K-factor

Measure a semiconductor laser's small-signal response at several bias currents and two numbers come out of each fit: the relaxation-oscillation frequency $f_r$ and the damping rate $\gamma$. Plot $\gamma$ against $f_r^2$ and the points fall on a line:

The slope $K$ \u2014 the K-factor, units of nanoseconds \u2014 is one of the standard figures of merit reported for any high-speed laser, because it encodes the device's intrinsic bandwidth ceiling. As bias rises, $f_r$ grows but damping grows faster (as $f_r^2$); at some point the resonance is so damped that the response rolls off before benefiting from further $f_r$. Working through the two-pole response, the damping-limited maximum 3-dB bandwidth is

(with $K$ in ns, $f$ in GHz). A K-factor of 0.3 ns caps the intrinsic response near 30 GHz no matter how hard the laser is driven.

Physically, $K = 4\pi^2(\tau_p + \varepsilon/v_g a)$: the photon lifetime plus a gain-compression term. Short cavities (small $\tau_p$) and low compression make fast lasers \u2014 the design logic behind short-cavity DFBs and high-speed VCSELs.

Two cautions when using quoted K-factors. Real devices frequently hit parasitic (RC) or thermal limits below the K-factor ceiling, so $8.9/K$ is an upper bound, not a prediction. And the offset $\gamma_0$ \u2014 dominated by the inverse differential carrier lifetime \u2014 matters at low bias; fits that ignore it skew K. The same $\gamma$–$f_r$ pairs can be pulled from RIN spectra when a network analyzer isn't available.