# Tailorable Brillouin Light Scattering in a Lithium Niobate Waveguide
## 2 Materials and Methods  
The forward stimulated Brillouin scattering (FSBS) waveguide represents a direct interaction between pump light, Stokes light or anti-Stokes light and sound waves [12,13]. Its basic physical mechanism is shown in Figure 1: a pump with frequency \(\omega_{p}\) and Stokes light with frequency \(\omega_{s}\) are coupled into the waveguide in the same direction to generate a beat signal. When the pump light propagates along the waveguide, a photonics effect will occur on the surface of the waveguide, which will generate an electrostriction force. In the meantime, the dielectric constant in the waveguide will also change, resulting in a periodically modulated refractive index [12, 13, 14]. The Stokes light and phonons produced by FSBS propagate in the same direction as the pump light, and the vibration direction of the sound field is lateral. More specifically, in the FSBS process, the light field propagates in the same direction, and the propagation direction of the phonons and the propagation direction of the light field during the scattering process are perpendicular to each other [17,18]. In the SBS process, the phase-matching condition should be met, i.e., the conservation of energy and momentum should meet the following conditions:  
\[\vec{k_{A}}=\vec{k_{p}}-\vec{k_{s}} \tag{1}\]  
\[\Omega=\omega_{p}-\omega_{s} \tag{2}\]  
where \(\vec{k_{A}}\), \(\vec{k_{p}}\) and \(\vec{k_{s}}\) are the phonon wave vector, pump wave vector and Stokes wave vector, respectively. Given the propagation direction of the Stokes wave, SBS can be divided into backward stimulated Brillouin scattering (BSBS) and forward stimulated Brillouin scattering (FSBS). For the FSBS process, we assume that the propagation directions of pump light and Stokes light are both along the z-axis, so the light wave field of pump light and Stokes light can be described as:  
\[E_{P}(z,t)=\widetilde{E}_{p}(x,y)\mathrm{e}^{\mathrm{i}(k_{p}z-\omega_{p}t)} \tag{3}\]  
\[E_{s}(z,t)=\widetilde{E}_{s}(x,y)\mathrm{e}^{\mathrm{i}(k_{s}z-\omega_{s}t)} \tag{4}\]  
\[E(z,t)=E_{p}(z,t)+E_{s}(z,t) \tag{5}\]  
where \(E_{P}(z,t)\) is the pump light field and \(E_{s}(z,t)\) is the Stokes light field. Using small signal approximation, assuming that the pump power in the waveguide is greater than the Stokes signal power, the coupling between the pump light and the Stokes signal light should meet the following requirements [18],  
\[\frac{dP_{p}}{dz}=-(\alpha+\beta P_{p}+\gamma P_{p}^{2})P_{p} \tag{6}\]  
\[\frac{dP_{s}}{dz}=(\alpha-gP_{p}+2\beta P_{p}+\gamma P_{p}^{2})P_{s} \tag{7}\]  
where \(P_{p}\) and \(P_{s}\) are the power of the pump light and Stokes light, respectively, \(\alpha\) is the linear loss of the light wave and \(\beta\) and \(\gamma\) are the nonlinear loss coefficient caused by two-photon absorption and the non-linear loss coefficient caused by free carrier absorption, respectively. In Equation (7), \(g(\Omega)\) is the SBS gain coefficient, which can be expressed by the Lorentz model as follows [19]:  
\[g(\Omega)=\sum_{m}G_{m}\frac{\left(\Gamma_{m}/2\right)^{2}}{\left(\Omega- \Omega_{m}\right)+\left(\Gamma_{m}/2\right)^{2}} \tag{8}\]  
where \(\Omega_{m}\) is the eigenfrequency satisfied by the \(u_{m}\) eigenequation of the acoustic mode without considering the acoustic loss; \(\Gamma_{m}\) is the sound film loss coefficient when sound loss  
Figure 1: FSBS effect in the waveguide. is included [1], which depends on the light quality factor \(Q_{m}\), and shall comply with the following relationship: \(Q_{m}=\frac{\Omega_{m}}{T_{m}}\)[20], where the subscript \(m\) is the m-order acoustic mode (\(m=1,2,3\dots\)) [1]. When considering the sound loss, each SBS gain coefficient can be simplified to [20]:  
\[\vec{k_{A}}=\vec{k_{p}}-\vec{k_{s}} \tag{1}\]0  
where \(V_{gs}\), \(\epsilon\) and \(\rho\) are the light group velocity, conductivity and density, respectively; \(f\) is the sum of the optical forces of pump light and Stokes light.