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Physical Review A

Theoretical description of two-photon phase conjugation in polar molecules

Claudine Katan, J. P. Lavoine, A. A. Villaeys

Published Aug 1, 1993• 18 citations
Publisher

Abstract

Degenerate four-wave-mixing phase conjugation is analyzed in the case of two-photon transitions between levels having unequal dipole moments. The theoretical treatment is based upon a nonperturbative solution of the density matrix and does not require the rotating-wave approximation. We give an analytical expression for the polarization component oscillating at the field frequency for a medium excited by stationary waves. In the weak-probe-beam limit, the contributions in the direction of the probe and conjugated beams are evaluated. For instance, by using this two-photon process, instead of the usual single-photon one which is independent of the permanent dipole moments even in the strong-field limit, it is shown that lower intensities are sufficient to reach higher reflectivities. The influences of the permanent dipole moments, relaxation rates, and field frequency are also discussed.

DOI
10.1103/physreva.48.1564

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Degenerate four-wave-mixing phase conjugation is analyzed in the case of two-photon transitions between levels having unequal dipole moments. The theoretical treatment is based upon a nonperturbative solution of the density matrix and does not require the rotating-wave approximation. We give an analytical expression for the polarization component oscillating at the field frequency for a medium excited by stationary waves. In the weak-probe-beam limit, the contributions in the direction of the probe and conjugated beams are evaluated. For instance, by using this two-photon process, instead of the usual single-photon one which is independent of the permanent dipole moments even in the strong-field limit, it is shown that lower intensities are sufficient to reach higher reflectivities. The influences of the permanent dipole moments, relaxation rates, and field frequency are also discussed.
The main goal of this paper is to analyze a two-photon degenerate four-wave-mixing process that occurs in an isotropic media modeled by homogeneously broadened two-level systems having unequal permanent dipole moments and subject to rotational diffusion. Our theoretical treatment is based upon a third-order perturbative solution of the density-matrix equations and does not require the rotating-wave approximation. The medium being excited by three stationary fields, we give an analytical expression of the macroscopic polarization up to the third order, including permanent dipole moments, orientational diffusion, and field polarizations. We apply these general results to the description of a degenerate four-wave-mixing process by selecting, in the expressions, the contributions in the direction ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$-${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$+${\mathbf{k}}_{\ensuremath{\gamma}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathbf{k}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathbf{k}}_{\ensuremath{\gamma}}$ being, respectively, the wave vectors of the exciting fields. We show that the two-photon degenerate four-wave mixing is dependent on the permanent dipole moments and is not present if these dipole moments are equal or zero. We analyze the influence of rotational diffusion. Different cases of field polarizations are also considered, and we discuss a comparison of the well-known one-photon degenerate four-wave-mixing process and this two-photon process. This process is also discussed in the particular case of phase conjugation.
We develop a semiclassical theory of the polarization properties of phase conjugation by two-photon resonant degenerate four-wave mixing. The theory includes the effects of saturation by the pump waves. We solve the density-matrix equations of motion in steady state for a nonlinear medium consisting of stationary atoms with a ground and excited state connected by two-photon transitions. As an illustration of the general results, we consider an S0{S}_{0}S0​\ensuremath{\rightarrow}S0{S}_{0}S0​ two-photon transition, which is known to lead to perfect polarization conjugation in the limit of third-order theory. We show that the fidelity of the polarization-conjugation process is degraded for excessively large pump intensities. The degradation can occur both due to transfer of population to the excited state and due to nonresonant Stark shifts. Theoretical results are compared to those of a recent experiment [Malcuit, Gauthier, and Boyd, Opt. Lett. 13, 663 (1988)].
Properties of the resonance peaks associated with the non-degenerate four-wave mixing signal in a two-level molecular system with non-zero permanent dipole moments are studied in the space defined by the pump ( omega 1 ) and probe ( omega 2 ) frequencies. An analytical expression for the Fourier components of the nonlinear homogeneous macroscopic polarization are obtained, where the terms responsible for the oscillation (at frequency omega 3 =2 omega 1 - omega 2 ) of the populations and coherences are also identified. Topological properties of the resulting nonlinear spectra are studied in frequency space using three-dimensional curves and contour plots. We have additionally identified the dominant contributions of the coherence and/or population that determine the intensity and the spectral linewidth. The critical quantities for the present analysis are the transition and permanent dipole moments when the rotating-wave approximation (RWA) is not considered.
We present in this paper a theoretical study of the angular dependence of phase-conjugate emission via resonant degenerate four-wave mixing (FWM) in Doppler-broadened gas media. The nonlinear optical polarization is calculated through an expansion of the atomic density matrix up to the third order in the incident fields. One finds that, when the incident frequency is resonant for a single-photon transition, both strength and lineshape of the phase-conjugate emission strongly depend on the angle 0 between standing pump wave and probe wave. The emission intensity, maximum at grazing incidences, decreases rapidly by several orders of magnitude when 0 increases. Simultaneously the emission linewidth increases from homogeneous to Doppler width. Similar effects are predicted for nearly-degenerate FWM (different pump and probe frequencies). In particular, one shows that the bandwidth of the equivalent optical reflection filter undergoes a Doppler-broadening linearly increasing with sin θ. On the opposite, none of the above effects is predicted in two-photon-resonant FWM which does not exhibit any angular dependence.
The theory of polarised light phase conjugation (PC) in non-degenerate four-wave mixing in a resonant medium has been developed in the irreducible tensorial formalism. The theory is applied to a wave with any polarisation in a resonant medium with levels of arbitrary momentum in the absence of pump-induced saturation effects. The formula for the non-linear susceptibility of the medium has been found and it is shown that the change of the polarisation at PC is essentially connected with the relaxation of the atomic multipole momenta (population, orientation and alignment). Wave equations were investigated for different pump-wave polarisations and three different physical mechanisms of the PC are discussed. The influence of the different relaxation rates of the atomic multipole momenta on the PC reflectivity has been investigated. The spectral dependence of the reflectivity and the spectral ranges which depend on the wave polarisation are found. For motionless atoms with small momenta (J 1 = 1 / 2 to J 2 = 1 / 2 , J 1 =1 to J 2 =0), exact formulae for the non-linear susceptibility of the medium as a function of pump intensity were found. The spontaneous decay of higher level to the lower one was taken into account. Because of optical pumping, saturation effects can be shown at considerably smaller intensities than in the usual two-level system. The exact expressions for the PC reflectivity at different polarisations of the pump waves are given.