Fabry-Perot Interferometer Simulator — Finesse and Transmission
Visualize the Airy transmission spectrum of light bouncing between two high-reflectivity mirrors. Compute free spectral range, finesse, FWHM, and Q factor in real time.
Parameters
Mirror spacing d
mm
Mirror reflectivity R
—
Wavelength λ
nm
Incidence angle θ
°
Air-filled (n = 1.0), lossless mirrors assumed. Default is a green laser line (λ = 532 nm).
Results
—
Free spectral range FSR
—
Finesse F* = π√R/(1-R)
—
FWHM
—
Q factor (f/FWHM)
Airy transmission spectrum T(ν)
Horizontal: frequency (±2 FSR around c/λ) — Vertical: transmittance T — Red dot: current wavelength — Arrow: FSR — Yellow band: FWHM
Theory & Key Formulas
Light undergoes multiple reflections between two parallel mirrors. With phase δ = (4πd/λ) cosθ, the transmittance is the Airy function:
$$T(\delta) = \frac{1}{1 + F\,\sin^2(\delta/2)}, \quad F = \frac{4R}{(1-R)^2}$$
Free spectral range (frequency spacing between adjacent peaks):
Resonance condition δ = 2πm gives λ_m = 2nd cosθ / m. The closer R is to 1, the larger F* and the sharper the peaks.
About the Fabry-Perot Interferometer Simulator
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I keep hearing about Fabry-Perot interferometers. What do they actually do? Just two mirrors facing each other?
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In a nutshell, two parallel high-reflectivity mirrors bounce light back and forth hundreds of times, building up multi-beam interference. Only wavelengths satisfying the resonance condition transmit sharply. With the default settings (d = 1 mm, R = 0.95) the free spectral range card reads 149.9 GHz — that is the frequency spacing between adjacent peaks.
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The finesse shows 61.2. What does that number mean?
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Finesse is a dimensionless sharpness measure: FSR divided by the peak FWHM. The formula $F^*=\pi\sqrt{R}/(1-R)$ depends only on mirror reflectivity. R = 0.95 gives 61, R = 0.99 gives about 313, R = 0.999 gives about 3140. Finesse rockets upward as R approaches unity — that is how spectrum analyzers resolve sub-MHz features.
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Let me push the R slider up to 0.99… wow, the peaks become razor-thin!
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That is multi-beam interference at work. Higher R means more effective round-trips, so more partial waves interfere. Tiny detuning from resonance triggers strong destructive interference. The factor $F=4R/(1-R)^2$ in $T=1/(1+F\sin^2(\delta/2))$ controls this. Going R = 0.95 → 0.99 boosts F from 1520 to 39600 — a 26× sharpening.
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Changing mirror spacing d changes the FSR — at d = 10 mm I see 15 GHz.
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Exactly: FSR = c/(2nd cosθ). Longer d means slower round-trips, hence a smaller FSR. In an optical spectrum analyzer you pick d to match the band you want to resolve: short d (wide FSR) to look at a narrow laser line, long d (narrow FWHM) to pull apart closely spaced cavity modes. Real instruments scan d with a piezo at nanometer precision and read transmission peaks as an electrical signal.
Frequently Asked Questions
Almost every laser uses a Fabry-Perot cavity — a gain medium between two mirrors (output coupler and high reflector). The cavity FSR sets the longitudinal mode spacing and the finesse sets the mode linewidth. For example a 300 mm cavity has FSR ≈ 500 MHz, and with 99% mirrors the finesse ≈ 313 gives a mode linewidth ≈ 1.6 MHz — the theoretical lower bound for that laser's linewidth before noise broadens it further.
Fabry-Perot etalons make superb ultra-narrowband filters, but the transmission peaks repeat periodically, so a separate coarse filter is needed to pick out a single one. They are also angle-sensitive because the resonance condition is 2nd cosθ = mλ. Good collimation, mechanical stability, and temperature control (often with Invar spacers) are essential to keep d and θ constant in the field.
Ring resonators such as silicon microrings let light recirculate around a closed loop in one direction. Their transmission spectra are also a comb of Lorentzian peaks described by FSR and finesse. The differences: no back-reflection toward the source, and an optional drop port that taps a single resonance wavelength. They are widely used in silicon photonic WDM multiplexers and biosensors.
Anti-reflection coatings and dielectric multilayers use the same multi-beam interference principle, but the films are sub-wavelength thick and the per-interface reflectivities are low. The resulting finesse is only 1–a few, producing smooth reflectance curves rather than sharp peaks. Stacked dielectric mirrors with R > 99.9% are then used as the high-finesse mirrors of a Fabry-Perot interferometer, so the two technologies are deeply intertwined.
Real-World Applications
Laser cavities: Nearly all semiconductor, solid-state, and gas lasers are built around a Fabry-Perot cavity. Cavity length sets the lasing wavelength and longitudinal mode spacing (FSR); mirror reflectivity sets the trade-off between linewidth and output coupling. Precision lasers add an external Fabry-Perot etalon to select a single longitudinal mode and suppress mode hopping.
Optical spectrum analyzers and wavemeters: Scanning Fabry-Perot interferometers (with piezo-driven mirror spacing) are the workhorse instruments for measuring laser linewidth, mode structure, and frequency noise. Models with finesse above 10,000 resolve features at the few hundred kHz level. DWDM telecom systems use FP etalons for wavelength locking and channel monitoring on the 100 GHz grid.
Gravitational-wave detectors (LIGO, KAGRA): Laser-interferometric gravitational-wave detectors embed kilometer-scale Fabry-Perot cavities into each arm of a Michelson interferometer, multiplying the effective optical path by hundreds. A finesse around 450 and quality factor near 10^13 enable detection of length changes down to 10^-19 m — smaller than a proton — making thermal and quantum noise control the central engineering challenge.
Optical communications and spectroscopy: FP etalons referenced to the 100 GHz WDM grid stabilize transmitter wavelengths. In Raman and fluorescence spectroscopy, high-finesse FP filters reject the Rayleigh line right next to weak signal peaks. Atomic physics laboratories rely on ultra-stable FP cavities (finesse > 100,000) to resolve and lock to atomic transitions only megahertz wide.
Common Misconceptions and Pitfalls
The most frequent misconception is to assume that raising mirror reflectivity reduces transmission. Off resonance, yes — but the peak transmittance for a lossless interferometer stays at T = 1 regardless of R. Push the R slider from 0.5 up to 0.99: the peaks become narrower but their height does not drop. That is because on resonance (δ = 2πm) all the transmitted partial waves add up in phase, no matter how many round-trips contribute. Real mirrors with absorption and scatter do lower the peak height, but that is a separate loss term outside the ideal Airy model.
Another common slip is to confuse FSR with FWHM. FSR is the spacing between adjacent peaks; FWHM is the full width at half maximum of a single peak; finesse is the ratio of the two. In the displayed spectrum, the horizontal tick spacing is FSR and the yellow band is FWHM. With R = 0.95 we have FSR = 149.9 GHz but FWHM = 2.45 GHz — a 61× gap. Reading "resolution = FWHM" and "measurement range = FSR" makes it clear that finesse describes the trade-off between the two.
Finally, do not overlook the incidence angle θ. The resonance condition 2nd cosθ = mλ contains θ, so even a small tilt shifts the resonance wavelength. Push the θ slider from 0° to 10° and watch the FSR change slightly as cosθ drops. In real instruments, the beam divergence sets an "angular finesse" that limits the achievable spectral resolution, which is why collimation and aperture stops are essential. Conversely, the angular dependence creates the famous circular fringe pattern when an extended source illuminates the interferometer — the reason it is called an interferometer in the first place.