Optical Resonators and Cavity Modes

The resonator is quietly present in nearly every serious use of light. In metrology, ultrahigh-finesse Fabry–Pérot reference cavities — mirrors superpolished to reflectances beyond $0.9999$, finesses of $10^5$ to $10^6$ — are the flywheels that stabilise the world's best lasers: locking to such a cavity by the Pound–Drever–Hall technique transfers its length stability onto the light, giving the sub-hertz linewidths that run optical atomic clocks. The largest example is a gravitational-wave detector, where each 4 km arm of LIGO is an optical resonator and a passing strain is read out as a shift in the arm cavity's resonance. At the opposite scale, cavity ringdown spectroscopy times the decay of light leaking from a high-finesse cell; because that rate is set by the round-trip loss, a trace of absorbing gas lengthens the effective path to kilometres and reveals parts-per-trillion concentrations. And in every laser on every bench, the same two mirrors choose which longitudinal modes fall under the gain, select the $\mathrm{TEM}_{00}$ beam a fibre will accept, and set the loss the gain must beat to reach threshold.

Launch a wave between two mirrors and watch it bounce. After one round trip it returns, superposes on the stored field, and either reinforces or fights it. If the round-trip phase has slipped by an exact multiple of $2\pi$ the returning wave lands in step and the amplitude builds; at any other frequency successive round trips dephase and average to nothing. Only a discrete comb of frequencies survives this filter, and their spacing — the free spectral range — depends on one thing only, the optical length the light must retrace. This is the origin of longitudinal modes, and why a metre-long cavity combs the spectrum into teeth 150 MHz apart even though the light itself oscillates at nearly $5\times10^{14}$ Hz.

Transversely the story is about focusing, not phase. A well-built cavity is a periodic sequence of lenses — the curved mirrors — that keep catching a spreading beam and throwing it back to the same size and curvature. Exactly one family of profiles survives this endless refocusing without collapsing or blowing up: the Hermite–Gaussian modes, with the plain Gaussian $\mathrm{TEM}_{00}$ as the lowest-loss member. If the mirrors focus too weakly or too strongly no such self-reproducing beam exists; a ray then climbs a little higher off the axis each pass until it clears the mirror edge and is lost. Whether a pair of mirrors traps light or spills it is decided entirely by geometry, packaged into the two numbers $g_1$ and $g_2$ and the band $0 \le g_1 g_2 \le 1$.

The resonator does two jobs at once for a laser: it recycles photons through the gain medium — the feedback that lets a modest single-pass gain build into full oscillation — and it dictates the allowed outputs, both frequency comb and beam shape. A useful way to picture the losses is to count bounces. A cavity in which light survives roughly $N$ round trips before leaking away has a finesse of order $N$, resonances $N$ times narrower than their spacing, and a photon lifetime $\tau_p$. High reflectivity means many bounces, sharp teeth, and a long optical memory — the physics behind reference cavities and cavity ringdown alike.

The most basic quantisation comes from the axial phase condition. A wave must return in phase after retracing the optical path $2nL$, forcing the round-trip phase $2knL$ to equal $2\pi q$ for an integer $q$; solving for frequency turns a continuous spectrum into a comb.

where: $\nu_q$ — frequency of the $q$-th longitudinal mode; $q$ — large positive integer, the half-wavelengths in one pass; $c$ — vacuum speed of light; $n$ — intracavity index; $L$ — one-way separation, so the round-trip path is $2nL$; $\nu_F$ — free spectral range. The comb is perfectly uniform: $\nu_F$ is independent of $q$ and set by optical length alone, never by the reflectivities. Whether any mode exists is a question of geometry, decided by iterating the round-trip ray matrix. A paraxial ray stays bounded only if that matrix rotates it — eigenvalues on the unit circle — rather than magnifies it, and the trace carries the verdict.

where: $g_i = 1 - L/R_i$ — stability parameter of mirror $i$; $R_i$ — radius of curvature, positive when concave toward the cavity; $L$ — separation. Inside the band the cavity refocuses light onto itself; on the edges it is marginally stable; outside it a ray diverges and no confined mode survives. Inside the stable band the transverse eigenmode is a Gaussian beam, written through its complex beam parameter $q(z)$. Self-consistency under one round trip means $q(z)$ must be unchanged by the round-trip ABCD map — a fixed-point condition that fixes both the spot size and the wavefront curvature everywhere in the cavity.

where: $q(z)$ — complex beam parameter; $R(z)$ — wavefront radius of curvature; $w(z)$ — $1/e^2$ intensity spot radius; $\lambda$ — vacuum wavelength; $n$ — refractive index. The imaginary part pins the beam to a finite width: the physical fixed-point root has $\mathrm{Im}\,q > 0$ whenever the cavity is stable, which is why stability and a confined mode are the same statement. Carrying that fixed-point condition through the mirror geometry gives the beam waist in closed form — for a two-mirror spherical cavity it depends only on $L$, $\lambda$, and the two $g$-parameters.

where: $w_0$ — Gaussian beam waist radius; $g_1,g_2$ — mirror stability parameters; $L$ — separation; $\lambda$ — wavelength in the medium. The waist shrinks toward zero at the stability edges and is largest deep inside the band — a plane wave has an infinite waist, a tightly focused confocal mode a small one. The transverse profile also shifts the comb. Higher-order Hermite–Gaussian modes accumulate extra Gouy phase each pass, so a $\mathrm{TEM}_{mn}$ mode sits not at the pure longitudinal frequency but displaced by an amount proportional to $(m+n+1)$.

where: $m,n$ — transverse Hermite–Gaussian indices; $q$ — longitudinal index; the $\arccos\sqrt{g_1g_2}$ factor is the round-trip Gouy phase. The transverse modes interleave between the longitudinal teeth; in the confocal case the Gouy term is exactly $\pi/2$, collapsing the observed spacing to $c/4L$. The profiles are the Hermite–Gaussian functions $E_{mn}\propto H_m(\sqrt2\,x/w)\,H_n(\sqrt2\,y/w)\,e^{-(x^2+y^2)/w^2}$: the fundamental $\mathrm{TEM}_{00}$ is a clean Gaussian of smallest extent and lowest loss, while raising $m$ or $n$ adds off-axis lobes that clip on any aperture. Real mirrors leak, so each sharp mode broadens into a Lorentzian. The width follows from the Airy transmission of the Fabry–Pérot etalon and is quoted through the finesse — the ratio of mode spacing to mode width.

where: $\mathcal{F}$ — finesse; $\delta\nu$ — FWHM linewidth of one resonance; $R_1,R_2$ — mirror intensity reflectances. For equal mirrors this reduces to $\mathcal{F} = \pi\sqrt{R}/(1-R)$; finesse is essentially the round trips a photon completes before leaking out, diverging as $R\to 1$. The same losses set a decay time and a quality factor. An exponentially decaying stored field has a Lorentzian spectrum, tying photon lifetime, linewidth, and $Q$ into one chain.

where: $Q$ — quality factor, stored energy over energy lost per radian; $\nu_0$ — resonance frequency; $\tau_p$ — photon lifetime; $\delta\nu$ — linewidth. Finesse counts round trips, $Q$ counts optical radians; they differ by $\nu_0/\nu_F = q$, so $Q = q\,\mathcal{F}$ with $q\sim 10^6$.

The behaviour of a resonator turns sharply on its geometry and its loss. The four configurations below anchor the extremes.

1 · A helium–neon laser cavity, end to end (confidence builder)

A HeNe laser has a 0.30 m air cavity with mirrors of reflectance $R_1 = R_2 = 0.99$ at $\lambda = 633\ \mathrm{nm}$. Find the free spectral range, finesse, linewidth, photon lifetime, and $Q$, and estimate how many longitudinal modes oscillate under the 1.5 GHz Doppler gain bandwidth.

Step 1. Free spectral range from equation (1): $\nu_F = c/2L = (3.00\times10^8)/(2\times 0.30) = \mathbf{5.00\times10^8\ Hz = 500\ MHz}$, the tooth spacing set by length alone.

Step 2. Finesse from equation (6) with equal mirrors, $\mathcal{F} = \pi\sqrt R/(1-R) = \pi\times 0.995/0.01 = \mathbf{313}$ — the photon makes of order 313 round trips before leaking away.

Step 3. Linewidth from $\delta\nu = \nu_F/\mathcal{F} = 500\ \mathrm{MHz}/313 = \mathbf{1.60\ MHz}$ — a single resonance about 300 times narrower than the spacing.

Step 4. Photon lifetime two ways, as a cross-check. From equation (7), $\tau_p = 1/(2\pi\,\delta\nu) = 1/(2\pi\times 1.60\times10^6) = \mathbf{99.6\ ns}$. From the round-trip picture, $t_{RT} = 2L/c = 2.0\ \mathrm{ns}$ and $\ell \approx 1 - R^2 = 0.0199$, so $\tau_p = t_{RT}/\ell = 100\ \mathrm{ns}$ — the two agree.

Step 5. Quality factor: $\nu_0 = c/\lambda = 4.74\times10^{14}\ \mathrm{Hz}$, so $Q = \nu_0/\delta\nu = \mathbf{3.0\times10^8}$, consistent with $Q = 2\pi\nu_0\tau_p$ and with $Q = q\,\mathcal{F}$ at longitudinal index $q = 2L/\lambda \approx 9.5\times10^5$.

Step 6. Modes under the gain curve: $1.5\ \mathrm{GHz}/500\ \mathrm{MHz} = \mathbf{3}$. A typical short HeNe runs on about three longitudinal modes at once — which is why single-frequency versions use a shorter cavity to push $\nu_F$ past the gain bandwidth. ν_F = 500 MHz, F = 313, δν = 1.60 MHz, τ_p ≈ 100 ns, Q ≈ 3.0×10⁸, ~3 longitudinal modes.

2 · Is a symmetric cavity stable, and how big is its mode?

A symmetric cavity uses two mirrors of radius $R = 1.0\ \mathrm{m}$ separated by $L = 0.50\ \mathrm{m}$ at $\lambda = 1064\ \mathrm{nm}$ (Nd:YAG). Check its stability and find the beam waist.

Step 1. Compute the $g$-parameters: $g_1 = g_2 = 1 - L/R = 1 - 0.50/1.0 = 0.50$, so $g_1 g_2 = 0.25$. Since $0 \le 0.25 \le 1$, the cavity is comfortably stable, well inside the band, halfway between confocal and plane–parallel.

Step 2. Apply the waist formula (4), which for a symmetric cavity simplifies to $w_0^2 = \dfrac{\lambda L}{2\pi}\sqrt{\dfrac{1+g}{1-g}}$ with $g = 0.5$, so the square-root factor is $\sqrt{1.5/0.5} = \sqrt 3 = 1.732$.

Step 3. Insert the numbers: $\lambda L/2\pi = (1.064\times10^{-6})(0.50)/2\pi = 8.47\times10^{-8}\ \mathrm{m^2}$, so $w_0^2 = 8.47\times10^{-8}\times 1.732 = 1.47\times10^{-7}\ \mathrm{m^2}$ and $w_0 = \mathbf{0.383\ mm}$.

Step 4. Sanity-check the scale: the Rayleigh range is $z_R = \pi w_0^2/\lambda = 0.43\ \mathrm{m}$, comparable to the cavity length — just what you expect for a mode that fills a half-metre cavity without collapsing to a point or spilling off the mirrors. The cavity is stable (g₁g₂ = 0.25) with a waist w₀ ≈ 0.38 mm at the centre.

The optical resonator arrived long before its use: the two-mirror etalon was a precision spectroscopy tool for half a century before anyone saw it was the missing feedback element of a laser.

Texts. Siegman, Lasers (1986), Ch. 19–21 — the definitive treatment of resonator modes and stability · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 11 · Hecht, Optics, 5th ed., Ch. 9 and Ch. 13 · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 and 7 · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 5 · Foot, Atomic Physics, Ch. 7. The founding papers are Fox and Li (1961), Boyd and Gordon (1961), and Kogelnik and Li, Applied Optics (1966).

Provenance. Every equation is a standard textbook result; a verifier should recompute: (1) the comb and free spectral range $c/2nL$ (Hecht Ch. 9; Saleh and Teich Ch. 11), check $c/(2\times 1\ \mathrm{m}) = 150\ \mathrm{MHz}$; (2) the band $0 \le g_1 g_2 \le 1$ from the round-trip half-trace $2g_1g_2 - 1$ (Siegman Ch. 19); (3) the finesse $\mathcal{F} = \pi\sqrt R/(1-R)$ and $\delta\nu = \nu_F/\mathcal{F}$ from the Airy function; (4) $Q = 2\pi\nu_0\tau_p$, $\tau_p = 1/(2\pi\delta\nu)$ (Loudon Ch. 1). Numerical check: the 0.30 m HeNe cavity gives $\nu_F = 500\ \mathrm{MHz}$, $\mathcal{F} = 313$, $\delta\nu = 1.60\ \mathrm{MHz}$, $\tau_p \approx 100\ \mathrm{ns}$, $Q \approx 3.0\times10^8$; the symmetric $R = 1\ \mathrm{m}$, $L = 0.5\ \mathrm{m}$ cavity has $g_1g_2 = 0.25$ and waist $w_0 \approx 0.38\ \mathrm{mm}$ at 1064 nm.

Prerequisites. Gaussian Beams · ABCD Matrix Beam Propagation · Stimulated Emission and the Gain Coefficient. Leads to. Longitudinal Modes and Free Spectral Range · Transverse Cavity Modes (Hermite–Gaussian) · Resonator Stability and the g-Parameter Diagram · Cavity Q Factor and Photon Lifetime · Laser Threshold Condition.

Ask the tutor: Give me a wavelength, a length, and your mirror curvatures, and I will walk you through the cavity's stability, waist, mode comb, finesse, and photon lifetime step by step.