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IEEE Journal of Quantum Electronics

Phase conjugation and degenerate four-wave mixing in three-level systems

Govind Prasad Agrawal

Published Dec 1, 1981• 18 citations
Publisher

Abstract

Considers phase conjugation through degenerate four-wave mixing in a Λ-type three-level system under certain simplifying assumptions, the most restrictive among them being the assumption of symmetrical detuning of the pump from the two lower lying sublevels. When compared to an absorbing resonant two-level system, it is found that significantly higher values of the phase-conjugate reflectivity are possible at pump intensities well below saturation provided the sublevel splitting is less than a homogeneous linewidth. The physical mechanism responsible for phase conjugation is coherent population trapping arising from two-photon-induced atomic coherence between the two sublevels.

DOI
10.1109/jqe.1981.1071047

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Considers phase conjugation through degenerate four-wave mixing in a Λ-type three-level system under certain simplifying assumptions, the most restrictive among them being the assumption of symmetrical detuning of the pump from the two lower lying sublevels. When compared to an absorbing resonant two-level system, it is found that significantly higher values of the phase-conjugate reflectivity are possible at pump intensities well below saturation provided the sublevel splitting is less than a homogeneous linewidth. The physical mechanism responsible for phase conjugation is coherent population trapping arising from two-photon-induced atomic coherence between the two sublevels.
The effect of Zeeman coherence on phase conjugation is investigated in the saturation regime by using a nonperturbative analysis of resonant degenerate four-wave mixing with orthogonally polarized pump beams in a A-type homogeneously broadened three-level system. The dependence of the phase-conjugation efficiency on phase-interrupting collisions and pump imbalance is particularly studied. Significant phase-conjugate reflectivities can be obtained at pump intensities much below those required for a two-level system.
Nondegenerate four-wave mixing on the j=0\ensuremath{\rightarrow}j=1 transition, in a ``V '' system, using orthogonal, circularly polarized pump beams is used to study theoretically the generation of the phase-conjugate signals. The efficiency of generation of the conjugate wave is studied as a function of the various system parameters such as pump-probe detuning, Zeeman splitting, pump-atom detuning, collisional parameters, and the intensities of the pump beams. The origin of the various resonances in the reflectivity is explained. For weak pump fields, some resonances are shown to have subnatural linewidths. In the degenerate case, the reflectivity as a function of pump-atom detuning has, in contrast to a two-level system, a doublet structure for moderate absorption-length parameters and field strengths. The reflectivity is shown to have additional resonances due to pump imbalance. The phase-conjugate signal is also shown to exhibit the pressure-induced extra resonances.
Current experimental studies of degenerate four-wave mixing in saturable absorbers1,2 have shown that the dependence of the reflectivity on the pump intensity exhibits a double maxima. This phenomenon can arise from the fact that the medium approximates a set of four-level quantum systems. We have carried out a detailed analysis of the nonlinear processes of degenerate four-wave mixing in a four-level system. Because resonant saturable absorbers are being considered for systems applications, e.g., optical phase conjugation and wide FOV filters, the results presented are relevant to various optical phase conjugation devices. There are two distinct regimes which are of interest. The first case is when the intramolecular relaxation is very weak or absent. The second is when the intramolecular relaxation is strong. For the weak case the medium behaves like a set of two distinct dipole radiation sources which can generate interference effects.
The use of resonantly enhanced four-wave mixing to generate high reflectivity phase-conjugate waves is discussed. A physical understanding of the mechanisms underlying the nonlinear response for a simple two-level system is presented followed by a discussion of the complicating effects of pump absorption, pump depletion, unbalanced pumps, and thermal motion. Modifications of the theory to describe four-level behavior are presented and compared with experiments. The application of nearly degenerate four-wave mixing to narrow-band filtering complicated by the A. C. Stark effect is dis-cussed along with examples of experimental observations.
An analytic nonperturbative treatment of degenerate four-wave mixing is presented to study Zeeman-coherence effects in phase conjugation. The counterpropagating pump waves are taken to be orthogonally circularly polarized and interact with different Zeeman sublevels. The nonlinear medium is modeled in terms of a homogeneously broadened, folded three-level system. Particular attention is paid to the phase-conjugate spectrum obtained by varying the Larmor frequency associated with the Zeeman splitting. A level-crossing resonance is found to occur in the saturation regime and manifests itself as a narrow peak or dip depending on various level decay rates. Even for a purely absorptive system, the Zeeman coupling of the ground-state degenerate sublevels is shown to produce significant phase-conjugate reflectivities at pump intensities much lower than those required for a two-level system.
A general formalism is developed to study intracavity phase conjugation through resonant degenerate four-wave mixing. The effects of spatial hole burning, arising from interference of the counterpropagating pump waves, are fully incorporated. The analysis is applied to discuss intracavity phase conjugation in nonlinear media that can be modeled using either a two-level system or a Λ-type three-level system. In both cases, the phase-conjugate reflectivity exhibits bistability for an appropriate choice of the input parameters. For a two-level system, the tuning characteristics of the Fabry–Perot cavity significantly affect the phase-conjugate spectrum that displays hysteresis when the cavity is detuned from the applied laser frequency.