The NLO interaction we consider is second harmonic generation (SHG) from 1550 nm to 775 nm, using fundamental TE-like modes with the cross-section of Fig. \ref{908186}a, where the x-cut LN is a 300 nm film. Following the derivation of \cite{Yariv1973}, the electric field in the hybrid waveguide is \textbf{E}$(x,y,z) = \mathrm{A}(y)\mathbfcal{E}(x,z)e^{j(\omega t-\mathrm{k}y)} \hat{z}$ for a TE-like mode. Note that the coordinate system used here follows that of the LN crystal axes (Fig.~\ref{SiCFig}a), so that the $\hat{y}$-axis is the direction of propagation (positive or negative) instead of the more conventional $\hat{z}$-axis. $\mathrm{A}(y)$ is the longitudinal amplitude of the mode, and and $\mathbfcal{E}(x,z)$ is the transverse mode profile. $\omega = \frac{2\pi c}{\lambda}$, and k$ = \frac{2\pi\mathrm{n}}{\lambda}$; n and $\lambda$ are the effective index and wavelength of the mode, respectively. $c$ is the speed of light in vacuum. The coupled mode equations in this interaction are: