Absorption coefficient

The absorption coefficient $\alpha$ is defined by the exponential decay of power in an absorbing medium, $P(z) = P_0 e^{-\alpha z}$, the Beer-Lambert law. Its inverse, $1/\alpha$, is the absorption depth, the distance over which power falls to $1/e$ (37%). Converted to decibels, a coefficient of 1 cm⁻¹ is 4.34 dB/cm, the same conversion used for waveguide propagation loss, where $\alpha$ lumps scattering together with absorption. The coefficient is tied to the imaginary part $\kappa$ of the complex refractive index $n + i\kappa$ by $\alpha = 4\pi\kappa/\lambda$.

In semiconductors $\alpha$ follows the bandgap. Photons with energy below the gap pass through nearly unabsorbed; above it, absorption rises steeply in a direct-gap material such as InGaAs or GaAs, reaching $10^4$ cm⁻¹ and more within a small fraction of an electron-volt of the edge. In an indirect-gap material such as silicon or germanium, absorption near the edge needs a phonon to conserve momentum and rises more gradually, so thin layers are poor absorbers at wavelengths just above the gap. Germanium's direct transition lies close to its indirect gap, which is what makes germanium-on-silicon photodetectors usable across the telecom bands, with their response weakening at the long-wavelength end of the L band. Below the gap, free-carrier absorption and defect absorption set a small residual $\alpha$ that matters for waveguides and cavities rather than detectors.

For a detector the coefficient sets the thickness. A photodiode whose absorbing layer has thickness $W$ absorbs the fraction $1 - e^{-\alpha W}$ of the light that enters it, so the internal quantum efficiency of a surface-illuminated device is bounded by that fraction.

The thicker layer that a weak absorber needs costs speed, because carriers take longer to cross it, which is the bandwidth-efficiency trade of surface-normal photodiodes. Waveguide photodetectors escape it by absorbing along the direction of propagation: light travels tens of micrometers through a thin absorber, collecting the full $\alpha L$ while the carriers cross only the thin dimension. Near an absorption edge $\alpha$ also depends on temperature and doping, because the gap shifts with both, so detector responsivity at the edge of a material's range is a temperature-sensitive specification.

References: S. M. Sze and K. K. Ng, Physics of Semiconductor Devices, 3rd ed. (Wiley, 2007); B. E. A. Saleh and M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).