Cavity Q Factor and Photon Lifetime

Trap a beam of light between two mirrors and it does not stay forever. Every round trip a fixed slice of the energy leaks through the partially transmitting mirrors, scatters off coating imperfections, or is absorbed in the beam path — so once the drive is cut, the stored field rings down exponentially. Two numbers pin down how well a resonator hoards its light: the dimensionless quality factor $Q$ and the photon lifetime $\tau_p$ — the same fact in two languages, one in the frequency domain (how sharp the resonance) and one in the time domain (how long the light survives). This article takes its raw material from the resonator chapters before it. From optical resonators and cavity modes it inherits the standing-wave modes and the round-trip survival factor; from longitudinal modes and free spectral range, the mode comb, its spacing, and the finesse that measures how sharp each tooth is. And underneath it borrows one mathematical result: a damped harmonic oscillator has a Lorentzian spectrum, so an exponentially decaying cavity field shows up as a Lorentzian resonance whose width is fixed by the decay time. $Q$ and $\tau_p$ simply translate the sharpness of that comb into energy-storage and decay language. Because an optical mode oscillates at $10^{14}$ – $10^{15}$ Hz, even a nanosecond of storage is millions of periods — which is why optical cavities are among the highest- $Q$ resonators humans build, and why they make narrow-linewidth lasers, optical clocks, and cavity QED possible at all.

Key takeaways

The quality factor $Q=\omega_0\,W/P_\mathrm{diss}=\omega_0\tau_p$ compares the energy stored in a mode to the energy lost per radian of oscillation — it counts cycles the light survives.
The photon lifetime $\tau_p$ is the $1/e$ decay time of the stored energy once the source is removed; it fixes the cold-cavity linewidth through $\Delta\omega_c=1/\tau_p$ .
For a two-mirror cavity $\tau_p=t_r/\delta$ , the round-trip time divided by the fractional round-trip loss — a bigger bucket or a smaller leak stores light longer.
Finesse $\mathcal{F}=\mathrm{FSR}/\Delta\nu_c\approx2\pi/\delta$ measures resolving power and depends only on the mirrors; $Q=q\,\mathcal{F}$ adds the mode number $q$ , so length matters for $Q$ but not for $\mathcal{F}$ .
A typical laser cavity reaches $Q\sim10^7$ – $10^8$ (nanosecond lifetimes); whispering-gallery microresonators exceed $Q>10^9$ (microsecond lifetimes).
Key idea: $Q$ counts oscillation cycles before the energy decays and $\tau_p$ counts seconds; they are locked together by $Q=\omega_0\tau_p$ , and both are set by the total fractional loss per round trip.

The core insight

$Q$ counts oscillation cycles before the energy decays; the photon lifetime $\tau_p$ counts seconds. They are linked by $Q=\omega_0\tau_p$ , and both are fixed by the total fractional loss per round trip.

Where this lives today

Nowhere is the photon lifetime read more literally than in cavity ring-down spectroscopy. A laser pulse is injected into a high-finesse cell, the drive is cut, and a detector times the exponential decay of the leaking light: the decay constant is $\tau_p$ , and any trace gas shortens it by adding absorption to the loss budget, shifting $1/\tau_p$ by $c\,\alpha_\mathrm{gas}/n$ . Because it reads a time constant rather than an absolute intensity, the method shrugs off laser-power drift and resolves absorptions of $10^{-9}$ per pass. The same physics runs in reverse when stabilising a laser: locking it to a Fabry–Pérot cavity of finesse $\sim10^5$ (Pound–Drever–Hall locking) transfers the cavity's sub-hertz linewidth $\Delta\nu_c=1/2\pi\tau_p$ onto the laser, producing the references behind optical atomic clocks and the arm cavities of gravitational-wave detectors like LIGO. At the small extreme, whispering-gallery microspheres reach $Q>10^9$ in structures you could balance on an eyelash: at $Q=10^9$ and $\lambda=1.55\ \mu$ m the lifetime is $\tau_p=Q/\omega_0\approx0.82\ \mu$ s and the light travels $c\,\tau_p\approx247$ m — a million round trips inside a sphere tens of micrometres across — enabling single-atom strong coupling in cavity QED and single-molecule biosensing.

The physical picture

Picture the cavity as a leaky bucket of light. Each round trip a fixed fraction $\delta$ of the photons escapes or is absorbed, so the bucket empties by the same proportion every transit — and proportional emptying is exponential emptying. The stored energy falls as $W(t)=W_0\,e^{-t/\tau_p}$ , and the time constant $\tau_p$ is longer for a bigger bucket (a longer cavity, longer round-trip time) or a smaller leak (higher reflectivity). Cut the drive and watch the transmitted light fade — that fade is the photon lifetime, and it is what a ring-down measurement times directly.

Figure

Cavity ring-down in the time domain: after the drive is cut, the stored energy decays as W = W₀ exp(−t/τ_p) with τ_p = 100 ns, reaching 1/e ≈ 0.37 of its value at t = τ_p, while the field amplitude decays half as fast so its envelope ∝ exp(−t/2τ_p) reaches 1/e only at 200 ns

$Q$ reframes the same leak in units natural to a wave: instead of "how many seconds," it asks "how many oscillation cycles" the light survives — energy stored divided by energy lost per radian. Because optical frequencies are enormous, even a short-lived photon completes millions of cycles, so optical $Q$ values dwarf those of a pendulum or an RLC circuit by many orders of magnitude — the 100 ns lifetime above already gives $Q$ near $2\times10^8$ . Now flip to the frequency domain. An exponential decay in time is a Lorentzian peak in frequency — the two are Fourier partners — and the narrower the peak, the longer the ringing. That is the whole content of the duality $\Delta\omega_c\,\tau_p=1$ : reading a razor-thin transmission line is reading, directly, a long photon lifetime. The finesse strips the length back out and reports each mode's sharpness relative to the comb spacing, so identical mirrors give identical finesse even when the lengths differ.

Figure

The same decay seen in the frequency domain: each cavity mode is a Lorentzian of width Δν_c = 1/2πτ_p = 1.6 MHz, and the modes are spaced by the free spectral range FSR = c/2L = 500 MHz, so the finesse ℱ = FSR/Δν_c ≈ 313 counts the resolvable modes — a long photon lifetime is a sharp resonance

The core equations

The quality factor is defined for any resonator — pendulum, RLC circuit, atomic line, optical mode — as the stored energy divided by the energy lost per radian of oscillation. Because a decaying mode loses power at the rate $W/\tau_p$ , this universal definition collapses into a one-line bridge between $Q$ and $\tau_p$ .

Q \equiv \omega_0\,\frac{W}{P_\mathrm{diss}} = \omega_0\,\tau_p \qquad (1)

where: $Q$ — dimensionless quality factor; $\omega_0$ — angular resonance frequency (rad s⁻¹); $W$ — energy stored in the mode (J); $P_\mathrm{diss}$ — power dissipated (W); $\tau_p$ — photon lifetime, the energy $1/e$ decay time (s). Read it as: $Q$ is the resonance frequency in units of the decay rate $1/\tau_p$ — a slow leak and a fast oscillation both push $Q$ up. The stored energy decays as a simple exponential — and here the subject's most common trap hides. The energy decays with time constant $\tau_p$ , but the field amplitude decays with time constant $2\tau_p$ , because energy is proportional to amplitude squared. Keeping this factor of two straight is what makes the linewidth right.

W(t) = W_0\,e^{-t/\tau_p}, \qquad E(t) \propto e^{-t/2\tau_p}\,e^{-i\omega_0 t} \qquad (2)

where: $W_0$ — initial stored energy (J); $E(t)$ — cavity field amplitude. The amplitude decays half as fast as the energy, and it is the amplitude that Fourier-transforms into the line. To connect $\tau_p$ to hardware, count the loss per pass. The fractional loss $\delta$ per round trip, spread over the transit time $t_r$ , sets the decay rate — the master formula tying the lifetime to mirror reflectivity and cavity length.

\tau_p = \frac{t_r}{\delta} = \frac{2nL}{c\,\delta}, \qquad t_r = \frac{2nL}{c}\ \ (\text{linear cavity}) \qquad (3)

where: $t_r$ — round-trip transit time (s); $\delta$ — fractional energy loss per round trip; $n$ — refractive index of the intracavity medium; $L$ — mirror separation (m); $c$ — vacuum speed of light. A longer cavity or a smaller leak both lengthen $\tau_p$ . The loss $\delta$ bundles every escape channel. Formally it is minus the logarithm of the survival factor, but for high-reflectivity mirrors it linearises and the channels simply add — two mirror transmissions plus whatever the medium absorbs and scatters per pass.

\delta = -\ln\!\big(R_1 R_2\,T_\mathrm{int}\big) \approx (1-R_1) + (1-R_2) + \alpha_\mathrm{int}\,\ell \qquad (4)

where: $R_1, R_2$ — power reflectivities of the two mirrors; $T_\mathrm{int}$ — single-pass internal transmission; $\alpha_\mathrm{int}$ — internal absorption-plus-scatter coefficient (m⁻¹); $\ell$ — single-pass path length in the lossy medium (m). For a symmetric high- $R$ cavity with no internal loss, $\delta\approx 2(1-R)$ . The lifetime immediately fixes the cold-cavity linewidth. An exponentially damped oscillation has a Lorentzian spectrum whose FWHM is exactly the energy decay rate — the Fourier duality that makes a sharp resonance and a long-lived mode the same statement.

\Delta\omega_c = \frac{1}{\tau_p}, \qquad \Delta\nu_c = \frac{1}{2\pi\tau_p} \qquad (5)

where: $\Delta\omega_c$ — FWHM angular linewidth (rad s⁻¹); $\Delta\nu_c$ — FWHM linewidth (Hz). Because the field decays at $2\tau_p$ , its Fourier line has half-width $1/2\tau_p$ and full width $1/\tau_p$ — equation (2)'s factor of two reappears. The finesse is the mirror-only figure of merit: the ratio of mode spacing (free spectral range) to linewidth — how many resolvable resonances fit between neighbouring modes. For low loss it is just $2\pi/\delta$ , blind to the cavity length.

\mathcal{F} = \frac{\mathrm{FSR}}{\Delta\nu_c} = \frac{\pi\,(R_1 R_2 T_\mathrm{int})^{1/4}}{1 - (R_1 R_2 T_\mathrm{int})^{1/2}} \approx \frac{2\pi}{\delta} \qquad (6)

where: $\mathcal{F}$ — finesse (dimensionless); $\mathrm{FSR}=c/2nL$ — free spectral range (Hz). The middle form comes from the multiple-beam sum; the right form is its low-loss limit. Assemble $Q$ from finesse and geometry and the two roles separate: the finesse carries the mirror quality, the mode index $q$ carries the length in wavelengths, and their product is $Q$ — which is why a long, mediocre-mirror cavity can still reach an enormous $Q$ .

Q = \omega_0\tau_p = \frac{\nu_0}{\Delta\nu_c} = \mathcal{F}\,\frac{\nu_0}{\mathrm{FSR}} = q\,\mathcal{F} \qquad (7)

where: $\nu_0$ — resonance frequency (Hz); $q=\nu_0/\mathrm{FSR}=2nL/\lambda$ — longitudinal mode index (half-wavelengths in a round trip). Finesse is a mirror property; $q$ scales with length, so only $Q$ grows with length. Finally, when several loss mechanisms act at once their rates add, so the reciprocal $Q$ -values add: the lowest individual $Q$ dominates, and the intrinsic $Q$ sets an unreachable ceiling.

\frac{1}{Q} = \frac{1}{Q_\mathrm{out}} + \frac{1}{Q_\mathrm{abs}} + \frac{1}{Q_\mathrm{scat}} \qquad (8)

where: $Q_\mathrm{out}$ — coupling (mirror-transmission) quality factor; $Q_\mathrm{abs}$ — absorption-limited; $Q_\mathrm{scat}$ — scattering-limited. Just as $1/\tau_p=\sum_i 1/\tau_i$ , dividing by $\omega_0$ gives reciprocal- $Q$ addition.

Derivation APhoton lifetime from the round-trip loss ledger

The goal is to turn a discrete bookkeeping fact — that the intracavity intensity is multiplied by a fixed survival factor $s=R_1R_2T_\mathrm{int}=e^{-\delta}$ every round trip — into the smooth exponential decay $I(t)=I_0e^{-t/\tau_p}$ , and read off the lifetime. The trick: "the same fraction lost per trip" is geometric decay in the trip number, and that becomes exponential decay once we map the integer trip count onto continuous time using the transit time $t_r$ .

Step 1. After $m$ round trips the intensity is $I_m = I_0\,s^{\,m} = I_0\,e^{-m\delta}$ , since each trip applies the same survival factor. Why: proportional loss per trip is multiplicative — the signature of exponential, not linear, decay.

Step 2. Each round trip consumes the transit time $t_r=2nL/c$ , so trip number and elapsed time are locked by $t=m\,t_r$ , giving $m=t/t_r$ . Why: this bridges a per-trip ledger to a clock reading, legitimate whenever the light survives many trips so the staircase in $m$ looks smooth in $t$ .

Step 3. Substitute $m=t/t_r$ into the intensity to get $I(t)=I_0\,e^{-\delta t/t_r}\equiv I_0\,e^{-t/\tau_p}$ , matching exponents. Why: identifying $\delta/t_r$ as $1/\tau_p$ is the whole point — the fractional loss and the transit time together fix the decay time.

Step 4. Read off $\tau_p=t_r/\delta=2nL/(c\,\delta)$ ; for a symmetric high-reflectivity cavity $\delta\approx 2(1-R)$ , so $\tau_p\approx nL/[c\,(1-R)]$ . Why: this exhibits the two levers — lengthen the cavity or push $R$ toward one and the lifetime grows without bound, capped only by residual absorption and scatter. Dimensional check: $[L/c]$ is seconds and $\delta$ dimensionless, so $\tau_p$ is in seconds, consistent with Saleh and Teich Ch. 11.

Go deeper · Derivation B · 1 minCold-cavity linewidth from the ringing field

Derivation BCold-cavity linewidth from the ringing field

The aim is to derive the resonance lineshape and its width straight from the decaying field: a cavity ringing down after excitation is a damped harmonic oscillator whose power spectrum is a Lorentzian of full width $1/\tau_p$ . This delivers equation (5) and exposes where the notorious factor of two enters — the field decays at $2\tau_p$ , yet the energy linewidth is $1/\tau_p$ , because the field spectrum's half-width $1/2\tau_p$ doubles into the full width.

Step 1. Write the switched-on damped field $E(t)=E_0\,e^{-t/2\tau_p}\,e^{-i\omega_0 t}\,\Theta(t)$ , with $\Theta(t)$ enforcing excitation at $t=0$ . Why: the amplitude time constant is $2\tau_p$ (from equation 2) — start with the wrong constant and every number downstream is off by two.

Step 2. Fourier transform: $\tilde E(\omega)=E_0\int_0^\infty e^{-(1/2\tau_p + i(\omega_0-\omega))t}\,\mathrm{d}t = \dfrac{E_0}{1/2\tau_p + i(\omega_0-\omega)}$ . Why: the decay rate $1/2\tau_p$ becomes the real part of the denominator — the seed of the linewidth.

Step 3. The power spectrum is the modulus squared, $S(\omega)=|\tilde E(\omega)|^2=\dfrac{E_0^2}{(\omega-\omega_0)^2 + (1/2\tau_p)^2}$ , a Lorentzian centred at $\omega_0$ . Why: detectors see intensity, not field, so the measurable lineshape is this squared quantity.

Step 4. Half maximum occurs where $(\omega-\omega_0)^2=(1/2\tau_p)^2$ , i.e. at $\pm 1/2\tau_p$ , so the full width is $\Delta\omega_c=2\cdot(1/2\tau_p)=1/\tau_p$ . Why: the two half-power detunings are symmetric about $\omega_0$ , so the FWHM is twice the half-width, giving $\Delta\omega_c\tau_p=1$ and $Q=\omega_0/\Delta\omega_c$ — the universal resonator identity, matching Loudon Ch. 1 and Foot Ch. 7.

Go deeper · Derivation C · 1 minSeparating mirror quality from cavity length

Derivation CSeparating mirror quality from cavity length

The goal is to prove $Q=q\,\mathcal{F}$ , factorising the quality factor into a piece the mirrors own (the finesse) and a piece the geometry owns (the mode index $q$ ). This resolves the perennial confusion between finesse and $Q$ : two cavities with the same mirrors share a finesse, yet the longer one has the larger $Q$ . We start from $Q$ as an inverse fractional linewidth, eliminate the linewidth with the finesse, and recognise the leftover ratio as the mode order.

Step 1. Write $Q=\nu_0/\Delta\nu_c$ , the resonance frequency divided by its FWHM — identical to $\omega_0\tau_p$ since $\Delta\nu_c=1/2\pi\tau_p$ . Why: casting $Q$ as a fractional linewidth is what lets the finesse, itself a linewidth ratio, slot straight in.

Step 2. Rearrange finesse, $\mathcal{F}=\mathrm{FSR}/\Delta\nu_c$ , to $\Delta\nu_c=\mathrm{FSR}/\mathcal{F}$ , and substitute: $Q=\nu_0\,\mathcal{F}/\mathrm{FSR}$ . Why: this trades the linewidth for the mirror-only finesse times a bare frequency ratio.

Step 3. Evaluate the ratio: $\dfrac{\nu_0}{\mathrm{FSR}}=\dfrac{\nu_0}{c/2nL}=\dfrac{2nL}{\lambda}=q$ , the number of half-wavelengths in a round trip. Why: this is the longitudinal mode index, and it is where the cavity length enters, purely geometrically.

Step 4. Combine to get $Q=q\,\mathcal{F}$ . Why: the split is now explicit — better mirrors raise $\mathcal{F}$ , a longer cavity raises $q$ , and only their product is $Q$ . Sanity check: a 1 m cavity at 633 nm has $q\approx 3\times10^6$ ; with $\mathcal{F}=100$ , $Q\approx 3\times10^8$ , matching Saleh and Teich.

Limiting cases

Case	Result	Meaning
High reflectivity (low loss)	$R\to 1 \Rightarrow \tau_p\approx \dfrac{t_r}{2(1-R)}\to\infty$	Fractional round-trip loss $\delta\approx2(1-R)$ becomes tiny; $\tau_p$ and $Q$ grow without bound, capped in practice only by absorption and scatter.
Broadband / lossy cavity	$\delta\to 1 \Rightarrow \tau_p\to t_r,\ \mathcal{F}\to\pi$	Most light is lost each pass; the photon barely survives one round trip, finesse falls to order unity, and neighbouring resonances broaden into overlap.
Long cavity, fixed mirrors	$L\uparrow \Rightarrow Q=\dfrac{2\pi L\mathcal{F}}{\lambda}\propto L,\ \mathcal{F}=\text{const}$	Lengthening raises $Q$ (more wavelengths stored per decay and a longer $t_r$ ) but leaves finesse untouched — sharpness relative to the FSR is a mirror property.
Empty vs. amplifying medium	$g\to\alpha_\mathrm{loss} \Rightarrow \tau_p^{\mathrm{eff}}\to\infty$	Intracavity gain shrinks the net loss; at threshold gain exactly cancels loss and the cold-cavity lifetime gives way to the dynamics of the laser rate equations.

Common trapFinesse is not Q, and the field does not decay at τ_p

Two traps live side by side. First, do not equate finesse with $Q$ . The finesse depends only on the round-trip loss set by the mirrors and internal medium — blind to the cavity length — whereas $Q=q\,\mathcal{F}$ picks up the mode number $q\propto L$ . A long, low-finesse cavity can therefore carry a very high $Q$ , so a finesse tells you nothing about $Q$ until you know the length in wavelengths. Second, $\tau_p$ is the energy ( $1/e$ ) decay time; the field amplitude decays half as fast, at $2\tau_p$ . Forget this and the linewidth comes out a factor of two wrong: it is the amplitude constant $2\tau_p$ that Fourier-transforms into a line of half-width $1/2\tau_p$ , hence full width $\Delta\omega_c=1/\tau_p$ .

Worked examples

1 · Photon lifetime and linewidth of a real cavity (confidence builder)

A 30 cm linear cavity in vacuum ( $n=1$ ) has two mirrors of reflectivity $R=0.99$ and negligible internal loss. Find the round-trip time, the photon lifetime, and the cold-cavity linewidth.

Step 1. The fractional round-trip loss is set by the two mirror transmissions: $\delta\approx 2(1-R)=2\times0.01=\mathbf{0.02}$ — one percent at each mirror.

Step 2. The round-trip time is the there-and-back transit: $t_r=2L/c=(2\times0.30)/(2.998\times10^8)=\mathbf{2.0\ ns}$ .

Step 3. The photon lifetime is the round-trip time over the loss: $\tau_p=t_r/\delta=2.0\ \text{ns}/0.02=\mathbf{100\ ns}$ — the energy falls to $1/e$ after about fifty round trips, the exact curve in the first figure.

Step 4. The linewidth follows from the Fourier duality: $\Delta\nu_c=1/(2\pi\tau_p)=1/(2\pi\times10^{-7})=\mathbf{1.6\ MHz}$ . The mode spacing is $\mathrm{FSR}=c/2L=500$ MHz, so $\mathcal{F}=\mathrm{FSR}/\Delta\nu_c\approx 313$ — the second figure's numbers, all from this one cavity. τ_p = 100 ns, Δν_c = 1.6 MHz, ℱ ≈ 313.

2 · Quality factor, two independent ways

The same cavity operates at $\lambda=1064$ nm. Find $Q$ from $Q=\omega_0\tau_p$ , then confirm it from $Q=q\,\mathcal{F}$ .

Step 1. The angular frequency is $\omega_0=2\pi c/\lambda=2\pi\times(2.998\times10^8)/(1.064\times10^{-6})=2\pi\times2.82\times10^{14}=\mathbf{1.77\times10^{15}\ rad/s}$ .

Step 2. Direct route: $Q=\omega_0\tau_p=(1.77\times10^{15})\times(10^{-7})=\mathbf{1.8\times10^{8}}$ — about 28 million full oscillation cycles before the energy drains.

Step 3. Geometric route: the finesse is $\mathcal{F}=2\pi/\delta=2\pi/0.02\approx 314$ , and the mode index is $q=2L/\lambda=0.60/(1.064\times10^{-6})\approx 5.6\times10^{5}$ . Their product is $q\,\mathcal{F}\approx(5.6\times10^5)\times314\approx\mathbf{1.8\times10^{8}}$ — the two routes agree, confirming $Q=q\,\mathcal{F}$ . Q ≈ 1.8×10⁸ by both methods.

Physics of the portrait

The photon lifetime and the quality factor are, at bottom, statements about a sharp resonator — and the archetypal sharp optical resonator is the one Charles Fabry helped build.

Origins · 1 minCharles Fabry and how this was discovered

Charles Fabry

1867–1945 · multiple-beam interferometry and the high-finesse optical resonator

Charles Fabry was a French physicist whose name is attached to the single most important optical resonator ever built — the Fabry–Pérot interferometer, developed with Alfred Pérot in Marseille at the close of the nineteenth century. Their insight was that letting light bounce many times between two closely spaced, highly reflective surfaces makes the transmitted fringes vastly sharper than the two-beam fringes of a Michelson interferometer: the multiple-beam sum concentrates transmission into razor-thin peaks whose sharpness is exactly what we now call the finesse. And that sharpness is the direct expression of a long photon lifetime — the more times a photon circulates before escaping, the narrower the resonance and the higher the $Q$ . Fabry went on to become a superb spectroscopist, using the etalon to resolve hyperfine structure and, most famously, co-discovering the atmospheric ozone layer in 1913. The instrument he co-created is the workhorse behind laser cavities, optical spectrum analysers, gravitational-wave arm cavities, and the reference cavities that stabilise the world's best clocks — every one quoted in the language of finesse, $Q$ , and photon lifetime that this article develops.

History

1899

— Charles Fabry and Alfred Pérot describe the multiple-beam interferometer, the first high-finesse optical resonator

1920

— K. S. Johnson of Western Electric introduces the letter Q for the quality factor of a resonant circuit

1946

— Purcell notes that a resonant cavity alters spontaneous-emission rates, tying emission to Q and mode volume

1958

— Schawlow and Townes derive the cavity-limited laser linewidth from the photon lifetime

1988

— O'Keefe and Deacon demonstrate cavity ring-down spectroscopy, timing τ_p directly

Going further

Texts. Hecht, Optics, 5th ed., Ch. 9 (Fabry–Pérot and finesse) · Saleh and Teich, Fundamentals of Photonics, 3rd ed., Ch. 11 (resonator quality factor and photon lifetime) · Foot, Atomic Physics, Ch. 7 (cavity Q and linewidth) · Loudon, The Quantum Theory of Light, 3rd ed., Ch. 1 (cavity decay) · Demtröder, Laser Spectroscopy, Vol. 1, Ch. 5 (cavity losses and finesse).

Provenance. Every relation here is standard (Saleh and Teich Ch. 11; Hecht Ch. 9; Foot Ch. 7; Loudon Ch. 1; Demtröder Ch. 5). The mode comb, free spectral range, and finesse are recalled by name from optical resonators and cavity modes and longitudinal modes and free spectral range, not re-derived. A verifier should confirm: (1) $\tau_p=t_r/\delta$ with $\delta\approx2(1-R)$ gives $\tau_p=100$ ns, $\Delta\nu_c=1.6$ MHz for the worked cavity; (2) $Q=\omega_0\tau_p\approx1.8\times10^8$ at 1064 nm equals $q\,\mathcal{F}$ ( $q\approx5.6\times10^5$ , $\mathcal{F}\approx314$ ); (3) the damped-field Fourier transform gives a Lorentzian of FWHM $1/\tau_p$ ; (4) reciprocal- $Q$ addition of loss channels; (5) the microsphere numbers $\tau_p\approx0.82\ \mu$ s, $\Delta\nu_c\approx193$ kHz at $Q=10^9$ , $\lambda=1.55\ \mu$ m. $c=2.998\times10^8$ m/s throughout.

Prerequisites. Optical Resonators and Cavity Modes · Longitudinal Modes and Free Spectral Range · Gaussian Beams. Leads to. The Laser Threshold Condition · Laser Rate Equations · Stimulated Emission and the Gain Coefficient.

Ask the tutor: Want to see how the photon lifetime $1/\tau_p$ becomes the loss term in the laser rate equations and sets the threshold inversion? Ask me to walk from this article into the threshold condition.

Is Q the same thing as finesse?

No, and conflating them is the most common error in the subject. Finesse $\mathcal{F}\approx2\pi/\delta$ depends only on the fractional loss per round trip — a property of the mirrors and intracavity medium — and is blind to the cavity length. The quality factor $Q=q\,\mathcal{F}$ adds the mode index $q=2nL/\lambda$ , so it also grows with length: a metre-long cavity with modest mirrors ( $\mathcal{F}=100$ ) already has $Q\approx3\times10^8$ , while a micron-scale cavity with the same mirrors has a $Q$ tens of thousands of times smaller. Finesse answers "how sharp is each mode relative to its neighbours"; $Q$ answers "how many cycles does the light survive."

Why does the energy decay at τ_p but the field at 2τ_p?

Because energy is proportional to the square of the field amplitude. If $W\propto e^{-t/\tau_p}$ and $W\propto|E|^2$ , then $|E|\propto e^{-t/2\tau_p}$ — the amplitude decays at half the rate of the energy. This is not a subtlety you can ignore, because it is the amplitude that Fourier-transforms into the resonance line: its slower decay ( $2\tau_p$ ) gives a spectral half-width $1/2\tau_p$ , and doubling that for the FWHM gives $\Delta\omega_c=1/\tau_p$ . Drop the factor of two and your predicted linewidth is wrong by two.

What exactly counts as "loss" in the round-trip loss δ?

Everything that removes energy from the mode in one round trip. The dominant terms are usually the two mirror transmissions $(1-R_1)$ and $(1-R_2)$ — light that leaks out to be used or wasted — but $\delta$ also includes absorption in the coatings and substrates, scatter off surface roughness, absorption and scatter in any intracavity medium ( $\alpha_\mathrm{int}\ell$ ), and diffraction past finite mirrors. Formally $\delta=-\ln(R_1R_2T_\mathrm{int})$ , but for high- $R$ mirrors these channels simply add. Their reciprocal $Q$ -values add too, so the single worst channel sets the ceiling — raising mirror reflectivity stops helping once absorption or scatter dominates.

Can Q really reach a billion, and what limits it?

Yes. Whispering-gallery microspheres and toroids routinely exceed $Q=10^9$ , and record silica and crystalline microresonators approach $10^{11}$ . These confine light by total internal reflection at a smooth curved surface, not by discrete mirrors, so there is no mirror-transmission loss at all — the leak is set by residual material absorption, scattering off sub-nanometre roughness, and evanescent tunnelling. Once reflectivity losses vanish the intrinsic $Q$ is capped by the material: ultrapure silica's near-infrared transparency sets a ceiling near $10^{10}$ – $10^{11}$ , beyond which adsorbed surface water and thermal effects intrude — which is why record cavities are often cryogenic crystals rather than room-temperature glass.

How does the photon lifetime set the laser threshold?

Lasing begins when the round-trip gain first equals the round-trip loss — equivalently, when the stimulated-emission rate matches the cavity decay rate $1/\tau_p$ . A higher- $Q$ cavity has a smaller $1/\tau_p$ , so less gain, and hence a smaller population inversion, is needed to reach threshold: the photons make more passes through the gain medium before escaping. This is why $1/\tau_p$ appears explicitly as the loss term in the laser rate equations, and why lowering cavity loss is one of the two levers (the other being pump power) for reducing threshold.

Is a high-Q cavity always what I want?

Not always — it trades against bandwidth and speed. A high $Q$ means a narrow linewidth $\Delta\nu_c=1/2\pi\tau_p$ and long storage, exactly right for frequency references, narrow-linewidth lasers, and building up huge circulating intensities. But the same long lifetime slows the response: a mode storing light for a microsecond cannot follow a signal changing faster than a few hundred kilohertz, so high- $Q$ resonators make poor fast modulators or broadband filters, and mode-locked lasers actually need low enough $Q$ to support a wide comb of modes at once. The right $Q$ is set by the job.