= Three-level ground-state inversion threshold
{title2=$R\tau_{21}>1$}
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.
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