Relaxation oscillations

A laser's photon number and carrier number form a coupled oscillator. Bump the carrier density and the gain rises, the photon density surges, stimulated emission burns the carriers back down, the gain falls, the photons decay, the carriers recover, and the cycle repeats at the relaxation-oscillation frequency $f_r$, damped over a few periods. The oscillation is a property of any laser whose upper-state lifetime is longer than its cavity photon lifetime, which covers interband diode lasers and solid-state lasers (quantum cascade lasers, with picosecond upper-state lifetimes, are the exception), and it sets both the ringing seen on a pulse edge and the bandwidth available for direct modulation.

The frequency

For a semiconductor laser the small-signal rate equations give

$$f_r=\frac{1}{2\pi }\sqrt{\frac{v_{g}aS}{{\tau }_p}}$$

with $v_g$ the group velocity, $a$ the differential gain $\mathrm{\partial }g\mathrm{/}\mathrm{\partial }N$, $S$ the photon density and ${\tau }_p$ the photon lifetime. Since $S$ grows linearly with the current above threshold, the practical form is

$$f_r=D\sqrt{I-I_{th}}$$

where the D-factor, in GHz per √mA, collects the differential gain, the confinement factor, the injection efficiency and the active volume: $D=(1\mathrm{/}2\pi )\sqrt{\mathrm{\Gamma }{v}_{g}a{\eta }_i\mathrm{/}(qV)}$. A telecom DFB with $D=2$ GHz/√mA resonates at 10 GHz when biased 25 mA above threshold; a datacom VCSEL, with a mode volume ten to twenty times smaller and a correspondingly larger $D$ of 5 to 10 GHz/√mA, reaches the same frequency a few milliamps above its sub-milliamp threshold, which is the physics behind its low drive power. Higher differential gain (quantum wells, strain, p-doping) and a smaller mode volume both raise $D$. The photon lifetime cancels out of $D$; a shorter one (shorter cavity, lower facet reflectivity) lowers the K-factor instead.

Damping and the modulation response

The resonance is damped at a rate $\gamma$ that itself grows with the photon density, through gain compression and the carrier lifetime:

$$\gamma =K{f}_r^2+{\gamma }_0,K\approx 4{\pi }^2({\tau }_p+\frac{ϵ}{v_{g}a})$$

where $ϵ$ is the gain-compression factor and ${\gamma }_0\approx 1\mathrm{/}{\tau }_n$ the damping offset set by the carrier lifetime. The K-factor, typically 0.2 to 0.5 ns in good DFBs, fixes the maximum intrinsic bandwidth $f_{3dB,max}=2\sqrt{2}\pi \mathrm{/}K$: 30 GHz for $K=0.3$ ns. The small-signal intensity response to current modulation at frequency $f$ is the second-order transfer function

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

which peaks near $f_r$ when the damping is light and gives a 3 dB modulation bandwidth of about $1.55f_r$ in that limit; as the bias rises the damping catches up, the peak flattens, and the bandwidth saturates at the K-limited value or, more often, at the value set by the parasitic capacitance and the thermal roll-over of the differential gain.

Where it shows up

The resonance appears in three measurements, and each is a way of reading $f_r$ and $\gamma$. In the small-signal response it is the peak before roll-off, fitted to $H(f)$ to extract both parameters and, from several bias points, $D$ and $K$. In the time domain any step in drive current launches ringing at $f_r$ that decays in a time $2\mathrm{/}\gamma$: a laser switched on from below threshold overshoots by a factor that can reach two before the ringing settles, and the overshoot on the leading edge of a directly modulated NRZ pulse, together with the chirp it carries, is what limits direct modulation to a few tens of gigabits per second. In the RIN spectrum it is the characteristic peak, since intensity noise concentrates at $f_r$ and the peak height reads the damping; the RIN measurement needs no modulation and is the passive way to find the resonance. Because $f_r$ tracks $\sqrt{I-I_{th}}$, plotting $f_r^2$ against current also gives the threshold, and the frequency is one of the standard outputs of the DFB characterization workflow.

In solid-state and other class-B lasers

The same physics governs Nd:YAG, fiber and other lasers whose upper-state lifetime $\tau$ far exceeds the cavity photon lifetime ${\tau }_c$. With the pump ratio $r$ above threshold,

$$f_r=\frac{1}{2\pi }\sqrt{\frac{r-1}{\tau {\tau }_c}}$$

which for a Nd:YAG laser pumped at twice threshold, with $\tau =230$ µs and ${\tau }_c=20$ ns, gives 74 kHz: the spikes seen when such a laser is switched on and the low-frequency noise peak in its output are relaxation oscillations, five orders of magnitude slower than in a diode laser because the upper-state lifetime is a hundred thousand times longer and the photon lifetime ten thousand times longer. Most gas lasers (helium-neon, argon-ion) and dye lasers, whose upper-state lifetimes are comparable to or shorter than the photon lifetime, are class-A and show no relaxation oscillation at all.

References: L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012), chapter 5. R. S. Tucker, "High-speed modulation of semiconductor lasers," Journal of Lightwave Technology 3, 1180 (1985). R. Olshansky, P. Hill, V. Lanzisera, and W. Powazinik, "Frequency response of 1.3 μm InGaAsP high speed semiconductor lasers," IEEE Journal of Quantum Electronics 23, 1410 (1987). A. E. Siegman, Lasers (University Science Books, 1986), chapter 25, for the class-B rate-equation treatment.