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A laser generates amplified light through stimulated emission. Its name abbreviates “light amplification by stimulated emission of radiation”. Lasers supply the intense beams used in optical tweezers; optical coherence also makes them useful for wave interference measurements. ^[ref].

In the usual material picture, a gain medium stores energy supplied by pumping, for example by incident light or an electric current. Stimulated emission transfers that stored energy to a selected optical normal mode, while competing photon absorption removes light. Producing net optical gain requires a suitable population distribution, such as population inversion in the simple model below.

A familiar laser oscillator uses mirrors to return light through the gain medium. This optical feedback favours resonant normal modes; a partly transmitting mirror lets useful power escape. Feedback and mode selection have different implementations, including ring arrangements: every laser need not resemble the same two-mirror device. The laser threshold example treats just that device. ^{[ref] [ref]}.

The useful combination is concentrated power and controlled optical normal modes. It gives optical tweezers a localized source of force and wave interference measurements a source with useful optical coherence. A laser can nevertheless emit several frequencies; “laser” alone does not guarantee a single frequency. ^[ref].

An original experimental landmark is T. H. Maiman's 1960 report of stimulated optical radiation from optically pumped ruby at about $694.3\mathrm{nm}$. His paper describes both the pumping path and the observed emission, connecting stored material energy with amplified light. ^[ref].

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 $0<R_1,R_2\le 1$. One round trip has multiplierTwo traversals give the distance $2L$; multiplying the two mirror factors completes the bookkeeping. Setting $M=1$ and taking a logarithm givesThe 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. ^[ref].

For $L=0.50\mathrm{m}$, $R_1=1$, $R_2=0.90$ and $\alpha =0.10{\mathrm{m}}^{-1}$, the threshold is $g_{\mathrm{th}}\approx 0.2053605{\mathrm{m}}^{-1}$. At $g=0.15{\mathrm{m}}^{-1}$, $M\approx 0.946144$: a small circulating signal decays even though the material has positive optical gain. At $g=0.30{\mathrm{m}}^{-1}$, $M\approx 1.099262$: the signal initially grows. Raising passive loss or lowering mirror reflectivity raises laser threshold.

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. ^[ref].

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 obeysin 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. ^[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 $e^{0.10}\approx 1.10517$. 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.