Theoretical analysis of nonlinear optical phenomena taking into account the beating vibration of the electron density in semiconductor lasers

Nonlinear optical phenomena in semiconductor lasers are induced by two types of mechanisms. The first mechanism is beating vibration (i.e., pulsing modulation) on spectral distributions of injected carriers (i.e., electrons and holes) due to lasing frequencies, which is observed as the spectral hole burning effect and whose relaxation is characterized with the intraband relaxation time on the order of 10−13 s. This effect occurs even when the numbers of injected carriers are constant. The second mechanism is beating vibration on the number of injected carriers, whose relaxation is characterized with the electron lifetime on the order of 10−9 s due to band-to-band transition. Master equations including these two nonlinear phenomena are obtained in this paper in a general form based on the density matrix formalism. As examples of the equations, characteristics of the saturated gain profile in a laser oscillator and the four-wave mixing effect (or generation of the phase-conjugate wave) in a traveling-wave laser amplifier are analyzed. A saturated gain profile on optical frequency in a laser oscillator contains both symmetric and asymmetric properties induced by the first and the second mechanisms, respectively, where the strengths of these properties are almost the same. While in the case of the four-wave mixing effect in a traveling-wave laser amplifier, the second mechanism is more effective than the first mechanism. These results coincide well with previous analyses by other authors which treated the above-mentioned phenomena individually.