Analytical formulas for the optical gain of quantum wells

Analytical expressions for the quantized energy levels in quantum wells, the optical gain, the differential optical gain, and the linewidth enhancement factor are presented based on a simple parabolic-band gain model. Explicit formulas show clearly the dependence of these factors on well width, doping, and photon energy. The optical gain in the form of g=g₀ In(N/N₀) is derived using explicit approximations in the Fermi functions, where g₀ is the proportionality constant, N is the injected carrier density, and N ₀ is the transparency carrier density. The approximate formulas are shown to provide not only an efficient way of computing the gain-related parameters but also a convenient way of getting physical insights into the overall interplay of quantum well parameters