= Solution
Let the <aperture> occupy the $z=0$ plane, with coordinates $(x,y)$ and opening $\Sigma$. Write $\psi_\Sigma(x,y)$ for the incident scalar <complex amplitude> and $h(x,y)$ for its dimensionless amplitude transmission, including any imposed <phase>. The observation point has transverse coordinates $(X,Y)$, corresponding to the PDF's $(x_0,y_0)$, and axial distance $R$. Also $k=2\pi/\lambda$ is the <wavenumber> and $i^2=-1$.
The <Huygens-Fresnel principle> expresses the propagated field, up to common prefactors and an approximately constant obliquity factor, as
$$
\psi_P\propto\iint_\Sigma \psi_\Sigma h\,\frac{e^{ikr}}r\,dx\,dy,\qquad
r=\sqrt{R^2+(X-x)^2+(Y-y)^2}.
$$
In the <paraxial approximation>, expand the path as
$$
r\simeq R+\frac{X^2+Y^2}{2R}-\frac{Xx+Yy}{R}+\frac{x^2+y^2}{2R}.
$$
The last term is negligible in the <Fraunhofer diffraction> limit, while the first two terms and $1/r\simeq1/R$ are common across the <aperture>. Factoring them out leaves
$$
\boxed{\psi_P\propto\iint_\Sigma\psi_\Sigma(x,y)h(x,y)e^{-ik(Xx+Yy)/R}\,dx\,dy}.
$$
It is the two-dimensional <Fourier transform> of the transmitted <aperture> field, evaluated at transverse <wavevector> $(kX/R,kY/R)$. For appreciable observation angles the corresponding exact directional components are used rather than $X/R,Y/R$.
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