Self-pulsation is a transient phenomenon in continuous-wave lasers. Self-pulsation takes place at the beginning of laser action. As the pump is switched on, the gain in the active medium rises and exceeds the steady-state value. The number of photons in the cavity increases, depleting the gain below the steady-state value, and so on. The laser pulsates; the output power at the peaks can be orders of magnitude larger than that between pulses. After several strong peaks, the amplitude of pulsation reduces, and the system behaves as a linear oscillator with damping. Then the pulsation decays; this is the beginning of the continuous-wave operation.
Equations The simple model of self-pulsation deals with number X {\displaystyle X} of photons in the laser cavity and number Y {\displaystyle ~Y~} of excitations in the gain medium. The evolution can be described with equations:
d X / d t = K X Y − U X d Y / d t = − K X Y − V Y + W {\displaystyle ~{\begin{aligned}{{\rm {d}}X}/{{\rm {d}}t}&=KXY-UX\\{{\rm {d}}Y}/{{\rm {d}}t}&=-KXY-VY+W\end{aligned}}}
where
K = σ / ( s t r ) {\displaystyle ~K=\sigma /(st_{\rm {r}})~} is coupling constant,
U = θ L {\displaystyle ~U=\theta L~} is rate of relaxation of photons in the laser cavity,
V = 1 / τ {\displaystyle ~V=1/\tau ~} is rate of relaxation of excitation of the gain medium,
W = P p / ( ℏ ω p ) {\displaystyle ~W=P_{\rm {p}}/({\hbar \omega _{\rm {p}}})~} is the pumping rate;
t r {\displaystyle ~t_{\rm {r}}~} is the round-trip time of light in the laser resonator,
s {\displaystyle ~s~} is area of the pumped region (good mode matching is assumed);
σ {\displaystyle ~\sigma ~} is the emission cross-section at the signal frequency ω s {\displaystyle ~\omega _{\rm {s}}~} .
θ {\displaystyle ~\theta ~} is the transmission coefficient of the output coupler.
τ {\displaystyle ~\tau ~} is the lifetime of excitation of the gain medium.
P p {\displaystyle P_{\rm {p}}} is power of pump absorbed in the gain medium (which is assumed to be constant). Such equations appear in the similar form (with various notations for variables) in textbooks on laser physics, for example, the monography by A.Siegman.
Steady-state solution
X 0 = W U − V K Y 0 = U K {\displaystyle {\begin{aligned}X_{0}&={\frac {W}{U}}-{\frac {V}{K}}\\Y_{0}&={\frac {U}{K}}\end{aligned}}}
Weak pulsation Decay of small pulsation occurs with rate
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