Fabry–Pérot Interferometer and Etalon

Take the two-surface interference of a soap film and push it to its logical extreme: make both surfaces highly reflective, hold them rigidly parallel, and let a wave rattle back and forth hundreds of times before it leaks out. Every bounce sheds a faint transmitted copy of the wave; the copies march out in a parallel stream, each one lagging the last by the same round-trip phase, and their sum is the sharpest thing in classical optics. Where an ordinary two-beam fringe is a lazy cosine, the Fabry–Pérot transmits almost nothing — except in razor-thin spikes at the wavelengths for which every one of the many copies arrives in step. Those spikes are what make this the highest-resolution passive instrument in the optical toolbox. The device is really two knobs turning independently. The spacing of the transmission spikes is set by the optical thickness $nd$ between the mirrors — pack more wavelengths into a round trip and the spikes crowd together. The sharpness of each spike is set by the mirror reflectivity alone — the more bounces the light survives, the more copies interfere and the more brutally off-resonance light cancels. This article owns the first knob: the resonance condition $2nd\cos\theta = m\lambda$ , the free spectral range $c/2nd$ that separates the orders, and the ring pattern an extended source throws on a screen. The Airy transmission function that shapes each peak is recalled by name from aq-optics-multiple-beam-interference-airy, and the finesse and resolving power that quantify the second knob are developed in aq-optics-finesse-free-spectral-range. The same resonant cavity, turned into an oscillator, is the longitudinal-mode structure of every laser — so the Fabry–Pérot is at once a spectrometer and the archetype of the optical resonator.

Key takeaways

A Fabry–Pérot interferometer traps light between two highly reflective parallel surfaces; the many transmitted partial waves interfere, giving sharp transmission peaks wherever the round trip is a whole number of wavelengths, $2nd\cos\theta = m\lambda$ .
The transmitted lineshape is the Airy function $T = 1/(1 + F\sin^2(\delta/2))$ — recalled from multiple-beam interference; this article uses it to place and space the peaks.
Adjacent orders are separated by the free spectral range $\Delta\nu_{\text{FSR}} = c/2nd$ in frequency, or $\lambda^2/2nd$ in wavelength — the unambiguous window of the instrument.
A fixed-gap device is an etalon (a spectral filter or laser mode-selector); scanning the spacing or the gas pressure turns it into a spectrometer that maps intensity versus path difference.
The transmission peaks fall at the equally spaced frequencies $\nu_m = m\,c/2nd$ — the same comb of longitudinal modes that a laser cavity supports.
Key idea: the optical thickness $nd$ alone fixes where the transmission peaks sit and how far apart they are ( $\Delta\nu_{\text{FSR}} = c/2nd$ ), while the reflectivity alone fixes how sharp they are — the resonance condition is the etalon's owned physics; the peak shape and its finesse are recalled from the multiple-beam Airy treatment.

The core insight

Two highly reflective parallel surfaces form a resonant cavity that transmits light only in narrow spikes, at the wavelengths for which one round trip $2nd\cos\theta$ is an integer number of wavelengths $m\lambda$ . The spikes repeat every free spectral range $\Delta\nu_{\text{FSR}} = c/2nd$ ; between them the transmission is nearly zero. Where the peaks sit and how they are spaced is the geometry owned here; how sharp each peak is comes from the multiple-beam Airy function.

Where this lives today

The Fabry–Pérot is the quiet workhorse of precision optics. In high-resolution spectroscopy a scanning etalon resolves hyperfine structure and isotope shifts that a grating smears together, its free spectral range chosen just wider than the feature of interest. In the laser world, a thin intracavity etalon forces single-frequency operation, while an external ultra-high-finesse reference cavity is the ruler a laser is locked to — by the Pound–Drever–Hall technique — to reach sub-hertz linewidths for optical atomic clocks. In fibre telecommunications, etalon filters and interleavers slice the dense wavelength-division grid into its individual channels. At the extremes the same object changes character entirely. Shrink the mirror spacing and push the finesse to millions and the cavity becomes the resonator of cavity quantum electrodynamics, where a single atom and a single photon exchange energy coherently. Stretch it to four kilometres and the Fabry–Pérot becomes the arm cavity of a gravitational-wave detector, folding the light path so a passing wave's femtometre stretch registers as a measurable phase shift. One resonance condition, $2nd\cos\theta = m\lambda$ , spans that entire range.

The physical picture

Picture a single wave slipped into the gap between two nearly perfect mirrors. Most of it reflects at the far mirror and starts back; a thin sliver escapes. Then most of that reflects at the near mirror and heads out again, shedding another sliver — and so on, dozens or hundreds of times. All the escaping slivers travel out parallel to one another, and each one has made exactly one more round trip than the sliver before it, so it lags by the same fixed phase $\delta$ . If that round-trip phase is a whole multiple of $2\pi$ , every sliver steps out in lock-step with all the others and they pile up into nearly complete transmission. If the phase is off by even a little, the slivers fan out across all phases — and because there are so many of them when the mirrors are good — they cancel almost perfectly. That is the whole instrument: sharp transmission where the round trip resonates, darkness everywhere else.

Figure

A Fabry–Pérot cavity: a wave bounces between two reflective surfaces spaced by d, shedding a parallel stream of transmitted partial waves that a lens focuses into concentric Haidinger rings

Now think about the stream of escaping slivers as a tally. When the mirrors are poor, only two or three slivers carry any real amplitude, and interference of a few beams gives a broad, gentle fringe — essentially the thin-film cosine. When the mirrors are excellent, dozens of slivers contribute, and the situation is exactly that of a diffraction grating with a large number of slits: the transmission collapses into needle-sharp maxima. The reflectivity therefore acts as an effective beam count. Crank it up and the peaks sharpen without moving; the resonance wavelengths stay pinned to $2nd\cos\theta = m\lambda$ . The next figure shows what a detector sees as you sweep the frequency across such a cavity.

Figure

Transmitted intensity versus frequency for a 1 cm air-spaced etalon: sharp peaks separated by the free spectral range FSR = c/2d = 15.0 GHz, each far narrower than the spacing

Resonance in a Fabry–Pérot is really light storage. On resonance the wave interferes constructively with its own earlier round trips, so the field between the mirrors builds up far above the incident field, and a photon lingers for many bounces before it leaks away. A narrow transmission peak and a long storage time are two faces of the same coin: a cavity that holds light for a long time can only accept a narrow band of frequencies, because only a pure enough tone survives all those round trips in phase. This is why the highest-finesse reference cavities — millions of bounces — pass only a sliver of a hertz.

The core equations

Everything hinges on the phase one round trip stamps on the wave. A ray crossing the gap at internal angle $\theta$ and returning accumulates an optical path $2nd\cos\theta$ ; multiplying by the vacuum wavenumber turns it into a phase, and constructive transmission demands that phase be a whole multiple of $2\pi$ .

\delta = \frac{2\pi}{\lambda_0}\,2nd\cos\theta, \qquad 2nd\cos\theta = m\lambda_0 \;\;(m \in \mathbb{Z}) \qquad (1)

where: $\delta$ — round-trip phase difference between successive transmitted partial waves; $\lambda_0$ — vacuum wavelength; $n$ — index of the medium in the gap; $d$ — mirror separation; $\theta$ — internal ray angle to the mirror normal; $m$ — the (large) integer interference order. The transmission peaks sit exactly where $\delta = 2\pi m$ , i.e. where the round trip is a whole number of wavelengths. The shape of each peak is the Airy transmission function of multiple-beam interference. It is derived and owned in aq-optics-multiple-beam-interference-airy by summing the geometric series of partial waves; here it is recalled and used to locate and space the maxima.

T(\delta) = \frac{1}{1 + F\sin^2(\delta/2)}, \qquad F = \frac{4R}{(1-R)^2} \qquad (2)

where: $T$ — transmitted intensity fraction (unity at a peak, for lossless mirrors); $F$ — coefficient of finesse; $R$ — intensity reflectivity of each surface. The crucial reading: $T = 1$ at every resonance $\delta = 2\pi m$ regardless of $R$ — high reflectivity does not dim the peaks, it only narrows them — while between resonances $T$ falls to $1/(1+F)$ , which becomes vanishingly small as $R \to 1$ . Successive orders differ by one full wavelength in the round trip, $\Delta m = 1$ . Converting that to a frequency (or wavelength) interval gives the free spectral range — the spacing of the comb, and the widest span the instrument can measure without adjacent orders overlapping.

\Delta\nu_{\text{FSR}} = \frac{c}{2nd}, \qquad \Delta\lambda_{\text{FSR}} = \frac{\lambda^2}{2nd} \qquad (3)

where: $\Delta\nu_{\text{FSR}}$ — free spectral range in frequency; $\Delta\lambda_{\text{FSR}}$ — the same interval in wavelength; $c$ — vacuum speed of light. Only the optical thickness $nd$ enters — not the reflectivity. A thick, high-order cavity has a small FSR (a crowded comb) but, as aq-optics-finesse-free-spectral-range shows, a correspondingly large resolving power.

Derivation AResonance condition and free spectral range of the etalon

The goal is to pin down, from the round-trip phase alone, exactly which frequencies the etalon transmits and how far apart those transmitted frequencies lie — the two facts that define the instrument's dispersion before any question of peak sharpness enters. We start from the geometric path a tilted ray traces on one round trip, convert it to a phase, impose constructive interference of the transmitted partial waves, and then ask how much the frequency must change to step from one transmitted order to the next.

Step 1. A plane wave inside the gap propagating at internal angle $\theta$ to the normal covers a slant path $2d/\cos\theta$ on a round trip, but the wavefront tilt claws back a projection $2d\sin\theta\tan\theta$ ; the net optical path difference between one emerging partial wave and the next is $\Delta = n\,(2d/\cos\theta - 2d\sin\theta\tan\theta) = 2nd\cos\theta$ . Why: it is the difference in optical path between successive escaping slivers that decides whether they add or cancel, and the trigonometry collapses to the clean $2nd\cos\theta$ .

Step 2. Multiply by the vacuum wavenumber $k_0 = 2\pi/\lambda_0$ to get the round-trip phase $\delta = (2\pi/\lambda_0)\,2nd\cos\theta$ . The Airy function (recalled, Eq. 2) is maximal precisely when $\sin(\delta/2) = 0$ , i.e. $\delta = 2\pi m$ , which is the resonance condition $2nd\cos\theta = m\lambda_0$ . Why: constructive interference of all the partial waves is the statement that the round trip closes on itself in phase, and that is an integer number of wavelengths.

Step 3. Work at normal incidence ( $\cos\theta = 1$ ) and write the order as $m = 2nd\nu/c$ using $\nu = c/\lambda_0$ . Two adjacent transmitted orders differ by $\Delta m = 1$ , so the frequency step between them is $\Delta\nu_{\text{FSR}} = c/2nd$ . Why: the comb spacing is fixed by how much you must change the frequency to add exactly one extra wavelength to the round trip.

Step 4. Convert to wavelength with $|\Delta\lambda| = (\lambda^2/c)\,\Delta\nu$ , giving $\Delta\lambda_{\text{FSR}} = \lambda^2/2nd$ . Why: spectroscopists quote the free spectral range in whichever unit matches their dispersing element, and the two forms are related by the usual $\Delta\lambda = -(\lambda^2/c)\Delta\nu$ . *Verified: $2nd\cos\theta = m\lambda$ and $\Delta\nu_{\text{FSR}} = c/2nd$ checked symbolically, and the numbers $d = 1$ cm air $\Rightarrow \Delta\nu_{\text{FSR}} = 14.99$ GHz, $m = 33\,939$ at $589.3$ nm, in verify_aq-optics-fabry-perot-interferometer.py.*

Go deeper · Derivation B · 1 minThe transmission comb and the modes of the cavity

Derivation BThe transmission comb and the modes of the cavity

Here the aim is to recast the resonance condition as a set of equally spaced frequencies rather than angles, because that is the form in which the Fabry–Pérot meets its most important modern relatives — the longitudinal modes of a laser and the teeth of a filter comb. We hold the geometry fixed at normal incidence, solve the resonance condition for the transmitted frequencies, and show they form an arithmetic progression whose spacing is the free spectral range.

Step 1. At normal incidence the resonance condition $2nd = m\lambda_0 = m\,c/\nu$ solves for the transmitted frequencies $\nu_m = m\,c/2nd$ . Why: fixing the mirror spacing turns "an integer number of wavelengths fit the round trip" into a discrete ladder of allowed frequencies.

Step 2. The spacing between neighbours is $\nu_{m+1} - \nu_m = c/2nd = \Delta\nu_{\text{FSR}}$ , independent of $m$ — the comb is perfectly even across the whole spectrum. Why: because $\nu_m$ is linear in the integer $m$ , the transmitted frequencies are uniformly spaced, which is exactly what makes the etalon a frequency ruler.

Step 3. These are the same standing-wave conditions $d = m\,(\lambda/2n)$ that fix the longitudinal modes of a laser resonator of length $d$ : a Fabry–Pérot passes light at precisely the frequencies its own cavity would oscillate at. Why: transmitting a frequency and supporting a standing wave at that frequency are the same resonance seen from outside and inside, so the spectrometer and the laser cavity share one mode structure.

Step 4. An etalon placed inside a laser therefore filters the gain: only frequencies that are simultaneously a cavity mode and an etalon transmission peak survive, which is how a thin intracavity etalon forces single-longitudinal-mode operation. Why: superposing two combs of different spacing leaves transmission only where teeth coincide, giving a much sparser set of surviving modes. *Verified: $\nu_m = m\,c/2nd$ evenly spaced by $c/2nd$ , and the solid fused-silica etalon $n = 1.4585$ , $d = 5$ mm $\Rightarrow \Delta\nu_{\text{FSR}} = 20.56$ GHz, in verify_aq-optics-fabry-perot-interferometer.py.*

Limiting cases

Case	Result	Meaning
Low reflectivity $R \to 0$	$F \to 0$ , $T \approx 1 - 2R(1-\cos\delta)$	the comb dissolves into the shallow two-beam thin-film cosine — only a couple of partial waves matter
High reflectivity $R \to 1$	peaks $\to$ needles at $\delta = 2\pi m$	ideal frequency filter; the peaks stay pinned while the gaps go dark ( $T_{\min} = 1/(1+F) \to 0$ )
On resonance $\delta = 2\pi m$	$T = 1$ (lossless)	perfect transmission at any $R$ — reflectivity narrows the peaks, it does not lower them
Thick / high-order etalon (large $nd$ )	small $\Delta\nu_{\text{FSR}} = c/2nd$	a crowded comb — high resolving power but a narrow unambiguous window
Normal vs. oblique incidence	order $m$ drops as $\cos\theta$ falls	tilting the etalon tunes the transmitted wavelength — the basis of ring formation and angle-tuned filters

Common trap“A higher-reflectivity Fabry–Pérot always resolves better, so just keep raising R”

Reflectivity sharpens the peaks, but it never moves them or widens the window between them. The free spectral range $\Delta\nu_{\text{FSR}} = c/2nd$ is fixed by the mirror spacing alone; raising $R$ shrinks each peak's width without changing how far apart the peaks are, so the ratio of window to resolution — the finesse — climbs, but the window itself does not. Push the resolution too far and any spectral feature wider than one shrinking FSR will overlap the next order, and you can no longer tell which peak came from which wavelength. Real high-resolution work therefore pairs a Fabry–Pérot with a coarse pre-disperser (a grating or a second, lower-order etalon) to sort the orders — and remembers that the achievable finesse is capped by mirror flatness and parallelism long before reflectivity runs out (that ceiling is quantified in aq-optics-finesse-free-spectral-range). Keep the two knobs straight: $nd$ sets spacing, $R$ sets sharpness, and resolving power needs both.

Worked examples

1 · An air-spaced etalon — free spectral range and order (confidence builder)

Two mirrors are held $d = 1.00$ cm apart in air ( $n = 1$ ) and illuminated at the sodium line $\lambda = 589.3$ nm. Find the free spectral range and the interference order at the centre of the pattern.

Step 1. Free spectral range straight from Eq. 3: $\Delta\nu_{\text{FSR}} = c/2nd = (2.998\times10^{8})/(2\times1\times0.0100) = 1.499\times10^{10}\ \text{Hz} = \mathbf{15.0\ GHz}$ . In wavelength, $\Delta\lambda_{\text{FSR}} = \lambda^2/2nd = (589.3\ \text{nm})^2/(2\times10^{7}\ \text{nm}) = \mathbf{0.0174\ nm}$ — a tiny window, which is exactly why an etalon out-resolves a grating.

Step 2. Interference order at normal incidence: $m = 2nd/\lambda = (2\times0.0100)/(589.3\times10^{-9}) = \mathbf{33\,939}$ . The etalon works in a very high order — over thirty thousand wavelengths fit a round trip — and it is that large order that, multiplied by the finesse, delivers a resolving power above $10^6$ .

Step 3. Reality check on the peaks: they recur every $15.0$ GHz across the whole spectrum. A source whose entire spectrum sits within a $15$ GHz span can be measured unambiguously; anything broader needs a pre-filter so that only one order reaches the detector at a time. FSR = 15.0 GHz, order m = 33 939.

2 · A solid fused-silica etalon as a laser mode filter (exam level)

A single-mode laser mirror is coated onto a slab of fused silica ( $n = 1.4585$ ) of thickness $d = 5.00$ mm, forming a solid etalon. Find its free spectral range and explain how it selects one laser mode.

Step 1. Now the index multiplies the thickness: $\Delta\nu_{\text{FSR}} = c/2nd = (2.998\times10^{8})/(2\times1.4585\times0.00500) = \mathbf{20.6\ GHz}$ . The optical thickness $nd = 7.29$ mm, not the physical $5$ mm, sets the comb — a point worth double-checking on every solid etalon.

Step 2. A typical gas laser's gain bandwidth spans a few gigahertz and supports several longitudinal modes of its metre-scale cavity (spaced by hundreds of MHz). Insert this etalon and only frequencies lying on both combs survive; with the etalon's $20.6$ GHz teeth far sparser than the laser modes, tilting it slightly to align one etalon peak with one gain-centred laser mode kills all the neighbours.

Step 3. The mechanism is loss selection: off-peak modes see high round-trip loss at the etalon and never reach threshold, while the aligned mode sees near-unit transmission. Temperature stabilisation of $nd$ then keeps the chosen peak locked in place. FSR = 20.6 GHz; single-mode operation by aligning one etalon peak with the gain-centred cavity mode.

Physics of the portrait

The Fabry–Pérot carries two names because it was two people's instrument — and the partnership that turned a laboratory curiosity into the standard of high-resolution spectroscopy is the story behind it.

Origins · 1 minCharles Fabry and how this was discovered

Charles Fabry

1867–1945 · Marseille and Paris (Institut d'Optique)

Charles Fabry was a French physicist who, working at Marseille through the late 1890s with the engineer Alfred Pérot, developed the multiple-beam interferometer that now bears both their names. Pérot's mechanical skill produced the rigid, precisely parallel silvered plates the method demanded; Fabry supplied the interferometric theory and the measurements. Between 1897 and 1901 they showed that the instrument could measure wavelengths and length standards to a precision far beyond any grating of the day, and it rapidly became the tool of high-resolution spectroscopy. Fabry's most celebrated result came in 1913, when with Henri Buisson he used the interferometer to detect the ozone layer in the upper atmosphere by its ultraviolet absorption — a measurement of lasting geophysical importance. He later became the first director of the Institut d'Optique in Paris and a professor at the Sorbonne. The two-mirror resonant cavity he helped invent is now ubiquitous: it is the lineshape filter in spectrometers, the frequency reference that stabilises lasers, and, scaled to kilometres, the optical cavity in the arms of gravitational-wave detectors.

History

1899

— Fabry and Pérot describe their multiple-beam interferometer and its theory

1901

— the Fabry–Pérot interferometer is established as a tool for precision wavelength metrology

1913

— Fabry and Buisson detect atmospheric ozone by its ultraviolet absorption using the interferometer

1960

— the same resonant-cavity physics underlies the optical resonator of the first lasers

2015

— kilometre-scale Fabry–Pérot arm cavities in LIGO help detect gravitational waves

Going further

Texts. Hecht, Optics, 5th ed., §9.6 (Multiple-beam interference; the Fabry–Pérot interferometer) · Born & Wolf, Principles of Optics, 7th ed., §7.6 (multiple-beam interference) · Demtröder, Laser Spectroscopy, Vol. 1, §4.2 (interferometers) · Foot, Atomic Physics, §8.3–8.4 (the Fabry–Pérot etalon) · Saleh & Teich, Fundamentals of Photonics, ch. on optical resonators.

Provenance. The resonance condition $2nd\cos\theta = m\lambda$ and free spectral range $c/2nd$ are standard (Hecht §9.6, Born & Wolf §7.6); the Airy transmission function used here is owned and derived in aq-optics-multiple-beam-interference-airy; the finesse, resolving power, and effective-finesse relations are developed in aq-optics-finesse-free-spectral-range. The intracavity build-up ( $\sim\mathcal{F}/\pi$ ) and photon-lifetime relations quoted in passing follow Saleh & Teich's resonator chapter. Every number here — the $15.0$ GHz and $20.6$ GHz free spectral ranges, the order $m = 33\,939$ — is recomputed in verify_aq-optics-fabry-perot-interferometer.py.

Prerequisites. Superposition Principle of Electromagnetic Waves · Two-Wave Interference and the Interference Term · Optical Path Difference and Phase Retardation · Wavefront- and Amplitude-Splitting Interferometers · Thin-Film Interference · Temporal Coherence and Coherence Length. Leads to. Finesse, Free Spectral Range, and Resolving Power · Fringes of Equal Inclination and Equal Thickness · Fringe Visibility and Contrast.

What is the difference between an etalon and a Fabry–Pérot interferometer?

They are the same optics with different mechanics. An etalon has a fixed spacing — two coated flats on a rigid spacer, or a single coated slab of glass — and acts as a passive filter or laser mode-selector at whatever wavelengths its geometry happens to transmit. A Fabry–Pérot interferometer has an adjustable spacing (or an adjustable gas pressure that changes $n$ ); scanning it sweeps the transmission comb across the spectrum so a detector records intensity versus path difference. The physics — the Airy transmission function, the resonance condition, the free spectral range — is identical; only whether you tune it differs. "Etalon" comes from the French for a measuring gauge, which is exactly how the fixed device is used.

Why does the peak transmission reach 100% even though the mirrors reflect almost all the light?

It seems paradoxical that mirrors reflecting 99% of the light can transmit 100% of it on resonance, but it is genuine and it is interference. On resonance the wave stored between the mirrors builds up to a large circulating intensity — roughly $\mathcal{F}/\pi$ times the incident intensity — and the small fraction that this large field leaks through the far mirror adds up to the full incident power. Meanwhile the promptly reflected wave and the leakage back out the front mirror interfere destructively, so nothing is reflected. Energy is conserved throughout; the cavity is not a barrier but a resonator, and at resonance it hands all the incident power to the transmitted side. It works only for lossless mirrors — any absorption breaks the perfect cancellation and pulls the peak below unity.

Why do I see rings when I look through a Fabry–Pérot with an extended source?

Because the resonance condition $2nd\cos\theta = m\lambda$ depends on the angle $\theta$ . An extended (diffuse) source sends rays into the etalon at every angle at once; only those angles that satisfy the resonance for some integer $m$ are transmitted, and all rays at a given $\theta$ lie on a cone, which a lens focuses to a circle. The result is a set of concentric bright rings — fringes of equal inclination, the same Haidinger rings a thick plate makes — with the highest order at the centre ( $\theta = 0$ ) and successive orders stepping outward. Measuring the ring radii is a standard way to read off small wavelength differences, because two nearby wavelengths make two interleaved sets of rings.

How is the Fabry–Pérot related to a laser cavity?

Intimately — they are the same resonant cavity viewed from opposite sides. A Fabry–Pérot transmits light at the frequencies $\nu_m = mc/2nd$ for which a round trip closes in phase; a laser cavity of the same length oscillates at exactly those frequencies, its longitudinal modes. So the transmission comb of the etalon and the mode comb of the laser are one and the same. This is why a small etalon dropped inside a laser cavity selects a single mode (only frequencies on both combs survive), and why the whole language of finesse, free spectral range, and photon lifetime carries over unchanged from the spectrometer to the resonator. Learn the Fabry–Pérot and you have already learned the mode structure of the laser.

Does tilting the etalon change the transmitted wavelength?

Yes, and it is a common tuning trick. The resonance condition carries a $\cos\theta$ , so as you tilt the etalon away from normal incidence the effective optical thickness $nd\cos\theta$ shrinks and every transmission peak slides to a shorter wavelength. A small tilt is enough to walk one peak across a laser's gain profile and pick a different mode, which is how intracavity etalons are fine-tuned. The cost is that large tilts also walk the beam off itself between bounces — the successive reflections no longer perfectly overlap — which lowers the effective finesse. Practical angle-tuning therefore stays within a degree or two of normal.

Why does a thicker etalon give higher resolving power but is harder to use?

Resolving power is the order times the finesse, $m\mathcal{F} = (2nd/\lambda)\mathcal{F}$ , so it grows in proportion to the optical thickness $nd$ through the order $m$ . A thick etalon works in a very high order and resolves extremely fine spectral detail. The catch is the free spectral range $c/2nd$ , which shrinks in the same proportion: the unambiguous window closes up as the resolution sharpens. A thick etalon therefore resolves a hair-fine feature but only over a tiny spectral span, so it must be pre-filtered to isolate one order, and it demands ever-better mirror flatness and mechanical stability to hold $nd$ constant to a fraction of a wavelength across the beam. Resolution and usable range are the two ends of one lever.

What actually limits the finesse of a real etalon?

Not the reflectivity, usually. The reflectivity finesse $\pi\sqrt{R}/(1-R)$ can be enormous, but three other imperfections cap the real instrument: surface flatness (a mirror figured only to $\lambda/M$ smears the resonance across the beam and limits the finesse to about $M/2$ ), parallelism (a wedge between the mirrors detunes one side relative to the other), and absorption in the coatings (which also pulls the peak transmission below unity). These combine with the reflectivity finesse in a root-sum-square, so the smallest of them wins. This is why reaching a finesse of even a few hundred demands mirrors flat to $\lambda/100$ or better — and why the full accounting is done in aq-optics-finesse-free-spectral-range rather than from reflectivity alone.