Fabry–Pérot: Resonance Between Two Mirrors

Two mirrors, one cavity, infinite reflections

A Fabry–Pérot interferometer is two closely spaced, highly reflective, parallel mirrors facing each other. Light entering the cavity bounces back and forth between them, and a small fraction leaks out through each mirror on every bounce. Because all those leaked beams are coherent copies of the same input, spaced apart by the same round-trip path difference, they interfere with each other outside the cavity — and depending on the exact spacing and wavelength, that interference is constructive (the cavity transmits almost everything) or destructive (it transmits almost nothing).

The Airy function

Summing the infinite series of internally reflected beams (a geometric series in the round-trip reflectivity) gives the transmitted intensity as a function of the round-trip phase δ — the Airy function:

T(δ) = 1 / [ 1 + F · sin²(δ/2) ]

F  = 4R / (1 - R)²        (the coefficient of finesse)
δ  = (4π / λ) · n · L · cos(θ)   (round-trip phase)
R  = mirror reflectivity (both mirrors assumed equal)
L  = mirror spacing, n = refractive index inside the cavity, θ = angle of incidence

When δ is a multiple of 2π the sine term vanishes and T reaches exactly 1 — perfect transmission, regardless of how reflective the mirrors are, because on resonance every one of the infinitely many internally reflected beams exits in phase. Off resonance, transmission drops rapidly, and how rapidly is set entirely by F.

Finesse: the sharpness dial

Finesse (F, distinct from the lowercase F above — conventionally written 𝓕) is the ratio of the spacing between transmission peaks to their width, and it is set purely by mirror reflectivity: 𝓕 = π√F / 2 = π√R / (1−R). Push R from 90% to 99% and 𝓕 roughly triples, from about 30 to about 312 — the transmission peaks get dramatically narrower without moving. This is the entire design lever of a Fabry–Pérot cavity: better mirrors buy you a sharper, more selective filter, at the cost of a dimmer peak (higher-finesse cavities also take longer to build up their internal field, an effect that becomes important for pulsed light).

Free spectral range and linewidth

Transmission peaks repeat periodically in frequency, spaced by the free spectral range (FSR) — the frequency interval corresponding to one extra round trip's worth of phase:

FSR = c / (2 n L)              (in frequency)
δν (FWHM linewidth) = FSR / 𝓕

FSR depends only on the cavity's physical length and the medium inside it — double the mirror spacing and the peaks halve in spacing. The individual peak's linewidth then depends on both FSR and finesse: a long cavity with modest finesse and a short cavity with very high finesse can have the same absolute linewidth, but very different FSR, which matters enormously in laser design, because a laser cavity's mode spacing (its FSR) determines how many longitudinal modes fit under the gain bandwidth.

Where it actually gets used

Fabry–Pérot cavities are the resonator inside most laser designs — the gain medium sits between the mirrors and only wavelengths satisfying the resonance condition build up enough round trips to reach lasing threshold. High-finesse Fabry–Pérot etalons are used as narrow-band optical filters in telecom (selecting one WDM channel from many), as the frequency reference in laser stabilisation, and — pushed to extraordinary finesse values in the millions — as the core sensing element of LIGO's gravitational-wave interferometer, where a sub-attometre change in mirror spacing has to shift the transmitted phase measurably.

Frequently asked questions

Try it live

Everything above runs in your browser — open Fabry-Pérot Interferometer and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.