= Laser threshold
{title2=$g_{\mathrm{th}}$}
In a simple two-mirror oscillator, <laser threshold> is the small-signal <optical gain> at which a resonant optical <normal mode> exactly replaces its round-trip intensity loss. Assume constant intensity coefficients $g$ and $\alpha$ throughout a one-way distance $L$, and mirror intensity reflectivities $0<R_1,R_2\leq1$. One round trip has multiplier
$$
M=R_1R_2\exp\bigl(2(g-\alpha)L\bigr),
\qquad
I_{n+1}=MI_n.
$$
Two traversals give the distance $2L$; multiplying the two mirror factors completes the bookkeeping. Setting $M=1$ and taking a <logarithm> gives
$$
g_{\mathrm{th}}=\alpha-\frac{\ln(R_1R_2)}{2L}.
$$
The reflectivities here are intensity fractions, not field-amplitude reflectivities. A shorter gain region or nonuniform loss requires replacing the path exponent by the appropriate path integral. The balance also presumes a permitted resonant optical <normal mode>. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/18f30fad63a62ef4dd894d3752b55a60_chapter7.pdf[Franz Kärtner, MIT 6.974, §§7.1 and 7.5: round-trip balance and steady state; intensity convention specified here]{ref}.
For $L=0.50\,\mathrm m$, $R_1=1$, $R_2=0.90$ and $\alpha=0.10\,\mathrm{m^{-1}}$, the threshold is $g_{\mathrm{th}}\approx0.2053605\,\mathrm{m^{-1}}$. At $g=0.15\,\mathrm{m^{-1}}$, $M\approx0.946144$: a small circulating signal decays even though the material has positive <optical gain>. At $g=0.30\,\mathrm{m^{-1}}$, $M\approx1.099262$: the signal initially grows. Raising passive loss or lowering mirror reflectivity raises <laser threshold>.
\Image[/laser-round-trip-gain-threshold.png]
{title=Round-trip intensity balance and <laser threshold> in a two-mirror model}
{description=The curve uses $M=R_1R_2e^{2(g-\alpha)L}$ with intensity reflectivities $R_1=1$, $R_2=0.90$, one-way length $L=0.50\,\mathrm m$ and passive intensity loss $\alpha=0.10\,\mathrm{m^{-1}}$. The two worked cases straddle $M=1$. The right panel shows how a smaller output-mirror reflectivity raises the required small-signal intensity gain.}
For $M>1$, the fixed-gain recurrence predicts startup amplification, not infinite actual output. Growing <light intensity> depletes the available <population inversion>, reducing <optical gain> through saturation. In the simplest steady oscillator, the saturated round-trip gain balances loss and pumping supplies the outgoing <power> and dissipated <energy>. The small-signal model locates the onset; it does not determine the final output <power>. https://ocw.mit.edu/courses/6-974-fundamentals-of-photonics-quantum-electronics-spring-2006/18f30fad63a62ef4dd894d3752b55a60_chapter7.pdf[Franz Kärtner, MIT 6.974, §7.5: gain saturation and steady output]{ref}.
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