Fabry-Perot Interferometers & Optical Resonators

Take two parallel mirrors and bounce light between them. That's a Fabry-Perot interferometer, and it's one of the most useful devices in optics. Depending on the mirror reflectance and spacing, it can be a precision wavelength filter, a laser cavity, or a spectrum analyzer that resolves individual laser modes just megahertz apart.

The physics is simple: light bouncing between two mirrors interferes with itself. At wavelengths where the round-trip path is an exact integer number of wavelengths, the interference is constructive and the cavity transmits. At all other wavelengths, the multiply-reflected beams cancel and the cavity reflects. The sharpness of this wavelength selection depends entirely on how reflective the mirrors are.

The Surprise: 100% Transmission Through 99% Mirrors

Here's something counterintuitive. Put two mirrors with 99% reflectance facing each other and shine light at the right wavelength. You might expect 99% of the light to bounce off the first mirror and almost nothing to get through. Instead, 100% of the light transmits through both mirrors. Every photon makes it through.

This happens because at resonance, the reflected beams from the two mirrors interfere destructively. The reflections cancel perfectly, and all the light ends up on the transmitted side. This works at any reflectance: 90%, 99%, even 99.99%. The condition is that the round-trip optical path ($2nL$, where n is the refractive index and L is the mirror spacing) equals an exact integer number of wavelengths.

What about a plain glass plate? A 1 mm BK7 plate has $R=4.2$ at each surface. The transmission varies between 100% and 84.5%, a contrast of just 1.18:1. You'd barely notice the fringes. This is why Fabry-Perot etalons need high-reflectance coatings: without them, there's no useful filtering.

The Airy function captures this mathematically:

where $\delta =4\pi nL\mathrm{/}\lambda$ is the round-trip phase, $F=4R_{\text{eff}}\mathrm{/}(1-R_{\text{eff}}{)}^2$ is the coefficient of finesse, and $R_{\text{eff}}=\sqrt{R_1{R}_2}$ for mirrors with reflectances $R_1$ and $R_2$. For lossless identical mirrors, $T_{\max }=1$. Between resonances, $T$ drops to $T_{\min }=1\mathrm{/}(1+F)$. For $R=90$, $F=360$ and $T_{\min }=0.28$, a contrast ratio of 361:1 (25.6 dB).

Free Spectral Range, Finesse, and Linewidth

The transmission peaks are equally spaced in frequency. The spacing is the free spectral range (FSR):

For an air-spaced etalon with $L=10$ mm: FSR = 14.99 GHz. In wavelength units at 550 nm: $\text{FSR}={\lambda }^2\mathrm{/}(2nL)=15.1$ pm. These peaks correspond to successive longitudinal modes of the cavity, differing by one wavelength in the round trip.

The finesse measures how sharp the peaks are relative to the spacing:

The full width at half maximum (FWHM) of each peak is $\text{FWHM}=\text{FSR}\mathrm{/}\mathcal{F}$.

Here's a practical table for the $R=90$, $L=10$ mm air-spaced etalon:

The Q-factor (quality factor) is $Q={\nu }_0\mathrm{/}\text{FWHM}$, where ${\nu }_0$ is the optical frequency. The photon lifetime is the average time a photon bounces inside the cavity before escaping. Higher finesse means longer photon lifetime, which means more stored energy at resonance.

Finesse and Reflectance

The relationship between mirror reflectance and finesse is steep and nonlinear:

Below about $R=30$, the peaks are so broad they overlap and the etalon barely filters at all. Above $R=99$, the peaks become extraordinarily narrow.

In practice, the achievable finesse is limited by mirror quality, not just reflectance. Surface flatness, alignment, and beam divergence each contribute an effective finesse, and the total is approximately the harmonic mean of squares:

A cavity with $R=99.9$ mirrors (${\mathcal{F}}_{\text{mirror}}=\mathrm{3,140}$) and modest surface quality (${\mathcal{F}}_{\text{surface}}\approx 100$) achieves only ${\mathcal{F}}_{\text{eff}}\approx 100$. The mirrors are the easy part; the surfaces are the hard part. This is why high-finesse cavity mirrors are super-polished to better than $\lambda \mathrm{/}1000$ and cost thousands of dollars each.

Absorption and Loss

Remember the 100% transmission at resonance? That's only true for lossless mirrors. Real mirrors absorb a small fraction $A$ of the incident light, and the peak transmission drops:

For $R=95$ mirrors with $A=0.2$ absorption per surface: $T_{\max }$ drops to 92.4%. The "missing" 7.6% is absorbed inside the cavity, amplified by the multiple bounces. For high-finesse cavities ($R=99.9$, $A=10$ ppm), even tiny absorption matters: it limits $T_{\max }$ to about 98%. This is why the thin-film coating quality directly determines etalon performance, and why the best cavity mirrors use ion-beam-sputtered coatings with sub-ppm absorption.

From Etalon to Laser: Optical Resonator Stability

Everything above treats the Fabry-Perot as a spectral filter. But the same two-mirror cavity is also the foundation of every laser, and for lasers we need to worry about whether the beam stays inside the cavity.

Not every pair of mirrors makes a stable resonator. If the beam walks off the mirrors after many round trips, the cavity is unstable and won't sustain a laser mode. Stability depends on the g-parameters:

where $R_1$ and $R_2$ are the radii of curvature (positive for concave, negative for convex, infinity for flat). The cavity is stable when:

This defines the shaded region in the $g_1$–$g_2$ stability diagram. Several classic configurations sit on the boundaries or inside:

Worked example: A He-Ne laser with a flat mirror and a 1-meter concave mirror, $L=300$ mm:

$g_1=1$ (flat), $g_2=1-300\mathrm{/}1000=0.70$, $g_1{g}_2=0.70$ — stable. The beam waist sits at the flat mirror ($w_0=304;\mu \text{m}$) and the beam is slightly larger at the curved mirror ($w=363;\mu \text{m}$). The FSR is $c\mathrm{/}(2L)=500$ MHz.

Applications

Laser cavities. Every laser is a Fabry-Perot resonator. The cavity selects which longitudinal modes oscillate (only those within the gain bandwidth and at multiples of FSR), and the mode spacing determines whether the laser runs single-mode or multi-mode. A 300 mm He-Ne cavity has FSR = 500 MHz, and the neon gain bandwidth is about 1.5 GHz, so roughly 3 longitudinal modes can oscillate simultaneously. Telecom etalon filters use the same physics in reverse: a fused-silica etalon with FSR matched to the 50 GHz channel spacing selects one DWDM channel and rejects the rest.

Gravitational wave detection. LIGO uses Fabry-Perot cavities with 4-km arm lengths and finesse around 450 to detect length changes of ${10}^{-18}$ meters. That's one ten-thousandth the diameter of a proton. Each photon bounces roughly 300 times before escaping, with a storage time of about 1 ms. The sensitivity of the entire detector comes down to cavity finesse: more bounces mean more accumulated phase shift per gravitational wave.

Going Further

The Fabry-Perot principle extends beyond two flat mirrors. Ring cavities, multi-mirror cavities, and coupled-cavity systems build on the same interference physics. The Photizon Resonator Calculator handles both etalon mode (spectral analysis) and resonator mode (cavity stability and Gaussian beam analysis). Try loading the presets and adjusting mirror reflectance to see how finesse, linewidth, and peak transmission change in real time.

References

1. G. Hernandez, Fabry-Perot Interferometers (Cambridge University Press, 1986). The comprehensive monograph on etalon design and applications.
2. A. E. Siegman, Lasers (University Science Books, 1986), Ch. 11–14. Definitive treatment of optical resonators, stability, and Gaussian beam modes.
3. B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019), Ch. 9–10. Clear pedagogical treatment of resonators and laser theory.
4. E. Hecht, Optics, 5th ed. (Pearson, 2017), Ch. 9. Undergraduate-level treatment of multiple-beam interference and the Fabry-Perot interferometer.

Keep Reading

Next: how to compute the Q-factor of a cavity →

Ring resonators are curved Fabry-Perot cavities without mirrors → Ring Resonators in Photonics