= Fresnel diffraction by a circular aperture
{c}
For a circular aperture of radius $a$, paraxial plane-wave illumination and propagation distance $z$, define $N_F=a^2/(\lambda z)$. On axis, the radial <Fresnel diffraction> integral normalized by the unobstructed field gives
$$
\frac{U_a}{U_0}=1-e^{i\pi N_F},\qquad
\frac{I_a}{I_0}=4\sin^2\left(\frac{\pi N_F}{2}\right).
$$
For a finite axial point source a distance $s$ before the aperture and detector a distance $r$ after it, replace $z$ by $sr/(s+r)$ in the phase. The formula assumes illumination amplitudes and obliquity factors approximately constant across the aperture. Odd and even numbers of <Fresnel half-period zones> give bright and dark axial extrema; large local intensity is compatible with global <conservation of energy>.
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