I'm considering about the leading term in the asymptotic expansion of the following integral as $k\to\infty$ $$ I_k(x,z) = \int_{\mathbb{R}} S(x-y) H_0^{(1)}\left(k\sqrt{y^2+z^2}\right) \mathrm{d} y $$ with $S:\mathbb{R}\to\mathbb{R}$ being a smooth and compactly supported function. $H_0^{(1)}$ is the zero order Hankel function of the first kind.
This integral comes from the solution $u(x,z)$ to 2D Helmholtz equation with source concentrated on the line $\{z = 0\}$. $$ \Delta u + k^2 u = S(x)\delta(z) \quad (x,z)\in\mathbb{R}^2 $$
My intuition is that it would be something like $$ I_k \sim \frac{c}{k^p}e^{\mathrm{i} k |z|} S(x) + \mathcal{o}\left( \frac{1}{k^p} \right) $$ for some $p>0$, but I'm not sure how to derive it. Any hint will be appreciated.