Fabry–Pérot etalons: a spacer between two mirrors

Put a transparent layer between two partially reflecting mirrors and light bounces back and forth inside it. Where the round trip adds a whole number of wavelengths, every pass adds in phase and the cavity transmits strongly, even though each mirror alone reflects most of the light. Elsewhere the passes cancel and the cavity reflects. Fabry and Pérot built it in 1897 as a spectroscope; it is the resonator of every laser, the spacer of every narrowband filter, and the simplest case in which the multiple-reflection sum can be done by hand.

The Airy function

Let the two mirrors have amplitude coefficients $r_1, t_1$ and $r_2, t_2$ seen from inside the spacer, and let the round-trip phase be $\delta = \frac{4\pi}{\lambda} n d\cos\theta$. The transmitted amplitude is the sum over the number of round trips, a geometric series,

which is exactly what the transfer matrix returns for one layer, now with its multiple-beam origin visible. With equal lossless mirrors of reflectance $R = |r|^2$ the transmittance is the Airy function

where $\phi$ collects the mirrors' reflection phases. $F$ is the coefficient of finesse; for $R = 0.9$ it is 360, and the peaks, of unit height, are separated by valleys of depth $1/(1 + F)$. The Verification page's first film row compares the engine with this formula for a plain slab.

The resonance condition

For a spacer of index $n$ and thickness $d$ transmission peaks where the total round-trip phase is a multiple of $2\pi$,

For dielectric mirrors at their design wavelength the reflection phases are 0 or $\pi$, and the condition reduces to the spacer being a whole number of half waves. For metal mirrors they are neither: silver's reflection phase in the visible is around $-150°$ from the glass side, so every order sits a fraction of a free spectral range away from the half-wave position and the spacer that resonates at a given wavelength is thinner than $m\lambda/2n$. Dielectric mirrors have the complementary subtlety: their phase varies with wavelength as if the cavity were longer by the penetration depth of the mirror, which shifts the mode spacing.

Free spectral range, finesse, resolving power

Neighbouring orders are separated in wavelength by

so a thicker cavity packs its peaks closer (a 1 mm air gap has 150 GHz between orders, a 1 cm laser cavity 15 GHz). The width of each peak relative to that spacing is set by the mirrors alone, through the finesse

30 for $R = 0.9$, 300 for $R = 0.99$. The resolving power of the etalon as a spectroscope is $\lambda/\delta\lambda = m\mathcal{F}$, the order times the finesse, which is how a centimetre-thick etalon with modest mirrors resolves hyperfine structure that a grating cannot. Real etalons fall short of $\mathcal{F}$ through the flatness of their plates and the parallelism of the beam: a plate flat to $\lambda/N$ limits the finesse to about $N/2$, and a beam of half-angle $\alpha$ smears $\cos\theta$ by $\alpha^2/2$.

What absorption costs

For an ideal lossless cavity with equal mirrors the peaks reach $T = 1$ whatever $R$ is. Metal mirrors are not lossless. If each mirror absorbs a fraction $A$ with $R + T_\mathrm{m} + A = 1$, the peak transmittance becomes

the square of the mirrors' potential transmittance: what matters is the absorption relative to what the mirror lets through, not the absorption itself. A silver mirror with $R = 0.9$ and $A = 0.04$ limits the peak to 36 %; with $R = 0.97$ and the same $A$, to 2 %. Better metal mirrors sharpen the peaks but lower them, which is the dead end that led to all-dielectric filters.

import numpy as np
from pystratify import planar
from pystratify.docs import material  # n + ik of a tabulated page (refractiveindex.info format)

n = lambda wl: [1.0, material('docs/data/tutorials/Ag_Johnson.yml', wl), material('docs/data/tutorials/SiO2_Malitson.yml', wl), material('docs/data/tutorials/Ag_Johnson.yml', wl), 1.52]  # incident medium, layers, substrate
d = [np.inf, 0.025, 0.2, 0.025, np.inf]  # thicknesses, µm
lam = (400 + 2 * np.arange(201)) * 1e-3  # wavelength, µm
th = np.radians(0)
R, T = {}, {}
for pol in ('s', 'p'):
    rows = [planar.coh_tmm(pol, n(wl), d, th, wl) for wl in lam]
    R[pol], T[pol] = np.array([r['R'] for r in rows]), np.array([r['T'] for r in rows])
R_avg, T_avg = (R['s'] + R['p']) / 2, (T['s'] + T['p']) / 2  # unpolarized

The plotted peaks stay below 1 because the silver absorbs; the check below the figure confirms that what is not reflected or transmitted is exactly what the layers absorb. Tilting the beam shortens $d\cos\theta$ and moves every peak to shorter wavelengths, which is how an etalon is angle-tuned.

Angle, étendue and the Jacquinot advantage

Since $\delta$ depends on $\cos\theta$, an etalon lit by a cone of light transmits each wavelength in a ring of angles: the Haidinger fringes seen through a plate held to the eye, and the reason an etalon spectrometer records a ring pattern. For a small tilt the peak shifts by

always to shorter wavelengths, and it is this dependence on $\theta^2$ rather than $\theta$ that gives the etalon its large throughput for a given resolution, the Jacquinot advantage over a slit instrument. The same formula says how far a tilted bandpass filter moves, and why a filter in a converging beam broadens.

The cavity as a resonator: lifetime and Q

Seen in time rather than frequency, a photon in the cavity survives on average $\tau = \frac{n d}{c\,(1 - R)}$ round trips' worth of time (for $R$ close to 1), and the quality factor $Q = \omega\tau = m\mathcal{F}$ is again the order times the finesse. A 1 mm cavity with $R = 0.99$ stores light for about a nanosecond. The field inside at resonance is enhanced by $1/(1 - R)$ in intensity over the incident wave, which is what a dye molecule or a nonlinear crystal placed in the spacer sees, and why cavity enhancement is a tool of spectroscopy and of frequency conversion.

The Gires–Tournois etalon

Make the back mirror perfect. Nothing is transmitted, $|r| = 1$ at every wavelength, and all the physics is in the phase of the reflected wave, which sweeps through $2\pi$ around each resonance the more rapidly the higher the front mirror's reflectance. Such a Gires–Tournois etalon (1964) is a pure phase element: a delay that depends on wavelength, used to compensate dispersion in femtosecond lasers and fibre links, and the simplest demonstration that a lossless structure can have a strongly frequency-dependent group delay with a perfectly flat reflectance.

From etalons to filters

A Fabry–Pérot cavity is the core of almost every narrowband filter. Replacing the metal mirrors by quarter-wave stacks removes the absorption and raises the reflectance at will, which is the all-dielectric bandpass filter.