= Solution
For a circular structure, cylindrical symmetry reduces the scalar <Fresnel-Kirchhoff diffraction integral> to concentric annuli. With plane-wave illumination, radius $a$ and distance $z$, the relevant <Fresnel number> is $N_F=a^2/(\lambda z)$. Successive <Fresnel half-period zones> have radii $r_j^2\simeq j\lambda z$: their areas are approximately equal while their phases alternate by $\pi$, causing neighbouring-zone cancellation.
For <Fresnel diffraction by a circular aperture>, the on-axis field follows directly from the radial <integral>,
$$
\frac{U_{\rm ap}}{U_{\rm free}}
=\frac1{i\lambda z}\int_0^a e^{i\pi r^2/(\lambda z)}2\pi r\,dr
=1-e^{i\pi N_F},\qquad
\boxed{I_{\rm ap}/I_{\rm free}=4\sin^2(\pi N_F/2)}.
$$
Odd zone counts reinforce on axis; even counts cancel in this ideal scalar model. An annular opening gives the difference of the corresponding two edge exponentials. A <zone plate> alternately blocks or phase-shifts zones so that otherwise cancelling contributions reinforce at a focus, with $r_j^2\simeq j\lambda f$ for its principal focal distance.
<Babinet's principle> gives $U_{\rm disk}=U_{\rm free}-U_{\rm ap}=U_{\rm free}e^{i\pi N_F}$ for <Fresnel diffraction by an opaque disk>. Thus \b[its ideal on-axis intensity equals the unobstructed intensity], the <Arago spot>, even within the geometric shadow. This concerns a narrow axial feature, not the whole shadow. Finite source size, imperfect circularity, <bandwidth> and alignment reduce its visibility. When $N_F\ll1$, the circular-aperture far-field pattern becomes the <Airy disk> intensity pattern $[2J_1(u)/u]^2$, with $u=ka\sin\theta$ and first minimum near $\theta=1.22\lambda/(2a)$.
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2013/paper-2-optical-notes.png]
{title=Original calculated circular Fresnel, Fabry–Pérot and finite-grating intensity patterns under their stated idealizations}
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