 Three-level ground-state inversion threshold

In a three-level laser whose lower lasing level is the ground state, fast relaxation from a pumped higher level feeds a long-lived upper lasing level. With equal lasing-level degeneracies, negligible pump-level population and pump rate R per ground-state emitter, the no-light stationary rate equation gives $N_2/N_1=R{\tau }_{21}$. Positive population inversion therefore requires more than half the active emitters to leave the ground state. This is a transparency threshold; optical resonator loss imposes an additional positive-gain requirement. Degenerate sublevels require population comparisons weighted by their statistical weights and transition strengths.