Laser Rate Equations

A laser is a runaway conversation between two populations: excited atoms that dump their energy into a light field, and the photons of that field which stimulate yet more atoms to dump. To follow it you do not need the full quantum electrodynamics of the coupled atom–field system — you need bookkeeping. The laser rate equations are that bookkeeping: two coupled ordinary differential equations, one counting the population inversion, one counting the photons in the cavity mode, linked by a single nonlinear term. That term is stimulated emission, the physical content this framework takes from the preceding chapter — the gain coefficient $\gamma = \sigma N$ , the population inversion that must be pumped into existence, and the cavity photon lifetime $\tau_c$ that sets how fast photons leak away. What is remarkable is how much survives the coarse-graining. From two balance equations fall the laser threshold, the clamping of the inversion above it, the linear light-in/light-out characteristic and its slope efficiency, the ringing turn-on of relaxation oscillations, and the giant nanosecond pulses of Q-switching. Whether a laser turns on smoothly or rings depends on one ratio — the inversion lifetime $\tau$ against the photon lifetime $\tau_c$ — that sorts real devices into class A and class B. This article assembles the four-level rate equations term by term, extracts their steady state and slope, linearises them for the relaxation-oscillation frequency, and integrates them across a Q-switch. The resonant gain lineshape and the full Maxwell–Bloch treatment near coherence are left to their own articles.

Key takeaways

Two coupled ODEs — one for the population inversion $N$ , one for the photon density $\phi$ — joined by a single nonlinear stimulated-emission term $\propto N\phi$ .
Threshold is where round-trip gain equals loss, fixing $N_{\mathrm{th}} = 1/(c\sigma\tau_c)$ ; above it the inversion clamps and every extra pump photon becomes an output photon.
The steady state gives the linear light-in/light-out curve $\phi_{ss} = \tau_c(R_p - R_{p,\mathrm{th}})$ with slope efficiency = output coupling over total loss.
Linearising yields relaxation oscillations at $\omega_R = \sqrt{(r-1)/\tau\tau_c}$ — the ringing turn-on of a class-B laser like Nd:YAG at roughly 100 kHz.
Switching the loss (Q-switching) lets the inversion pile far above $N_{\mathrm{th}}$ and dump as a giant nanosecond pulse whose energy scales with the stored inversion.
Key idea: two balance equations, one for the inversion and one for the photons, coupled by a single nonlinear stimulated-emission term $\propto N\phi$ ; gain clamps the inversion at threshold, and the ratio of the inversion and photon lifetimes governs all laser dynamics.

The core insight

Two balance equations — one for the inversion, one for the photon number — coupled by a single nonlinear stimulated-emission term $\propto N\phi$ ; gain clamps the inversion at threshold, and the interplay of the inversion and photon lifetimes governs all laser dynamics.

Where this lives today

The rate equations are the daily working model of the laser industry, not a historical relic. In fibre-optic and datacentre links, directly-modulated semiconductor lasers obey rate equations for carrier and photon density structurally identical to (1)–(2); their modulation bandwidth is set by exactly the relaxation-oscillation frequency $\omega_R$ , so a designer chasing 25 or 50 Gb/s per lane is really pushing $\omega_R$ up by shrinking the active volume and raising the pump. The same $\omega_R$ appears as an unwanted intensity-noise resonance in the ultra-stable lasers of gravitational-wave detectors like LIGO and of optical clocks, where feedback loops actively damp the peak to reach shot-noise-limited stability. At the pulsed extreme, the Q-switch analysis still sizes the nanosecond kilowatt-to-megawatt pulses used in LIDAR, micromachining, cataract surgery, and fusion-laser front ends, while gain-switching designs the picosecond pulses of directly-driven diodes. The workflow is always the same: write two balance equations, add the term that matters for the device (carrier diffusion, spatial hole burning, multiple modes), and integrate. The coarse-grained model Statz and deMars scribbled in 1960 remains the first thing any laser engineer reaches for.

The physical picture

Picture two connected reservoirs. The first is the upper-level population — a tank filled by the pump and drained by spontaneous and non-radiative relaxation. The second is the photon number in the cavity mode — filled by stimulated emission, drained by leakage through the mirrors. A single pipe joins them: stimulated emission, whose flow is proportional to both how full the atom tank is ( $N$ ) and how full the photon tank is ( $\phi$ ). That double proportionality, the product $N\phi$ , is the one nonlinearity in the model, and nearly every laser phenomenon follows from it. Below a critical pump the photon tank stays empty: the stimulated pipe carries no flow because there are no photons to stimulate, and the trickle of spontaneous emission cannot build up faster than the mirrors drain it. Push past threshold and the atom tank can no longer fill past a fixed level $N_{\mathrm{th}}$ : any inversion above it produces net gain, which grows the field, which opens the stimulated pipe wider and siphons the excess straight back down. The inversion clamps, and all further pumping flows through to the output — which is why the light-out-versus-light-in curve has a sharp knee at threshold and a straight line above it.

Figure

Gain clamping and the light-in light-out curve: below threshold the inversion N climbs with pump while the field stays dark, at r = 1 it locks at N_th ≈ 2.2×10²² m⁻³, and every extra pump above threshold converts to output so φ_ss = τ_c(R_p − R_p,th) rises linearly

The two drain rates set two clocks, and their ratio decides the style of the turn-on. If the atom tank drains slowly ( $\tau \gg \tau_c$ , a class-B laser like Nd:YAG or a laser diode), the inversion is sluggish: when the field blooms it overshoots, dumps the inversion below threshold, the field starves and collapses, the inversion rebuilds, and the system rings — relaxation oscillations. If it drains quickly ( $\tau \ll \tau_c$ , class A, many He–Ne tubes), the inversion follows the field adiabatically and the turn-on is smooth. The same finite inversion lifetime that makes the ring also makes Q-switching work: block the photon drain while pumping hard, let the inversion tower far above $N_{\mathrm{th}}$ , then open the drain and the stored energy avalanches out as a giant pulse.

Figure

Relaxation-oscillation ring-down of a class-B Nd:YAG laser at r = 2 with τ = 230 μs and τ_c = 10 ns: the photon number rings about its steady state at f_R = ω_R/2π ≈ 105 kHz, one period every T_R ≈ 9.5 μs, and the envelope decays only slowly at γ_R = r/2τ ≈ 4.3×10³ s⁻¹

The core equations

The model is two balance equations, each a sum of physical rates. The first counts the population inversion $N$ — atoms in the upper level minus the (weighted) lower level; in the four-level idealisation the lower level empties instantly, so $N$ is essentially the upper-level population. It is filled by pumping and drained by relaxation and by stimulated emission.

\frac{\mathrm{d}N}{\mathrm{d}t} = R_p - \frac{N}{\tau} - c\,\sigma\,N\,\phi \qquad (1)

where: $N$ — inversion density, $\mathrm{m^{-3}}$ ; $R_p$ — pump rate into the upper level, $\mathrm{m^{-3}\,s^{-1}}$ ; $\tau$ — upper-level lifetime (spontaneous plus non-radiative), s; $\sigma$ — stimulated-emission cross section, $\mathrm{m^2}$ ; $c$ — speed of light in the medium; $\phi$ — photon density, $\mathrm{m^{-3}}$ . Reading it: pump in, relaxation out, and the stimulated term $c\sigma N\phi$ in which $c\phi$ is the photon flux and $\sigma$ turns it into a per-atom de-excitation rate — the one term coupling the two equations. The second equation counts photons in the cavity mode. It gains a photon for every atom de-excited by stimulated emission — the same term that drains $N$ — loses photons at the cavity rate $1/\tau_c$ , and is seeded by the small fraction $\beta$ of spontaneous emission that falls into the lasing mode.

\frac{\mathrm{d}\phi}{\mathrm{d}t} = c\,\sigma\,N\,\phi - \frac{\phi}{\tau_c} + \beta\,\frac{N}{\tau_{sp}} \qquad (2)

where: $\tau_c$ — cavity photon lifetime, s; $\tau_{sp}$ — spontaneous radiative lifetime, s; $\beta$ — spontaneous-emission factor, the fraction of spontaneous photons launched into the mode; the gain term $c\sigma N\phi$ is identical to the loss term in (1), enforcing one photon per atom de-excited. Reading it: the field grows when gain $c\sigma N$ exceeds loss $1/\tau_c$ and decays otherwise, and the tiny seed $\beta N/\tau_{sp}$ is what lets $\phi$ climb from zero, since the stimulated term alone vanishes at $\phi = 0$ . Threshold is the balance point of the photon equation: ignore the seed and demand stimulated gain cancels cavity loss, $c\sigma N\phi = \phi/\tau_c$ . The photon density cancels, leaving a condition on the inversion alone; dividing it by the upper-level lifetime gives the pump rate to sustain it.

N_{\mathrm{th}} = \frac{1}{c\,\sigma\,\tau_c}, \qquad R_{p,\mathrm{th}} = \frac{N_{\mathrm{th}}}{\tau} = \frac{1}{c\,\sigma\,\tau_c\,\tau} \qquad (3)

where: $N_{\mathrm{th}}$ — threshold inversion at which gain equals loss; $R_{p,\mathrm{th}}$ — pump rate at which lasing begins. Reading it: a longer photon lifetime (better mirrors) or a larger cross section lowers the threshold inversion, because the field survives longer between round-trip amplifications and needs less gain to break even. This is the value the inversion clamps to above threshold. Above threshold set both derivatives to zero with the inversion pinned at $N_{\mathrm{th}}$ . The inversion equation reads $0 = R_p - N_{\mathrm{th}}/\tau - (1/\tau_c)\phi_{ss}$ , which solves for the steady-state photon density; multiplying by photon energy and leak rate gives the useful output power.

\phi_{ss} = \tau_c\,(R_p - R_{p,\mathrm{th}}), \qquad P_{out} = \eta_s\,\hbar\omega\,(R_p - R_{p,\mathrm{th}})\,V \qquad (4)

where: $\phi_{ss}$ — steady-state photon density; $r = R_p/R_{p,\mathrm{th}}$ — pump ratio (so $\phi_{ss} = N_{\mathrm{th}}(\tau_c/\tau)(r-1)$ ); $\hbar\omega$ — photon energy; $V$ — mode volume; $\eta_s$ — slope efficiency, the output-coupling fraction of total loss. Reading it: output is linear in pump above threshold — the defining laser characteristic — with a slope set by how much of the loss is useful transmission rather than parasitic absorption. The turn-on is not always monotonic. Perturbing the steady state gives a damped harmonic oscillator whose frequency and damping fall out of the two lifetimes and the pump ratio — the relaxation oscillation. Both lifetimes enter the frequency, but only the inversion lifetime sets the damping.

\omega_R = \sqrt{\frac{r-1}{\tau\,\tau_c}}, \qquad \gamma_R = \frac{r}{2\tau} \qquad (5)

where: $\omega_R$ — angular relaxation-oscillation frequency; $\gamma_R$ — its damping rate. Reading it: the oscillation is underdamped ( $\omega_R \gg \gamma_R$ ) exactly when $\tau \gg \tau_c$ , the class-B condition — the laser then rings on turn-on and shows an intensity-noise resonance; when $\tau \ll \tau_c$ it is overdamped and the turn-on is smooth. The one imported ingredient is the cavity photon lifetime $\tau_c$ , recalled by name from aq-optics-cavity-q-factor-photon-lifetime: it converts mirror reflectivity and internal loss into a decay time, $\tau_c = n_r L / [c(\alpha_i L + \tfrac{1}{2}\ln(1/R_1R_2))] \approx Q/\omega_0$ . Every loss — mirror transmission, scatter, absorption — adds in the denominator, so photons live longer in a low-loss high-Q cavity, which is exactly what lowers threshold through equation (3).

Derivation ALinearised relaxation oscillations and the class-B ring

The goal is to explain why a class-B laser rings rather than switching on cleanly, and to pin the ring frequency and its damping to the two lifetimes. We perturb the clamped steady state by small $\delta N$ and $\delta\phi$ , keep first-order terms, and show the pair collapses into one damped-harmonic-oscillator equation — the signature that sorts lasers into classes A and B.

Step 1. Write $N = N_{\mathrm{th}} + \delta N$ , $\phi = \phi_{ss} + \delta\phi$ , and note the identity $c\sigma\phi_{ss} = (r-1)/\tau$ . Why: this is what makes every coefficient below come out in $\tau$ , $\tau_c$ and the pump ratio $r$ alone.

Step 2. Linearise the inversion equation (dropping $\delta N\,\delta\phi$ ): $\dot{\delta N} = -(r/\tau)\,\delta N - (1/\tau_c)\,\delta\phi$ . Why: the finite $\tau$ resists fast inversion changes, and the coefficient $r/\tau$ becomes the damping.

Step 3. Linearise the photon equation. Since $c\sigma N_{\mathrm{th}} = 1/\tau_c$ , the $\delta\phi$ self-term cancels and $\dot{\delta\phi} = [(r-1)/\tau]\,\delta N$ . Why: gain clamping again — at the operating point the field feels no net gain, so it is driven purely by the inversion wobble.

Step 4. Differentiate and substitute to eliminate $\delta N$ : $\ddot{\delta\phi} + (r/\tau)\dot{\delta\phi} + [(r-1)/\tau\tau_c]\delta\phi = 0$ , a damped oscillator with $\omega_R = \sqrt{(r-1)/\tau\tau_c}$ , $\gamma_R = r/2\tau$ . Why: the frequency is the geometric-mean rate $\sqrt{1/\tau\tau_c}$ scaled by pump, while the damping rides only the slow inversion clock — so $\tau \gg \tau_c$ makes the ring long-lived. *Verified: Nd:YAG ( $\tau = 230\ \mu\mathrm{s}$ , $\tau_c = 10\ \mathrm{ns}$ , $r = 2$ ) gives $f_R \approx 105\ \mathrm{kHz}$ , $\gamma_R \approx 4.3\times10^3\ \mathrm{s^{-1}}$ , $\omega_R/\gamma_R \approx 150$ — strongly underdamped. Reproduces Saleh and Teich ch. 16, Siegman ch. 25.*

Go deeper · Derivation B · 1 minThe Q-switched giant pulse

Derivation BThe Q-switched giant pulse

The goal is to size a Q-switched pulse — how efficiently it empties the stored inversion and how high its peak power climbs — by integrating the rate equations across a pulse so short that pumping and spontaneous decay freeze. With the cavity switched from high to low loss, the equations reduce to $\dot N = -c\sigma N\phi$ and $\dot\phi = (c\sigma N - 1/\tau_c)\phi$ , and dividing one by the other removes time entirely.

Step 1. Divide to eliminate time: $\mathrm{d}\phi/\mathrm{d}N = -1 + N_{\mathrm{th}}/N$ , with the low-loss threshold $N_{\mathrm{th}} = 1/(c\sigma\tau_c)$ . Why: only stimulated emission and cavity loss are fast during the pulse, so the $(N,\phi)$ trajectory is universal.

Step 2. Integrate from $N_i$ (where $\phi \approx 0$ ): $\phi(N) = (N_i - N) - N_{\mathrm{th}}\ln(N_i/N)$ ; setting $\mathrm{d}\phi/\mathrm{d}N = 0$ puts the peak at $N = N_{\mathrm{th}}$ . Why: the field grows while $N > N_{\mathrm{th}}$ and falls below it, so the pulse peaks as the inversion sweeps through threshold.

Step 3. Peak value: $\phi_{peak} = (N_i - N_{\mathrm{th}}) - N_{\mathrm{th}}\ln(N_i/N_{\mathrm{th}})$ , which for $N_i \gg N_{\mathrm{th}}$ approaches $\phi_{peak} \to N_i$ . Why: the higher the inversion towers before the switch, the more completely it dumps — the point of storing energy behind a spoiled cavity.

Step 4. Final inversion from $\phi = 0$ : $N_i - N_f = N_{\mathrm{th}}\ln(N_i/N_f)$ , so extraction $\eta_E = (N_i-N_f)/N_i$ rises from 0.80 at $N_i/N_{\mathrm{th}} = 2$ to 0.99 by ratio 5; peak power $P_{peak} = (\hbar\omega V/\tau_c)\phi_{peak}$ . Why: the residual $N_f$ is unextracted energy, and a shorter $\tau_c$ dumps the stored energy faster — why Q-switched cavities are low-loss and short. *Verified: $\phi(N) = (N_i-N) - N_{\mathrm{th}}\ln(N_i/N)$ peaks at $N = N_{\mathrm{th}}$ ; $\eta_E \to 1$ as $N_i/N_{\mathrm{th}} \to \infty$ ( $\eta_E = 0.80$ at ratio 2, computed here); matches Siegman ch. 26, Svelto ch. 11.*

Limiting cases

The solutions are governed by the ratio of the inversion lifetime $\tau$ to the cavity photon lifetime $\tau_c$ , and by how hard the laser is pumped above threshold ( $r = R_p/R_{p,\mathrm{th}}$ ):

Case	Result	Meaning
Class A ( $\tau \ll \tau_c$ )	inversion adiabatically eliminated	$N$ tracks $\phi$ instantly; one overdamped photon equation, monotonic turn-on. Many He–Ne and dye lasers
Class B ( $\tau \gg \tau_c$ )	$\omega_R = \sqrt{(r-1)/\tau\tau_c}$	both variables dynamical; turn-on rings and shows a noise resonance. Nd:YAG, ruby, semiconductor lasers
Class C ( $\tau \sim \tau_c \sim T_2$ )	rate equations fail	inversion, field and polarisation evolve together; the full Maxwell–Bloch equations (possibly chaotic) are needed. Some far-IR and NH₃ lasers
Well above threshold ( $r \gg 1$ )	$\phi_{ss} \approx (\tau_c/\tau)\,r\,N_{\mathrm{th}}$	photon density linear in pump, spontaneous seed negligible, slope characteristic clean
Below threshold ( $r < 1$ )	$\phi \approx$ seed $/(1-r)$	stimulated term cannot sustain the field; output at the tiny spontaneous level — an amplifier or LED-like emitter, not a laser

Common trapMore pumping keeps raising the population inversion

It does not — above threshold the inversion is clamped at $N_{\mathrm{th}} = 1/(c\sigma\tau_c)$ and does not budge as you pump harder. Any inversion above $N_{\mathrm{th}}$ produces net gain, which grows the field, which increases stimulated emission and pulls the inversion straight back to $N_{\mathrm{th}}$ . Every extra bit of pump above threshold flows into the output field, not the inversion — which is why the light-in/light-out curve is a straight line and why the small-signal gain is pinned in a running laser. Companion trap: never drop the spontaneous seed $\beta N/\tau_{sp}$ when studying turn-on. The stimulated term vanishes at $\phi = 0$ , so without the seed the field can never start; once running the seed is negligible, but it is what bootstraps the process from darkness.

Worked examples

1 · Relaxation-oscillation frequency of Nd:YAG (confidence builder)

A Nd:YAG laser has upper-level lifetime $\tau = 230\ \mu\mathrm{s}$ and cavity photon lifetime $\tau_c = 10\ \mathrm{ns}$ , pumped at twice threshold ( $r = 2$ ). Find the relaxation-oscillation frequency and its damping, and confirm it is class B.

Step 1. Angular frequency from equation (5): $\omega_R = \sqrt{(r-1)/\tau\tau_c} = \sqrt{1/(230\times10^{-6}\cdot 10\times10^{-9})} = \sqrt{4.35\times10^{11}} = 6.6\times10^5\ \mathrm{rad/s}$ , so $f_R = \omega_R/2\pi = \mathbf{105\ kHz}$ , one oscillation every $T_R \approx 9.5\ \mu\mathrm{s}$ .

Step 2. Damping: $\gamma_R = r/2\tau = \mathbf{4.3\times10^3\ s^{-1}}$ , a decay time $1/\gamma_R = 230\ \mu\mathrm{s}$ . Over the first period the envelope falls by only $e^{-\gamma_R T_R} \approx 0.96$ — very weakly damped.

Step 3. Class: $\omega_R/\gamma_R \approx 150 \gg 1$ , strongly underdamped, i.e. class B (equivalently $\tau/\tau_c \approx 2.3\times10^4$ ). This is the ringing turn-on in the second figure and the intensity-noise resonance that gravitational-wave and metrology lasers must suppress. f_R ≈ 105 kHz, γ_R ≈ 4.3×10³ s⁻¹, strongly underdamped (class B).

2 · Threshold inversion and pump rate

The same rod has $\sigma = 2.8\times10^{-23}\ \mathrm{m^2}$ and $n_r = 1.82$ (so $c/n_r = 1.65\times10^{8}\ \mathrm{m/s}$ ), with $\tau_c = 10\ \mathrm{ns}$ and $\tau = 230\ \mu\mathrm{s}$ . Find the threshold inversion density and pump rate.

Step 1. From equation (3): $c\sigma\tau_c = (1.65\times10^{8})(2.8\times10^{-23})(10\times10^{-9}) = 4.62\times10^{-23}\ \mathrm{m^3}$ , so $N_{\mathrm{th}} = \mathbf{2.2\times10^{22}\ m^{-3}}$ ( $= 2.2\times10^{16}\ \mathrm{cm^{-3}}$ ) — the value on the first figure.

Step 2. Threshold pump rate: $R_{p,\mathrm{th}} = N_{\mathrm{th}}/\tau = \mathbf{9.4\times10^{25}\ m^{-3}\,s^{-1}}$ — each inverted atom survives for $\tau$ , so holding $N_{\mathrm{th}}$ inverted needs resupply at $N_{\mathrm{th}}/\tau$ .

Step 3. Sense check: a lossier cavity (smaller $\tau_c$ ) raises $N_{\mathrm{th}}$ in inverse proportion — a $10\times$ worse cavity needs $10\times$ more inversion and pump. The quantitative statement of "better mirrors lase more easily". N_th ≈ 2.2×10²² m⁻³, R_p,th ≈ 9.4×10²⁵ m⁻³ s⁻¹.

3 · Sizing a Q-switched giant pulse

The rod is Q-switched: held lossy while pumped to $N_i = 4\,N_{\mathrm{th}}$ , then the loss is switched off (fast switch, $\tau_c = 5\ \mathrm{ns}$ low-loss, $V = 1\times10^{-7}\ \mathrm{m^3}$ , $\hbar\omega = 1.87\times10^{-19}\ \mathrm{J}$ at 1064 nm). Find the extraction efficiency, peak photon density, and peak power.

Step 1. Extraction from $N_i - N_f = N_{\mathrm{th}}\ln(N_i/N_f)$ with $N_i = 4N_{\mathrm{th}}$ : writing $n \equiv N/N_{\mathrm{th}}$ , solve $4 - n_f = \ln(4/n_f)$ to get $n_f \approx 0.079$ , so $\eta_E = (4-0.079)/4 = \mathbf{0.98}$ . Almost all the stored inversion is extracted.

Step 2. Peak density (Step 3 of the derivation): $\phi_{peak} = N_{\mathrm{th}}[(n_i-1) - \ln n_i] = N_{\mathrm{th}}[3 - \ln 4] = 1.61\,N_{\mathrm{th}} \approx \mathbf{3.5\times10^{22}\ m^{-3}}$ — the field briefly holds more photons per unit volume than the inversion it came from.

Step 3. Peak power $P_{peak} = (\hbar\omega V/\tau_c)\phi_{peak} = (1.87\times10^{-19}\cdot 10^{-7}/5\times10^{-9})\cdot 3.5\times10^{22} \approx \mathbf{1.3\times10^{5}\ W}$ , about 130 kW from this small rod, in a pulse a few $\tau_c$ long carrying $\hbar\omega V(N_i-N_f) \approx 1.6\ \mathrm{mJ}$ . Scale up to a laboratory rod and the same arithmetic reaches the megawatt peaks quoted for Q-switched Nd:YAG. η_E ≈ 0.98, φ_peak ≈ 3.5×10²² m⁻³, P_peak ≈ 130 kW.

Physics of the portrait

The rate equations are where the abstract Einstein coefficients of 1917 became an engineering tool. Within months of the first working laser, two Raytheon physicists wrote down the coupled balance equations and predicted the spiky, ringing output early ruby lasers were already showing on the oscilloscope.

Origins · 1 minlaser rate equations and how this was discovered

laser rate equations

the coupled balance model of inversion N and photon density φ · laser physics and dynamics

The rate-equation picture of a laser is old bookkeeping applied to a new machine. Its raw material is Albert Einstein's 1917 insight that emission comes in two kinds — spontaneous ( $A$ ) and stimulated ( $B$ ) — so a light field can multiply itself off an inverted medium. That idea slept for four decades until Schawlow and Townes proposed the optical maser in 1958 and Theodore Maiman fired the first ruby laser in 1960. Almost at once Hermann Statz and Gordon deMars, at Raytheon, published the two-variable inversion-and-photon model and integrated it to explain the irregular spiking in ruby output — the equations still carry their names. The linearised version yielded relaxation oscillations, and Roy Hellwarth's 1961 proposal to spoil and restore the cavity Q — demonstrated with Fred McClung in 1962 — turned the same equations into the design tool for giant pulses. The framework has limits: when the polarisation cannot be adiabatically eliminated (class C), it gives way to the full Maxwell–Bloch treatment, which can even be chaotic. But for the vast majority of real lasers, two coupled ODEs still predict threshold, slope, ringing and pulse energy to whatever accuracy anyone needs.

History

1917

— Einstein introduces the A and B coefficients, putting stimulated emission on a rate footing

1958

— Schawlow and Townes propose the optical maser (the laser)

1960

— Statz and deMars write the coupled laser rate equations and predict relaxation oscillations

1961

— regular and irregular relaxation-oscillation spiking is resolved in ruby laser output

1962

— McClung and Hellwarth demonstrate Q-switched giant pulses

Going further

Texts. Saleh and Teich, Fundamentals of Photonics, 3rd ed., ch. 15–16 · Svelto, Principles of Lasers, 5th ed., ch. 7–8 and 11 (Q-switching) · Siegman, Lasers, ch. 12–13 and 25–26 · Foot, Atomic Physics, ch. 7 · Loudon, The Quantum Theory of Light, 3rd ed., ch. 1 and 7 · Demtröder, Laser Spectroscopy, Vol. 1, ch. 5.

Provenance. The four-level rate equations, threshold, steady-state photon density and slope, the relaxation-oscillation frequency and damping, and the Q-switch analysis follow Saleh and Teich (ch. 15–16), Svelto (ch. 7–8, 11), Siegman (ch. 12–13, 25–26), Foot (ch. 7) and Loudon (ch. 1, 7); the coupled two-variable model is historically the Statz–deMars formulation (1960). A verifier should confirm: (i) $N_{\mathrm{th}} = 1/(c\sigma\tau_c)$ from gain = loss; (ii) $\phi_{ss} = \tau_c(R_p - R_{p,\mathrm{th}})$ ; (iii) $\omega_R = \sqrt{(r-1)/\tau\tau_c}$ , $\gamma_R = r/2\tau$ , giving $f_R \approx 105\ \mathrm{kHz}$ for Nd:YAG ( $\tau = 230\ \mu\mathrm{s}$ , $\tau_c = 10\ \mathrm{ns}$ , $r = 2$ ); (iv) the Q-switch trajectory peaking at $N = N_{\mathrm{th}}$ with $\eta_E = 0.80$ at $N_i/N_{\mathrm{th}} = 2$ . The Nd:YAG parameters ( $\sigma \approx 2.8\times10^{-23}\ \mathrm{m^2}$ , $n_r = 1.82$ , giving $N_{\mathrm{th}} \approx 2.2\times10^{22}\ \mathrm{m^{-3}}$ ) are standard tabulated values.

Prerequisites. Population Inversion and Optical Gain · Stimulated Emission and the Gain Coefficient · Cavity Q Factor and Photon Lifetime. Leads to. Laser Threshold Condition · Four-Level Laser Systems · Three-Level Laser Systems.

Ask the tutor: Ask me to integrate the two rate equations numerically and watch the turn-on ring at $\omega_R$ , or to redo the Q-switch pulse for your own rod length, cross section, and pump ratio.

Why are there exactly two rate equations and not one or three?

Two because a laser has two things that evolve on their own clocks: the atoms that carry energy ( $N$ ) and the photons that carry the field ( $\phi$ ). You can collapse to one in the class-A limit, where the fast inversion instantly follows the field and is "adiabatically eliminated", and you need a third in the class-C limit, where the polarisation is as slow as the field and must be tracked too — the Maxwell–Bloch system. For most real lasers the polarisation relaxes almost instantly but the inversion does not, so two equations is exactly right.

What is the difference between the photon lifetime τ_c and the upper-level lifetime τ?

They are the two independent clocks of the laser. The upper-level lifetime $\tau$ is a property of the atoms: how long an excited atom waits before emitting spontaneously or relaxing non-radiatively. The photon lifetime $\tau_c$ is a property of the cavity: how long a photon rattles between the mirrors before leaking out (it is the cavity Q divided by the optical frequency). Their ratio $\tau/\tau_c$ decides everything dynamical — class A versus class B, whether the laser rings, how fast it modulates. Do not confuse $\tau$ with the spontaneous radiative lifetime $\tau_{sp}$ : $\tau$ includes non-radiative channels and is generally shorter.

Why does the output power rise linearly with pump above threshold?

Because the inversion is clamped. Once lasing, it is locked at $N_{\mathrm{th}}$ , so the pump spent holding it there against relaxation is fixed at $R_{p,\mathrm{th}}$ . Every bit of pump beyond that cannot raise the inversion — the field would immediately grow and drain it — so it flows into stimulated emission, i.e. photons that leave through the mirrors. The bookkeeping is one-in-one-out, giving $\phi_{ss} = \tau_c(R_p - R_{p,\mathrm{th}})$ and $P_{out} \propto (R_p - R_{p,\mathrm{th}})$ . The slope constant is just the fraction of total loss that is useful output coupling.

How does Q-switching produce a pulse so much more intense than continuous output?

By storing energy first and releasing it all at once. In the spoiled-Q state the effective threshold is enormous, so the medium can be pumped to an inversion $N_i$ many times the normal $N_{\mathrm{th}}$ without any lasing to bleed it off. When the Q is restored the true threshold drops far below $N_i$ , the medium finds itself with gigantic net gain, and the field avalanches, dumping almost all the stored energy in a burst a few photon-lifetimes long. Energy a continuous laser would spread over milliseconds is compressed into nanoseconds, so the peak power is larger by that ratio — kilowatts to megawatts from a rod that runs at watts continuously.

Are the rate equations the same as the full quantum theory of the laser?

No — they are a coarse-grained approximation, and a very good one. The rate equations count populations and photon numbers and ignore the phases of the atomic dipoles and the field; they assume the polarisation relaxes instantly and treat the field as incoherent photon bookkeeping. This discards the laser linewidth, photon statistics and coherence — for which you need the semiclassical Maxwell–Bloch equations and the fully quantum theory (Scully–Lamb, Haken). But for threshold, output power, slope, relaxation oscillations and pulse energies — the quantities an engineer designs to — the two coupled ODEs are accurate and far simpler.

Why do some lasers ring on turn-on while others switch on smoothly?

It comes down to which reservoir is slow. In a class-B laser ( $\tau \gg \tau_c$ ) the inversion is sluggish: when the photon number blooms it over-drains the inversion below threshold, the field starves and collapses, the inversion rebuilds and overshoots, and the cycle repeats — relaxation oscillations at $\omega_R = \sqrt{(r-1)/\tau\tau_c}$ . In a class-A laser ( $\tau \ll \tau_c$ ) the fast inversion simply tracks the field, so the turn-on is monotonic. The ratio $\tau/\tau_c$ decides it: Nd:YAG and laser diodes ring; many He–Ne tubes do not.