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IEEE Journal of Quantum Electronics

Theory of mode hopping and phase conjugation characteristics in degenerate four-wave mixing

K. Tajima, Hsiung Hsu

Published Sep 1, 1988• 0 citations
Publisher

Abstract

A large-signal theory of noncollinear degenerate four-wave mixing (DFWM) is presented, taking into account the pump depletion and the full nonlinear polarization density appropriate for the noncollinear geometry. Phases of all of the interacting fields, as well as the amplitudes, are obtained as functions of the spatial coordinate. Owing to the completeness of the set of analytic solutions, considerable physical insights into the physics of DFWM are obtained. The DFWM exhibits a variety of rich behaviors, including hysteresis and bistability, multistability, and multiple isolated branches. The authors show that all of these behaviors can be explained in a unified way, that is, in terms of mode hopping characteristics.>

DOI
10.1109/3.7132

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Physical Review A•Nov 1, 1989•52 citations
A large-signal theory of noncollinear degenerate four-wave mixing (DFWM) is presented, taking into account the pump depletion and the full nonlinear polarization density appropriate for the noncollinear geometry. Phases of all of the interacting fields, as well as the amplitudes, are obtained as functions of the spatial coordinate. Owing to the completeness of the set of analytic solutions, considerable physical insights into the physics of DFWM are obtained. The DFWM exhibits a variety of rich behaviors, including hysteresis and bistability, multistability, and multiple isolated branches. The authors show that all of these behaviors can be explained in a unified way, that is, in terms of mode hopping characteristics.>
Large signal analyses for degenerate four-wave mixing (DFWM) have been reported for sometime(1-3). It was nevertheless not obvious why, in specific examples(1,2). DFWM exhibits hysteresis and bistability characteristics, whereas in the case of ideal phase conjugation(3), there is no such instability. The objective of this paper is to show that hysteresis and bistability are caused by phase variations in the interaction. These phase variations are detrimental to phase conjugation and may Indeed up-set the basic characteristies of phase conjugation.
The exact theory of degenerate four-wave mixing in a Kerr-like medium in the Bragg scattering regime is used to study the reflectivity of phase-conjugate mirrors pumped by intense laser beams. The analysis takes into account pump depletion, nonlinear phase modulation and the relaxation time of the nonlinearity. The steady-state response exhibits multivalued reflectivity, hysteresis and potential bistability; however, the transient response shows that at most one value of the phase-conjugate reflectivity is stable. Periodic or chaotic pulsations are predicted beyond threshold values of pump intensity. The onset of a natural oscillation in the Kerr-like medium is also predicted in the absence of a continuous probe beam; the resulting field amplitude is stable and weakly sensitive upon the presence of external noise.
Co-polarised degenerate four-wave mixing (DFWM), in which all the four interacting beams have the same direction of polarisation, has been utilised to generate high phase conjugate reflectivities (~ 2500) recently [1]. In the co-polarised geometries six gratings are written in the material in which each beam forms a grating with another beam. Several theoretical models have been used to describe this co-polarised DFWM process in gain media [2]-[3]. However, all of them involve the use of small probe approximation and hence not all gratings are present in these models. Moreover, the effect of pump depletion has been neglected. A novel approach using the underlying symmetry is described recently to solve the co-polarised DFWM problem [4]. This approach allows us to take into account the effect of pump depletion, although the two standing gratings between the pumps and between the probe and the conjugate beams are ignored.
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 revision of the theory of finite laser bandwidth effects on degenerate four-wave mixing (DFWM) of Alber, Cooper, and Ewart (ACE) [Phys. Rev. A 31, 2344 (1985)]. The same model is used for intense, broad-bandwidth pump lasers interacting with a weak monochromatic probe in a medium composed of two-level atoms. The density-matrix equations describing the time evolution of the atomic polarization coupled to fluctuating fields are solved using an appropriate decorrelation approximation. A steady-state analytical solution is found for resonant and near-resonant DFWM, for intensities that do not saturate the medium. For more intense fields it is found to be necessary to form a more complicated set of differential equations for all products of the field and atomic density-matrix elements. The results show that as in the ACE theory, increasing the bandwidth b leads to an increased effective saturation intensity. However, we find the DFWM reflectivity scales as 1/b2{b}^{2}b2 when the probe is monochromatic and not 1/b as in the ACE theory. Furthermore, the saturation behavior of nonresonant DFWM is found to differ from the predictions of ACE. Atomic motion effects are shown to yield a Doppler-broadened line shape and also to affect the saturation behavior of the signal.