• Spot of Arago pixel-art physics T-shirt, white tee, front view

Index波、混沌與物質Tee

Arago

The Spot of Arago Physics

\[N_F = \frac{a^{2}}{\lambda L}\]
$29.99 USD
$29.99 USD
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Size guide S–5XL · inches

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

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.

N_F = a² / (λ L) and, on axis, I ≈ I₀

  • N_F: Fresnel number (dimensionless)
  • a: disk radius (m)
  • λ: wavelength (m)
  • L: disk-to-screen distance (m)
  • I₀: unobstructed intensity at the same point (W/m²)

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.

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

On the shirt

Front: Arago, the pixel-art character, with the title. Back: the Fresnel number, the on-axis result, the 1818–1819 prize story and the earlier sightings. Printed on a unisex tee.

Shipping calculated at checkout by destinationMade to order, ships in 2–5 business days

Details

Size & fitS–5XL

Unisex heavy cotton (Gildan 5000), classic fit.

SizeWidthLengthSleeve
S182815.1
M202916.5
L223018
XL243119.5
2XL263221
3XL283322.4
4XL303423.7
5XL323525

參考尺度：M 碼寬 20 in（50.80 cm），約為 3.143 × 1034 個普朗克長度（lP = 1.616255 × 10−35 m）。用捲尺量更快。

Measurements in inches, ±1.5 in tolerance.

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Returns & replacements21 days

Every item is printed to order, so we can’t accept returns or exchanges for change of mind, or if you ordered the wrong size or colour.

Print defect, misprint, damage or the wrong item? We’ll send a free replacement, or a refund if you prefer. Report it within 21 days of delivery with a photo; no need to send it back.

Full refund & replacement policy

The physics

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

In 1818 Augustin Fresnel submitted his wave theory of diffraction to the French Academy's prize. Siméon Poisson, one of the judges, worked out a consequence he thought absurd: behind an opaque disk lit by a point source, waves bending round the rim all arrive in step on the axis, so the centre of the shadow should be about as bright as if the disk were not there. François Arago put a disk in the beam and saw the spot. Fresnel won the 1819 prize.

The Fresnel number a²/(λL) tells you which regime you are in, and real disks need very round, smooth edges, hence the ≈. The spot had been seen before, by Delisle in 1715 and Maraldi in 1723, and passed unnoticed. It is also called Poisson's spot or Fresnel's spot.

Equations

\[N_F = \frac{a^{2}}{\lambda L}\]
\[I_{\text{on axis}} \approx I_0\]
SymbolMeaningUnit
\(N_F\) Fresnel number \(\mathrm{1}\)
\(a\) disk radius \(\mathrm{m}\)
\(\lambda\) wavelength \(\mathrm{m}\)
\(L\) disk-to-screen distance \(\mathrm{m}\)
\(I_{\text{on axis}}\) intensity at the centre of the shadow \(\mathrm{W/m²}\)
\(I_0\) unobstructed intensity at the same point \(\mathrm{W/m²}\)

Sources

  1. Lucke (2006) Rayleigh–Sommerfeld diffraction and Poisson's spot, Eur. J. Phys. 27, 193
  2. Born & Wolf (1999) Principles of Optics, 7th ed., Cambridge University Press, ch. 8

Field notes

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