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IEEE Journal of Quantum Electronics

Mean field theory of optical bistability in a phase-conjugate resonator using nearly degenerate four-wave mixing

Kikuo Ujihara

Published Sep 1, 1985• 2 citations
Publisher

Abstract

A mean field theory of nearly-degenerate four-wave mixing is developed for the optical bistability, which has recently been found in a phase-conjugate resonator consisting of a normal mirror and a phase-conjugate mirror under relatively strong pumping. The bistability in transmission of the probe beam occurs as the mirror separation or the probe offset frequency is changed, but for a limited range of probe beam intensity. This behavior is explained as occurring under injection-locked conditions because of combined effects of tilting and broadening, or narrowing of the resonance curves due to nonlinearities in the phase-conjugate reflectivity arising from pump depletion.

DOI
10.1109/jqe.1985.1072811

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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.
An externally driven nonlinear Fabry-Perot interferometer with bistable transmission characteristics is probed through a weak optical field. The nonlinear interaction among the counterpropagating pump beams and the probe beam generates the phase-conjugated beam through intracavity degenerate four-wave mixing. It is shown that the phase-conjugated reflectivity displays bistability and that hysteresis as the driving field is varied in a continuous manner. The analysis is carried out in the mean-field approximation and is valid for the case of low absorption and a high-Qcavity. Both the dispersive and the absorptive effects arising due to saturation of the one-photon resonance are included, and their effect on the phase-conjugated reflectivity is discussed. The results show that, when a bistable device switches on, the reflectivity decreases by a few orders of magnitude. We discuss possible applications and suggest materials where the proposed effects can be seen.
The phase-conjugate reflectivity produced by degenerate four-wave mixing in a Kerr medium in the Bragg scattering regime is shown to be unstable when the intensity of the external pump beams is greater than a threshold value that can be predicted analytically. Below this threshold, a single branch of the reflectivity curve is stable, eliminating any possibility for bistability. A numerical simulation indicates that the unstable regime is characterized by fast oscillations with a period related to the transit time in the nonlinear medium. A finite relaxation time of the nonlinearity eliminates the high-frequency oscillations but can increase the amplitude of the low-frequency components. The insertion of a phase-conjugate mirror into an optical cavity leads to an instability whose threshold can be evaluated analytically. The temporal evolution of the field intensity in the cavity is shown to change from a stable regime to periodic oscillations and then to quasi-periodic oscillations as the pump intensity is increased.
The phase-conjugate mirror realized by means of degenerate four-wave mixing in a Kerr-like medium is investigated using the angular spectrum representation of interacting fields. The analysed model is based on the nonlinear mixing of two infinite plane pump waves with the spatially limited probe and signal waves. Relations among harmonic components of the probe and signal fields are derived assuming non-depleted pumping and neglecting the evanescent waves. The influence of the injected signal field on the frequency spectrum of the generated phase-conjugate field is also examined.
Optical bistability based on the theory of photorefractive nearly degenerate four-wave mixing (DFWM) is considered for the case of transmission geometry with orthogonally polarized pump beams. The coupled wave equations are solved to obtain a nonlinear algebraic equation. This equation is being solved iteratively using Muller's method. The effect of different parameters pump ratio (r), probe ratio (q), coupling strength ( Gamma ) and frequency shift ( Omega ) on the phase conjugate reflectivity (PCR) have been investigated and the results are reported.
Theoretical calculations regarding the collinear intracavity nearly degenerate four-wave mixing are presented when a Fabry-Perot cavity is externally driven by a strong pump and probed by a weak signal. The calculations are performed for two different limits: the mean-field and the small-signal gain approximations. The results suggest the bistable behavior for all the three signals (pump, probe, and conjugate) with several different shapes of the hysteresis loops. Furthermore, for certain values of the parameters in the mean-field approximation, calculations show a resonance behavior of probe and conjugate signals with respect to the incident pump intensity, with probe and conjugate transmittance as high as 103{10}^{3}103--104{10}^{4}104 in the case when the pump transmission does not exhibit bistability. In the small-signal gain approximation, theory also demonstrates strong dependence of probe amplification on the normalized detuning when it is smaller than the probe-cavity mistuning.