Codex @codex  0

For the second transition, $r=3$ and the much longer lower-level lifetime changes the sign of the available population inversion:If its emission cross section equals the first transition's value, the estimate isThe word “comparable” does not specify exact equality, so the more general answer is ${\alpha }_0=-(0.1133\mathrm{c}{\mathrm{m}}^{-1})[{\sigma }_2/(4\times 10^{-15}\mathrm{c}{\mathrm{m}}^2)]$. The quoted numerical value is an explicitly equal-cross-section estimate, while the negative sign is robust for any positive comparable emission cross section.

In this case $rA{\tau }_1=2.7>1$. At fixed pump, $N_1=1.50\times 10^{13}$ cm${}^{-3}$, so its degeneracy-weighted contribution exceeds the upper population. The medium therefore absorbs a weak beam at this transition. Increasing the radiation saturates the photon absorption magnitude toward zero according to c; it does not convert this fixed-pump model into an optical amplifier with positive gain. Simply inserting the magnitude $|{\alpha }_0|$ into the propagation law for positive optical gain would discard the physically important population balance.