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A laser rate equation approximates coupled evolution of population inversion and intracavity photon number, including pumping, spontaneous emission, stimulated emission and optical cavity losses. A simple model is $\dot{S}=[g(N)-1/{\tau }_c]S$, $\dot{N}=R_p-N/{\tau }_u-g(N)S$, where compatible normalizations absorb mode-volume factors. Large excess optical gain drives rapid field growth until inversion depletion reduces it below threshold.

A laser relaxation oscillation is the damped exchange between stored upper-level excitation and intracavity photon number after a small perturbation of a lasing steady state. Extra photons increase stimulated depletion of the upper level; the resulting smaller gain then reduces the photon population. Pumping subsequently replenishes the inversion, so the two populations overshoot their equilibrium values before damping settles them.

Laser spiking is the corresponding large-signal sequence of intense output pulses, often during switch-on. Excitation first accumulates while the optical field is small. Once gain exceeds loss, rapid stimulated emission releases this stored energy in a spike, depleting the inversion below threshold. Recovery can produce further spikes. This is distinct from an externally imposed Q-switching cycle, although both draw on stored excitation.

Many solid-state gain media have upper-state lifetimes far longer than the cavity photon lifetime. The optical polarization relaxes rapidly, leaving the two slow rate-equation variables of a class-B laser. A long-lived population reservoir and a much faster photon response make underdamped recovery and pronounced startup spiking common. The criterion and frequency follow from the linearization in part c; they are not universal consequences of the word “solid-state”.

Write $N=N^{\ast }$, $N_i=N_i^{\ast }$ and $N_f=N_f^{\ast }$ during this derivation. Eliminating the stimulated term between the two laser rate equations gives the photon balanceIntegrate over the pulse, taking negligible initial and final stored photon densities:The photons lost from the cavity carry total energy $\hslash {\omega }_L{V}_c\int n/{\tau }_{c}dt$. Since $f_c{V}_c=V_g$,This identifies the supplied energy formula under the ideal extraction convention that all photon loss is useful output. If only a fraction $\zeta ={\kappa }_{\mathrm{out}}/{\kappa }_{\mathrm{total}}$ of the loss exits the output coupler, the measured output energy is $\zeta E$. The rate equations by themselves do not distinguish useful output from parasitic loss.

For a nonzero photon density, divide the laser rate equations to remove time:Integrating from the initial inversion and retaining a small seed density $n_i$ givesThe source's pulse approximation neglects $n_i$ compared with the macroscopic pulse density, givingA positive seed, or a spontaneous-emission source omitted from these equations, is nevertheless needed to start the pulse. Setting $n(0)=0$ exactly in the unforced equations gives the counterexample $n(t)=0$, $N^{\ast }(t)=N_i^{\ast }$ for all time, even above threshold. The displayed formula is therefore used in the negligible-seed limit, rather than as an assertion that a pulse starts from an identically zero field.

In Q-switching, hold the optical resonator's quality factor low while pumping builds population inversion, then rapidly reduce the loss. The stored excitation produces a large optical gain, the intracavity light intensity grows, and stimulated emission depletes the population inversion until the optical gain falls below the laser threshold. The resulting optical pulse is a transient discharge of stored excitation.

Its duration depends on the high-quality factor photon lifetime, initial population inversion, saturation dynamics and switching time. For example, a simple laser rate equation model iswhere $S$ is intracavity photon number, $N$ is the available inversion in compatible units, $R_p$ is the pump rate and ${\tau }_c$ is the photon lifetime. During the short discharge, pumping and upper-state decay can often be neglected. The build-up and depletion rates determine the width; there is no universal equality between the width and ${\tau }_c$. The initially stored energy and the output coupling also matter. Typical Q-switching produces nanosecond-scale optical pulses.

Define $\sigma (\omega )$ as the emission cross section for an upper-level atom, averaged over the participating sublevels, and let $W=\sigma I/(\hslash \omega )$. Detailed balance gives the upward stimulated rate per lower-level atom as $rW$, where $r=g_2/g_1$. With $D=N_2-r{N}_1$ the complete laser rate equations areAdding them eliminates the internal transitions. Therefore steady state has $N_1=R_2{\tau }_1$ even when the radiation is strong. Substitution into the upper equation givesOne net stimulated transition changes the beam energy by $\hslash \omega$, so the coefficient of optical gain for intensity is $\alpha (\omega ,I)=\sigma (\omega )D$. At line center, with ${\sigma }_0=\sigma ({\omega }_{21})$,Here ${\alpha }_0$ is the unsaturated intensity optical gain per length, which may be negative, and $I_s$ is the intensity at which its magnitude is halved in this model with homogeneous broadening. The cancellation of ${\tau }_1$ from $I_s$ follows from exclusive upper-to-lower decay and constant upper-state pumping. Keeping $N_1=A{\tau }_1{N}_2$ in the stimulated problem would discard the stimulated feeding term and give an incorrect saturation scale. Other decay branches or different pump constraints lead to different expressions.

For a traveling beam with uniform pump, no additional loss and local steady populations,If ${\alpha }_0>0$ and $I\gg I_s$, the increment becomes $dI/dz\simeq {\alpha }_0{I}_s=\hslash {\omega }_{21}{R}_2(1-rA{\tau }_1)$, approximately constant. Intensity then grows linearly, rather than exponentially, with distance while that strong-saturation regime persists. Negative ${\alpha }_0$ gives saturable absorption and a decreasing beam.