acoustic velocity itself serves in this capacity (vertical scale in Figure 1), but an \"acoustic index\" also may be utilized. Usually, this is defined as the cladding acoustic velocity divided by the material acoustic velocity in the radial direction [31]. Therefore, similar to the optical waveguide, the core should have an acoustic index that is greater than that of the cladding to satisfy some analogous TIR in the acoustic realm. Logically, since the acoustic wave is key to the sensing mechanism, one may quickly conclude that guiding them might somehow improve system performance. With this in mind, the fiber possesses two important profiles: (1) the optical RIP; and (2) the acoustic velocity profiles. [PERSON] et al. [32] describes a procedure by which, depending on the dopants used (in silica), a glass fibers' optical and acoustic profiles may independently be designed. In other words, one can realize, for example, a fiber that guides optical waves, but is anti-guiding to acoustic waves. More complicated structures, such as fibers with cores whose optical index or acoustic velocity vary in the radial direction, also become possible. As a result, charts such as that shown in Table 1 typically are used to show the effects of the dopant on the optical (\(n_{o}\)) or acoustic (\(n_{a}\)) index of the silicate glass. They assist in the selection of materials for a specific fiber design. Interestingly, in Table 1, all but alumina raise the acoustic index (reduce the acoustic velocity). Only recently has it been confirmed in fiber form that both MgO [33] and Li\({}_{2}\)O [34] act in a manner similar to alumina, in raising the acoustic velocity when added to silica. An extensive study on these and several other relevant acoustic properties can be found in [35]. The next section provides additional details regarding the material and enumerate of the effects of the various dopants. Determination of the acoustic phase velocity (or the acoustic modes), therefore, requires solving an acoustic wave equation subject to the relevant boundary conditions. The acoustic waves obey a damped wave equation given by [36; 37]  
\[\rho\ddot{u}-\
abla\bullet[\overline{\xi}\
abla u+\overline{\eta}\
abla u]=-\frac {1}{2}\overline{\
abla}\bullet[\overline{\overline{\gamma}}E_{k}E_{l}] \tag{2}\]  
where the electrostrictive coefficients are given by a fourth rank tensor in \(\overline{\overline{\gamma}}\), the damping term, \(\overline{\overline{\eta}}\), is a tensor of rank four, and \(\overline{\overline{\epsilon}}\) is the rank-four elastic modulus tensor. \(\overline{u}\) is the displacement field (and \(\dot{u}\) is its time derivative) associated with the sound wave and can take on values in any of the directions in cylindrical coordinates. [PERSON] et al. [38] provided a detailed mathematical and experimental analysis of the scattering of light waves from acoustic modes of an optical fiber whose cladding velocity is less than that of the core (acoustically anti-guiding, pure silica core with a borosilicate cladding). Roughly 10 years later, [PERSON] and colleagues [39; 40; 41] offered thorough mathematical analyses of fibers configured to be acoustically guiding. It was about the same time that [PERSON] et al. [42; 43] made some of the first observations of multi-peaked Brillouin spectra, which they attributed to the presence of multiple guided acoustic modes.