Next term in the Stationary Phase Lemma expansion in dimension 2

Question

Consider two functions $f\in\mathcal{S}(\mathbb{R}^2)$ and $\phi\in C^\infty(\mathbb{R}^2)$ satisfying that $\nabla\phi(x_0,y_0)=0$ and $\mathrm{det}\text{ }\mathrm{Hess}\text{ }\phi(x_0,y_0)\neq 0$, and that this is the only critical point of $\phi$. Consider $$ I(\lambda)=\iint_{\mathbb{R}^2} e^{i\lambda\phi(x,y)}f(x,y)dxdy $$ An application of the stationary phase lemma shows that, for $\lambda>0$ large, at first order $$ I(\lambda)= \dfrac{2\pi e^{i\frac{\pi}{4} s}}{\sqrt{\vert\mathrm{det}\text{ } \mathrm{Hess}\text{ }\phi}\vert }\dfrac{e^{i\lambda \phi(x_0,y_0)}}{\lambda}f(x_0,y_0)+O\left(\dfrac{1}{\lambda^{2}}\right), $$ where $s=\mathrm{sign}\text{ }\mathrm{Hess}\text{ }\phi$. Now, my issue is, I have a function $f(x,y)$ satisfying that $f(x_0,y_0)=0$, so the leading order term in this expansion vanishes. I know that the next term in the expansion should be of order $\lambda^{-2}$, but I was wondering if there was an explicit formula for the coefficient in front of said term. I was looking for references but all of them stop the expansion at first order, since most of the time that is enough. Does anyone know how to explicitly compute the next coefficient in dimension 2?