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Physical Review A

Topological nature of four-wave-mixing polarization bistability

Junji Yumoto, Kenju Otsuka

Published Nov 1, 1986• 7 citations
Publisher

Abstract

A novel four-wave-mixing polarization bistability is predicted in a system consisting of an anisotropic crystal and one partial reflection mirror (retroreflection configuration). The polarization input-output characteristics at constant input power are shown to exhibit not only conventional ``S'' but also ``seesaw''- and ``isola''-type bistability. The global topological nature of this polarization bistability is depicted in the entire phase space in terms of the polarization state of light.

DOI
10.1103/physreva.34.4445

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Journal of the Optical Society of America B•Oct 1, 1996•2 citations
A novel four-wave-mixing polarization bistability is predicted in a system consisting of an anisotropic crystal and one partial reflection mirror (retroreflection configuration). The polarization input-output characteristics at constant input power are shown to exhibit not only conventional S'' but also seesaw''- and ``isola''-type bistability. The global topological nature of this polarization bistability is depicted in the entire phase space in terms of the polarization state of light.
Optical bistability based on the cross-interaction of two counterpropagating plane waves in a Kerr-like medium is considered. It is shown that the "input-output" characteristics of such a system have not only multivalued hysteretic response (which was demonstrated by Winful and Marburger in 1980), but also multiple isolated branches (so called isolas) with nonhysteretic jumps from these branches to the main curve (but not vice versa). The isolas were found both in the case in which the problem can be solved analytically (particularly for electrostriction mechanism of optical nonlinearity) and in the general case of arbitrary light-induced anisotropy.
Collinear degenerated four-wave mixing can demonstrate optical bistability with hysteretic response.1 For this system, we predict the existence of multistable isolated nonhysteretic branches of solution (so-called isolas). We consider two optical beams with the same frequency counterpropagating along the z axis in a Kerrlike medium. Their total electric field is E = [E1 exp (ikz) + E2 exp(−ikz)] exp(−iωt), where E1 and E2 are the vectorial envelope of forward and backward waves, respectively. The influence of the Kerrlike medium enters through the nonlinear polarization2 PNL = χ [AE(E · E*) + BE*(E · E)], where χ is a constant of nonlinear interaction for a single linearly polarized plane wave; A and B(A + B= 1) are dimensionless constants dependent on the particular mechanism of nonlinearity. The wave equations3 for the electric fields Ej(j = 1,2) in general can be solved in terms of an elliptic integral. For special cases B=0 (which corresponds to the electrostriction mechanism of nonlinearity) and B = 0.5, analytical solutions can be found. Both analytical solutions (for B = 0 or B = 0.5) and numerical solutions4 (for other values of B) exhibit a new remarkable feature of optical multi-stability in four-wave mixing: the existence of multiple isolated branches of solutions (isolas), the total number of which depends on the intensity of the waves. The results of these calculations are shown at Figs. 1 and 2 for the case when one of the beams (e.g., the second one) is circularly polarized at the plane of incidence.
The anisotropic four-wave mixing in the scheme recently proposed by S.I. Stepanov and M.P. Petrov (Opt. Commun. vol.53, p.64-8, 1985) is studied. The main feature of this interaction arrangement is that the same phase hologram (index grating) has opposite contrasts for orthogonally polarized, counterpropagating waves. The coupled nonlinear equations, in the case of transmission and reflection gratings, for the pi /2 photorefractive phase shift are exactly solved, and their properties are discussed in detail. The numerical evidence of the effects of bistability and self-oscillation is included.>
The steady-state polarizations of fields counterpropagating in nonlinear Kerr media can exhibit complicated behavior.1 We examine the temporal behavior for the case of an isotropic nonlinear Kerr medium and show that the polarizations are temporally unstable above a certain threshold intensity, even when the input polarizations are linear and parallel to each other. This polarization configuration is known to be spatially stable and is often chosen for the two strong pump waves in phase conjugation by degenerate four-wave mixing. We find that the threshold intensity for the polarization instability is approximately the intensity at which the phase conjugate reflectivity is equal to unity. Depending on the ratio of the two input intensities, this instability just above threshold can manifest itself as an abrupt rotation of the output polarizations to a new steady-state value or as a periodic oscillation of the output polarizations. At higher intensities the polarization can evolve chaotically in time, and depending on the ratio of the transit time through the medium to the response time of the medium the output intensity can fluctuate chaotically as well.
The effects of linear absorption on the degenerate and nondegenerate self-oscillation of a photorefractive phase-conjugate mirror under condition of anisotropic four-wave mixing are considered. The phase conjugation results from nonlinear mixing of two strong undepleted orthogonally polarized counterpropagating pump waves with a probe wave. We consider two types of situation for orthogonally polarized pump beams corresponding to two crystal families: one such as the Bi12SiO20 crystal family (sillenite family) and the other such as the BaTiO3 and strontium barium niobate crystal family. In both of these crystal families the degenerate and nondegenerate self-oscillations are shown to occur in the presence of linear absorption, which tends to increase the value of the coupling coefficient required for self-oscillation. Numerical results are presented that exhibit various features of degenerate and nondegenerate self-oscillation in both crystal families for different values of the phase ϕb of the backward grating.