## 3 The Material  
The previous section reviewed the acoustic waveguide properties of the optical fiber. In this section, the focus shifts to the material and the influence of dopants on those physical properties of silica glass which are relevant to Brillouin scattering. From the perspective of the waveguide, the acoustic velocity values are most important. The longitudinal and shear velocity values are related to the Lame constants (\(\lambda_{L}\) and \(\mu_{L}\) where the subscript \(L\) has been added for disambiguation) as \(V_{L}=\sqrt{(\lambda_{L}+2\mu_{L})/\rho}\) and \(V_{S}=\sqrt{\mu_{L}/\rho}\), respectively. The numerators within square root are the longitudinal and shear elastic moduli. For the sake of simplicity, the acoustic velocities, rather than the elastic moduli, are considered the key design parameter. To form the acoustic (or optical) waveguide, some amount of dopant should be added to the base silica glass. For the present analysis, these come in the form of oxide and F additives. To form an acoustic waveguide, a material that reduces the acoustic velocity when added to silica is desired. Optical guidance requires that this dopant also raise the refractive index. [PERSON] et al. [54; 55] offered a detailed ultrasonic and optical analysis of doped fiber preforms, giving rise to the empirical design coefficients provided in Table 2. These coefficients suggest a linear dependence of the parameters on the dopant concentration in percentage by weight. This relationship does not necessarily hold for  
Figure 3: Simulated Brillouin gain spectrum (BGS) calculated (1534 nm) from the profiles in Figure 2. The positions of the longitudinal mode interactions are also shown. The insets plot the LP\({}_{01}\) optical mode intensity (red) and the acoustic displacement \(u_{z}(r)\) (blue). Vertical dashed lines in the insets delineate the core region. all dopants and parameters. See, for example, the index of binary phosphosilicate glass [56] where linearity in the mole % (mol%) of P\({}_{2}\)O\({}_{5}\) was observed up to 40 mol%. There is not a linear relationship between P\({}_{2}\)O\({}_{5}\) concentration expressed in weight % (wt%) and in mol%. Regardless, these linear values can hold over small, but significant, compositional ranges. Several more such empirical design equations have been provided over the years. For instance, [PERSON] et al. [45] summarized data from [57; 58; 59] giving the following  
\[n=1.458\Big{(}1+1.0\times 10^{-3}[GeO_{2}]_{wt\%}-3.3\times 10^{-3}[F]_{wt\%} \Big{)} \tag{11}\]  
\[V_{L}=5944\Big{(}1-7.2\times 10^{-3}[GeO_{2}]_{wt\%}-2.7\times 10^{-2}[F]_{wt\% }\Big{)}\ (\rm{m/s}) \tag{12}\]  
\[V_{S}=3749\Big{(}1-6.4\times 10^{-3}[GeO_{2}]_{wt\%}-2.7\times 10^{-2}[F]_{wt \%}\Big{)}\ (\rm{m/s}) \tag{13}\]  
\[\rho=2202\Big{(}1+6.4\times 10^{-3}[GeO_{2}]_{wt\%}-3.4\times 10^{-3}[F]_{wt \%}\Big{)}\ \Big{(}\rm{kg/m^{3}}\Big{)} \tag{14}\]  
where the number in brackets is the composition in wt%. These values appear to have been at least partly adopted by others [60; 61; 62]. What is particularly interesting is that this set of equations now enables the design of ternary glass compositions. Presumably, any number of additional dopants could be appended to these equations. Using them, the composition of the core necessary to achieve the example profiles in Figure 2 is 0.5 wt% F and 4.4 wt% GeO\({}_{2}\) in silica (but assuming that pure silica has an acoustic velocity of 5970 m/s [35]). Rather than utilize simple linear fittings to experimental data, the authors here have been employing a somewhat different approach; one that gives some access to glass design over wider compositional ranges and properties. For the parameters described thus far in this section, a modern form of the Winkelmann-Schott [63] addition model is used to calculate the refractive index (and other properties) of a mixture of glass materials, using volume fraction as the additivity parameter [64].