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A three-level laser uses a pump transition from ground level1 to a higher level3, rapid relaxation $3\to 2$, and a long-lived upper lasing level2. The laser transition is $2\to 1$, so its lower level is the ground state. Fast feeding and slow upper-state decay allow excitation to accumulate, unlike steady resonant pumping of a simple two-level transition.

Stimulated emission adds photons to the stimulating optical mode, but ground-state photon absorption competes with it. The population inversion condition is $N_2/g_2>N_1/g_1$ when the relevant transition strengths obey the usual Einstein relations. For equal degeneracies and negligible $N_3$, it reduces to $N_2>N_1$, requiring more than half the active population in level2.

For a simple pump rate R per ground-state emitter, fast $3\to 2$ relaxation and negligible laser intensity, the three-level laser rate equation reduces toThis three-level ground-state inversion threshold is only transparency; positive gain must also overcome optical losses. For an illustrative uniformly filled cavity of length $\ell$, equal absorption/emission cross-section $\sigma$, internal intensity-loss coefficient ${\alpha }_i$ and mirror power reflectivities $R_1,R_2$, the threshold condition isHere $n_i$ are number densities, unlike the total populations $N_i$. The optical resonator supplies feedback and selects modes; a partially transmitting mirror couples out light. Spontaneous emission seeds the field, and stimulated transitions deplete the inversion until saturated gain balances loss. A ruby laser is the standard example. Real level degeneracies, branching losses and pump saturation modify the simple equations. ^[ref]

A ruby laser uses chromium ions in ruby as its amplifying medium. Maiman's report identifies a metastable state fed by optical excitation and red emission near $694.3\mathrm{nm}$.^[ref] The pump stores excitation in the medium; stimulated emission transfers it into the optical field. The simplified three-level laser rate equation explains why the stored population and the output pulse evolve together.

A short optical pulse can draw on previously stored excitation. In the original calculation below, $C=100$, $K=20$, $\beta =10^{-4}$, $u(0)=s(0)=0$, and $r=3$ for $0\le x<4$, followed by $r=0$. The cavity field initially remains small while pumping fills the upper state. Once sufficient gain develops, the photon population grows and depletes the stored inversion. Continuing pump action can refill it, giving relaxation oscillations in this simple model. This is not a fit to Maiman's recorded output.

The equations expose the excitation budget underlying conservation of energy. Adding them cancels the stimulated exchange:Pumping supplies excitations; nonlasing decay and cavity loss remove them. Integrating gives the boundOnly part of the cavity loss may be useful output; passive absorption is also a loss. The pulse's instantaneous power can draw on stored excitation, while its total extracted energy still obeys the integrated budget. The model also preserves $0\le u\le 1$ and $s\ge 0$: at each boundary its derivatives point inward or along the boundary.

There is a further energy cost in the pump cycle. For one absorbed pump photon producing at most one emitted laser photon, the ideal energy-conversion bound isChosen wavelengths $550\mathrm{nm}$ and $694.3\mathrm{nm}$ give about $0.7922$ before other losses. The missing energy can leave through nonradiative relaxation and heating. Actual efficiency also depends on how much pump light is absorbed, unwanted decays and optical losses; a population model alone cannot predict it.