Past exam of the physics A course of the University of Cambridge / 2013 / Paper 2 / Section D / D13 / a / 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,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 $U_{\mathrm{disk}}=U_{\mathrm{free}}-U_{\mathrm{ap}}=U_{\mathrm{free}}{e}^{i\pi N_F}$ for Fresnel diffraction by an opaque disk. Thus 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 $N_F\ll 1$, 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 $\theta =1.22\lambda /(2a)$.