Differential gain

Differential gain is the slope of the material gain of a semiconductor with respect to the carrier density at a fixed wavelength, $a = \partial g/\partial N$. Gain is in cm⁻¹ and density in cm⁻³, so $a$ has units of cm². Bulk InGaAsP active regions have $a$ of roughly 2–3 × 10⁻¹⁶ cm²; unstrained and strained quantum wells typically reach 5 × 10⁻¹⁶ to 1 × 10⁻¹⁵ cm². These are typical ranges at the densities where lasers operate, and a quoted value means little without the carrier density and wavelength at which it was taken. The quantity matters because a laser is modulated by changing its carrier density, and $a$ converts that change into a change of gain.

Dependence on carrier density

For quantum wells the gain near its peak is well described by a logarithmic fit, $g = g_0 \ln(N/N_\text{tr})$, which gives

$$a = \frac{dg}{dN} = \frac{g_0}{N}.$$

The differential gain therefore falls as the density rises. With $g_0$ = 1500 cm⁻¹ and $N_\text{tr}$ = 1.5 × 10¹⁸ cm⁻³ ($g_0$ is the illustrative value of the modal gain entry), $a$ is 1.0 × 10⁻¹⁵ cm² at transparency. A 300 µm cleaved laser with a confinement factor of 0.05 needs a material gain of 1003 cm⁻¹ at threshold (threshold condition), reached at $N_\text{th}$ = 1.95 $N_\text{tr}$, where $a$ has dropped to 5.1 × 10⁻¹⁶ cm². Halving the cavity to 150 µm raises the threshold gain to 1805 cm⁻¹ and the density to 3.33 $N_\text{tr}$, and $a$ falls to 3.0 × 10⁻¹⁶ cm². This is why designs that keep the threshold gain low, with lower losses, more wells or a larger confinement factor, also tend to be faster.

Relaxation-oscillation frequency

In the small-signal solution of the laser rate equations the differential gain sets the resonance of the coupled carrier and photon populations:

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

with $v_g$ the group velocity, $S$ the photon density in the mode and $\tau_p$ the photon lifetime (relaxation oscillations). For the 300 µm laser above, with $n_g$ = 3.6 ($v_g$ = 8.33 × 10⁹ cm/s), $\tau_p$ = 2.4 ps, $a$ = 5 × 10⁻¹⁶ cm² and $S$ = 5 × 10¹⁴ cm⁻³ (about 10 mW total output from both facets), $f_R$ = 4.7 GHz. Since $f_R \propto \sqrt{a}$, doubling $a$ to 1 × 10⁻¹⁵ cm² raises it to 6.6 GHz at the same photon density, and a bulk value of 2.5 × 10⁻¹⁶ cm² lowers it to 3.3 GHz. Differential gain also enters the D-factor, the slope of $f_R$ against $\sqrt{I - I_\text{th}}$, and the compression term $\varepsilon/(v_g a)$ of the K-factor, so a larger $a$ raises the resonance and reduces the damping penalty of a given gain compression factor.

How it is raised

Compressive strain in the wells lowers the heavy-hole effective mass and brings the valence-band density of states closer to that of the conduction band, so population inversion builds with fewer carriers and the gain rises more steeply. Modest p-type doping of the wells or barriers raises $a$ at the cost of more free-carrier absorption. Quantum-dot active regions can have very high differential gain at low density, although their gain saturates early. In DFB lasers the grating can be placed on the short-wavelength side of the gain peak (detuned loading), where band filling makes $dg/dN$ larger than at the peak.

Measurement

Two routes are common. In the first, the relaxation-oscillation frequency is measured from the small-signal modulation response or the intensity-noise spectrum at several bias currents; the slope of $f_R^2$ against current gives the square of the D-factor, and $a$ follows once the confinement factor, injection efficiency and active volume are known. In the second, sub-threshold gain spectra (Hakki–Paoli or segmented-contact methods) are recorded at several currents and converted to carrier density with a measured carrier lifetime; the slope of gain against density at the lasing wavelength is $a$. The first route returns an effective value at the operating point, which also depends on the assumed confinement factor and volume; the second returns the modal value $\Gamma a$, which gives the material value once $\Gamma$ is known.

Pitfalls

Differential gain is easily confused with the modal quantity $\Gamma a$, which is smaller by the confinement factor. Some texts use a gain coefficient in s⁻¹ per carrier, $G_N = \Gamma v_g a/V$, which includes the active volume and has different units. In the linewidth enhancement factor, $\alpha = -(4\pi/\lambda)(dn/dN)/(dg/dN)$, the same $a$ appears in the denominator, so a higher differential gain also reduces chirp.

Common questions

What is a typical differential gain for a quantum-well laser?

About 5 × 10⁻¹⁶ to 1 × 10⁻¹⁵ cm² at typical threshold densities for InGaAsP and InGaAs quantum wells, with strained wells at the upper end; bulk active regions are around 2–3 × 10⁻¹⁶ cm².

Why does differential gain fall at high carrier density?

The gain saturates as the bands fill: each additional carrier occupies a higher energy state that contributes less to the gain at the lasing wavelength, so $dg/dN$ decreases roughly as $1/N$.

Why does higher differential gain give a faster laser?

The relaxation-oscillation frequency scales as $\sqrt{a}$ at fixed photon density, and the intrinsic modulation bandwidth scales with it; higher $a$ also lowers the gain-compression contribution to damping.

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). B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).