= Optical gain
{title2=$g$}
<Optical gain> is amplification of <light> by a medium. Here the small-signal intensity gain coefficient $g$ has units $\mathrm{m^{-1}}$: for fixed populations, a weak beam obeys
$$
\frac{dI}{dz}=gI,
\qquad I(z)=I(0)e^{gz},
\qquad g=\sigma(N_2-N_1)
$$
in the nondegenerate resonant two-level model, with common transition cross-section $\sigma$. Thus $g$ includes the competition between <stimulated emission> and <photon absorption> on that transition. Separate passive intensity loss $\alpha$ gives $I(z)=I(0)e^{(g-\alpha)z}$. Both coefficients describe <light intensity>, not <wave amplitude>; the corresponding field-amplitude exponent is half as large. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/2641db6431ea70989c29e8512d5d91a7_chapter6.pdf[Franz Kärtner, MIT 6.974, §§6.7–6.8: amplification and stimulated-transition cross-section]{ref}.
For $g=0.30\,\mathrm{m^{-1}}$, $\alpha=0.10\,\mathrm{m^{-1}}$ and $z=0.50\,\mathrm m$, the single-pass intensity multiplier is $e^{0.10}\approx1.10517$. This is amplification during one passage, before mirror losses. <Population inversion> can therefore yield positive <optical gain> while the complete oscillator remains below <laser threshold>.
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