Relaxation oscillations

A laser's photon number and carrier number form a coupled oscillator. Bump the carrier density and gain rises, photon density surges, stimulated emission burns carriers back down, gain falls, photons decay, carriers recover. The cycle rings at the relaxation-oscillation frequency $f_r$, damped over a few cycles.

The controlling relation:

$f_r = \frac{1}{2\pi}\sqrt{\frac{v_g\, a\, S}{\tau_p}}$

with $v_g$ group velocity, $a$ differential gain, $S$ photon density, $\tau_p$ photon lifetime. Since $S$ scales with drive above threshold, the practical form is $f_r \propto \sqrt{I - I_{th}}$: bias harder, resonate faster. The proportionality constant (MHz per √mA) is the D-factor, a standard extracted figure of merit.

The resonance shows up in three measurements. In the small-signal response, it is the peak before roll-off, tying directly to modulation bandwidth ($f_{3dB} \approx 1.55 f_r$, low damping). In the time domain, any step in drive current launches visible ringing at $f_r$, the overshoot on pulse edges. In the RIN spectrum, it is the characteristic noise peak: intensity noise concentrates at $f_r$, which makes the RIN plot a convenient, purely passive way to read the resonance and its damping.

Damping grows with photon density (gain compression, $\epsilon S$ terms) and eventually caps the useful bandwidth: an overdamped laser rolls off before its nominal $f_r$ would suggest. VCSELs, with tiny mode volumes and high photon densities, reach multi-GHz $f_r$ at sub-mA currents, the physics behind their datacom dominance. Typical numbers: telecom DFBs, 5–20 GHz at operating bias; datacom VCSELs, similar with far less current.

For the measurement procedures (extracting $f_r$, damping, and D-factor from response fits), see the DFB characterization workflow article.