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Physical Review A

Theory of energy exchange and conversion via four-wave mixing in a nondissipative χ(3) material

Boris Kryzhanovsky, Artak Karapetyan, B. Glushko

Published Nov 1, 1991• 12 citations
Publisher

Abstract

Without making the nondepleted pump approximation, we solve the problem of four-wave mixing (FWM) in a nondissipative ${\mathrm{\ensuremath{\chi}}}^{(3)}$ medium. Instead of describing the dynamics of FWM in terms of coupled wave amplitudes, we base our solution on canonical equations that describe the propagation of the field's intensities. This structure clearly identifies the conservative exchange of energy in the FWM process. Consequently, analysis of the FWM process is reduced to a single propagation equation that describes the energy exchange between the pump and amplified waves. This equation yields an elliptic integral solution. The conversion efficiency reduces to the simple analysis of a fourth-order polynomial, which we analyze to determine the conditions for optimization. The destructive influence of the optical Kerr effect on the phase-matching condition is shown to be eliminated by proper choice of nonzero initial wave-vector mismatch, dependent on the input intensity. Furthermore, we show that the conversion efficiency for pulses is limited by the splitting of the pulses into a set of subpulses.

DOI
10.1103/physreva.44.6036

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Without making the nondepleted pump approximation, we solve the problem of four-wave mixing (FWM) in a nondissipative \ensuremathχ(3){\mathrm{\ensuremath{\chi}}}^{(3)}\ensuremathχ(3) medium. Instead of describing the dynamics of FWM in terms of coupled wave amplitudes, we base our solution on canonical equations that describe the propagation of the field's intensities. This structure clearly identifies the conservative exchange of energy in the FWM process. Consequently, analysis of the FWM process is reduced to a single propagation equation that describes the energy exchange between the pump and amplified waves. This equation yields an elliptic integral solution. The conversion efficiency reduces to the simple analysis of a fourth-order polynomial, which we analyze to determine the conditions for optimization. The destructive influence of the optical Kerr effect on the phase-matching condition is shown to be eliminated by proper choice of nonzero initial wave-vector mismatch, dependent on the input intensity. Furthermore, we show that the conversion efficiency for pulses is limited by the splitting of the pulses into a set of subpulses.
Without making the nondepleted-pump approximation, we solve the problem of four-wave mixing (FWM) in a nondissipative ${\mathrm{\ensuremath{\chi}}}^{(3)}$ medium. Instead of describing the dynamics of FWM in terms of coupled wave amplitudes, we base our solution on canonical equations that describe the propagation of the field's intensities. This structure clearly identifies the conservative exchange of energy in the FWM process. Consequently, analysis of the FWM process is reduced to a single propagation equation that describes the energy exchange between the pump and amplified waves. This equation yields an elliptic-integral solution. The conversion efficiency reduces to the simple analysis of a fourth-order polynomial, which we analyze to determine the conditions for optimization. The destructive influence of the optical Kerr effect on the phase-matching condition is shown to be eliminated by proper choice of nonzero initial wave-vector mismatch, dependent on the input intensity. The effect of the nonuniform transverse pump-beam intensity profile on the process of conversion is considered. The direction of the energy exchange is shown to be a periodic function of the propagation distance. The transverse intensity distribution of the generating waves provides the characteristic spatial structure (the set of coaxial rings with or without the central spot; the number of rings is explicitly determined by the length of interaction). Several methods of process optimization are discussed, and nonuniformity in the pump-beam profile is shown to be the main reason that complete energy transfer is not achieved.
The exact solution of the self-consistent problem of four-wave mixing in a resonant medium of four-level atoms (molecules) is obtained. The pump depletion, level-population movement, and atomic-transition saturation are explicitly taken into account. It is shown that (i) in the process of transformation the generated pulse is split into a train of subpulses, and their number is strongly related to the length of propagation; in the case of saturation, this splitting does not occur; (ii) the conversion-efficiency definition problem is reduced to solution of a third-order polynomial; (iii) by variation of the initial value of the wave-vector mismatch \ensuremath{\Delta}k0{\mathit{k}}_{0}k0​, the processes of conversion may lead to the complete depletion of one of the pump beams; (iv) a prediction is made of the intersection of quasienergetic terms in the process of energy transfer; this phenomenon is shown to be responsible for the experimentally observed disruption of generation.
Four-wave coherent mixing models for two quasi-monochromatic pumping fields and pulses of a two-component Stokes field with an elliptic polarization and a duration on the order of the period of oscillations have been derived for a two-level medium with a forbidden dipole transition. It is shown that, under the unidirectional wave propagation conditions and in the absence of depletion of pumping, the system of Maxwell-Bloch equations can be reduced to a new completely integrable system of equations. Nonsoliton radiation dynamics of generation of Stokes field pulses is studied in the framework of the integrable reduction of this model. The apparatus for the inverse problem algorithm corresponding to the solvable problem is developed. An approximate asymptotic expression for the leading front of the pulse packet being generated is obtained for various initial and boundary conditions. The application of these results for describing parametric processes involving various types of waves is discussed.
This thesis describes the theoretical solution and experimental verification of phase conjugation via nondegenerate four-wave mixing in resonant media. The theoretical work models the resonant medium as a two-level atomic system with the lower state of the system being the ground state of the atom. Working initially with an ensemble of stationary atoms, the density matrix equations are solved by third-order perturbation theory in the presence of the four applied electro-magnetic fields which are assumed to be nearly resonant with the atomic transition. Two of the applied fields are assumed to be non-depleted counterpropagating pump waves while the third wave is an incident signal wave. The fourth wave is the phase conjugate wave which is generated by the interaction of the three previous waves with the nonlinear medium. The solution of the density matrix equations gives the local polarization of the atom. The polarization is used in Maxwell's equations as a source term to solve for the propagation and generation of the signal wave and phase conjugate wave through the nonlinear medium. Studying the dependence of the phase conjugate signal on the various parameters such as frequency, we show how an ultrahigh-Q isotropically sensitive optical filter can be constructed using the phase conjugation process. In many cases the pump waves may saturate the resonant medium so we also present another solution to the density matrix equations which is correct to all orders in the amplitude of the pump waves since the third-order solution is correct only to first-order in each of the field amplitudes. In the saturated regime, we predict several new phenomena associated with degenerate four-wave mixing and also describe the ac Stark effect and how it modifies the frequency response of the filtering process. We also show how a narrow bandwidth optical filter with an efficiency greater than unity can be constructed. In many atomic systems the atoms are moving at significant velocities such that the Doppler linewidth of the system is larger than the homogeneous linewidth. The latter linewidth dominates the response of the ensemble of stationary atoms. To better understand this case the density matrix equations are solved to third-order by perturbation theory for an atom of velocity v. The solution for the polarization is then integrated over the velocity distribution of the macroscopic system which is assumed to be a gaussian distribution of velocities since that is an excellent model of many real systems. Using the Doppler broadened system, we explain how a tunable optical filter can be constructed whose bandwidth is limited by the homogeneous linewidth of the atom while the tuning range of the filter extends over the entire Doppler profile. Since it is a resonant system, sodium vapor is used as the nonlinear medium in our experiments. The relevant properties of sodium are discussed in great detail. In particular, the wavefunctions of the 3S and 3P states are analyzed and a discussion of how the 3S-3P transition models a two-level system is given. Using sodium as the nonlinear medium we demonstrate an ultrahigh-Q optical filter using phase conjugation via nondegenerate four-wave mixing as the filtering process. The filter has a FWHM bandwidth of 41 MHz and a maximum efficiency of 4 x 10-3. However, our theoretical work and other experimental work with sodium suggest that an efficient filter with both gain and a narrower bandwidth should be quite feasible.
This paper presents an exact solution to the slowly varying envelope equations for the probe and signal fluxes in degenerate four-wave mixing in Kerr media. The analysis is performed for a noncollinear geometry, and all beams have the same linear polarization in the nonlinear medium. It is assumed that the angle separating the directions of the counterpropagating pumps and probe and signal beams is large enough so that cross-coupled waves do not contribute to the envelope equations describing the spatial evolution of the pump, probe, and signal beams. Multiple values of the transmitted probe are obtained for equal pump fluxes above a threshold determined by the nonlinear phase shift in the medium. Significant pump depletion occurs in this regime, but it can also occur without the appearance of multiple solutions. Since exact optical phase conjugation occurs only in the limit where the pumps are undepleted in Kerr media, this result implies that there are fundamental upper limits on the pump intensity that can be used to increase signal reflectivities in optical phase conjugation.
The theory of collinear degenerate four-wave mixing in a lossless, isotropic Kerr medium is described and solved numerically. Multiple solutions to the equations governing the spatial evolution of the probe and signal fluxes are found when the input fluxes exceed a critical intensity determined by the electric susceptibility and the interaction length of the medium. This treatment includes all phase-matched contributions to the equations of motion of all fields, including the cross-coupled waves appearing in the collinear geometry.