## 2 The Waveguide  
Optical fiber has gained the status of omnipresence in the modern, technologically-connected world. Even for those without scientific training, the operating principle behind optical fiber can intuitively be understood. Indeed, more than 150 years ago, [PERSON] left an indelible mark with his public lectures demonstrating the guidance of light through a laminar flow of water [26]. Most school children in a science class have witnessed some form of this demonstration, and the notion of total internal reflection (TIR) and guidance can conceptually be understood through observation. Although calculations of modes, propagation constants, dispersion coefficients, etc. mathematically are somewhat more complex [27], the simple ray propagation picture suffices to illustratively explain how a thin strand of glass can propagate a signal with losses now as low as 0.14 dB/km [28]. As [PERSON] and [PERSON] predicted, however, this achievement largely has been enabled by materials science and engineering [29]. An optical fiber is described by its refractive index profile (RIP). Figure 1 shows an example of a very simple RIP for a conventional, step-index optical fiber. The central part of the structure, the core, guides light. Its refractive index (\(n_{core}\)) is greater than that of the cladding (\(n_{clad}\)), therefore satisfying the main requirement for TIR. The cladding diameter (2b, typically 125 \(\upmu\)m as a standard for telecommunications fiber) is considered to be infinite in most mathematical treatments of the fiber. It is coated with a buffer material, serving both to provide mechanical strength to the fiber, and to provide filtering/removal of any light propagating in the cladding due to scattering, bending, or other losses attributable to optical coupling at interfaces. Hence, the buffer usually has a refractive index higher than that of the cladding. Such a structure has perfect guidance, whose propagation constants are purely real-valued. Figure 1: Refractive index profile (RIP) for a conventional step-index optical fiber. Within this structure, the optical wave interacts with one or more acoustic waves, forming the basis for Brillouin scattering in optical fiber. Therefore, it is the optical mode, its effective index (determined from the propagation constant), and its spatial distribution in the fiber (or in other words, how the mode overlaps with different regions of the waveguide) that give rise to the measurements made in the process of distributed sensing. The modes are calculated by solving [PERSON]'s Wave Equation subject to the boundary conditions in cylindrical symmetry [30] in the usual way. The effective area, relating to the field intensity, is given by  
\[A_{eff}=\frac{\left(\int\limits_{0}^{2\pi}\int\limits_{0}^{\infty}rE(r,\theta )E^{*}(r,\theta)drd\theta\right)^{2}}{\int\limits_{0}^{2\pi}\int\limits_{0}^{ \infty}r(E(r,\theta)E^{*}(r,\theta))^{2}drd\theta} \tag{1}\]  
where \(E(r,\theta)\) is the electric field distribution (in the radial and azimuthal directions) and \(E^{*}(r,\theta)\) is the complex conjugate. Intensity is particularly important as it sets the absolute gain in a nonlinear process such as Brillouin scattering. Brillouin scattering typically manifests as an interaction between the optical wave and a longitudinal-type sound (or pressure) wave. The interaction itself arises from axially-periodic perturbations to the permittivity due to the pressure associated with the sound wave. It is governed by the Bragg condition \(\lambda_{a}=\lambda_{o}/2n_{eff}\) where \(\lambda_{a}\) is the acoustic wavelength, \(\lambda_{o}\) is the vacuum optical wavelength, and \(n_{eff}\) is the effective index of the optical mode. The acoustic wavelength also may be expressed in terms of the acoustic velocity, \(V_{a}\), via \(V_{a}=\lambda_{a}v_{a}\) where \(v_{a}\) is the frequency of the acoustic wave (in the hypersonic range). It is important to recognize that the optical and acoustic wavelengths are of comparable magnitude (indeed, only a factor of 2 between them).