Two mirrors face each other and trap the light. Each pass adds another reflected beam, and the infinite sum interferes — smearing everywhere except at the resonances, where it stacks into a razor-sharp spike. The transmission is the Airy function T = 1/(1 + F·sin²(δ/2)): broad valleys and needle peaks whose sharpness is the finesse π√R/(1−R). Down the center, light goes in, the etalon selects, the comb comes out — the heart of the laser and the spectrometer. The blue team builds it; the red team attacks it.
source C. Fabry & A. Perot, Théorie et applications d'une nouvelle méthode de spectroscopie interférentielle (1899), Ann. Chim. Phys. 7ᵉ sér., 16, 115 — no stable open DOI; cited author/title/year. Rendered, not quoted.
The sharpness is not painted on; it falls out of the algebra. Summing every reflected beam gives:
T = 1/(1 + F·sin²(δ/2)) with coefficient F = 4R/(1−R)² and round-trip phase δ = 4πnd·cosθ/λ. Resonance (δ a multiple of 2π) ⇒ T = 1; between peaks T falls to 1/(1+F). The peak width sets the finesse = π√F/2 = π√R/(1−R).
For the current mirror reflectivity, live:
| quantity | value |
|---|
Young's two slits interfere two beams — soft cosine fringes. Fabry-Perot interferes infinitely many: each round trip loses a factor R, and the geometric sum turns the gentle cosine into the peaked Airy function.
Two beams give contrast; N beams give resolution. Push R → 1 and the finesse climbs, the peaks narrow toward a delta comb — the laser cavity and the interferential spectrometer. Each sphere is the next one's premise: the double-slit's fringe is this etalon's soft limit at R → 0.
The blue team's live check: re-derive the resonance peak, the trough, the finesse ordering, and the mean-transmission integral from scratch and confirm them against the known laws. If red tampers, this badge is where it shows.
Feed the etalon: light of wavelength λ striking two parallel mirrors of reflectivity R, separated by an optical path n·d, at angle θ. Two numbers fix everything downstream:
| input | controls |
|---|---|
| reflectivity R | the finesse — peak sharpness |
| spacing n·d | the free spectral range — peak spacing |
| phase δ = 4πnd·cosθ/λ | which order is on resonance |
R alone sets how sharp; the spacing sets how far apart. That is the whole game — and it is what you feed the panel below.
The comb below is computed live from the Airy function — never looked up.
Transmission T across four free-spectral-ranges. Raise R and watch the peaks narrow.
Move any control — the finesse, the FSR and the peak width are computed from the four laws on the spot, never stored.
What the machine produces, proven: a comb of transmission peaks at δ = 2πm, each reaching T = 1, dropping to 1/(1+F) between; equally spaced in frequency by the free spectral range c/(2nd); each of width FWHM = FSR/finesse with finesse π√R/(1−R) — narrowing as R → 1 into laser-grade resonances.
The blue team's witness (left) re-derives these live; the red team (right) tries to make them wrong.
The reflectivity finesse π√R/(1−R) is a ceiling, not a promise. It is the first, cleanest term — the one Fabry and Perot could compute — not the whole instrument.
"Finesse depends on the mirror spacing." Cut. Finesse depends only on R (π√R/(1−R)). The spacing sets the FSR — how far apart the peaks sit — not how sharp.
"An etalon transmits one wavelength." Cut. It transmits a comb — infinitely many equally spaced resonances. You isolate one order with a coarse pre-filter.
"Higher R is always better." Kept, corrected. Higher R narrows peaks but slashes throughput and is capped by mirror loss and flatness — resolution and brightness trade off.
The red team's move: drop the sin² from the phase term — use a linear T = 1/(1 + F·δ). The transmission stops being periodic; the sharp, equally spaced resonances vanish. The blue team's witness (window 7) is watching.
Without sin², T no longer returns to 1 every 2π — the comb collapses into a single lopsided decay. The witness recomputes, fails the resonance/periodicity check, and turns red. Nothing is faked; the attack is real and it is caught.