Past exam of the laser science and quantum information processing course of the University of Oxford / 2017 / 1

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”.

Let $v_g\simeq c/n_r$ be the light's group velocity in the fibre, with unit mode overlap and a uniform transverse mode area. The fibre-laser gain-volume rate equations follow because the upper-level density is defined per doped volume, whereas the photon density is defined per full cavity volume. Stimulated emission therefore removes upper-level ions at rate $v_g{\sigma }_{21}{N}_{2}n$ per doped volume, but supplies cavity photons at the volume-averaged rate $f{v}_g{\sigma }_{21}{N}_{2}n$. ThusAt threshold the stimulated photon growth rate balances the photon loss rate, givingUsing this relation yields the stated equations, including the essential $1/f$ in the ion-depletion term:Above threshold, the nonzero stationary photon density requires $N_{2,\mathrm{ss}}=N_{\mathrm{th}}$. The remaining stationary equation then givesThe parameter $r$ is the pump rate in units of its lasing-threshold value. The above-threshold branch requires $r>1$; below threshold the ideal stationary solution is $n=0$, $N_2=S{\tau }_2$. Spontaneous emission into the lasing mode has been omitted in this deterministic model, so an exactly zero initial field would need a seed or noise to start, even when the zero-field branch is unstable.

Write $N_2=N_{\mathrm{th}}+u$ and $n=n_{\mathrm{ss}}+w$. Keeping terms linear in the perturbations givesEquivalently,The characteristic exponents areConsequently the above-threshold steady state has damped oscillations precisely whenTheir ordinary frequency isThe approximation neglects the damping correction when $r^2{\tau }_c\ll 4(r-1){\tau }_2$. The exponential envelope decays on time $2{\tau }_2/r$. The doped-volume fraction cancels out of this linear frequency once the pump is expressed through $r$. ^[ref]

The upper-state density is clamped near threshold once the laser operates, so excess pumping mainly increases photon density and output power. For approximately pump-independent absorption efficiency, $r=P_p/P_{p,\mathrm{th}}$, and both $P_{\mathrm{out}}$ and $f_{\mathrm{ro}}^2$ should rise approximately linearly with $P_p-P_{p,\mathrm{th}}$.

The left ordinate is the square of the frequency, in ${\mathrm{kHz}}^2$, rather than frequency in kHz. Reading the original triangles gives a threshold near $12$–$13\mathrm{mW}$ and a slope about $1.0{\mathrm{kHz}}^2/\mathrm{mW}$. A representative straight-line fit to visually read marker centers gives $P_{p,\mathrm{th}}\simeq 12.2\mathrm{mW}$ and $A=d{f}_{\mathrm{ro}}^2/d{P}_p\simeq 1.02\times 10^6{\mathrm{Hz}}^2/\mathrm{mW}$. ThereforeThe displayed precision is limited by graph reading and scatter; a lifetime around $0.15\mu \mathrm{s}$ is the meaningful result. The output squares have a compatible threshold and an approximately constant slope above it.

The round-trip time isIf mirror transmission is the only appreciable passive loss, energy decays by $R_1{R}_2=e^{-t_{\mathrm{rt}}/{\tau }_c}$ per round trip. For $R_1=1$,a power reflectance of roughly 65%. The exact exponential is preferable to replacing $1-R_2$ by $t_{\mathrm{rt}}/{\tau }_c$, since the inferred round-trip loss is not tiny. If distributed fibre loss is appreciable, the lifetime instead fixes the total round-trip retention; without a separate loss measurement, the bare mirror reflectance is not uniquely determined and is at least this retention factor.