Silicon Photonics: Fundamentals and Devices

5.6 Optical Processes in QWRs

The calculation of gain or absorption coefficient may be performed as usual by assuming the k-conservation condition to be valid along the direction of free motion (x-direction). The absorption coefficient is expressed as

(5.70)

where we have dropped the subscript x to the wave vector. The symbol E_g denotes the effective gap which is the bulk band gap plus the electron and hole subband energies. The summation over k may be converted into an integral and, assuming f_e − f_h = 1, the integration may be carried out to yield 29

(5.71)

where

(5.72)

The coefficients A_1D and C_1D have similar meanings to A_mn and C_mn in QWs, and A is the cross-sectional area of the wire. The polarization dependence of the matrix element is discussed in 1.

Eq. (5.71) shows that the absorption coefficient is proportional to the joint density-of-states function. The absorption coefficient shows a singularity at photon energy equal to the effective band gap energy and falls with increasing photon energy. The sharp feature is smoothed out by including the effect of broadening. The expression for the gain is written by replacing the δ-function in Eq. (5.70) by a Lorentzian with an intraband ...