Gain compression factor (ε)

The gain compression factor $\epsilon$ describes how the gain of a semiconductor laser falls as the density of photons in its cavity rises, over and above the dependence of the gain on the carrier density:

$$g(N,S)=\frac{g_0(N)}{1+\epsilon S}\approx g_0(N)(1-\epsilon S),$$

where $g_0(N)$ is the gain set by the carrier density $N$ and $S$ the photon density in the lasing mode. It is also called the nonlinear gain coefficient. In InGaAsP quantum-well lasers $\epsilon$ is of the order of ${10}^{-17}$ cm³, so at the photon densities of normal operation the gain is compressed by about 1%. That small reduction nonetheless supplies most of the damping of the relaxation oscillations, and with it the upper limit on how fast a directly modulated laser can be.

Where it comes from

At a high photon density, stimulated emission removes carriers from the energies of the lasing transition faster than scattering among the carriers refills them, leaving a dip in the carrier distribution at the lasing energy: spectral hole burning within the bands. It also heats the carriers, because it removes those near the band edge, colder than the average, and because free-carrier absorption gives energy to the rest. Both effects relax in less than a picosecond, much faster than the carrier density itself changes, so on the time scale of the laser's dynamics they act as an immediate loss of gain in proportion to the photon density, and the single parameter $\epsilon$ represents them. Slower effects that fits of measured data absorb into the same parameter include the capture of carriers into the quantum wells and spatial hole burning along the cavity.

How large it is

The 300 µm cleaved laser of the Schawlow–Townes linewidth entry holds $2.4\times {10}^5$ photons when it emits 10 mW. Taking a mode volume of $4.8\times {10}^{-10}$ cm³, that of a 2 µm wide stripe with 40 nm of quantum wells and a confinement factor of 0.05, the photon density is $5\times {10}^{14}$ cm⁻³, and with $\epsilon =2\times {10}^{-17}$ cm³ the product $\epsilon S$ is 0.01. To make up the lost 1% of gain, the carrier density above threshold keeps rising slightly with power instead of clamping exactly. The same $\epsilon$ compresses the gain further in lasers with small mode volumes, such as VCSELs, and at high power.

Damping and bandwidth

The damping rate of the relaxation oscillations grows in proportion to the square of their frequency, $\gamma =K{f}_r^2+{\gamma }_0$, with the K-factor

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

where ${\tau }_p$ is the photon lifetime, $v_g$ the group velocity and $a=\mathrm{\partial }g\mathrm{/}\mathrm{\partial }N$ the differential gain. The K-factor sets the largest intrinsic modulation bandwidth, $f_{\text{3dB,max}}=2\sqrt{2}\pi \mathrm{/}K$. For the example laser, with ${\tau }_p$ = 2.4 ps, a group index of 3.6 and an assumed differential gain of $5\times {10}^{-16}$ cm², the compression term $\epsilon \mathrm{/}(v_{g}a)$ is 4.8 ps, twice the photon lifetime. $K$ is then 0.29 ns and the bandwidth limit 31 GHz; without gain compression $K$ would be 0.097 ns and the limit 92 GHz. Because $\epsilon$ enters divided by the differential gain, lasers with a higher differential gain, from strained or p-doped quantum wells, reduce the compression term even when $\epsilon$ is unchanged. In practice the bandwidth of most lasers is limited first by electrical parasitics or heating, but the K-factor remains the ceiling (modulation bandwidth).

Chirp

Because the carrier density above threshold rises with the photon density, the refractive index and with it the optical frequency shift with the output power, through the linewidth enhancement factor $\alpha$. The steady shift is

$$\mathrm{\Delta }\nu =\frac{\alpha \epsilon S}{4\pi {\tau }_p},$$

the adiabatic term $\kappa P$ in the chirp of a directly modulated laser, whose connection to gain compression Koch and Linke showed. For the example laser with $\alpha =4$ it is 1.3 GHz at 10 mW, so the high and low levels of a modulated signal leave the laser at frequencies about a gigahertz apart, which fiber dispersion then turns into pulse distortion (frequency chirp).

Measuring it

$\epsilon$ is extracted rather than looked up. The usual route is the K-factor: the relaxation frequency and damping are fitted to the small-signal modulation response or the relative intensity noise spectrum at several bias currents, $\gamma$ is plotted against $f_r^2$, and $\epsilon =v_{g}a(K\mathrm{/}4{\pi }^2-{\tau }_p)$, with $a$ taken from the growth of $f_r^2$ with photon density (D-factor) and ${\tau }_p$ from the cavity. The adiabatic chirp gives a second estimate when $\alpha$ is known. Quoted values differ between conventions: some authors apply the compression to $g$ and others to $v_{g}a$, some write it against the output power in W⁻¹, and the photon density may be defined over the mode or over the active region, so values from different papers are comparable only after checking the rate equations behind them.

Common questions

What is the gain compression factor?

The coefficient $\epsilon$ in $g=g_0\mathrm{/}(1+\epsilon S)$, which describes how the gain of a semiconductor laser falls as the photon density $S$ rises. In quantum-well lasers it is of the order of ${10}^{-17}$ cm³.

Why does gain compression limit the modulation bandwidth?

It adds to the damping of the relaxation oscillations a term that grows with the photon density as fast as the oscillation frequency squared, so at high bias the resonance is damped out before the response can extend further. In the example above it lowers the intrinsic limit from 92 GHz to 31 GHz.

Is gain compression the same as gain saturation?

No. Gain saturation is the fall of gain as stimulated emission depletes the carrier density, which in a laser above threshold holds the gain at the cavity loss. Gain compression is an additional fall at a fixed carrier density, from hole burning and carrier heating, which makes the carrier density rise slightly with power instead.

How is the gain compression factor measured?

From the K-factor, the slope of the damping rate against the square of the relaxation frequency, measured at several bias currents; subtracting the photon lifetime and multiplying by $v_{g}a$ gives $\epsilon$.

References: L. A. Coldren, S. W. Corzine, and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). T. L. Koch and R. A. Linke, "Effect of nonlinear gain reduction on semiconductor laser wavelength chirping," Applied Physics Letters 48, 613 (1986). G. P. Agrawal and N. K. Dutta, Semiconductor Lasers, 2nd ed. (Van Nostrand Reinhold, 1993).