Multiple-Beam Interference and the Airy Function

Two-wave interference, the kind that paints a soap film or a Young's pattern, blends exactly two beams and gives a gentle cosine ripple. But let the two surfaces be good mirrors and the light no longer bounces once — it rattles between them dozens of times, leaking a little at each pass. Now dozens of coherent beams must be added, each a fixed round-trip phase behind the last, and the sum is something far sharper than a cosine: the Airy function, a comb of narrow bright resonances separated by wide dark gaps. The single knob that decides how sharp is the surface reflectivity $R$ — the fraction of intensity each mirror throws back. That one number is the whole story. Nudge $R$ from the $0.04$ of a bare glass face up to the $0.95$ of a coated mirror and the transmission peaks collapse from a lazy sinusoid into spikes a hundred times narrower than their spacing. This is why a Fabry–Pérot etalon can split a spectral line finer than any prism, why a laser cavity selects one colour from a continuum, and why a dielectric mirror stack can hand back $99.99\%$ of the light: all of them are multiple-beam interferometers, and all of them live or die by how reflective their surfaces are. The mathematics is a single geometric series — but the physics it unlocks is the entire art of building a resonator. This article owns the Airy function and its reflectivity-set sharpness; the cavity that applies it (Fabry–Pérot) and the figures of merit that grade it (finesse, resolving power) are taken up by name in the articles that follow.

Key takeaways

A pair of partially reflecting surfaces returns not two but an infinite ladder of beams, each delayed by the same round-trip phase $\delta$ and weighted by $R$ per bounce; summing the geometric series gives the transmitted Airy function.
Transmittance $T = 1/[1 + F\sin^2(\delta/2)]$ with the coefficient of finesse $F = 4R/(1-R)^2$ — a comb of peaks at $\delta = 2\pi m$ that sharpen without limit as $R \to 1$ .
On resonance a lossless cavity transmits everything, $T = 1$ , no matter how reflective the mirrors — high $R$ narrows the peaks, it does not dim them.
The peaks' full width is $\Delta\delta_{1/2} = 4/\sqrt{F}$ , so the reflectivity finesse $\mathcal{F} = \pi\sqrt{R}/(1-R)$ counts how many resolvable peaks fit between orders: $29.8$ at $R = 0.9$ , $61$ at $R = 0.95$ , $313$ at $R = 0.99$ .
Drop $R$ toward zero and the Airy function melts back into the two-wave cosine of a thin film — multiple-beam interference contains two-beam interference as its faint-mirror limit.
Key idea: summing the coherent geometric series of repeatedly reflected beams yields the Airy function $T = 1/[1 + F\sin^2(\delta/2)]$ , and the surface reflectivity $R$ — through $F = 4R/(1-R)^2$ — is the sole knob that turns broad two-wave cosine fringes into razor-sharp cavity resonances.

The core insight

When light is trapped between two partially reflecting surfaces it escapes as an infinite ladder of beams, each one round-trip phase $\delta$ and one factor of $R$ behind the last. Adding that geometric series and squaring gives the Airy transmission function $T = 1/[1 + F\sin^2(\delta/2)]$ , where the coefficient of finesse $F = 4R/(1-R)^2$ is fixed entirely by the reflectivity $R$ . Small $R$ gives soft cosine fringes; $R$ near one gives a comb of needle-sharp resonances. Reflectivity is the control parameter of the whole phenomenon.

Where this lives today

The Airy function is the design equation for every optical resonator in a modern lab. Laser physics rests on it: a laser cavity is a multiple-beam interferometer, and its longitudinal modes are precisely the Airy transmission peaks, so mirror reflectivity sets both which frequencies oscillate and how narrow each mode can be. Telecom systems slice dense wavelength grids with etalon filters whose passbands are single Airy resonances. And the most extreme instance of the physics — a kilometre-scale Fabry–Pérot arm cavity of finesse in the hundreds, its mirrors coated to reflect all but a few parts per million — is what lets a gravitational-wave detector sense a mirror displacement a thousandth the width of a proton, by folding the light back on itself finesse-times and letting the circulating power build. Metrology and spectroscopy use the sharpness directly: a scanning Fabry–Pérot resolves the hyperfine splittings and Doppler profiles of atomic lines that a grating cannot, and ultra-stable reference cavities with finesse above $100{,}000$ are the frequency flywheels that stabilise the world's best lasers and optical clocks. In every one of these, the engineer's first move is to choose a reflectivity — and the Airy function turns that choice into a linewidth.

The physical picture

Send a wave into the gap between two mirrors and watch what escapes on the far side. The first beam passes straight through — one transmission in, one out. But a fraction $R$ was reflected at the exit and sent back; it crosses the gap, reflects again off the entrance mirror, and arrives at the exit a full round trip — one extra $2nd\cos\theta$ of path, one extra phase $\delta$ — later, where it too partly escapes. Then a third, a fourth, each a copy of the last shrunk by the factor $R$ (two more reflections) and retarded by another $\delta$ . The transmitted light is this whole descending ladder of beams added together.

Figure

The escaping ladder of beams: each round trip inside the gap loses a factor R in amplitude and gains a round-trip phase δ, so the transmitted field is a geometric series

When the round-trip phase $\delta$ is a whole multiple of $2\pi$ , every rung of the ladder arrives in step and they add to their fullest — transmission is total. Detune $\delta$ a hair and the many beams fan out in phase and destroy one another. Here is the crucial point: the more beams there are carrying real amplitude — that is, the higher the reflectivity — the more violently a small detuning wrecks the sum, because now a hundred vectors must stay aligned rather than two. So high $R$ does not change where the peaks sit; it makes them narrow. The next figure shows exactly that transformation.

Figure

The Airy transmittance T(δ): at R = 0.04 a soft two-wave cosine, at R = 0.9 a comb of spikes 30× narrower than their spacing — coefficient of finesse F = 360, finesse 𝓕 = 29.8

Both curves touch $T = 1$ at every $\delta = 2\pi m$ — that never changes, and it is a consequence of the cavity being lossless. What reflectivity buys is width, not height: the amber $R = 0.04$ curve barely dips to $0.85$ between peaks, while the blue $R = 0.9$ comb crashes to near zero and returns only in a spike a thirtieth of an order wide. Reflectivity is the dial that sets how many beams effectively join the sum, and therefore how sharp the resonance.

The core equations

The whole phenomenon is one geometric series. Label the mirror's amplitude coefficients $r, t$ for a wave arriving from outside and $r', t'$ for one arriving from inside the gap (the internal/external pair related by the Stokes reversibility relations, recalled below). Every escaping beam is the previous one multiplied by $r'^2$ — two internal reflections — and by the round-trip phase factor $e^{i\delta}$ . The transmitted field is their sum.

E_t = E_0\,t t' \sum_{n=0}^{\infty} \left(r'^2 e^{i\delta}\right)^n = \frac{E_0\,t t'}{1 - r'^2 e^{i\delta}} \qquad (1)

where: $E_0$ — incident amplitude; $t, t'$ — amplitude transmission at the two mirror crossings; $r'$ — internal amplitude reflection, with intensity reflectivity $R = |r'|^2$ ; $\delta$ — phase gained in one round trip; $n$ — the round-trip counter. The series converges because $|r'| < 1$ , so light genuinely leaks out and the ladder terminates. Squaring the summed field and using the Stokes relations for lossless mirrors ( $t t' = 1 - R$ , and the phase bookkeeping $r'^2 = R$ ) collapses the modulus into the compact form named after Airy.

T = \frac{I_t}{I_0} = \frac{(1-R)^2}{(1-R)^2 + 4R\sin^2(\delta/2)} = \frac{1}{1 + F\sin^2(\delta/2)} \qquad (2)

where: $I_t, I_0$ — transmitted and incident intensity; $T$ — transmittance; $F$ — the coefficient of finesse. Read it as a resonance: the transmittance is $1$ wherever $\sin(\delta/2) = 0$ , i.e. $\delta = 2\pi m$ , and dives toward zero in between at a rate governed entirely by $F$ .

F = \frac{4R}{(1-R)^2} \qquad (3)

where: $F$ — coefficient of finesse, a pure function of reflectivity that fixes both the peak-to-trough contrast ( $T_{\max}/T_{\min} = 1 + F$ ) and the peak width. It is the single dimensionless number in which all the sharpness lives: $F = 0.17$ for a bare glass face, $360$ at $R = 0.9$ , nearly $40{,}000$ at $R = 0.99$ . The round-trip phase carries the geometry of the gap, and it is the variable a spectrometer actually scans — by tilting the plate ( $\theta$ ), changing its spacing ( $d$ ), or letting the wavelength ( $\lambda_0$ ) vary.

\delta = \frac{2\pi}{\lambda_0}\,2 n d\cos\theta \qquad (4)

where: $\lambda_0$ — vacuum wavelength; $n$ — index of the gap; $d$ — gap thickness; $\theta$ — internal angle. The optical path per round trip is $2nd\cos\theta$ ; multiplying by the vacuum wavenumber $2\pi/\lambda_0$ turns it into phase. How the peak spacing in this variable becomes a free spectral range, and how the resolving power follows, is the business of the finesse and Fabry–Pérot articles that build on this one.

Derivation AThe Airy transmission function from the geometric series

The goal is to turn the intuitive ladder of reflected beams into an exact intensity law, showing that the coherent sum of infinitely many partial waves is not a mess but the clean single-parameter Airy function. We enumerate the escaping beams, recognise them as a geometric series, sum it in closed form, and square the modulus — using only that the mirrors are lossless — to reach $T = 1/[1 + F\sin^2(\delta/2)]$ with $F = 4R/(1-R)^2$ .

Step 1. Enumerate the transmitted beams. The first escapes with amplitude $E_0 t t'$ ; the second has made one round trip, so it is smaller by two internal reflections $r'^2$ and retarded by $e^{i\delta}$ ; the $n$ th carries $E_0 t t'(r'^2 e^{i\delta})^n$ . Why: each extra round trip is one identical loop — the same two reflections and the same added path — so the beams differ only by a constant multiplicative factor, which is the definition of a geometric progression.

Step 2. Sum the series: $E_t = E_0 t t'\sum_{n\ge0}(r'^2 e^{i\delta})^n = E_0 t t'/(1 - r'^2 e^{i\delta})$ . Why: the common ratio $r'^2 e^{i\delta}$ has modulus $R < 1$ , so the infinite sum converges to $1/(1 - \text{ratio})$ — physically, the light that keeps leaking out guarantees the tail of the ladder vanishes.

Step 3. Bring in the Stokes reversibility relations, recalled by name from the phase-change-on-reflection article: for lossless surfaces $t t' = 1 - R$ and $r' = -r$ , so $r'^2 = R$ is real. The transmitted intensity is $I_t = |E_t|^2 = I_0(1-R)^2/|1 - R e^{i\delta}|^2$ . Why: without a relation tying $tt'$ to $R$ the amplitudes would be free parameters; reversibility fixes them, and it is what makes the on-resonance transmittance come out to exactly one.

Step 4. Expand the denominator: $|1 - R e^{i\delta}|^2 = (1-R)^2 + 2R(1-\cos\delta) = (1-R)^2 + 4R\sin^2(\delta/2)$ , using $1 - \cos\delta = 2\sin^2(\delta/2)$ . Dividing through gives $T = 1/[1 + F\sin^2(\delta/2)]$ with $F = 4R/(1-R)^2$ . Why: the half-angle identity converts the raw interference term into $\sin^2(\delta/2)$ , which vanishes precisely at the resonances $\delta = 2\pi m$ — locating the peaks and isolating $F$ as the lone sharpness parameter. *Two checks confirm the result: the closed-form geometric sum and modulus expansion are exact, and numerically the peaks reach $T = 1$ at $\delta = 2\pi m$ while $R \to 0$ recovers the two-wave cosine.*

Go deeper · Derivation B · 1 minReflectivity finesse from the peak width

Derivation BReflectivity finesse from the peak width

Here we quantify sharpness. The goal is to find the full width of an Airy peak and express, as a single reflectivity-only number, how many such peaks fit in the $2\pi$ interval between orders — the reflectivity finesse $\mathcal{F} = \pi\sqrt{R}/(1-R)$ . This is the figure that makes multiple-beam interference a precision tool, and it follows from the Airy function by asking where the transmittance falls to one half.

Step 1. Set the half-maximum condition. $T = 1/2$ requires the denominator to equal $2$ , i.e. $F\sin^2(\delta/2) = 1$ . Why: the peak sits at $T = 1$ ; the width we want is the span between the two flanking points where it has dropped to half, and those are exactly where the interference term equals unity.

Step 2. Expand near a resonance $\delta = 2\pi m + \varepsilon$ with $\varepsilon$ small: $\sin(\delta/2) \approx \varepsilon/2$ , so $F(\varepsilon/2)^2 = 1$ gives $\varepsilon = \pm 2/\sqrt{F}$ and a full width $\Delta\delta_{1/2} = 4/\sqrt{F}$ . Why: for a high-reflectivity cavity the peak is narrow, so only the immediate neighbourhood of the resonance matters and the sine may be linearised — the approximation improves as the peak sharpens, exactly where we care.

Step 3. Form the finesse as the ratio of peak spacing to peak width. Adjacent orders are $2\pi$ apart in $\delta$ , so $\mathcal{F} = 2\pi/\Delta\delta_{1/2} = 2\pi\sqrt{F}/4 = \tfrac{\pi}{2}\sqrt{F}$ . Substituting $F = 4R/(1-R)^2$ yields $\mathcal{F} = \pi\sqrt{R}/(1-R)$ . Why: finesse is deliberately defined as a ratio so it depends only on the mirrors, not on the plate spacing — it is a property of the surfaces, portable to any cavity built from them. *Both results — $\Delta\delta_{1/2} = 4/\sqrt{F}$ and $\mathcal{F} = \pi\sqrt{R}/(1-R)$ — agree with direct half-maximum roots of the Airy function, giving $\mathcal{F} = 29.8, 61.2, 312.6$ at $R = 0.9, 0.95, 0.99$ .*

Go deeper · Derivation C · 1 minContrast and the two-wave limit

Derivation CContrast and the two-wave limit

This card connects the sharp Airy comb back to the gentle cosine of ordinary two-beam interference, proving that multiple-beam interference is the general case and two-wave the faint-mirror corner of it. The goal is the peak-to-trough contrast $T_{\max}/T_{\min} = 1 + F$ and its low-reflectivity limit, which pins down when the two-beam picture is a legitimate shortcut.

Step 1. Read the extremes off the Airy function. The maximum is $T_{\max} = 1$ at $\sin(\delta/2) = 0$ ; the minimum is $T_{\min} = 1/(1+F)$ at $\sin(\delta/2) = 1$ . Their ratio is $T_{\max}/T_{\min} = 1 + F$ . Why: contrast is set by how deep the troughs go, and the deepest possible trough is fixed by $F$ alone — a direct, measurable stand-in for the reflectivity.

Step 2. Take $R \to 0$ , so $F = 4R/(1-R)^2 \to 4R \to 0$ . Then $T \approx 1 - F\sin^2(\delta/2) = 1 - \tfrac{F}{2}(1 - \cos\delta)$ , a pure cosine ripple of vanishing depth. Why: with faint mirrors only the first two beams carry real amplitude — the third is down by $R^2 \ll R$ — so the sum truncates to two waves and the Airy function must reproduce the two-wave cosine, which it does.

Step 3. Compare regimes numerically: $R = 0.04$ gives $F = 0.17$ and contrast $1.17$ (a barely-there sinusoid), while $R = 0.9$ gives $F = 360$ and contrast $361$ (a stark comb). Why: the same formula spans a factor of $300$ in contrast as reflectivity climbs, which is the quantitative statement that "how many beams participate" is entirely a reflectivity question. *All three results hold: the contrast formula $T_{\max}/T_{\min} = 1 + F$ , the two-wave cosine limit as $R \to 0$ , and the specific contrast values $1.17$ at $R = 0.04$ and $361$ at $R = 0.9$ .*

Limiting cases

Case	Result	Meaning
$R \to 0$ (faint mirrors)	$F \to 0$ , $T \approx 1 - F\sin^2(\delta/2)$	only two beams survive — the Airy comb relaxes into the two-wave cosine of a thin film
$R \to 1$ (near-perfect mirrors)	$\mathcal{F} = \pi\sqrt{R}/(1-R) \to \infty$	peaks become needle-sharp resonances; unbounded finesse and resolving power (until real defects intervene)
on resonance $\delta = 2\pi m$	$T = 1$ for any $R$	a lossless cavity transmits everything at resonance — reflectivity narrows the peaks, never dims them
midway $\delta = (2m{+}1)\pi$	$T_{\min} = 1/(1+F)$	the darkest point; contrast $T_{\max}/T_{\min} = 1 + F$ deepens with reflectivity
absorbing mirrors ( $A$ loss/bounce)	$T_{\max} = [1 - A/(1-R)]^2 < 1$	absorption caps the peak below unity — the first place a real etalon departs from the ideal

Common trap“Very reflective mirrors block most of the light, so a high-finesse cavity is dim”

Backwards. On resonance a lossless Fabry–Pérot transmits all of the incident light, $T = 1$ , no matter whether the mirrors reflect $90\%$ or $99.99\%$ — the circulating field builds up until the little that leaks through the exit mirror equals everything coming in. What high reflectivity changes is the width of the resonances, not their height: it makes them narrow, not weak. The reason a high-finesse etalon looks dark in white light is that only a razor-thin slice of wavelengths sits on resonance at once; every one of those slices is transmitted in full. Confusing "reflective" with "opaque" hides the entire point of the device — you build the mirrors reflective precisely so that the transmission peaks get sharper while staying bright.

Worked examples

1 · The reflectivity dial: from R = 0.9 to R = 0.95 (confidence builder)

A Fabry–Pérot etalon uses matched mirrors. Find the coefficient of finesse, the reflectivity finesse, and the fractional peak width for $R = 0.90$ , then for $R = 0.95$ . What does each buy you?

Step 1. At $R = 0.90$ : $F = 4R/(1-R)^2 = 4(0.90)/(0.10)^2 = 3.6/0.01 = \mathbf{360}$ . The finesse is $\mathcal{F} = \pi\sqrt{R}/(1-R) = \pi\sqrt{0.90}/0.10 = \mathbf{29.8}$ — about thirty resolvable peaks between orders.

Step 2. The peak's full width in phase is $\Delta\delta_{1/2} = 4/\sqrt{F} = 4/\sqrt{360} = \mathbf{0.211}$ rad, a fraction $\Delta\delta_{1/2}/2\pi = 1/\mathcal{F} = 0.0336$ of the spacing between orders. Off resonance by even a thirtieth of an order, the transmission has already halved.

Step 3. Push to $R = 0.95$ : $F = 4(0.95)/(0.05)^2 = 3.8/0.0025 = \mathbf{1520}$ , and $\mathcal{F} = \pi\sqrt{0.95}/0.05 = \mathbf{61.2}$ . A five-percentage-point rise in reflectivity has doubled the finesse.

Step 4. The lesson is the leverage near $R = 1$ : because $\mathcal{F} \propto 1/(1-R)$ , the sharpness is governed by the transmission $1-R$ , not the reflectivity itself. Halving the loss from $10\%$ to $5\%$ halves the peak width. This is why cavity builders chase the last fraction of a percent of mirror reflectivity. R = 0.90 → 𝓕 = 29.8; R = 0.95 → 𝓕 = 61.2.

2 · Contrast, and where two-wave interference is good enough (exam level)

Compare the peak-to-trough contrast of the multiply-reflected pattern from a coated plate ( $R = 0.90$ ) with that from a single bare glass surface ( $R = 0.04$ ). When is the simple two-wave cosine an acceptable description?

Step 1. Contrast is $T_{\max}/T_{\min} = 1 + F$ . For $R = 0.90$ , $F = 360$ , so the contrast is $\mathbf{361}$ — peaks $361$ times brighter than the troughs between them, a stark comb.

Step 2. For a bare glass face, $R = (n-1)^2/(n+1)^2 \approx 0.04$ : $F = 4(0.04)/(0.96)^2 = 0.174$ , so the contrast is only $1 + 0.174 = \mathbf{1.17}$ . The "fringes" are a $17\%$ ripple on a bright background — essentially the two-wave cosine of thin-film interference.

Step 3. The two-wave approximation is legitimate exactly when $F \ll 1$ , i.e. $R$ small, because then the third beam ( $\propto R^2$ ) is negligible beside the second ( $\propto R$ ). Uncoated glass ( $R = 0.04$ , $R^2 = 0.0016$ ) sits safely there; a metallised or dielectric-coated surface does not.

Step 4. So the same plate wears two faces. Uncoated, it is a soap-film-like two-beam interferometer with soft cosine fringes; coat it to $R = 0.9$ and it becomes a sharp multiple-beam resonator — one physical object, moved across the $F \ll 1$ to $F \gg 1$ boundary purely by its reflectivity. Contrast 361 (R = 0.9) versus 1.17 (R = 0.04); two-wave is valid only for $F \ll 1$ .

Physics of the portrait

Every quantity in this article — the coefficient of finesse, the peak width, the contrast, the finesse itself — is a function of one number, the surface reflectivity $R$ . It deserves the portrait.

Origins · 1 minreflectivity and how this was discovered

reflectivity

the intensity fraction returned at a surface, R = · r

Reflectivity $R$ is the fraction of incident intensity a surface throws back, the square of the amplitude reflection coefficient, $R = |r|^2$ . For a bare dielectric interface it is fixed by the index contrast — Fresnel's $R = [(n_1-n_2)/(n_1+n_2)]^2$ gives the familiar $4\%$ of an air–glass face — but the whole technology of thin-film coatings exists to engineer $R$ to order: a single quarter-wave layer can push it toward zero (anti-reflection), a stack of them toward one (a dielectric mirror). In multiple-beam interference $R$ is the sole control parameter: it sets how many partial waves carry appreciable amplitude, and through the coefficient of finesse $F = 4R/(1-R)^2$ it alone decides whether a plate produces lazy cosine ripples or a comb of sharp resonances. The sensitive combination is not $R$ but the loss $1 - R$ : because finesse scales as $1/(1-R)$ , the difference between a $99\%$ and a $99.9\%$ mirror is a tenfold change in sharpness. That is why the history of high-resolution interferometry, from Fabry and Pérot's silvered plates to the multilayer mirrors of a gravitational-wave detector, is really a history of pushing reflectivity — and suppressing the absorption that secretly limits it — ever closer to unity.

History

1831

— Airy computes the intensity distribution transmitted by a plane-parallel plate, the function that now bears his name

1893

— Boulouch studies multiple-beam interference in low-loss plates

1899

— Fabry and Pérot build the multiple-beam interferometer from a pair of silvered surfaces

1903

— Lummer and Gehrcke introduce their multiple-beam plate for high-resolution spectroscopy

1948

— Tolansky develops multiple-beam interferometry for measuring surface topography to nanometre precision

Going further

Texts. Hecht, Optics, 5th ed., §9.6 (Multiple-Beam Interference) · Born & Wolf, Principles of Optics, §7.6 (multiple-beam interference and the Airy formula) · Saleh & Teich, Fundamentals of Photonics, 3rd ed., §2.5 and §11.1 (resonators) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 2 (cavity modes) · Demtröder, Laser Spectroscopy, Vol. 1, §4.2 (interferometers). The key figures are: $\mathcal{F} = 29.8, 61.2, 312.6$ at $R = 0.9, 0.95, 0.99$ ; contrast $1 + F$ ; half-width $4/\sqrt{F}$ . The reflectivity-only figure of merit $\mathcal{F} = \pi\sqrt{R}/(1-R)$ is derived in this article; the geometry-dependent free spectral range and resolving power are taken up in the Fabry–Pérot and finesse articles. The Stokes relations $t t' = 1 - R$ , $r' = -r$ are recalled by name from the phase-change-on-reflection article, which owns them.

Prerequisites. Superposition Principle of Electromagnetic Waves · Two-Wave Interference and the Interference Term · Thin-Film Interference · Optical Path Difference and Phase Retardation · Temporal Coherence and Coherence Length. Leads to. Fabry–Pérot Interferometer and Etalon · Finesse, Free Spectral Range, and Resolving Power · Fringe Visibility and Contrast.

Why do more beams make the peaks sharper rather than just brighter?

Because sharpness is about how fast the sum falls apart off resonance, and that is a game of alignment. On resonance every beam points the same way in the complex plane and they add to the full amount — as they do for any reflectivity, which is why the peaks all reach $T = 1$ . Detune slightly and each beam picks up a small extra phase relative to its predecessor; with only two beams the sum sags gently (a cosine), but with a hundred beams the last one is nearly antiphase to the first and the vector sum collapses almost at once. High reflectivity keeps more beams in play, so the collapse is faster and the resonance narrower. The height is set by the resonant, in-phase sum; the width is set by how quickly detuning spoils it.

What is the difference between the coefficient of finesse F and the finesse 𝓕?

They measure related but distinct things. The coefficient of finesse $F = 4R/(1-R)^2$ is the number that appears inside the Airy function; it sets the contrast directly, $T_{\max}/T_{\min} = 1 + F$ , and grows quadratically in $1/(1-R)$ . The reflectivity finesse $\mathcal{F} = \pi\sqrt{R}/(1-R) = \tfrac{\pi}{2}\sqrt{F}$ is a ratio of widths — the peak spacing (one order, $2\pi$ in phase) divided by the peak's full width at half maximum. It answers "how many separate peaks can I fit between orders?", which is the practically useful quantity. A quick sanity check: $R = 0.9$ gives $F = 360$ but $\mathcal{F} \approx 30$ , so contrast and count are very different numbers.

Where do the Stokes relations enter, and could I do this without them?

They fix the amplitudes so the answer comes out physical. Summing the geometric series leaves the transmitted intensity proportional to $(tt')^2$ , and without a relation tying that to the reflectivity it would be an undetermined constant. The Stokes reversibility relations, recalled by name from the phase-change-on-reflection article, supply $tt' = 1 - R$ (and $r' = -r$ ) for a lossless surface, and it is exactly this that makes the on-resonance transmittance equal one rather than some smaller number. You could instead carry the Fresnel amplitude coefficients explicitly, but the reversibility relations are convention-independent and cleaner — they are the reason "lossless cavity transmits everything on resonance" is a theorem, not a coincidence.

Real etalons never quite reach T = 1 on resonance — why?

Because real mirrors absorb and scatter a little. The lossless Airy function assumes every photon is either transmitted or reflected; introduce a per-bounce absorption or scattering loss $A$ and the peak transmittance drops to $T_{\max} = [1 - A/(1-R)]^2$ , which can be well below one even though the peaks stay narrow. The insidious part is that the penalty scales as $A/(1-R)$ : the higher the reflectivity you chase for sharpness, the smaller the transmission window $1-R$ becomes and the more a fixed absorption eats into the peak. This is why cavity mirrors are specified not just by reflectivity but by parts-per-million loss, and why the finesse of the very best cavities is limited by absorption, not by how reflective the coatings are.

Is each Airy peak really a Lorentzian, like a resonance linewidth?

Near its top, yes. Expand the Airy function about a resonance $\delta = 2\pi m + \varepsilon$ : with $\sin(\delta/2) \approx \varepsilon/2$ it becomes $T \approx 1/[1 + (F/4)\varepsilon^2]$ , which is exactly a Lorentzian of half-width $\Delta\delta_{1/2} = 4/\sqrt{F}$ . That is more than a curve-fitting coincidence — the cavity is a damped harmonic resonator for the light, and its transmission line shape is the same Lorentzian you get for any weakly damped oscillator. The frequency width $\Delta\nu_{1/2}$ then relates to the photon storage time $\tau$ of the cavity by $\Delta\nu_{1/2} = 1/(2\pi\tau)$ : sharper peaks mean light is stored longer, which is the resonator viewpoint that laser and gravitational-wave physics lean on.

How is this different from a diffraction grating, which also gives sharp lines?

Both sharpen by adding many coherent beams, but they arrange the beams differently. A grating splits one wavefront into $N$ equal-amplitude beams side by side in space; its peak width scales as $1/N$ , set by how many slits you illuminate. A multiple-beam interferometer splits one beam into a ladder of beams in time, of geometrically decreasing amplitude $R^n$ ; its effective number of participating beams — and hence its sharpness — is set by the reflectivity through the finesse, roughly $\mathcal{F}$ beams. The grating's resolving power grows with slit count and order; the etalon's grows with reflectivity and order. They are complementary tools, and high-resolution spectrographs sometimes cross a grating with an etalon to get the best of both.