## 3 The Material
This results from the fact that the addition of P\({}_{2}\)O\({}_{5}\) decreases the acoustic velocity, and since the acoustic wavelength is fixed to the optical wavelength via the Bragg condition, the frequency also decreases. Most materials have a spectral width that typically is proportional to the square of the acoustic frequency [101], though this is suggested not to be the case for B\({}_{2}\)O\({}_{3}\)[102] (more on this topic below). Referring to Figure 5, \(g_{B}\) initially drops from that of pure SiO\({}_{2}\) due to the large, relative acoustic attenuation (large \(\Delta v_{B}\)) coefficient brought by P\({}_{2}\)O\({}_{5}\). However, as the P\({}_{2}\)O\({}_{5}\) concentration increases, and the acoustic velocity value decreases (decreasing \(v_{B}\)), \(\Delta v_{B}\) ultimately narrows to a point where that effect dominates, raising \(g_{B}\). This results in the absolute minimum in \(g_{B}\)([P\({}_{2}\)O\({}_{5}\)]) in Figure 5. In the case of materials with negative \(p_{12}\), such as alumina, a singular binary silicate composition where \(g_{B}\) = 0 is encountered [103]. What is more, materials that raise the acoustic velocity result in an increasing (not decreasing) spectral width with increasing concentration. Turning now to the thermo-mechanical response, the Bragg condition may be rewritten in terms of the acoustic velocity as  
\[v_{B}=\frac{2n_{eff}V_{a}}{\lambda_{a}} \tag{17}\]  
Figure 5: \(g_{B}\) for the binary phosphosilicate glass as a function of mol % of P\({}_{2}\)O\({}_{5}\) relative to that of pure SiO\({}_{2}\) in units of dB. An absolute minimum is observed in the plot. Even a few mol % of P\({}_{2}\)O\({}_{5}\) can cause a significant drop in \(g_{B}\). \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(n\) & \(V_{L}\) **(m/s)** & \(\rho\) **(kg/m\({}^{3}\))** & \(p_{12}\) & \(\Delta v_{B}\) **(MHz @ 11 GHz)** \\ \hline SiO\({}_{2}\) & 1.444 & 5970 & 2204 & 0.252 & 17.0 \\ P\({}_{2}\)O\({}_{5}\) & 1.488 & 3936 & 2390 & 0.226 & 176.8 \\ \hline \end{tabular}
\end{table} 
Table 3: Material properties used to calculate \(g_{B}\) for the binary phosphosilicate glass. where the index is the effective index of the optical mode, and \(V_{a}\) is the acoustic mode velocity. If the fiber undergoes strain (\(\epsilon\)) or a change in temperature (\(T\)), the fiber response may be found by differentiating Equation (17), giving rise to  
\[\frac{d\
u_{B}}{d(T,\epsilon)}=\frac{2}{\lambda_{o}}\bigg{(}V_{a}\frac{dn_{eff} }{d(T,\epsilon)}+n_{eff}\frac{dV_{a}}{d(T,\epsilon)}\bigg{)}. \tag{18}\]  
The values for pure silica (bulk material, not waveguide) are summarized in Table 4. The expression for \(\mathrm{d}n/\mathrm{d}\epsilon\) depends upon the unstrained index, \(n_{o}\). At a wavelength of 1534 nm, the index is 1.444 [50], so this term is 0.262 \(\epsilon^{-1}\). Evaluating the responses at said wavelength using the parameters provided previously and in Table 4 gives \(\mathrm{d}\
u_{B}/\mathrm{d}T=1.13\) MHz/K and \(\mathrm{d}\
u_{B}/\mathrm{d}\epsilon=57.1\) GHz/\(\epsilon\) (the thermal and strain coefficients, respectively). In the case of pure silica, the term possessing the derivative of \(V_{a}\) is more than one order of magnitude larger than the other. Regardless, the advantage of utilizing shorter optical wavelengths is obvious. As a quick note on \(\mathrm{d}n/\mathrm{d}T\), [PERSON] [104] argued that \(\mathrm{d}n/\mathrm{d}T\) encompasses two competing processes: (a) a decrease in density associated with thermal expansion, resulting in a decreased refractive index; and (b) an increase in polarizability as the electrons become more loosely bound to their nuclei, thus increasing the refractive index. If the latter process dominates, then \(\mathrm{d}n/\mathrm{d}T\) is positive. A list of \(\mathrm{d}n/\mathrm{d}T\) for numerous silica additives can be found in [35], but, in short, F, B\({}_{2}\)O\({}_{3}\), P\({}_{2}\)O\({}_{5}\), and SrO have been shown to bring negative values of \(\mathrm{d}n/\mathrm{d}T\) to the mixture.