## 2 The Waveguide
Next, the boundary conditions are applied, from which the imaginary part of the propagation and the acoustic phase velocity are found. The modes then can be plotted to determine the overlap integral given by Equation (8) and the BGS can be constructed using Equation (9). The result is shown in \[A_{eff}=\frac{\left(\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}rE(r,\theta )E^{*}(r,\theta)drd\theta\right)^{2}}{\int\limits_{0}^{2\pi}\int\limits_{0}^{ \infty}r(E(r,\theta)E^{*}(r,\theta))^{2}drd\theta} \tag{1}\]1 The relative gain coefficient in this case is dominated by the overlap between the acoustic and optical modes. As can be seen by the inset, visually, the L\({}_{01}\) acoustic mode clearly has the highest overlap with the LP\({}_{01}\) optical mode (optical intensity is shown in red). Higher order acoustic modes (HOAMs) become more oscillatory, and therefore the overlap steadily decreases with increasing mode number, with the LP\({}_{01}\)-L\({}_{04}\) interaction no longer visible above the tails of the other interactions. The acoustic waveguide is clearly multimode, supporting a fundamental and three HOAMs in the L\({}_{0m}\) mode family. The optical and acoustic profiles were selected for this example since they give rise to HOAM interactions of reasonable strength. In this case, the LP\({}_{01}\)-L\({}_{01}\) overlap integral (Equation (8)) is about 87%. This \"relatively low\" value results from the acoustic mode possessing a smaller diameter than the optical mode, providing for enhanced overlap with the HOAMs. In a way fully analogous to the optical mode, bringing the core and cladding acoustic velocity values closer together widens the fundamental acoustic mode and enhances this overlap. Increasing the core acoustic velocity to 5920 m/s increases the overlap integral between the fundamental modes to roughly 93%, while at the same time L\({}_{03}\) and L\({}_{04}\) become cut off. When the velocity difference decreases to 35 m/s the L\({}_{02}\) mode becomes cut off. Focusing now on the core diameter, increasing it may result in an increase in the number of propagating modes. For instance, setting the core diameter to 9 \(\upmu\)m (similar to conventional single mode fiber, with all else kept the same) adds the L\({}_{05}\) and L\({}_{06}\) mode interactions, though the latter is near cutoff. This also results in an LP\({}_{01}\)-L\({}_{01}\) overlap integral of approximately 97%. The L\({}_{07}\) becomes guided when the core diameter is increased to 10.8 \(\upmu\)m, but the optical V-Number then is only just below cutoff for single mode behavior at 1534 nm. While a dominant acoustic mode may seem most desirable from the perspective of maximizing the Brillouin gain, there are cases where multiple peaks are instead desirable. The example presented  
\[A_{eff}=\frac{\left(\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}rE(r,\theta )E^{*}(r,\theta)drd\theta\right)^{2}}{\int\limits_{0}^{2\pi}\int\limits_{0}^{ \infty}r(E(r,\theta)E^{*}(r,\theta))^{2}drd\theta} \tag{1}\]2 RIP and longitudinal acoustic velocity profile for the fiber of the example. above is for a simple step-index fiber where the acoustic and optical waveguide boundaries coincide. This usually leads to a dominant L\({}_{01}\) mode interaction, where fiber optical properties are still well within typical requirements (single mode, low bending loss, etc.). However, given the wide range of materials available in the fabrication of fiber, the RIP and acoustic velocity profiles can be tailored independently, though with some restrictions. With that, it is therefore possible to design a fiber that is multi-layered, both optically and acoustically, and with boundaries that need not coincide, although this can render fabrication extremely, or even prohibitively, difficult. These configurations are discussed in Section 4.