= Ring laser
A \b[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>.<Fundamentals of Photonics and Quantum Electronics lecture notes by Franz X. Kärtner>[Chapter7, printedpp.313–314, Figure7.19 and equations7.10–7.12]{ref}
\Image[/ring-laser-closed-path.png]
{title=Original schematic of a three-mirror <ring laser>, a <laser gain medium>, two possible circulation directions and one output port}
{description=Idealized optical-path diagram rather than a reconstruction of the McNamara experiment. The arrows indicate possible travelling waves; their simultaneous stable lasing is not assumed.}
The phase condition explains the allowed <normal modes>. Neglect dispersion and take a constant round-trip reflection phase $\phi_r$. If the circuit's <optical path length> is $L_{\rm opt}$, resonance requires
$$
\frac{2\pi\nu L_{\rm opt}}c+\phi_r=2\pi m,\qquad m\in\mathbb Z.
$$
Subtracting the equations for consecutive modes gives the <free spectral range>
$$
\boxed{\Delta\nu=\frac c{L_{\rm opt}}.}
$$
More 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.<Fundamentals of Photonics and Quantum Electronics lecture notes by Franz X. Kärtner>[Chapter7, equations7.10–7.12: inverse round-trip-time spacing and distinction between linear and ring cavities]{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 is
$$
\mathcal M=G R_1R_2R_3=1.058508>1.
$$
The 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.<Observation of Stochastic Resonance in a Ring Laser by Bruce McNamara and others>[Publisher abstract, Physical Review Letters60, pp.2626–2629, 1988]{ref}
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