 Solution

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]