# Intramode Brillouin Scattering Properties of Single-Crystal Lithium Niobate Optical Fiber
## 2 Materials and Methods
When V < 2.405, only one mode (HE\({}_{11}\)) is allowed, where the number of modes is M = \(\frac{\text{V}^{2}}{2}\). For the BSBS process, we assume that the propagation directions of the pump light and the Stokes light are both along the z-axis, and the [001] symmetry direction of the waveguide coincides with the z-axis. The acoustic wave participating in BSBS is the acoustic wave transmitted along the waveguide axis along with the light wave, and its acoustic velocity is calculated as \(C_{33}=c_{33}+\frac{d_{33}}{\varepsilon_{33}}^{2}\), \(\upsilon_{l}=\sqrt{\frac{C_{33}}{\rho}}\), where \(\upsilon_{l}\) is the longitudinal acoustic velocity, \(d_{33}\) is the piezoelectric constant, \(\varepsilon_{33}\) is the dielectric constant, \(C_{33}\) is the elastic constant, and \(\rho\) is the density. Each curve in BS usually corresponds to a different acoustic mode in the fiber, and the BFS expression in FMF [28] is:  
\[v_{B}=\left(\frac{n_{i}}{\lambda_{i}}+\frac{n_{j}}{\lambda_{j}}\right)V_{a} \approx\frac{\text{V}_{a}}{\lambda}\big{(}n_{i}+n_{j}\big{)} \tag{4}\]  
where \(v_{B}\) is the acoustic frequency and \(n_{i(j)}\) and \(\lambda_{i(j)}\) are the effective refractive index (ERI) and wavelength of the ith (jth) mode, respectively. At a wavelength of 1550 nm, an ERI of 2.245, and a velocity of acoustic wave of about 7318 m/s, the calculated \(v_{B}\) is about 21 GHz. We used the elastic dynamics equation to model the observation of backward-stimulated Brillouin scattering directly from the classical quantum mechanics of the waveguide, according to the energy conservation, through the elastic dynamics equation to study the acoustic wave in the Brillouin optical-mechanical interaction. The frequency and displacement distribution of the elastodynamic equation can be defined as:  
\[\frac{\partial^{2}u_{i}}{\partial t^{2}}-\left(c_{ijkl}u_{k,l}\right)_{,j}=- \mathbf{T}_{ij,j}^{\text{es}} \tag{5}\]  
In the formula, \(c_{ijkl}\) is the elastic matrix, \(\mathbf{T}_{ij}^{\text{es}}\) is the electrostrictive stress tensor, and \(\mathbf{T}_{ij}^{\text{es}}=-\epsilon_{0}\lambda\mathbf{c}_{klij}\big{[} \mathbf{E}_{k}\mathbf{E}_{l}^{*}\big{]}\), where \(\mathbf{\chi}_{klij}\) is a fourth-order polarization tensor and \(\mathbf{\chi}_{klij}=\epsilon_{im}\epsilon_{jn}\mathbf{\rho}_{klnm}\mathcal{S}_ {B}\) is the Brillouin gain, defined as:  
\[g_{B}(\Omega)=\sum_{m}G_{m}\frac{\left(\frac{\Gamma_{m}}{2}\right)^{2}}{\left( \Omega-\Omega_{\text{m}}\right)^{2}-\left(\frac{\Gamma_{m}}{2}\right)^{2}} \tag{6}\]  
Summing all the acoustic modes, \(\Omega_{\text{m}}\) is the eigenfrequency of the \(m\)th acoustic mode without considering the acoustic loss, \(\Omega\) is the frequency difference between the pump wave and the Stokes wave, and \(\Gamma_{\text{m}}\) is the loss coefficient of the \(m\)th acoustic mode. It depends on the mechanical energy quality factor \(Q_{m}\) and follows the relationship \(Q_{m}=\Omega_{\text{m}}/\Gamma_{\text{m}}\)The Brillouin gain of each acoustic mode is a Lorentzian-shaped function, and when the acoustic loss is considered, the Brillouin scattering gain coefficient is defined as [34]:  
\[G_{m}=\frac{2\omega Q_{m}}{\Omega_{\mathrm{m}}{}^{2}\mathrm{V_{gp}}\mathrm{V_{ gs}}}\frac{\left|\left\langle f,u_{m}\right\rangle\right|^{2}}{\left\langle E_{p}, \varepsilon E_{p}\right\rangle\left\langle E_{s},\varepsilon E_{s}\right\rangle \left\langle u_{m},\rho u_{m}\right\rangle} \tag{7}\]  
In Equation (7), \(\mathrm{V_{gp}}\) (\(\mathrm{V_{gs}}\)), \(\varepsilon\), and \(\rho\) are the group velocity, conductivity, and density of the pump light (Stokes light), respectively. \(f\) is the sum of the optical forces of the pump light and the Stokes light. \(u_{m}\) is the displacement vector of the \(m\)th acoustic mode.