How a Laser Works

Pump up to 0.9. Atoms turn rose and fall back, spitting dots in every direction. Now and then one leaves along the axis and starts bouncing. Watch it: it passes more hollow atoms than rose ones, and a hollow atom absorbs it with the same probability a rose one would clone it. It dies within a few trips. The photon panel on the right is flat.

Cross 1.0. Now more atoms are up than down. The next axial photon leaves a copy behind, the copy leaves copies, and in a second or two the strip chart takes off. The wave-trains all have the same phase — the clone inherits it — which is what makes the output a beam rather than a glow.

Push to 3. The upper population in the chart stops climbing the moment threshold is passed, while the photon number keeps rising in a straight line. In the cartoon, count rose against hollow: the lead stays at a couple of atoms no matter how hard you pump, because every extra excitation is stripped by the light it helps create.

Switch to two-level and pump to 4. Nothing. The amber dots going up and down are pump photons knocking excited atoms back to the ground — the same coefficient that lifts them. The best the pump can do is a dead heat.

A photon passing an atom in the upper level can stimulate it to emit a second photon into the same mode: same direction, same frequency, same phase. A photon passing an atom in the lower level can be absorbed. Einstein showed in 1917 that the coefficients for those two processes are equal (B₁₂ = B₂₁ for non-degenerate levels), so the net gain per pass is proportional to N₂ − N₁. Unless more atoms are up than down, light is absorbed on average — which is the normal state of every material you have ever seen.

Population inversion is that abnormal state. Making it needs atoms lifted faster than they decay, and a place to park them that the pump cannot empty. Then a cavity turns gain into a threshold: light builds up when gain per round trip beats the mirror loss, and the inversion needed for that is ΔN_th = 1/(Kτ_c). Above threshold the light itself pins the inversion at exactly that value — this is gain clamping — and every further pump photon becomes an output photon. That is why the photon panel is a straight line: its slope is fixed by the cavity, not by the medium.

A₂₁ is the spontaneous rate: an excited atom alone in the dark decays at A₂₁ per second, in a random direction with a random phase. B₁₂ρ and B₂₁ρ are the absorption and stimulated emission rates in light of spectral energy density ρ. Einstein got the two relations between them by demanding that atoms in a thermal cavity settle into the Boltzmann populations while the light settles onto Planck’s curve. That only works if g₁B₁₂ = g₂B₂₁ and A₂₁/B₂₁ = 8πhν³/c³ — the coefficients are not independent, and this page never types B in: it takes A from the lifetime and derives B from it.

The ν³ tells you why the first lasers were optical rather than X-ray and why masers came first: at fixed B, spontaneous emission grows as the cube of the frequency, and spontaneous emission is loss — it goes everywhere except into the mode. The table’s last row is the stimulated-to-spontaneous ratio for this transition in a 300 K thermal field: around 10⁻³⁰. Nothing in nature has enough photons in one mode for stimulated emission to matter. A cavity is how you build that number up by hand.

Pump a two-level atom optically and you are driving the same transition you hope to get gain on. Absorption lifts atoms at rate W per ground atom; stimulated emission by the same pump light drops them at rate W per excited atom. In steady state W(N₁ − N₂) = A₂₁N₂, so N₂/N = W/(2W + A₂₁): it approaches one half and never reaches it. At best the medium goes transparent to its own pump. There is no inversion and the photon panel stays flat.

The fix is a third level: pump into a band that decays quickly and irreversibly to the upper laser level, faster than the pump can drive it back. The upper level then fills without the pump being able to empty it. Ruby’s pump bands decay in nanoseconds into a level that lives for three milliseconds — that ratio is the whole design.

In ruby the lower laser level is the ground state, so inversion means more than half of all the chromium ions are excited at once. The threshold in the table is a hair above the inversion pump rate whatever you do to the mirrors: the cavity asks for ΔN_th of a few 10¹⁶, the medium has 10¹⁹, and lifting the first 8 × 10¹⁸ is the real cost. For the cubic centimetre of ruby modelled here that is close to a kilowatt absorbed, continuously. Maiman did it in 1960 with a flashlamp pulse; making ruby run continuously took a water-cooled arc lamp and was rarely worth it.

A four-level laser (Nd:YAG, helium-neon, most diode lasers) puts the lower laser level above the ground state, so it is empty until the laser fills it: inversion starts with the first excited atom and the threshold is set by ΔN_th alone. Shorten the cavity lifetime on this page and watch the three-level threshold barely move until the cavity is so lossy that ΔN_th passes N — at which point it vanishes altogether.

The rate equations are the standard single-mode laser rate equations, solved exactly for their steady state with the stimulated rate per photon K = σc/V derived from the Einstein B coefficient. They are tested inlib/quantum/laser.test.tsagainst Einstein’s detailed-balance argument with a Planck field, the textbook cross-section formula, the analytic three-level threshold, the analytic slope above it, and a direct integration of the differential equations. The two-level result is checked the same way.

Simplified: one cavity mode at line centre, a top-hat lineshape of width Δν, no refractive index, no non-radiative decay from the upper level, and every cavity loss counted as output. The ruby cross-section that comes out is about eight times the measured 2.5 × 10⁻²⁴ m² (a real Lorentzian line and a refractive index of 1.76 account for most of the gap), which shifts ΔN_th by the same factor and changes nothing qualitative. Erbium in glass is quasi-three-level in reality (its lower level is a thermally spread manifold) and is treated here as the ideal case.

The cartoon is the same physics with 42 atoms and probabilities chosen to be watchable — absorption and stimulated emission share one probability per pass, spontaneous photons are emitted isotropically, and the mode only ever holds photons that got there by those two routes. Its threshold is roughly two atoms of inversion and it clamps there; the chart’s is 10¹⁶. It is not drawn from the chart, and the numbers it prints are its own.