Modal gain

For a guided optical mode propagating through a waveguide with an active region producing material gain $g$ (per unit length), the modal gain is

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

where $\Gamma$ is the confinement factor — the fraction of mode intensity overlapping the active region.

Modal gain replaces material gain in all waveguide-system analyses. The threshold condition for a Fabry-Pérot laser, for example:

$$g_\text{modal,th} \;=\; \alpha_i + \alpha_m,$$

where $\alpha_i$ is the internal modal loss (also confinement-factor-weighted) and $\alpha_m$ is the mirror loss per unit length. Solving for the threshold material gain: $g_\text{th} = (\alpha_i + \alpha_m) / \Gamma$.

Material gain dependence on carrier density. For an undegenerate semiconductor active region above transparency, the material gain follows the empirical form:

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

where $N$ is the carrier density, $N_\text{tr}$ is the transparency density (gain = 0), and $g_0$ is a characteristic gain coefficient.

Typical parameters at 1550 nm:

Active region	$g_0$ (cm$^{-1}$)	$N_\text{tr}$ (cm$^{-3}$)
Bulk InGaAsP/InP	$\sim 1200$	$1.2 \times 10^{18}$
InGaAsP/InP MQW	$\sim 1800$	$1.5 \times 10^{18}$
InGaAlAs/InP MQW	$\sim 2000$	$1.2 \times 10^{18}$
Compressively-strained MQW	$\sim 2500$	$1.0 \times 10^{18}$

Modal gain spectrum and lasing wavelength. Material gain has a roughly parabolic spectrum centered at the band-edge offset by carrier-density-dependent renormalization. Modal gain inherits this spectrum shape, weighted by the optical-mode wavelength dispersion. The lasing wavelength of a Fabry-Pérot laser is set by the peak of the modal gain spectrum at the operating carrier density. A DFB laser is designed so the Bragg wavelength sits within the gain bandwidth.

Differential gain. The derivative $dg_\text{modal}/dN$ is the modal differential gain. It directly determines:

• Modulation bandwidth of laser diodes (relaxation oscillation frequency $\omega_R \propto \sqrt{dg/dN}$)
• Linewidth enhancement factor — depends on $dn/dN$ relative to $dg/dN$
• Slope efficiency above threshold (combined with $\Gamma$ and outcoupling)

Higher differential gain is universally desirable in diode laser design. Modern compressively-strained MQW and quantum dot active regions are partly motivated by their higher differential gain compared to bulk DH lasers.

Measurement. Modal gain is extracted from the Hakki-Paoli method (analyzing the modulation depth of Fabry-Pérot fringes in the below-threshold ASE spectrum) or from inverse-length analysis of slope efficiency and threshold across a set of devices with varying cavity lengths (yields both internal modal gain coefficient and $\alpha_i$ simultaneously).