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Physical Review A

Quantum theory of nondegenerate four-wave mixing in semiconductors

A. E. Paul, Markus Lindberg, Sunghyuck An, M. Sargent, S. W. Koch

Published Aug 1, 1990• 15 citations
Publisher

Abstract

A quantum theory of nondegenerate four-wave mixing is developed and applied to study a highly excited semiconductor resonator operating below or near the laser threshold. It is found that total carrier-density pulsations allow coherent effects in semiconductors despite the rapid decay of the electron-hole correlations. The output spectrum for a nondegenerate four-wave mixing intracavity experiment is computed and squeezing in the sidemode spectrum is predicted.

DOI
10.1103/physreva.42.1725

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•Jun 9, 1986•0 citations
A quantum theory of nondegenerate four-wave mixing is developed and applied to study a highly excited semiconductor resonator operating below or near the laser threshold. It is found that total carrier-density pulsations allow coherent effects in semiconductors despite the rapid decay of the electron-hole correlations. The output spectrum for a nondegenerate four-wave mixing intracavity experiment is computed and squeezing in the sidemode spectrum is predicted.
The theory of nondegenerate four-wave mixing (NDFWM) in semiconductor lasers and amplifiers is presented with particular emphasis on the physical processes that lead to population pulsations. In the case of nearly degenerate four-wave mixing, modulation of the carrier density at the beat frequency Ω of the pump and probe waves creates a dynamic population grating whose effectiveness is governed by the spontaneous carrier lifetime τs. Such a grating affects both the gain and the refractive index of the probe wave. In particular, the probe gain exhibits features analogous to those observed in a detuned atomic system arising from the optical Stark effect. Both the gain grating and the index grating contribute to NDFWM, with the dominant contribution coming from the index grating. For detunings such that Ωτs ≫ 1, population pulsations correspond to modulation of the intraband population arising from spectral hole burning. Our results show that NDFWM is then limited by the phase-mismatch effects governed by the transit time τ rather than by the intraband population-relaxation time T1. Significant NDFWM is expected to occur for detunings up to about 300 GHz for typical transit-time values of 3 psec in semiconductor lasers.
A theory of four-wave mixing in semiconductor laser media is developed by considering the contributions of both the gain and index gratings created by the carrier-density modulation occurring at the beat frequency of the pump and the probe waves. The general formalism can be applied to semiconductor lasers operating below or above threshold. As an illustration, we consider the case in which the semiconductor laser is operated as a traveling-wave amplifier. The results show that the dominant contribution to the four-wave mixing process comes from the index grating. Further, the index grating makes the probe transmission asymmetric with respect to the pump-probe detuning.
A theory of degenerate four-wave mixing (DFWM) in the low density excitation limit at the semiconductor band edge is developed on the basis of the bosonic representation of the electron-hole Hamiltonian. By accounting for the exciton-exciton interactions of the lowest order we arrive at the practical few-level density matrix description of the third-order excitonic nonlinearity. The developed theory accounts phenomenologically for the phase space filling, and excitation induced dephasing effects and well reproduces the results of the recent DFWM experiments in high- and low-fineness microcavity. We show that polarization-sensitive four-wave mixing measurements enable to discriminate the contributions to the DFWM signal from the exciton-exciton interactions of different symmetry.