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Physical Review A

Exact theory of the four-wave-mixing process in a nondissipative medium with a large rate of conversion: Weak-field case

Boris Kryzhanovsky, B. Glushko

Published Apr 1, 1992• 18 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. 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.

DOI
10.1103/physreva.45.4979

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Symmetry•Feb 2, 2022•2 citations
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. 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.
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.
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.
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.
We present a theoretical treatment of four-wave mixing (FWM) in a Brillouin-active medium for the case in which the pump waves differ in frequency by approximately twice the Brillouin frequency shift of the medium and in which the probe-wave frequency is approximately the arithmetic mean of the frequencies of the two pump waves. Under these conditions, the conjugate wave produced by the FWM process has the desirable property of being at the same frequency as the probe. We derive the coupled amplitude equations describing this interaction. We solve these equations analytically in the limit of negligible pump depletion and find that large phase conjugate reflectivities are readily achievable. The coupled amplitude equations are solved numerically for the general case, and it is found that large power transfer from the pumps to the output wave is possible. The output wave is shown to be a nearly perfect phase conjugate of the probe wave, even far into the regime where pump depletion effects are important. Our formalism predicts the existence of a parametric instability in the propagation of the pump waves, but good performance is predicted before the onset of this instability.
A strong pump-power dependence of the four-wave mixing (FWM) signal for an aqueous solution of Malachite green is reported. The characteristics of the pump-power dependence of the nonlinear signal are reproduced by a theoretical model based on the coupling between pump-probe, considering signal fields and propagation effects. The effect of the intramolecular coupling on the nonlinear intensity of the FWM signal is studied using a model molecule consisting of two-coupled harmonic curves of electronic energies with minima displaced in energy and nuclear positions. Two-vibrational states are considered while including non-adiabatic effects for the two-state model. Moreover, the coupling among the field components, as well as the propagation effects, are studied by considering a constant pump-intensity. Our calculation scheme, considering both the intramolecular coupling effects in the description of the molecular structure and the effects produced by the propagation of the FWM signal along the optical length, allows the exponential dependence of the latter, as the intensity of the pumping beam increases. Our treatments do not require the inclusion of other non-resonant processes outside the RWA approximation, due to the consideration of an adiabatic basis.