The NLO interaction we consider is second harmonic generation (SHG) from 1550 nm to 775 nm, using fundamental TE-like modes with a cross-section of Fig. \ref{908186}a, where the x-cut LN is a 300 nm film. Following the derivation of \cite{Yariv_1973}, the electric field in the hybrid waveguide is \(\textbf{E}(x,y,z) = \mathrm{A}(y)\varepsilon(x,z)e^{j(\omega t-\mathrm{k}y)} \hat{z}\)for a TE-like mode. Note that the coordinate system used here follows that of the LN crystal axes (Fig. \ref{908186}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 \(\varepsilon(x,z)\) 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: