Longitudinal Modes and Free Spectral Range

Give light two mirrors and a length to run between, and the resonator stops behaving like empty space: it becomes a filter that admits only a discrete comb of frequencies. This article takes the round-trip self-consistency that aq-optics-optical-resonators-cavity-modes establishes for the general cavity field and follows it down the optic axis to its sharpest consequence. A field builds to macroscopic amplitude only if it reproduces its own phase after one complete circuit; every other frequency interferes with its own reflections and dies. The survivors, distinguished by the integer number $q$ of half-wavelengths packed between the mirrors, are the longitudinal (axial) modes. Impose that round-trip phase be a multiple of $2\pi$ and the allowed frequencies fall out as an evenly spaced ladder, $\nu_q = q\,c/(2nL)$ , whose rung-to-rung spacing is the free spectral range $\Delta\nu_{\mathrm{FSR}} = c/(2nL)$ . Because that spacing depends on nothing but the round-trip optical length, it is one of the most robust numbers in laser physics: a $30\,\mathrm{cm}$ air-spaced cavity gives $500\,\mathrm{MHz}$ , a short semiconductor diode over a hundred gigahertz. Which modes oscillate is settled by how the gain bandwidth compares with the FSR, and each surviving mode is not infinitely sharp — mirror leakage gives it a photon lifetime and a Lorentzian width, whose ratio to the FSR is the finesse. That comb is the frequency ruler behind Fabry–Perot analysers, reference cavities, and optical frequency combs. Losses and transverse structure belong to the companion pages; here we build the ideal axial comb from the ground up.

Key takeaways

A linear cavity of length $L$ supports only axial standing waves whose round-trip phase is an integer multiple of $2\pi$ — the longitudinal modes, indexed by a large integer $q \sim 10^5$ – $10^6$ .
The empty-cavity resonances form an evenly spaced comb, $\nu_q = q\,c/(2nL)$ .
Adjacent modes are separated by the free spectral range $\Delta\nu_{\mathrm{FSR}} = c/(2nL)$ , fixed by the round-trip optical length alone and equal to the circulation rate $1/\tau_{\mathrm{rt}}$ .
Which modes lase follows from the overlap of the fixed comb with the gain bandwidth: $N_{\mathrm{modes}} \approx \Delta\nu_{\mathrm{gain}}/\Delta\nu_{\mathrm{FSR}}$ .
Finite reflectivity gives each mode a Lorentzian width $\delta\nu = \Delta\nu_{\mathrm{FSR}}/\mathcal{F}$ ; the finesse $\mathcal{F}=\pi\sqrt{R}/(1-R)$ counts resolvable teeth.
Key idea: a cavity is a frequency filter that transmits — and, with gain, oscillates on — a comb of equally spaced longitudinal modes at $\nu_q = q\,c/(2nL)$ , whose constant spacing $\Delta\nu_{\mathrm{FSR}} = c/(2nL)$ is fixed entirely by the round-trip optical length.

The core insight

A cavity is a frequency filter: it transmits, and with gain oscillates on, a comb of equally spaced longitudinal modes at $\nu_q = q\,c/(2nL)$ , with the constant spacing $\Delta\nu_{\mathrm{FSR}} = c/(2nL)$ fixed entirely by the round-trip optical length.

Where this lives today

The longitudinal-mode comb has quietly become one of metrology's master tools. An optical frequency comb — a mode-locked laser whose spectrum is a comb of longitudinal modes spaced by exactly the free spectral range, the pulse repetition rate — turns a cavity's mode structure into a gear-train linking optical frequencies to the microwave clocks that count them. Each tooth $\nu_q = q\,f_{\mathrm{rep}} + f_0$ carries an integer label $q$ , so measuring an unknown laser against the nearest tooth reduces an optical-frequency measurement to counting beats — the machinery inside optical atomic clocks now reaching fractional uncertainties near $10^{-18}$ , and inside "astrocombs" that calibrate spectrographs hunting Earth-mass exoplanets. At the narrow-linewidth extreme, ultra-stable reference cavities push the same physics the other way. A rigid, low-expansion spacer holds two super-mirrors ( $R$ better than $0.9999$ , finesse above $10^{4}$ ) whose single longitudinal mode is so sharp — sub-hertz, photon lifetimes of many microseconds — that a laser locked to it by the Pound–Drever–Hall technique inherits that stability. Such lasers interrogate optical clock transitions and, scaled to kilometre arms, stabilize the light of gravitational-wave detectors like LIGO. In every case the principle is the one derived above: a fixed comb whose spacing is set by length alone, whose teeth are as sharp as the mirrors are good.

The physical picture

Think of the cavity as a one-dimensional box for light. A string clamped at both ends supports only harmonics with an integer number of half-wavelengths between the clamps; the optical field between two mirrors obeys the same rule, admitting only wavelengths that satisfy $L = q\,\lambda/2$ . Each allowed integer $q$ is one longitudinal mode — a standing wave with $q$ half-humps and nulls at the mirrors. For a visible laser $q$ is of order $10^6$ , because the cavity is a million wavelengths long, but the counting is exactly that of a vibrating string.

Figure

A one-dimensional box for light: q half-waves fit between the mirrors, and the allowed frequencies form an evenly spaced comb with tooth spacing Δν_FSR = c ÷ 2nL = 500 MHz for a 30 cm air cavity

Translate "an integer number of half-wavelengths" into frequency and the spacing becomes perfectly uniform: advancing $q \to q+1$ adds one more half-wave and always raises the frequency by the same step $c/(2nL)$ . That constant step is the free spectral range — the tooth-spacing of the comb. It also has a clean time-domain reading: a photon completes a round trip of optical path $2nL$ in $\tau_{\mathrm{rt}} = 2nL/c$ , so it circulates $c/(2nL)$ times per second, and that circulation frequency is the mode spacing. A short cavity circulates fast and spreads the teeth wide; a long cavity crowds them together. The cold-cavity comb is not the laser spectrum, though — only the menu. A real mode is not infinitely sharp: photons leak through the mirrors, so each resonance is a Lorentzian of width $\delta\nu$ , and $\Delta\nu_{\mathrm{FSR}}/\delta\nu$ is the finesse. And of the whole comb, only modes lying beneath the gain curve and above the loss line oscillate. The gain is a bandpass window sliding over the fixed comb; how many teeth it covers decides single-mode versus multimode operation, as the second figure shows for a Doppler-broadened gas laser.

Figure

Gain selects modes: the cold-cavity comb of stems at each ν_q is filtered by the Doppler gain envelope and the loss threshold, so only modes where gain exceeds loss oscillate — about 3 longitudinal modes for a He–Ne laser with Δν_FSR = 500 MHz under a 1.5 GHz gain width

The core equations

The whole spectrum descends from one geometric fact: a whole number of half-wavelengths must span the mirror gap. Written for the wavelength inside the medium, this is the axial standing-wave condition, the optical analogue of a clamped string.

L = q\,\frac{\lambda_q}{2}, \qquad q = 1, 2, 3, \ldots \qquad (1)

where: $L$ — mirror separation; $\lambda_q$ — wavelength of mode $q$ inside the medium (in vacuum $\lambda_q^{\text{vac}} = n\lambda_q$ ); $q$ — longitudinal mode index, a large integer of order $10^5$ – $10^6$ . Mode $q$ is a standing wave with $q$ antinodes and a node at each mirror; raising $q$ by one inserts one more half-hump. Converting the wavelength condition to frequency (with $\nu = c/\lambda_q^{\text{vac}}$ and $\lambda_q = \lambda_q^{\text{vac}}/n$ ) turns the integer count into an evenly spaced ladder of resonances — the empty-cavity comb.

\nu_q = q\,\frac{c}{2nL} \qquad (2)

where: $\nu_q$ — resonance frequency of mode $q$ ; $c$ — vacuum speed of light; $n$ — intracavity refractive index. Read it as a ruler with a million marks: each integer $q$ names one tooth, perfectly regular because $\nu_q$ is strictly linear in $q$ . The spacing between neighbouring teeth is the single most useful number attached to any cavity. Because $\nu_q$ is linear in $q$ , subtracting adjacent modes cancels $q$ entirely, leaving a spacing that is the same everywhere in the spectrum.

\Delta\nu_{\mathrm{FSR}} = \nu_{q+1}-\nu_q = \frac{c}{2nL} \qquad (3)

where: $\Delta\nu_{\mathrm{FSR}}$ — free spectral range, the spacing of adjacent longitudinal modes. It depends only on the round-trip optical length $2nL$ — not on $q$ , wavelength, or reflectivity — which is why a cavity makes such a dependable frequency ruler. The same quantity has a time-domain identity that is often the fastest estimate: the FSR is the rate at which a photon circulates the cavity — the round-trip path over the speed of light, inverted.

\Delta\nu_{\mathrm{FSR}} = \frac{1}{\tau_{\mathrm{rt}}}, \qquad \tau_{\mathrm{rt}} = \frac{2nL}{c} \qquad (4)

where: $\tau_{\mathrm{rt}}$ — round-trip transit time of a photon. This bridges the frequency and time pictures: mode-comb spacing (frequency domain) and pulse repetition rate of a mode-locked train (time domain) are literally the same number. Real mirrors are not perfect, so each tooth broadens into a Lorentzian of finite width. Summing the multiply-reflected partial waves gives the Airy transmission function, whose sharpness is captured by a single figure of merit, the finesse.

\mathcal{F} = \frac{\Delta\nu_{\mathrm{FSR}}}{\delta\nu} = \frac{\pi\sqrt{R}}{1-R} \qquad (5)

where: $\mathcal{F}$ — cavity finesse; $\delta\nu$ — full width at half maximum of one mode; $R$ — intensity reflectivity of each mirror ( $R_1=R_2=R$ ). The finesse counts resolvable teeth per FSR; it is a property of the mirrors alone, independent of cavity length. Finally, a mode's width is tied to how long a photon survives in the cavity — the frequency and time faces of the same loss. A narrower line means a longer-lived photon.

\delta\nu = \frac{\Delta\nu_{\mathrm{FSR}}}{\mathcal{F}} = \frac{1}{2\pi\tau_c} \qquad (6)

where: $\tau_c$ — cavity photon lifetime, the $1/e$ decay time of stored energy. A high-finesse cavity stores photons long and so has a very narrow linewidth; this one relation underlies both reference-cavity stabilization and cavity ringdown.

Derivation AResonance frequencies and the free spectral range

The goal is to extract the entire longitudinal spectrum — the resonance frequencies and their constant spacing — from a single physical demand: that the intracavity field reproduce its own phase after one complete round trip, so successive passes interfere constructively rather than cancelling. We start from a plane wave inside the medium, impose the round-trip phase condition, solve for the allowed frequencies, then difference neighbours to expose the free spectral range and its time-domain meaning.

Step 1. Let the field be $E \propto e^{i(kz - \omega t)}$ with $k = 2\pi n\nu/c$ in the medium, and compute the phase gained on one round trip: $\Phi_{\mathrm{rt}} = 2kL = 2(2\pi n\nu/c)L$ . Why: the field traverses $L$ twice per circuit; with real mirror reflectivity, any $\pi$ shift on reflection cancels over the round trip or merely relabels $q$ .

Step 2. Demand self-consistency: the round-trip phase must be an integer multiple of $2\pi$ , so $\Phi_{\mathrm{rt}} = 2\pi q$ with $q$ a large positive integer. Why: any other phase makes successive round-trip contributions rotate and average to zero — only the resonant frequencies survive and grow.

Step 3. Equate and solve for the frequency: $2(2\pi n\nu_q/c)L = 2\pi q$ gives $\nu_q = q\,c/(2nL)$ . Why: this is the comb — equally spaced resonances, one per integer $q$ , the frequency form of the counting $L = q\lambda_q/2$ .

Step 4. Difference adjacent modes: $\Delta\nu_{\mathrm{FSR}} = \nu_{q+1} - \nu_q = c/(2nL)$ , independent of $q$ . Why: the cancellation of $q$ makes the spacing uniform across the whole spectrum — a self-calibrating ruler, not scattered lines.

Step 5. Read it in time: since $\tau_{\mathrm{rt}} = 2nL/c$ , the spacing is $\Delta\nu_{\mathrm{FSR}} = 1/\tau_{\mathrm{rt}}$ , the photon circulation frequency. Why: the spacing on a spectrum analyser equals the rate a pulse bounces back and forth. Check: $n=1$ , $L=0.30\,\mathrm{m}$ gives $(3\times10^8)/(2\times0.30) = 500\,\mathrm{MHz}$ , the standard He–Ne value.

Go deeper · Derivation B · 1 minFinesse and linewidth from the Airy function

Derivation BFinesse and linewidth from the Airy function

The goal is to see why finite mirror reflectivity turns each infinitely thin comb tooth into a Lorentzian of measurable width, and to derive the finesse that relates that width to the FSR. We sum the multiply-reflected partial waves bouncing between two mirrors, obtain the Airy transmission function, locate its half-maximum near a peak, and form the ratio of spacing to width — a finesse that depends on the mirrors alone.

Step 1. Superpose the transmitted partial waves — each one round trip further, carrying a factor $R\,e^{i\phi}$ — as a geometric series, giving the Airy transmission $T(\phi) = (1-R)^2 / [(1-R)^2 + 4R\sin^2(\phi/2)]$ , with $\phi = 2kL$ the round-trip phase. Why: the coherent sum of many reflections is what sharpens transmission into peaks; without it there is no narrow resonance.

Step 2. Recast with the coefficient of finesse: $T = 1/[1 + F\sin^2(\pi\nu/\Delta\nu_{\mathrm{FSR}})]$ , $F = 4R/(1-R)^2$ , so $T=1$ at every $\nu_q$ and peaks recur once per FSR. Why: the phase $\pi\nu/\Delta\nu_{\mathrm{FSR}}$ makes the periodicity explicit — a comb of identical peaks spaced by the free spectral range.

Step 3. Set the denominator to $2$ for the half-maximum, $F\sin^2(\pi\,\delta\nu/2\Delta\nu_{\mathrm{FSR}}) = 1$ , and linearise ( $\sin x \approx x$ ) to get $\pi\,\delta\nu/(2\Delta\nu_{\mathrm{FSR}}) \approx 1/\sqrt{F}$ . Why: for narrow peaks the argument is small, so the sine equals its argument — the step that makes the linewidth analytic.

Step 4. Form the finesse as spacing over width: $\mathcal{F} \equiv \Delta\nu_{\mathrm{FSR}}/\delta\nu = \pi\sqrt{F}/2 = \pi\sqrt{R}/(1-R)$ . Why: substituting $F$ collapses everything to reflectivity — higher $R$ means higher finesse, sharper modes. Check: $R=0.99$ gives $\mathcal{F} \approx 313$ , so a $500\,\mathrm{MHz}$ FSR yields $\delta\nu \approx 1.6\,\mathrm{MHz}$ .

Go deeper · Derivation C · 1 minPhoton lifetime as the time-domain twin of linewidth

Derivation CPhoton lifetime as the time-domain twin of linewidth

The goal is to show that a mode's spectral width and a photon's survival time are two descriptions of one loss, connected by the Fourier relation $\delta\nu = 1/(2\pi\tau_c)$ . We model the stored field as an oscillation decaying because a fixed fraction of the energy leaks out each round trip, then Fourier-transform it to read off the linewidth — the relation that lets ringdown experiments measure a megahertz linewidth by timing a microsecond decay.

Step 1. Let the stored energy decay as $U(t) = U_0\,e^{-t/\tau_c}$ , where each round trip of duration $\tau_{\mathrm{rt}}$ loses a fraction set by the mirrors, so $\tau_c = \tau_{\mathrm{rt}}/(1-R)$ for two mirrors $R$ with no other loss. Why: the photon lifetime is the round-trip time over the fractional loss per trip — low-loss trips give a long lifetime.

Step 2. The amplitude then rings as $E(t) \propto e^{-t/2\tau_c}\,e^{-i2\pi\nu_q t}$ (energy $\propto$ amplitude squared, hence the factor of two), a damped sinusoid. Why: a decaying oscillation, not a pure tone, is what the leaky cavity stores, and its finite duration forbids an infinitely sharp line.

Step 3. Fourier-transform it: the power spectrum is a Lorentzian centred at $\nu_q$ with FWHM $\delta\nu = 1/(2\pi\tau_c)$ . Why: a shorter-lived oscillation spans a broader band — the same time–bandwidth reciprocity that governs every damped resonance.

Step 4. Close the loop with the finesse: $\delta\nu = \Delta\nu_{\mathrm{FSR}}/\mathcal{F} = 1/(2\pi\tau_c)$ , so $\tau_c = \mathcal{F}\,\tau_{\mathrm{rt}}/(2\pi)$ . Why: a high-finesse cavity stores each photon for $\mathcal{F}/2\pi$ round trips — long lifetime and narrow line together. Check: $\mathcal{F}=313$ , $\Delta\nu_{\mathrm{FSR}}=500\,\mathrm{MHz}$ gives $\delta\nu \approx 1.6\,\mathrm{MHz}$ , $\tau_c \approx 100\,\mathrm{ns}$ .

Limiting cases

Case	Result	Meaning
Short cavity (chip / microcavity)	$\Delta\nu_{\mathrm{FSR}} = \dfrac{c}{2nL} \gg \Delta\nu_{\mathrm{gain}}$	The FSR exceeds the gain bandwidth, so at most one mode fits under the gain — natural single-mode operation, as in short diode lasers.
Long cavity (gas laser)	$\Delta\nu_{\mathrm{FSR}} \ll \Delta\nu_{\mathrm{gain}}$	Many teeth lie within the gain; a Doppler-broadened He–Ne ( $1.5\,\mathrm{GHz}$ over $500\,\mathrm{MHz}$ ) runs on 2–3 modes.
Ring (travelling-wave) cavity	$\Delta\nu_{\mathrm{FSR}} = \dfrac{c}{n\mathcal{L}}$	Light circulates a perimeter $\mathcal{L}$ once per round trip, so the factor of 2 disappears.
High-finesse limit	$\mathcal{F} = \dfrac{\Delta\nu_{\mathrm{FSR}}}{\delta\nu} \gg 1$	Each linewidth becomes far smaller than the FSR — the basis of reference cavities, ringdown, and frequency combs.
Dispersive medium	$\Delta\nu_{\mathrm{FSR}} \to \dfrac{c}{2 n_g L}$	When $n$ varies across the band the group index $n_g$ sets the spacing, so the teeth are not perfectly uniform.

Worked examples

1 · The full comb of a He–Ne laser (confidence builder)

A helium–neon laser has an air cavity of length $L = 30\,\mathrm{cm}$ ( $n=1$ ), lases at $\lambda = 632.8\,\mathrm{nm}$ , and has Doppler gain width $\Delta\nu_{\mathrm{gain}} \approx 1.5\,\mathrm{GHz}$ . Find the free spectral range, round-trip time, mode index $q$ , and how many longitudinal modes oscillate.

Step 1. Free spectral range from equation (3): $\Delta\nu_{\mathrm{FSR}} = c/(2nL) = (3.00\times10^8)/(2\times1\times0.30) = \mathbf{5.00\times10^8\ Hz = 500\ MHz}$ — the tooth spacing of the whole comb.

Step 2. Round-trip time from equation (4): $\tau_{\mathrm{rt}} = 2nL/c = (2\times0.30)/(3.00\times10^8) = \mathbf{2.0\ ns}$ , and indeed $1/\tau_{\mathrm{rt}} = 5.0\times10^8\,\mathrm{Hz}$ reproduces the FSR: a photon circulates 500 million times a second.

Step 3. Mode index from equation (1), $q = 2nL/\lambda = (2\times0.30)/(632.8\times10^{-9}) = \mathbf{9.48\times10^5}$ . The standing wave carries roughly 948,000 half-wavelengths — which is why $q$ is never written out and only its spacing matters.

Step 4. Number of oscillating modes, $N_{\mathrm{modes}} \approx \Delta\nu_{\mathrm{gain}}/\Delta\nu_{\mathrm{FSR}} = 1.5\times10^9/5.00\times10^8 = \mathbf{3}$ . The gain window spans about three teeth, so the free-running laser emits on three longitudinal modes at once — the picture in the second figure. An intracavity etalon or a shorter tube would widen the effective spacing until one mode survives.

2 · Finesse, linewidth, and photon lifetime

The same $30\,\mathrm{cm}$ cavity is built with mirrors of intensity reflectivity $R = 0.99$ . Find the finesse, the cold-cavity linewidth of a single mode, and the photon lifetime.

Step 1. Finesse from equation (5): $\mathcal{F} = \pi\sqrt{R}/(1-R) = \pi\sqrt{0.99}/0.01 = \mathbf{313}$ resolvable teeth per free spectral range.

Step 2. Linewidth from equation (6): $\delta\nu = \Delta\nu_{\mathrm{FSR}}/\mathcal{F} = 500\,\mathrm{MHz}/313 = \mathbf{1.6\ MHz}$ — the cold-cavity width the empty cavity would show; gain narrowing sharpens the actual laser line far below it.

Step 3. Photon lifetime from $\delta\nu = 1/(2\pi\tau_c)$ : $\tau_c = 1/(2\pi\,\delta\nu) = 1/(2\pi\times1.6\times10^6) = \mathbf{1.0\times10^{-7}\ s \approx 100\ ns}$ — roughly $\mathcal{F}/2\pi \approx 50$ round trips, the narrow line and long-lived photon made concrete.

3 · A single-mode diode by geometry alone

A Fabry–Perot semiconductor laser has a cleaved chip of length $L = 300\,\mu\mathrm{m}$ and refractive index $n = 3.5$ . Find its free spectral range and explain why such short cavities favour single-mode operation.

Step 1. FSR from equation (3), with the semiconductor index: $\Delta\nu_{\mathrm{FSR}} = c/(2nL) = (3.00\times10^8)/(2\times3.5\times300\times10^{-6}) = \mathbf{1.4\times10^{11}\ Hz \approx 143\ GHz}$ — nearly 300 times wider than the He–Ne, because the cavity is a thousand times shorter and the high index only partly compensates.

Step 2. Even a semiconductor's broad gain (a few terahertz) now spans only a handful of teeth. Because $\Delta\nu_{\mathrm{FSR}}$ scales as $1/L$ , shrinking the cavity is the simplest route to single-mode output — the principle behind short-chip and distributed-feedback diode lasers, where geometry, not filtering, thins the comb.

Physics of the portrait

The longitudinal-mode comb is, historically, the Fabry–Perot interferometer read as a laser cavity: the same multiple-beam interference Fabry and Perot turned into the sharpest spectroscopic tool of their age is what quantises a laser's frequencies and defines its free spectral range and finesse.

Origins · 1 minCharles Fabry and how this was discovered

Charles Fabry

1867–1945 · French physicist, co-inventor of the interferometer that bears his name · interferometry and precision spectroscopy

Charles Fabry, working with Alfred Perot at Marseille in the late 1890s, built the multiple-beam interferometer that carries their names and, with it, the vocabulary of this article. Where Michelson's two-beam instrument gave broad sinusoidal fringes, the Fabry–Perot etalon — two parallel, highly reflective plates — makes light interfere with itself over hundreds of round trips, sharpening transmission into needle-thin peaks. Fabry and Perot introduced the two quantities that still define such a cavity: the free spectral range, the interval between transmission maxima, and the finesse, the ratio of that interval to a peak's width. They used the instrument to resolve hyperfine structure and measure wavelengths against the metre, and Fabry later co-discovered the atmospheric ozone layer with the same spectroscopy. When the laser arrived in 1960 its resonator was recognized at once as a Fabry–Perot etalon with gain inside: the longitudinal modes are its transmission peaks, the mode spacing its free spectral range, the linewidth-to-spacing ratio its finesse. The frequency comb of a modern optical clock is, in the end, the Fabry–Perot comb with a laser lit inside it.

History

1899

— Fabry and Perot build the multiple-beam interferometer, defining free spectral range and finesse

1958

— Schawlow and Townes analyze the resonant cavity modes required for an optical maser

1961

— Javan, Bennett and Herriott operate the first He–Ne laser, oscillating on several longitudinal modes

1964

— Hargrove, Fork and Pollack mode-lock a laser, phase-locking its longitudinal modes into a pulse train

2005

— Hall and Hänsch share the Nobel Prize for the optical frequency comb built on the cavity mode ruler

Going further

Texts. Hecht, Optics, 5th ed., Ch. 9 (Fabry–Perot) · Saleh and Teich, Fundamentals of Photonics, 2nd ed., Ch. 10 (Resonator Optics), Ch. 15 (Lasers) · Svelto, Principles of Lasers, 5th ed., Ch. 5 · Foot, Atomic Physics, Ch. 7 · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 5.

Provenance. The mode-comb and FSR results follow Saleh and Teich (Ch. 10) and Hecht (Ch. 9); the Airy finesse $\mathcal{F}=\pi\sqrt{R}/(1-R)$ and the linewidth–lifetime relation $\delta\nu = 1/(2\pi\tau_c)$ are standard. A verifier should confirm: (1) $\Delta\nu_{\mathrm{FSR}} = 500\,\mathrm{MHz}$ for $L=0.30\,\mathrm{m},\,n=1$ ; (2) $\mathcal{F}\approx313$ and $\delta\nu\approx1.6\,\mathrm{MHz}$ for $R=0.99$ ; (3) the He–Ne Doppler width $\approx1.5\,\mathrm{GHz}$ and $\sim3$ oscillating modes (Demtröder Ch. 5, Foot Ch. 7); (4) the diode FSR $\approx143\,\mathrm{GHz}$ for $L=300\,\mu\mathrm{m},\,n=3.5$ . No non-standard numbers are asserted.

Builds on: Optical Resonators and Cavity Modes · Gaussian Beams · Stimulated Emission and Gain. Leads to: Cavity Q Factor and Photon Lifetime · Transverse Cavity Modes · Laser Threshold Condition · Four-Level Laser Systems.

Ask the tutor: Give me a cavity length and mirror reflectivity, and we will build its comb, finesse, and linewidth from scratch.

Why is the mode spacing exactly constant across the whole spectrum?

Because the resonance frequencies $\nu_q = qc/(2nL)$ are strictly linear in the integer $q$ . Subtracting neighbours, $\nu_{q+1}-\nu_q$ , cancels $q$ and leaves $c/(2nL)$ — the tooth at $q=948{,}000$ sits the same distance from its neighbour as the one at $q=948{,}001$ . The one caveat is dispersion: if $n$ varies across the band, the group index $n_g$ sets the spacing and the teeth become very slightly non-uniform — negligible in air, but a real effect in a long fibre or broadband comb, where it must be engineered away.

What is the difference between a longitudinal mode and a transverse mode?

They are the two independent halves of the cavity field. A longitudinal (axial) mode counts how many half-wavelengths fit along the optic axis — the integer $q$ — and sets the frequency through $\nu_q = qc/(2nL)$ . A transverse mode describes the beam's shape across the axis, the Hermite–Gaussian pattern $(m,n)$ that governs the spot you see on a card. The two are not fully separate in frequency: the Gouy phase shifts each transverse family by an amount proportional to $(m+n+1)$ , so real spectra are interleaved combs. The fundamental $\mathrm{TEM}_{00}$ beam has one clean longitudinal comb, which is what this article treats.

Is the free spectral range the same as a laser's linewidth?

No — confusing them is a common error. The free spectral range is the spacing between adjacent modes, $\Delta\nu_{\mathrm{FSR}} = c/(2nL)$ , typically hundreds of megahertz to hundreds of gigahertz. The linewidth $\delta\nu$ is the width of one mode, smaller than the FSR by the finesse: $\delta\nu = \Delta\nu_{\mathrm{FSR}}/\mathcal{F}$ . For a $500\,\mathrm{MHz}$ FSR and finesse $313$ the cold-cavity linewidth is only $1.6\,\mathrm{MHz}$ . Their ratio, the finesse, counts resolvable teeth per FSR, so a good cavity keeps spacing and width orders of magnitude apart.

How does a laser end up running on a single longitudinal mode?

Two routes, both visible in the equations. First, geometry: since $\Delta\nu_{\mathrm{FSR}} = c/(2nL)$ scales as $1/L$ , a short enough cavity spreads the teeth wider than the gain bandwidth, so only one mode lies under the gain — why short diode chips are naturally single-mode. Second, filtering: a long cavity carries a frequency-selective element (etalon, birefringent filter, or grating) that adds loss to every tooth but one, or relies on homogeneous gain where the strongest mode suppresses the rest by competition. Reference-cavity lasers add active locking to hold one mode to sub-hertz precision.

Why does each mode have a Lorentzian shape rather than a sharp spike?

Because the cavity leaks. A lossless resonator would store a mode forever and show an infinitely sharp line, but real mirrors transmit a little on every bounce, so the stored field is a damped oscillation, not a pure tone. Fourier-transforming a decaying sinusoid gives a Lorentzian centred on the resonance, its width inversely proportional to the decay time: $\delta\nu = 1/(2\pi\tau_c)$ . The faster the photons leak (shorter $\tau_c$ , lower finesse), the broader the line — the same time–bandwidth reciprocity that broadens any damped resonance, from an RLC circuit to an atomic transition.

If q is around a million, how do we ever know which mode we are on?

Usually we do not need the absolute value of $q$ — only differences matter, and those come cleanly from the FSR. Beating two lasers together, or scanning a Fabry–Perot analyser, measures spacings of one or a few FSR without ever counting to a million. The exception is the optical frequency comb, whose whole point is the absolute tooth number. There the trick is self-referencing: measuring the offset $f_0$ and repetition rate $f_{\mathrm{rep}}$ , and knowing the optical frequency roughly, one solves $\nu = qf_{\mathrm{rep}} + f_0$ for the unique integer $q$ — converting a microwave count directly into an optical frequency.