# Intramode Brillouin Scattering Properties of Single-Crystal Lithium Niobate Optical Fiber
## 2 Materials and Methods  
In a typical Brillouin process, the pump light field generates an acoustic wave field through electrostriction, which, in turn, periodically modulates the refractive index of the medium to form a moving grating that scatters the pump light. In the SBS process,the phase matching condition should be satisfied, that is, the conservation of energy and momentum should satisfy the following conditions:  
\[\omega_{\rm p}=\omega_{\rm s}+\omega_{\rm a} \tag{1}\]  
\[\vec{k}_{p}^{\rightarrow}=\vec{k}_{s}+\vec{k}_{a}^{\rightarrow} \tag{2}\]  
where \(\vec{k}_{p}^{\rightarrow},\vec{k}_{s}^{\rightarrow},\vec{k}_{a}\) are the pump wave, Stokes wave, and acoustic wave vector, respectively, and \(\omega_{\rm p}\), \(\omega_{\rm s}\), and \(\omega_{\rm a}\) are their angular frequencies, respectively. Brillouin scattering is divided into forward scattering and back scattering according to the direction of propagation of the pump wave and Stokes wave. In forward-stimulated Brillouin scattering (FSBS), the pump wave and Stokes wave propagate in the same direction. In this case, the acoustic wave has a zero wave vector along the waveguide axis, which means that the acoustic wave propagates perpendicular to the axis and moves back and forth between the waveguide boundaries [32; 33]. For back-stimulated Brillouin scattering (BSBS), the pump and Stokes waves propagate in the opposite directions along the waveguide axis, and the associated acoustic wave propagates along the waveguide axis. The high-energy pump photons generate anti-Stokes photons and phonons through the third-order nonlinear process shown in Figure 1, to explore the contribution of longitudinal acoustic modes to backward Brillouin scattering. We considered a circular step-index fiber (SIF) with lithium niobate as the core material and silica as the cladding material. The high refractive index difference between the core and the cladding enabled strong optical field confinement to enhance the SBS. The structural and material parameters used in the simulations are listed in Table 1. \begin{table}
\begin{tabular}{c c c} \hline \hline  & \multicolumn{2}{c}{**Values**} \\ \hline
**Parameters** & **Core** & **Cladding** \\ \hline Radius (\(\mu\)m) & r\({}_{\rm core}=4.5\) & r\({}_{\rm clad}=20\) \\ Refractive index & n\({}_{\rm core}=2.245\) & n\({}_{\rm clad}=1.45\) \\ Mass density (kg/m\({}^{3}\)) & \(\rho_{\rm core}=4700\) & \(\rho_{\rm clad}=2203\) \\ Longitudinal acoustic velocity (m/s) & \(V_{\rm core}=7318\) & \(V_{\rm clad}=5950\) \\ Photo-elastic coefficients & \(p_{11}=-0.02\), \(p_{12}=0.08\), \(p_{44}=0.12\) \\ Transmission loss at 1550 nm (dB/cm)a & 0.89 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Structure and material parameters for the SIF. Figure 1: Schematic diagram of Brillouin scattering in lithium niobate optical fiber, the distribution of LP\({}_{01}\), LP\({}_{11}\), and LP\({}_{21}\) optical modes in the core region, and the distribution of L\({}_{01}\), L\({}_{11}\), and L\({}_{21}\) acoustic modes in the core and cladding regions. In the step-type lithium niobate optical fiber structure, the large relative refractive index difference between the core material and the cladding material is not perfectly consistent with the weakly guiding approximation. In order to simplify the analysis, we used the exact formulas of the linear polarization (LP) modes. The number of modes in multi-mode transmission is determined by its normalized frequency V, and the expression of the step-type multi-mode normalized frequency V is:  
\[\text{V}=\frac{2\pi a}{\lambda}\text{NA} \tag{3}\]  
In Equation (3), \(\lambda\) is the wavelength of the incident light and \(a\) is the radius of the fiber core, where NA is the numerical aperture \(\text{NA}=\sqrt{n_{i}^{2}-n_{j}^{2}}\); the normalized frequency V is an important parameter to characterize the mode transmission characteristics. The larger V is, the more guided modes are allowed to exist, and vice versa.