K-factor

The K-factor of a semiconductor laser is the slope of the damping rate $\gamma$ of its relaxation oscillations against the square of their frequency $f_r$:

$$\gamma =K{f}_r^2+{\gamma }_0\mathrm{.}$$

Both quantities come from fitting the small-signal modulation response, or the relative intensity noise spectrum, at several bias currents. Plotted as $\gamma$ against $f_r^2$ the points fall on a line whose slope, in nanoseconds, is $K$, and whose intercept ${\gamma }_0$ is set mainly by the differential carrier lifetime. The K-factor fixes the damping-limited maximum 3 dB modulation bandwidth,

$$f_{\text{3dB,max}}=\frac{2\sqrt{2}\pi }{K}\approx \frac{8.9}{K}$$

with $K$ in ns and $f$ in GHz, so a laser with $K$ = 0.3 ns cannot be modulated faster than about 30 GHz however hard it is driven.

Where the limit comes from

The small-signal response of a directly modulated laser is a damped resonance,

$$H(f)=\frac{f_r^2}{f_r^2-f^2+jf\gamma \mathrm{/}2\pi }\mathrm{.}$$

At low bias the damping is light, the response peaks near $f_r$, and the 3 dB bandwidth is about $1.55f_r$. The relaxation frequency grows as the square root of the photon density, but the damping grows as $f_r^2$, so as the bias rises the peak flattens and the bandwidth grows more slowly than $f_r$. It is largest when the damping ratio $\gamma \mathrm{/}(4\pi f_r)$ reaches $1\mathrm{/}\sqrt{2}$, the maximally flat response, where the bandwidth equals $f_r$ and both equal $2\sqrt{2}\pi \mathrm{/}K$; beyond that point more bias lowers the bandwidth. Computed from the response above:

What sets it

The rate equations give

$$K=4{\pi }^2({\tau }_p+\frac{\epsilon }{v_{g}a}),$$

the photon lifetime ${\tau }_p$ plus a gain compression term, with $\epsilon$ the gain compression factor, $v_g$ the group velocity and $a$ the differential gain. For the 300 µm laser used as the example in the gain compression entry, ${\tau }_p$ = 2.4 ps and, with $\epsilon =2\times {10}^{-17}$ cm³ and $a=5\times {10}^{-16}$ cm², the compression term is 4.8 ps. $K$ is then 0.29 ns and the limit 31 GHz, where the photon lifetime alone would allow 92 GHz. The routes to a smaller K-factor follow from the formula: a shorter photon lifetime, from a shorter cavity or less reflective mirrors; a higher differential gain, from strained or p-doped quantum wells; and less gain compression.

Reaching the limit

The limit is reached only if $f_r$ can be driven up to it. For the example laser, whose relaxation frequency computed with the same parameters is 4.6 GHz at 10 mW of output, reaching 31 GHz would take about 45 times the photon density if $f_r^2$ stayed proportional to it, and about 80 times once gain compression is counted; heating and the capacitance and resistance of the chip and package stop it long before. The measured bandwidth of most lasers is therefore set by how fast $f_r$ grows with current (the D-factor), by thermal rollover or by electrical parasitics, and $8.9\mathrm{/}K$ is an upper bound rather than a prediction. The K-factor matters most for lasers designed for speed, such as short-cavity DFBs and high-speed VCSELs, whose relaxation frequency can come close to it.

Fitting it

Each measured response is fitted with $f_r$, $\gamma$ and a parasitic pole $f_p$ as free parameters, multiplying $H(f)$ by $1\mathrm{/}(1+jf\mathrm{/}{f}_p)$ so that the roll-off of the package is not read as damping, which would make $K$ too large. Bias points from just above threshold to near thermal rollover give a well-defined line; the low-bias points fix ${\gamma }_0$, and a fit that forces ${\gamma }_0$ to zero skews $K$. The same pairs of $f_r$ and $\gamma$ can be read from relative intensity noise spectra, which need no modulation. From the fitted $K$ the gain compression factor follows as $\epsilon =v_{g}a(K\mathrm{/}4{\pi }^2-{\tau }_p)$.

Common questions

What is the K-factor of a laser?

The slope of the relaxation-oscillation damping rate against the square of the relaxation frequency, $\gamma =K{f}_r^2+{\gamma }_0$, measured at several bias currents. It is quoted in nanoseconds and sets the laser's intrinsic bandwidth limit.

How is the maximum modulation bandwidth calculated from the K-factor?

$f_{\text{3dB,max}}=2\sqrt{2}\pi \mathrm{/}K$, about $8.9\mathrm{/}K$ GHz with $K$ in nanoseconds: 30 GHz for 0.3 ns and 18 GHz for 0.5 ns.

Why is the K-factor limit rarely reached?

Because the relaxation frequency has to be driven up to the same value, which takes a photon density that heating and the device's electrical parasitics usually prevent. The limit is a ceiling on the intrinsic response, not an expected bandwidth.

What determines the K-factor?

The photon lifetime of the cavity and the gain compression factor divided by the differential gain, $K=4{\pi }^2({\tau }_p+\epsilon \mathrm{/}{v}_{g}a)$. In the example above the compression term is twice the photon lifetime.

References: L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).