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Fabry–Pérot and Passive Optical Resonators

Two mirrors, no gain. One round-trip factor sets the Airy spectrum, the finesse, the photon lifetime and the ring-down; one round-trip ray matrix sets stability, the Gaussian mode and the transverse-mode comb.

Click the spectrum to choose the probe frequency • Drag the cavity point on the g₁–g₂ diagram (or focus it and use the arrow keys)
FSR — Free spectral range
Δν — Linewidth (FWHM)
𝓕 — Finesse
τₚ — Photon lifetime
g₁g₂ — Stability
w₀ — TEM₀₀ waist

Cavity and TEM₀₀ beam paraxial, physical w(z)

1/e² intensity radius ±w(z) of the self-consistent TEM₀₀ mode, in µm, against axial position in mm (so the two axes use different scales). Mirror arcs are drawn with the correct sign of curvature but exaggerated sagitta.

Transmission and reflection spectrum Airy function

Power transmittance T, reflectance R and internal loss 1 − R − T for unit input power in a mode-matched TEM₀₀ (plane wave if no mode exists), against ν − νq,00. The white cursor is the probe frequency used by the standing-wave and ring-down panels: click or drag to move it (arrow keys when focused).

Transverse-mode comb over the longitudinal comb Hermite–Gauss TEMmn

Each tick is a resonance νq,mn = ν₀ + {[q + (m+n+1)ψ/π]·c/(2L) − nν₀}/ng of order m + n (row). The tick width is the FWHM linewidth (at least 2 px). Dashed verticals: TEM₀₀ longitudinal comb. Positions only: how strongly each mode is excited depends on mode matching, which is not modelled.

Intracavity standing wave at the probe frequency

Top: envelope of |E|²/|Ein|² over the whole cavity (max and min of the standing-wave pattern; the gap between them shows the travelling-wave part). Bottom: zoom over three medium wavelengths λ₀/n with the local amplitude ±|E|/|Ein| (solid) and the instantaneous field Re{E e−iωt} (dashed, slowed by ~10¹⁵).

Buildup and ring-down (round-trip sum)

Input switched on at t = 0 at the probe frequency and off after ≈ 8 τₚ. Transmitted power per unit input, sampled once per round trip (the pulses leave mirror 2 every Trt). Dashed: e−(t − toff)/τₚ with τₚ from the survival factor.

g₁–g₂ stability diagram round-trip ABCD

Shaded: stable, |(A + D)/2| < 1 ⇔ 0 < g₁g₂ < 1. Solid axes and the dashed hyperbola g₁g₂ = 1 are the marginal boundaries. The dotted line is the path the point follows if only L changes (fixed mirrors). Drag the point or use arrow keys (Shift for coarse steps); it sets Rc = L/(1 − g).

Mode and boundary readouts

Transverse-mode offsets from TEM₀₀ of the same q
m + n Δν Δν / FSR (mod 1) Degeneracy
FSR = c/(2ngL)
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Round-trip time Trt
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Round-trip survival S = R₁R₂(1−ℓ)²
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Finesse (exact)
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Finesse ≈ π√ρ/(1−ρ)
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Coefficient of finesse 4ρ/(1−ρ)²
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Linewidth Δν (FWHM)
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Photon lifetime τₚ = −Trt/ln S
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τₚ fitted to ring-down
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2π τₚ Δν
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Quality factor Q = ν/Δν
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T on resonance
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R on resonance
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Loss on resonance
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Circulating / input power
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Probe: T, R
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Longitudinal order q
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νq,00 (λ)
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g₁, g₂
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m = (A + D)/2
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Gouy phase ψ (one way)
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Transverse spacing FSR·ψ/π
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Waist w₀, position from M1
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Spot sizes w₁, w₂
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Rayleigh range zR (in medium)
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💡 How to use

Learn with this tool

Learning objectives

  • Derive the Airy transmission of a two-mirror cavity from a geometric sum of round trips, and read FSR, linewidth and finesse from it.
  • Show that linewidth, finesse, photon lifetime and ring-down all follow from one round-trip power survival factor S = R₁R₂(1 − ℓ)².
  • Decide cavity stability from the round-trip ABCD matrix, find the self-consistent Gaussian mode, and predict the transverse-mode frequencies from the Gouy phase.

Prerequisites

  • Standing waves and reflection phase (standing waves).
  • Two-beam and multiple-beam interference (interference, thin films).
  • Gaussian beams, q parameter and ABCD matrices (Gaussian beams).
  • Complex phasors with the convention E = Re{E exp[i(kz − ωt)]}.

Model

A monochromatic field of unit power enters through mirror 1 (z = 0) and leaves through mirror 2 (z = L). The gap has phase index n and, optionally, group index ng. Mirrors are lossless with power reflectance Ri; amplitudes are power-normalised, ti = √(1 − Ri), and the reflection seen from inside is −√Ri (so a perfect mirror is a field node, as in the standing-wave tool). Each pass keeps a fraction 1 − ℓ of the power. The round-trip field factor and the resulting responses are

g(ν) = ρ eiδ(ν), ρ = √(R₁R₂)(1 − ℓ), S = ρ² (round-trip power survival)

t = t₁t₂√(1−ℓ) eiδ/2 / (1 − g), T = Tmax / [1 + (4ρ/(1−ρ)²) sin²(δ/2)], Tmax = (1−R₁)(1−R₂)(1−ℓ)/(1−ρ)²

δ(ν) = (4πL/c)[nν₀ + ng(ν − ν₀)] − 2(m+n+1)ψ, FSR = c/(2ngL)

𝓕 = π / [2 arcsin((1−ρ)/(2√ρ))] ≈ π√ρ/(1−ρ), Δν = FSR/𝓕, τₚ = −Trt/ln S, Trt = 2ngL/c

M = M₁·P(L)·M₂·P(L), stable ⇔ |(A + D)/2| < 1 ⇔ 0 < g₁g₂ < 1, gi = 1 − L/Rci

νq,mn = (c/2nL)[q + (m+n+1) ψ/π], ψ = arccos(±√(g₁g₂)), + if g₁, g₂ > 0, − if g₁, g₂ < 0

R₁, R₂
mirror power reflectances (input, output); Ti = 1 − Ri
ℓ
internal power loss per single pass (distributed loss α = −ln(1 − ℓ)/L)
ρ, S
round-trip amplitude factor and power survival factor; S is the only loss parameter in 𝓕, Δν and τₚ
n, ng
phase index (fixes the absolute resonance order q) and group index (fixes the spacing and Trt)
δ
round-trip phase; resonance when δ = 2πq
𝓕, Δν, Q
finesse FSR/Δν, full width at half maximum, quality factor ν/Δν
τₚ
photon (energy) lifetime: stored energy decays as e−t/τₚ
Rc1, Rc2, g₁, g₂
mirror radii (> 0 concave toward the cavity) and g parameters
A, B, C, D
round-trip ray matrix for rays [y, θ] starting just inside mirror 1
ψ
one-way Gouy phase of TEM₀₀ between the mirrors, 0 ≤ ψ ≤ π
w₀, zR, q
waist 1/e² intensity radius, Rayleigh range π n w₀²/λ₀, complex beam parameter q = z + i zR

Assumptions and validity: linear, time-invariant, passive cavity (no gain); lossless mirrors whose reflection phase does not depend on frequency; paraxial beams with mirrors large compared with w (no diffraction loss); a linearised dispersion n(ν)ν = nν₀ + ng(ν − ν₀) over the plotted range; an input that is monochromatic and mode-matched to TEM₀₀ for the spectrum, standing wave and ring-down. With no bound Gaussian mode (planar, unstable) the spectrum is the plane-wave idealisation.

Derivation: Airy function, finesse and lifetime from one round-trip factor

After mirror 1 the forward field is t₁. Every round trip multiplies it by g = (−√R₂)(−√R₁)(1 − ℓ)eiδ = ρeiδ. The circulating field is the geometric series t₁(1 + g + g² + …) = t₁/(1 − g), and the transmitted field is that times t₂√(1 − ℓ)eiδ/2. Using |1 − ρeiδ|² = (1 − ρ)² + 4ρ sin²(δ/2) gives the Airy function above. The reflected field is √R₁ − (1 − R₁)√R₂(1 − ℓ)eiδ/(1 − g); for ℓ = 0 one finds R + T = 1 exactly.

Linewidth. T falls to Tmax/2 when sin(δ/2) = (1 − ρ)/(2√ρ), so the full width in phase is 4 arcsin[(1 − ρ)/(2√ρ)] and 𝓕 = 2π/(full width). For ρ → 1, arcsin x ≈ x gives 𝓕 ≈ π√ρ/(1 − ρ); for a symmetric lossless cavity ρ = R and 𝓕 ≈ π√R/(1 − R). Below ρ = 3 − 2√2 ≈ 0.17 the peaks never fall to half-contrast and 𝓕 is undefined.

Lifetime. Switching off the input leaves Em = E₀gm, so the power after m round trips is Sm = exp(m ln S) = exp(−t/τₚ) with t = mTrt and τₚ = −Trt/ln S. Since ln S = 2 ln ρ ≈ −2(1 − ρ), τₚ ≈ Trt/[2(1 − ρ)] and 2πτₚΔν → 1: the Airy line is the Fourier transform of the exponential ring-down.

Transverse modes. A Gaussian q reproduces itself after a round trip when q = (Aq + B)/(Cq + D), i.e. 1/q = (D − A)/(2B) − i√(1 − m²)/|B| with m = (A + D)/2; a real spot size needs |m| < 1. For two mirrors m = 2g₁g₂ − 1, so stability is 0 < g₁g₂ < 1. A Hermite–Gauss TEMmn accumulates Gouy phase (m + n + 1)ψ per pass, with cos 2ψ = m (round trip), i.e. cos ψ = ±√(g₁g₂); the sign is that of g₁ (and g₂). Resonance 2kL − 2(m+n+1)ψ = 2πq gives the frequency formula. The tool also computes ψ independently as arctan[(L − zw)/zR] + arctan(zw/zR) from the propagated beam.

Marginal boundaries. Planar (g₁ = g₂ = 1): ψ = 0, every TEMmn is degenerate and the beam is unbounded. Concentric (g₁ = g₂ = −1): ψ = π, the transverse shift equals one FSR (degenerate again) and the waist collapses to a point. Symmetric confocal (g₁ = g₂ = 0): ψ = π/2, M = −I, so even orders coincide with TEM₀₀ and odd orders sit half-way; the ray matrix no longer selects q, and the curvature-matched mode w₀² = λ₀L/(2πn) is shown. Elsewhere on the axes (g₁g₂ = 0) one spot size goes to zero and the other diverges.

Exercise 1: finesse and linewidth of a symmetric etalon

  1. For R₁ = R₂ = 90 %, no loss, L = 10 mm in air, predict FSR, 𝓕, Δν and T on resonance.
  2. Load Symmetric FP, then tick Zoom spectrum and press Go to half maximum.
  3. Read the cursor detuning at T = Tmax/2 and double it; compare with the Δν readout and with FSR/𝓕.
  4. Why does the lossless symmetric cavity transmit 100 % on resonance even though each mirror reflects 90 %?
Show answer

FSR = c/(2L) = 14.99 GHz; 𝓕 = 29.79 (the approximation π√0.9/0.1 gives 29.80); Δν = 503 MHz; T = 1. On resonance the field leaking back out through mirror 1 is exactly out of phase with the promptly reflected field and has the same amplitude, so the reflected waves cancel (R = 0); with no loss all the power must be transmitted. The circulating power is (1 − R₁)/(1 − R)² = 10 times the input.

Exercise 2: one survival factor for lifetime and linewidth

  1. R₁ = R₂ = 99.95 %, L = 500 mm. Predict τₚ. Then predict τₚ and T on resonance after adding ℓ = 0.05 % internal loss per pass.
  2. Load Ring-down; tick Logarithmic. Then type 0.05 into the loss box.
  3. Read the fitted ring-down lifetime, Δν, and T on resonance in both cases.
  4. Why does a loss much smaller than the mirror transmission halve τₚ and cut T to a quarter?
Show answer

Trt = 3.336 ns and S = 0.9995² = 0.99900, so τₚ = 3.33 µs (Δν = 47.7 kHz, 2πτₚΔν = 1.000). With ℓ = 0.05 %, S = 0.9995⁴ = 0.99800: τₚ = 1.67 µs, Δν = 95.5 kHz. Tmax = T₁T₂(1 − ℓ)/(1 − ρ)² falls to 0.25, R rises to 0.25 and 0.50 is absorbed. Internal loss is suffered on both passes of a round trip, so 2ℓ = 0.1 % equals the combined mirror transmission T₁ + T₂ = 0.1 % and doubles 1 − ρ; ring-down spectroscopy uses exactly this to measure tiny absorptions.

Exercise 3 (limiting cases): walk the stability boundaries

  1. With L = 100 mm, predict ψ and the TEM₀₁ and TEM₀₂ offsets (as fractions of the FSR) for Rc = 1000 mm, 100 mm (confocal) and 50.5 mm (near-concentric). What happens to w₀ in each case?
  2. Load Confocal and Near-concentric, then drag the g₁–g₂ point toward (1, 1) along the diagonal.
  3. Read the mode table, the Gouy phase and w₀; watch the comb collapse onto the longitudinal comb.
  4. Explain why the transverse modes become degenerate at both ends of the stable diagonal but not in the middle.
Show answer

Rc = 1000 mm: g = 0.9, ψ = arccos 0.9 = 25.8°, TEM₀₁ at 0.144 FSR, w₀ large (272 µm at 1064 nm). Confocal: ψ = 90°, TEM₀₁ at 0.5 FSR and TEM₀₂ at 1.0 FSR (degenerate with the next q), w₀ = √(λL/2π) = 130 µm. Near-concentric: g = −0.980, ψ = 168.6°, TEM₀₁ at 0.937 FSR, w₀ = 41 µm and w at the mirrors = 414 µm. At g → 1 the Gouy phase vanishes (plane waves); at g → −1 it tends to π, a full FSR. Either way all transverse orders fall on the longitudinal comb, which is why marginal cavities are hard to mode-match and very sensitive to misalignment.

Exercise 4: critical coupling

  1. R₂ = 99 % and ℓ = 2 % per pass. Which R₁ makes the reflected power vanish on resonance? Where does the power go?
  2. Load Critical coupling, then move R₁ a few percent either side.
  3. Read R, T and loss on resonance for each R₁.
  4. Why is R = 0 reached for R₁ = R₂(1 − ℓ)² and not for R₁ = R₂?
Show answer

The reflected field is √R₁ − (1 − R₁)ρ/(√R₁(1 − ρ)) on resonance, which vanishes when √R₁ = √R₂(1 − ℓ) = ρ/√R₁, i.e. R₁ = 0.99 × 0.98² = 95.08 %. Then T = 0.199 and 0.801 is absorbed inside. The input mirror's transmission must equal all the other round-trip losses (impedance matching): with R₁ larger the cavity is under-coupled, with R₁ smaller over-coupled, and R rises in both directions.

Worked example: a 10 cm Nd:YAG reference cavity

Two identical mirrors, R = 99 %, Rc = 200 mm, spaced L = 100 mm in air, probed at λ₀ = 1064 nm (preset Worked example).

  1. FSR = c/(2L) = 1.499 GHz; Trt = 2L/c = 0.667 ns.
  2. ρ = R = 0.99, so 𝓕 = π/[2 arcsin(0.01/(2√0.99))] = 312.58 (approximation 312.58); Δν = 1.499 GHz/312.6 = 4.80 MHz.
  3. S = 0.9801, τₚ = −0.667 ns/ln 0.9801 = 33.2 ns; 2πτₚΔν = 1.000.
  4. g = 1 − 100/200 = 0.5, g₁g₂ = 0.25: stable; m = (A + D)/2 = 2(0.25) − 1 = −0.5.
  5. ψ = arccos(0.5) = 60°, so TEMmn sit (m + n) × FSR/3 = (m + n) × 499.7 MHz above TEM₀₀; order 3 falls on the next longitudinal mode.
  6. w₀² = (λL/π)√[g²(1 − g²)/(2g − 2g²)²] = (3.387 × 10⁻⁸ m²)(0.866): w₀ = 171 µm at the centre; w₁ = w₂ = (λL/π)½[1/(1 − g²)]¼ = 198 µm.
  7. Q = ν/Δν = 2.818 × 10¹⁴/4.80 × 10⁶ = 5.9 × 10⁷.

Check every line in the readouts; the ring-down fit reproduces τₚ, and the transverse-mode table shows the exact 1/3-FSR spacing.

When the model fails

  • Real mirrors: dielectric stacks have a reflection phase that varies with frequency and a penetration depth that adds to L; metallic mirrors absorb. Model them with thin films and add the phase to δ.
  • Finite mirrors and misalignment: near the stability boundaries w on a mirror grows without bound, so aperture diffraction loss, tilt and figure errors dominate; the paraxial ABCD result then overestimates 𝓕.
  • Input beam: a real beam that is not mode-matched excites many TEMmn; the comb shows where they resonate, not how strongly. A laser linewidth comparable to Δν averages the Airy peak.
  • Strong dispersion: the linear ng model fails over bandwidths where group-velocity dispersion matters (see dispersion and pulses); the FSR then varies across the spectrum.
  • Intense or active cavities: thermal lensing, photothermal effects, birefringence and gain (laser cavity) are outside this passive, linear model.
  • Very fast switching: the ring-down assumes an instantaneous switch and samples once per round trip; for switching times shorter than Trt the pulse shape inside a round trip matters.

References

  • A. E. Siegman, Lasers, University Science Books (1986), chapters 11, 17 and 19 (Fabry–Pérot, Gaussian modes, stability diagram).
  • H. Kogelnik and T. Li, "Laser beams and resonators", Appl. Opt. 5, 1550 (1966).
  • N. Ismail, C. C. Kores, D. Geskus and M. Pollnau, "Fabry–Pérot resonator: spectral line shapes, generic and related Airy distributions, linewidths, finesses, and performance at low or frequency-dependent reflectivity", Opt. Express 24, 16366 (2016).
  • B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed., Wiley (2019), chapter 11 (resonator optics).
  • E. Hecht, Optics, 5th ed., Pearson (2017), §9.6 (multiple-beam interference).