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Keep the given emission cross section in square centimetres so that the density remains per cubic centimetre. For the first transition, $r=5/3$ andThusThe separate populations are $N_2^{(0)}=1.667\times 10^{13}$ cm${}^{-3}$ and $N_1^{(0)}=1.00\times 10^{12}$ cm${}^{-3}$. Subtracting $5N_1/3$, rather than $N_1$ alone, reproduces the population inversion used above.

Here $rA{\tau }_1=0.10<1$, so upper-state pumping and sufficiently rapid lower-level removal produce positive small-signal optical gain. This coefficient is an intensity optical gain; a weak field amplitude would grow with half this exponent. A positive single-pass coefficient also does not by itself establish laser oscillation without a cavity-loss budget. No transition wavelength was supplied for this numerical example, so a numerical saturation intensity is not needed or invented to calculate ${\alpha }_0$.

A ring laser is a laser whose optical cavity follows a closed circulating path through a laser gain medium. Mirrors can fold the path into a triangle or another polygon; guided structures can implement the same circulating feedback principle. A travelling optical wave can return to its starting position and direction after one full circuit. The laser threshold requires round-trip gain to balance the complete losses; a closed path alone is not enough to produce laser oscillation.^[ref]

The phase condition explains the allowed normal modes. Neglect dispersion and take a constant round-trip reflection phase ${\varphi }_r$. If the circuit's optical path length is $L_{\mathrm{opt}}$, resonance requiresSubtracting the equations for consecutive modes gives the free spectral rangeMore generally the spacing is approximately the inverse round-trip group delay. A vacuum circuit of full geometric perimeter $P=0.90\mathrm{m}$ has a round-trip time about $3.00\mathrm{ns}$ and mode spacing about $333\mathrm{MHz}$. There is no additional factor of two: $P$ already counts the whole returning route.^[ref]

Gain and phase impose different conditions. In an original startup model let the three mirror intensity reflectivities be $0.99$, $0.99$ and $0.90$, and let the intensity gain accumulated during one complete circuit be $G=1.20$, with other losses neglected. The round-trip intensity multiplier isThe modeled field can grow from a seed. Reducing $G$ to $1.05$ gives $\mathcal{M}=0.9261945<1$, so it decays. These are intensity quantities, not amplitude reflectivities. Saturation and competition between normal modes must determine real steady operation.

A reciprocal ideal ring supports possible clockwise and counterclockwise waves. Their simultaneous stable emission is a dynamical question because both can interact with the same laser gain medium. A ring laser gyroscope uses rotation-dependent differences between the two circulation directions through the Sagnac effect; ring geometry by itself does not guarantee a useful rotation measurement.

A different motivating experiment used the directional dynamics as a nonlinear sensor. In 1988 Bruce McNamara, Kurt Wiesenfeld and Rajarshi Roy periodically modulated the asymmetry between counter-rotating lasing modes and added controlled external noise. Their measured signal-to-noise ratio improved for injected noise relative to no external noise, demonstrating stochastic resonance. This observation concerns a particular modulated dynamical response, not a claim that adding noise improves every optical measurement.^[ref]