Even one opening spreads light. A single slit does not cast a sharp shadow of itself — it throws a bright center flanked by dimmer fringes that fade away, the sinc-squared envelope. The dark fringes fall exactly where a·sinθ = m·λ for nonzero m; the center at m=0 is the peak, not a null. Down the middle, light goes in, the engine computes the pattern, the proven result comes out. The blue team builds it; the red team tries to darken the center.
source J. Fraunhofer, "Neue Modifikation des Lichtes durch gegenseitige Einwirkung und Beugung der Strahlen…", Denkschriften der k. Akad. der Wiss. zu München, vol. 8 (1821–22) — no stable DOI; digitized copy at Wellcome Collection. Scalar theory after G. Kirchhoff (1882). AMBER — cited by author/title/year. Rendered, not quoted.
The pattern is not memorized; it falls out of one phase variable. Split the slit of width a into wavelets; the path spread edge-to-edge is β = πa·sinθ / λ. Summing them gives:
I(θ) = I₀ · ( sin β / β )²
R1 as θ→0, β→0 and sinβ/β→1, so I→I₀ — the center is the maximum. R2 I ≥ 0 everywhere. R3 dark fringes wherever sinβ=0 with β≠0, i.e. a·sinθ = m·λ, m=±1,±2,…. R4 a narrower slit → larger β for the same angle → the pattern spreads wider (width ~ λ/a).
| feature | rule | live value |
|---|
One slit still spreads: this is Fraunhofer diffraction of a single aperture. Its sinc² envelope — center twice as wide as each side lobe, first side lobe only ~4.7% of the peak — is not a curiosity. It is the modulation that shapes every richer pattern.
Add a second opening and you get the-double-slit: sharp cosine² fringes, but their brightness rides inside this exact envelope. Shrink the aperture to a circle and the first null becomes the-rayleigh-criterion (1.22λ/D) — the diffraction limit on what any lens can resolve. Each sphere is the next one's premise.
The blue team's live check: re-derive the central maximum, the first null, and the ~4.7% side lobe from the engine and confirm them. If red tampers the intensity law, this badge is where it shows.
Three numbers feed the engine: the slit width a, the wavelength λ, and the angle θ off-axis. From them the machine builds the single phase term β = πa·sinθ / λ and nothing else — every fringe follows.
| symbol | is | role |
|---|---|---|
| a | slit width | sets the spread (λ/a) |
| λ | wavelength | sets the color & scale |
| θ | viewing angle | where on the screen |
| β | πa·sinθ/λ | the one phase variable |
Pick a width and a color below; the panel builds β and draws the whole pattern.
The curve below is I(θ)=I₀(sinβ/β)² computed live — the central lobe, the side lobes, and the sinc² envelope over them.
Change a or λ — β, the first-null angle, and the side-lobe ratio are recomputed from the law on the spot, never looked up.
What the machine proves: the intensity is exactly I₀(sinβ/β)², the central maximum sits at m=0 (not a null), the dark fringes fall at a·sinθ = m·λ for nonzero m, the central lobe is twice as wide (in sinθ) as each side lobe, and the first side lobe is ~4.7% of the peak. Every one of these is asserted to a stated tolerance.
The blue team's witness (left) re-derives these live; the red team (right) tries to make the center dark.
"a·sinθ" also caps the physics: |sinθ|≤1 means once a<λ there are no dark fringes at all — the whole forward hemisphere is one broad lobe. The formula still returns minima "positions" that no longer exist. The engine flags when a<λ.
"The single slit gives a row of equally bright bands." Cut. Only the double slit gives near-equal fringes; a single slit's side lobes fall off fast — 4.7%, then ~1.7%, then ~0.8%.
"The center is where a·sinθ = 0, so m=0 is a minimum like the rest." Cut. m=0 is the maximum — sinβ/β→1 there, not 0. Only nonzero m are nulls.
"A wider slit spreads the light more." Kept, corrected. Backwards: width ~ λ/a, so a narrower slit spreads wider. Wide slit → tight bright line.
The red team's move: swap the intensity law to one with a minimum at the center — replace (sinβ/β)² with a plain sin²β so the bright center goes dark. The blue team's witness (window 7) is watching the peak.
Flip the law and I(0) becomes 0 — the central maximum turns into a null. The witness recomputes the peak, sees the center dark, disagrees with the known physics, and turns red. Nothing is faked; the attack is real and it is caught.