Modal gain

Modal gain is the gain per unit length that a guided optical mode actually experiences in a laser or amplifier waveguide. Only part of the mode travels inside the active layer, the fraction $\mathrm{\Gamma }$ given by the confinement factor, so the modal gain is the material gain $g$ of the active region scaled by that fraction:

$$g_{\text{modal}}=\mathrm{\Gamma }g\mathrm{.}$$

In a laser with five quantum wells, $\mathrm{\Gamma }$ is about 0.1, so the wells must supply ten times the gain the mode needs. Modal gain is the quantity that enters the threshold condition of a laser and the gain of a semiconductor optical amplifier, and it is what measurements on finished devices return; the material gain is inferred from it through $\mathrm{\Gamma }$.

Threshold condition

A Fabry-Perot laser reaches threshold when the modal gain replaces what is lost in a round trip:

$$\mathrm{\Gamma }{g}_{\text{th}}={\alpha }_i+{\alpha }_m,{\alpha }_m=\frac{1}{2L}\ln \frac{1}{R_1{R}_2},$$

where ${\alpha }_i$ is the internal modal loss, from free-carrier absorption and scattering, and ${\alpha }_m$ is the mirror loss spread over the cavity length $L$. For a chip with cleaved facets of reflectance 0.3, an internal loss of 10 cm⁻¹ and $\mathrm{\Gamma }=0.1$:

Short cavities need more gain because their mirror loss is spread over less length. In an amplifier, which has no mirrors, the net modal gain $\mathrm{\Gamma }g-{\alpha }_i$ sets the single-pass gain, $G=\exp [(\mathrm{\Gamma }g-{\alpha }_i)L]$; a 1 mm SOA with 25 dB of gain has a net modal gain of 57.6 cm⁻¹.

Material gain and carrier density

The material gain of a semiconductor rises with the carrier density $N$ once the density passes transparency, $N_{\text{tr}}$, where the material neither absorbs nor amplifies. For quantum wells a logarithmic fit describes it well:

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

with $g_0$ a fitted gain coefficient. The threshold density follows as $N_{\text{th}}=N_{\text{tr}}\exp (g_{\text{th}}\mathrm{/}{g}_0)$; taking $g_0=1500$ cm⁻¹ for illustration, the three lasers above reach threshold at 1.22, 1.40 and 1.83 times the transparency density, longest cavity first.

The logarithm flattens as $N$ rises, so the differential gain $dg\mathrm{/}dN=g_0\mathrm{/}N$ falls: in this example the 150 µm laser works at 23% lower differential gain than the 300 µm laser. Differential gain sets the relaxation oscillation frequency and so the modulation bandwidth, and it enters the linewidth enhancement factor, which is why designs that keep the threshold gain low, with more wells, a larger $\mathrm{\Gamma }$ or lower losses, are also faster and chirp less. The confinement factor entry discusses the trade-off in the number of wells.

Gain spectrum and polarization

Gain exists for photon energies between the bandgap and the separation of the quasi-Fermi levels, so the gain spectrum widens toward shorter wavelengths as the carrier density rises, and its peak moves the same way. Heating shrinks the bandgap and moves the peak to longer wavelengths, by 0.3 to 0.5 nm/K in InP-based lasers. A Fabry-Perot laser lases near the peak of its modal gain spectrum, and a DFB laser at the grating's wavelength, which is placed within that spectrum.

The modal gain also depends on polarization. In unstrained and compressively strained quantum wells the transitions involving heavy holes favor TE-polarized light, so the TE modal gain exceeds the TM modal gain; lasers exploit this to run in TE, while polarization-insensitive amplifiers add tensile strain or shape the waveguide until ${\mathrm{\Gamma }}_{\text{TE}}{g}_{\text{TE}}={\mathrm{\Gamma }}_{\text{TM}}{g}_{\text{TM}}$.

Measuring modal gain

• Hakki-Paoli method. Below threshold, the spontaneous emission spectrum of a Fabry-Perot chip carries fringes from its cavity modes. Their peak-to-valley ratio $r$ gives the net modal gain at each fringe, $\mathrm{\Gamma }g-{\alpha }_i=\frac{1}{L}\ln \frac{\sqrt{r}-1}{\sqrt{r}+1}+\frac{1}{2L}\ln \frac{1}{R_1{R}_2}$, so the gain spectrum is traced as the current rises. The method needs a spectrometer that resolves the fringes fully.
• Segmented contact method. A waveguide divided into separately contacted sections is driven one or two sections at a time, and the amplified spontaneous emission from lengths $L$ and $2L$ gives the net modal gain and the internal loss without any mirrors.
• Cavity-length series. Thresholds of lasers cut to several lengths, fitted against the mirror loss, give $g_0$ and the transparency current density; the slope efficiencies of the same set give ${\alpha }_i$ and the internal quantum efficiency.

Common questions

What is the difference between modal gain and material gain?

Material gain is the gain of the active material itself, per unit length of light traveling entirely within it. Modal gain is the gain of the guided mode, only part of which overlaps the active region: $g_{\text{modal}}=\mathrm{\Gamma }g$.

What modal gain does a laser need to reach threshold?

Enough to equal the internal loss plus the mirror loss, $\mathrm{\Gamma }{g}_{\text{th}}={\alpha }_i+{\alpha }_m$. A 300 µm chip with cleaved facets and 10 cm⁻¹ of internal loss needs about 50 cm⁻¹.

Why is modal gain smaller than material gain?

Because the confinement factor is less than one: the mode extends into the undoped waveguide and cladding layers, which do not amplify. Thin quantum wells confine only a few percent of the mode each.

References: L. A. Coldren, S. W. Corzine and M. L. Mašanović, Diode Lasers and Photonic Integrated Circuits, 2nd ed. (Wiley, 2012). B. W. Hakki and T. L. Paoli, "Gain spectra in GaAs double-heterostructure injection lasers," Journal of Applied Physics 46, 1299 (1975). P. Blood, G. M. Lewis, P. M. Smowton, H. Summers, J. Thomson and J. Lutti, "Characterization of semiconductor laser gain media by the segmented contact method," IEEE Journal of Selected Topics in Quantum Electronics 9, 1275 (2003).