Material gain

Material gain $g$ is the exponential gain coefficient of an active material, in cm⁻¹: the fractional increase in intensity per unit length that a plane wave would experience if it traveled entirely inside that material. It is the gain counterpart of the absorption coefficient, and in a semiconductor it is the same quantity with the opposite sign: the unpumped material absorbs, and injecting carriers past transparency turns the absorption into gain. Quantum-well active regions at laser threshold typically operate at a material gain of several hundred to about 2000 cm⁻¹, and peak values of 3000–5000 cm⁻¹ are reported at high injection. Solid-state and fiber gain media have far lower values: of order 0.6 cm⁻¹ for a strongly pumped Nd:YAG rod.

Material gain and modal gain

In a waveguide laser or amplifier only part of the guided mode overlaps the active layers. The gain the mode actually experiences is the modal gain,

$$g_\text{modal} = \Gamma g,$$

where $\Gamma$ is the confinement factor. For a quantum-well edge emitter $\Gamma$ is of order 0.01 for a single well and about 0.1 for several wells, so the wells must supply 10–100 times the gain that the mode needs. Device measurements (Hakki–Paoli fringes, segmented contacts, cavity-length series) return modal gain; the material gain is inferred by dividing by a calculated $\Gamma$, so its accuracy is limited by that calculation.

As a worked case, a 300 µm Fabry–Pérot chip with cleaved facets of reflectance 0.32 has a mirror loss of $\ln(1/R)/L = 38.0$ cm⁻¹. With an internal loss of 10 cm⁻¹ and $\Gamma = 0.1$, the threshold material gain is

$$g_\text{th} = \frac{38.0 + 10}{0.1} = 480\ \text{cm}^{-1}.$$

Gain versus carrier density

For quantum wells, material gain at the gain peak follows a logarithmic dependence on the carrier density $N$ to good accuracy:

$$g(N) = g_0 \ln\frac{N}{N_\text{tr}},$$

where $N_\text{tr}$ is the transparency density, at which the material neither absorbs nor amplifies, and $g_0$ is a fitted coefficient, typically 1000–3000 cm⁻¹ (see transparency current and threshold current density, which use the same model). With $g_0 = 1500$ cm⁻¹, the 480 cm⁻¹ above requires $N_\text{th} = N_\text{tr}\exp(480/1500) = 1.38\,N_\text{tr}$. Near transparency a linear form, $g = a(N - N_\text{tr})$, is often used instead, with the differential gain $a = dg/dN$ of order $5 \times 10^{-16}$ cm² for InGaAsP quantum wells. The logarithm flattens as $N$ rises, so $dg/dN = g_0/N$ falls, and a laser forced to a high material gain by a small $\Gamma$ or a high loss works at low differential gain, with lower modulation bandwidth and more chirp.

The upper bound on material gain at any photon energy is the absorption of the unpumped material at that energy, reached at complete inversion. For direct-gap III-V semiconductors the band-edge absorption is of order $10^4$ cm⁻¹, which is why gain in the thousands of cm⁻¹ is attainable, while in practice heating, Auger recombination and leakage flatten the gain well before that limit.

For a medium of discrete ions, the material gain is the stimulated-emission cross-section times the inversion density, $g = \sigma\,\Delta N$. With $\sigma = 2.8 \times 10^{-19}$ cm² for Nd:YAG, 0.6 cm⁻¹ requires $\Delta N = 2.1 \times 10^{18}$ cm⁻³, about 1.6% of the neodymium ions in 1 at.% doped crystal.

Spectrum and polarization

Material gain is a function of photon energy. In a semiconductor it is positive between the bandgap and the separation of the quasi-Fermi levels, so the gain spectrum broadens and its peak moves to shorter wavelength as the carrier density rises. In quantum wells it also depends on polarization: heavy-hole transitions favor TE light, and strain is used to adjust the TE and TM material gains.

Pitfalls

• Quoting material gain without stating whether it is peak gain or gain at a fixed wavelength, and at what carrier density or current.
• Comparing material gains extracted with different confinement-factor calculations, since $\Gamma$ for a few-nanometer well depends on the assumed layer indices and thicknesses.
• Mixing natural-log units (cm⁻¹) with dB/cm; $g$ in dB/cm is $4.343\,g$ in cm⁻¹.
• Treating the logarithmic model as physics far from its fitted range; it is an empirical fit to calculated or measured gain curves.

Common questions

What is the difference between material gain and modal gain?

Material gain belongs to the active material alone. Modal gain is what the guided mode sees: $\Gamma g$, smaller by the fraction of the mode that overlaps the active region.

What are typical material gain values for quantum wells?

Several hundred to about 2000 cm⁻¹ at the threshold of a typical laser, with peak values of 3000–5000 cm⁻¹ reported at high injection. The logarithmic fit coefficient $g_0$ is usually 1000–3000 cm⁻¹.

Why do lasers with more quantum wells need less material gain?

Each added well raises $\Gamma$, so the same modal gain is reached at a lower material gain per well, at a point on the logarithmic curve where the differential gain is higher.

References: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012); S. L. Chuang, Physics of Photonic Devices, 2nd ed. (Wiley, 2009); A. E. Siegman, Lasers (University Science Books, 1986).