## 3 The Material
Given the speed of sound in pure silica (Equations (12) and (13)), the cladding is roughly two phonon lifetimes wide in a standard 125 \(\upmu\)m fiber. It is, therefore, not clear what role the cladding boundary does play in acoustically anti-guiding fiber in the case of small or intermediate cladding diameters. This, likely, is very little in the large fibers encountered in high-power lasers [92] (a 400 \(\upmu\)m cladding diameter gives about 6 phonon lifetimes). While acoustically guiding fibers may be preferable in Brillouin-based distributed sensors, hopefully the preceding discussion is still important, as it sheds more light on the contributions to \(\Delta v_{B}\). To conclude, it has been shown that, for an acoustic mode occupying regions of the fiber with differing material \(\Delta v_{B}\), both FEM and boundary value computational methods can be used to determine a modal value [93; 94] spectral width. Attention now is turned towards the other materials parameters in Equation (10). First, the material acoustic velocity, refractive index, and mass density usually are set by the design of the waveguide. All likely will be within 10% of that of silica, with the refractive index difference probably much less than that for a single mode, weakly guiding, optical fiber. The photoelastic constant, p\({}_{12}\), remains and is doubly interesting since: (1) it can take on either positive or negative values [95]; and (2) \(g_{B}\) is proportional to the square of its value. It turns out that the common dopants, GeO\({}_{2}\)[96], B\({}_{2}\)O\({}_{3}\)[96], and P\({}_{2}\)O\({}_{5}\)[97] all have photoelastic constants that are comparable to that of SiO\({}_{2}\), with that of B\({}_{2}\)O\({}_{3}\) apparently slightly higher [96]. However, other dopants that have been applied, such as Al\({}_{2}\)O\({}_{3}\)[52], possess a negative-valued \(p_{12}\). This leads to a reduction in p\({}_{12}\) when added to silica, and coupled with any spectral broadening, this can significantly offset \(g_{B}\). It should be noted that \(p_{12}\) does not affect \(\Delta v_{B}\). Therefore, in some fiber designs, perhaps where absolute or relative Brillouin gain is important, consideration of the material in the simulation of the BGS is essential. Complete sets of materials models derived from Equation (15) are provided in the paper sequence in [35; 98; 99; 100] (and are not repeated here). Although the focus was on the suppression of nonlinearities, generally, they also may be applied to the calculation of \(g_{B}\) for sensing applications. As can be inferred from Table 1 in [35],however, it will be a challenge to start with pure silica and, then through a materials route, enhance \(g_{B}\). Stated another way, adding just about anything to SiO\({}_{2}\) will lower its material \(g_{B}\). This is not meant to discourage investigations into these areas for Brillouin-based distributed sensors, since it is the _response_ of the material to the thermo-mechanical environment that is most important. Fortunately, some interesting cases require displacing only lesser amounts of silica, such that \(g_{B}\) is not decreased to a value below system feasibility, should this be a design concern. To conclude the discussion of \(g_{B}\), an example calculation is presented. Binary P\({}_{2}\)O\({}_{5}\) in SiO\({}_{2}\) is the selected material system. This is of interest since P\({}_{2}\)O\({}_{5}\) has a negative thermo-optic coefficient that can give rise to fibers whose \(v_{B}\) are immune to changes in temperature [94]. Table 3 provides a summary of the numbers utilized in the calculation. The refractive index values are those at 1534 nm, and the remaining data are from [94; 97]. The spectral width is listed at a reference frequency of 11 GHz (arbitrarily selected). This is important since, as the P\({}_{2}\)O\({}_{5}\) content changes, so does \(\Delta v_{B}\).