Fabry-Perot Cavity

The resonance condition

Picture light entering a cavity of length L, refractive index n, bounded by two mirrors with amplitude reflection coefficient r = √R. A wave makes one round trip in time 2nL/c and accumulates phase

δ = (2π/λ) · 2nL = 4πnL/λ

For all the multiply transmitted contributions to add constructively at the output, this phase must be a multiple of 2π:

2nL = m·λ   ⇒   λ_m = 2nL/m   (m = 1, 2, 3, ...)

In frequency, ν_m = m·c/(2nL). Consecutive modes are spaced by the free spectral range FSR_ν = c/(2nL).

Airy transmission

Summing the infinite geometric series of partially transmitted beams gives the Airy formula:

T(δ) = T_max / [1 + (4R/(1-R)²)·sin²(δ/2)]

Peaks at δ = 2πm, valleys halfway between. Peak width (FWHM) in phase is Δδ = 2(1-R)/√R. The ratio of FSR to FWHM is the finesse:

F = π·√R / (1 - R)

That single number determines how good the cavity is.

Finesse by mirror reflectivity

Mirror R	Finesse F	Photon lifetime (10 cm cavity)	Q at 1064 nm
0.90	~30	~3 ns	5.6 × 10⁶
0.99	~313	~33 ns	5.9 × 10⁷
0.999	~3 140	~330 ns	5.9 × 10⁸
0.9999	~31 400	~3.3 µs	5.9 × 10⁹
0.999995	~628 000	~67 µs	1.2 × 10¹¹

Photon lifetime τ = (n·L)/(c·(1-R)) for one mirror loss; Q = 2π·ν·τ. The longer a photon survives in the cavity, the narrower the resonance.

Worked example — sizing an etalon for a 50 GHz DWDM filter

You need an etalon that isolates one ITU channel from its neighbours at 50 GHz spacing in the 1550 nm telecom window. Required suppression at the adjacent channel: better than -25 dB.

1. Set FSR_ν = 50 GHz to match the channel spacing. Solving c/(2nL) = 50 GHz with n = 1.5 (fused silica): L = c/(2·1.5·50 × 10⁹) = 2.0 mm.
2. For -25 dB rejection at the adjacent peak (one channel away = FSR/2 from any resonance midpoint? No — sin²(δ/2) = 1 there). Demand T_adj/T_peak ≤ 10⁻².⁵ → solve 1 + (4R/(1-R)²) = 10².⁵ → 4R/(1-R)² ≈ 315 → R ≈ 0.90.
3. Check finesse: F = π·√0.90/(0.10) = 29.8. Linewidth = 50 GHz / 29.8 ≈ 1.68 GHz — narrower than the typical 10 GHz channel allocation, ✓.
4. Mirror tolerance: ΔL/L = Δν/ν → for 1 GHz drift, ΔL = 1 GHz · 2 mm / 200 THz ≈ 10 pm. Use thermally compensated mount.

So a 2.0 mm silica etalon with 90% reflectors and 10 pm/Hz thermal sensitivity does the job.

Real-world Fabry-Perot systems

System	Role of the cavity
Laser resonator	Defines longitudinal mode spectrum; one mirror partially transmits the output
DWDM telecom filter	Selects one wavelength channel out of dozens in a single fibre
Optical spectrum analyser	Scanned cavity sweeps through ν — output trace is the spectrum
Laser line narrowing	Pound-Drever-Hall locks a laser to a high-F reference cavity, sub-Hz linewidth
LIGO arm cavities	R ≈ 0.99, ~280 round trips multiply effective arm by ~280×
Cavity-enhanced absorption	CRDS measures trace gases via ring-down lifetime in a high-F cavity
Atomic frequency reference	ULE cavity at 1.5 µm or 698 nm provides the local oscillator for optical clocks
Anti-reflection coating	Dielectric stack is a chain of micro Fabry-Perots designed for destructive reflection

JavaScript — Fabry-Perot calculations

const c = 2.99792458e8;

// Free spectral range (Hz)
function fsrHz(L_m, n = 1) { return c / (2 * n * L_m); }

console.log(`FSR for L=1 cm, n=1: ${(fsrHz(0.01) / 1e9).toFixed(2)} GHz`); // 15.0
console.log(`FSR for L=10 cm: ${(fsrHz(0.1) / 1e9).toFixed(2)} GHz`); // 1.50

// Finesse from mirror reflectivity
function finesse(R) { return Math.PI * Math.sqrt(R) / (1 - R); }

console.log(`F(R=0.90): ${finesse(0.90).toFixed(1)}`);   // 29.8
console.log(`F(R=0.99): ${finesse(0.99).toFixed(0)}`);   // 313
console.log(`F(R=0.9999): ${finesse(0.9999).toFixed(0)}`);// 31400

// Resonance wavelengths in the cavity
function resonantWavelengths(L_m, n = 1, m_min = 1, m_max = 5) {
  const list = [];
  for (let m = m_min; m <= m_max; m++) {
    list.push({ m, lambda_m: 2 * n * L_m / m });
  }
  return list;
}

// 10 cm cavity → modes around 1064 nm
const targetLambda = 1064e-9;
const L = 0.1;
const m0 = Math.round(2 * L / targetLambda);
console.log(`Mode m₀ near 1064 nm: ${m0}, λ = ${(2*L/m0 * 1e9).toFixed(6)} nm`); // ~187970

// Linewidth (FWHM) in Hz
function linewidthHz(L_m, R, n = 1) { return fsrHz(L_m, n) / finesse(R); }

console.log(`Linewidth (10 cm, R=0.9999): ${(linewidthHz(0.1, 0.9999) / 1e3).toFixed(2)} kHz`); // ~48 kHz

// Photon lifetime
function photonLifetime_s(L_m, R, n = 1) {
  return (n * L_m) / (c * (1 - R));
}
console.log(`τ (10 cm, R=0.999): ${(photonLifetime_s(0.1, 0.999) * 1e9).toFixed(1)} ns`); // ~334 ns

// Q factor
function qFactor(L_m, R, lambda_m, n = 1) {
  const nu = c / lambda_m;
  return 2 * Math.PI * nu * photonLifetime_s(L_m, R, n);
}
console.log(`Q (10 cm, R=0.999, 1064 nm): ${qFactor(0.1, 0.999, 1064e-9).toExponential(2)}`);
// ~5.9e8

// Airy transmission as function of detuning
function airyT(detuning_Hz, FSR_Hz, R) {
  const F = finesse(R);
  const delta = 2 * Math.PI * detuning_Hz / FSR_Hz;
  return 1 / (1 + (4 * R / (1 - R) ** 2) * Math.sin(delta / 2) ** 2);
}
// At resonance T=1, at FSR/4 detuning, mid-skirt:
console.log(`T@FSR/4 (R=0.99): ${airyT(fsrHz(0.1)/4, fsrHz(0.1), 0.99).toExponential(2)}`);

Where Fabry-Perot cavities matter

• Laser design. Every laser is a Fabry-Perot — sets mode spacing, output coupling, and longitudinal stability.
• Frequency standards. Optical atomic clocks lock their interrogation laser to a ULE Fabry-Perot at F > 10⁵.
• Telecom. DWDM mux/demux, gain-flattening filters, channel selection.
• Spectroscopy. Resolution-enhancing post-filter for diode lasers; cavity-enhanced absorption for trace gases.
• Sensors. Etalon-based pressure, temperature, refractive-index sensors — read by tracking peak position.
• Gravitational waves. LIGO/Virgo/KAGRA use kilometre-scale Fabry-Perot arms to multiply Michelson sensitivity.
• Quantum optics. Cavity QED experiments confine single photons with single atoms for strong coupling.

Common mistakes

• Forgetting the index n. Resonance condition is 2nL = mλ — a solid etalon with n = 1.5 has 1.5× the optical length of the same physical thickness in air.
• Mixing wavelength FSR and frequency FSR. They aren't equal. ΔνFSR = c/(2nL) is constant; Δλ_FSR = λ²/(2nL) is not (depends on λ).
• Conflating finesse and Q. Finesse counts how many half-widths fit in one FSR. Q counts how many radians the field survives before decaying. They differ by 2π·ν·(2nL/c).
• Treating walk-off as negligible. Off-axis rays walk out the side of a finite-aperture cavity within a few bounces, killing finesse. Curved mirrors (confocal/concentric) confine the mode transversely.
• Ignoring substrate absorption. Coating losses set an effective max finesse — even with R → 1, scatter and absorption at ~1 ppm cap modern cavities at ~10⁷.
• Forgetting drift sensitivity. ΔL = (1 nm) on a 10 cm cavity at 200 THz shifts modes by ν·ΔL/L = 2 GHz. Stability requires temperature-controlled spacers and vacuum.