Arago: Spot of Arago

Optics, Waves & Sound arc, page 111 of 193. Back to The Pixelated Physics Universe

Who they are

Arago raises a round shield against the beam. Dead centre of its shadow, a star still burns.

Arago is a character of the Optics, Waves & Sound arc: drawn in pixels, ruled by one piece of real physics, and never allowed to break it.

The physics

Light waves bending around the edge of a round, opaque disk all arrive in step on the axis behind it. In ideal Fresnel (paraxial) theory, with a point source, the centre of the shadow is about as bright as if the disk were not there. The Fresnel number tells you which regime you are in. In 1818 Fresnel submitted his diffraction memoir; Poisson, judging it, derived the "absurd" bright spot as an objection, Arago put a disk in the beam and saw it, and Fresnel won the 1819 prize.

$$ \begin{gathered} N_F = \frac{a^{2}}{\lambda L} \\ I_{\text{on axis}} \approx I_0 \end{gathered} $$

Symbol Meaning SI unit
$N_F$ Fresnel number 1
$a$ disk radius m
$\lambda$ wavelength m
$L$ disk-to-screen distance m
$I_{\text{on axis}}$ intensity at the centre of the shadow W/m²
$I_0$ unobstructed intensity at the same point W/m²

Honest caveat: Real disks need very round, smooth edges, hence the ≈. The spot had been seen earlier and passed unnoticed: Delisle (1715), Maraldi (1723).

Sources

Crossovers

Arago meets Interfera (Path Unchosen / Quantum Interference). Interfera's stripes come from probability amplitudes adding and cancelling. Arago's bright spot comes from light waves around a disc's edge adding in step. It is the same superposition, once for quantum amplitudes and once for classical light.

Arago meets Distalis (Fraunhofer Diffraction). Arago's spot is a near-field (Fresnel) effect, behind a disc. Distalis lives far away, where the pattern becomes the Fourier transform of the aperture. Same diffraction, different distances.

Wear the lore

Concept keywords: Spot of Arago, Arago, Optics, Waves & Sound, physics lore, Pixelated Physics