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International Meeting on Instabilities and Dynamics of Lasers and Nonlinear Optical Systems

Multiple Isolated Branches in Four-Wave Mixing Optical Bistability

A. E. Kaplan, Chiu Tai Law

Published Jan 1, 1985• 0 citations
Publisher

Abstract

H. G. Winful and J. H. Marburger 1 have first proposed an optical bistability with hysteretic response in collinear degenerated four-wave mixing. We show here the feasibility of multistable nonhysteretic behavior and isolated branches in the system considered by them. We consider two counter-propagating optical beams along z axis with the same frequency such that the total electric field is: E → = [ E → 1 exp ( ikz ) + E → 2 exp ( − ikz ) ] e − i ω t where E → 1 and E → 2 are envelope vectors of forward and backward waves respectively. Based on 2, the nonlinear component of electric displacement D → NL in a Kerr-like medium can be expressed as: D → NL = ϵ o χ [ A E → ( E → ⋅ E → * ) + B E → * ( E → ⋅ E → ) ] where ϵo is a linear electric permittivity of medium, χ is a constant of nonlinear interaction for a single linearly polarized plane wave, A and B are dimensionless constants dependent upon the particular mechanism of nonlinearity. Decomposing E → 1 and E → 2 into a sum of two circularly polarized components: E → j = 1 2 [ u j ( e ^ x + i e ^ y ) + v j ( e ^ x − i e ^ y ) ] , ( j = 1 , 2 ) and using the vectorial nonlinear wave equation for E → 1 and E → 2 (Ref. 3), one can obtain four conservation relationships, which have four invariants of motion: I1 and I2 (total intensities of both of the beams) and two additional invariants referred below as M and J.

DOI
10.1364/idlnos.1985.fb3

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H. G. Winful and J. H. Marburger 1 have first proposed an optical bistability with hysteretic response in collinear degenerated four-wave mixing. We show here the feasibility of multistable nonhysteretic behavior and isolated branches in the system considered by them. We consider two counter-propagating optical beams along z axis with the same frequency such that the total electric field is: E → = [ E → 1 exp ( ikz ) + E → 2 exp ( − ikz ) ] e − i ω t where E → 1 and E → 2 are envelope vectors of forward and backward waves respectively. Based on 2, the nonlinear component of electric displacement D → NL in a Kerr-like medium can be expressed as: D → NL = ϵ o χ [ A E → ( E → ⋅ E → * ) + B E → * ( E → ⋅ E → ) ] where ϵo is a linear electric permittivity of medium, χ is a constant of nonlinear interaction for a single linearly polarized plane wave, A and B are dimensionless constants dependent upon the particular mechanism of nonlinearity. Decomposing E → 1 and E → 2 into a sum of two circularly polarized components: E → j = 1 2 [ u j ( e ^ x + i e ^ y ) + v j ( e ^ x − i e ^ y ) ] , ( j = 1 , 2 ) and using the vectorial nonlinear wave equation for E → 1 and E → 2 (Ref. 3), one can obtain four conservation relationships, which have four invariants of motion: I1 and I2 (total intensities of both of the beams) and two additional invariants referred below as M and J.
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.
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.
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.
We consider degenerate four-wave mixing in a Kerr medium with a standard noncollinear geometry and the polarization directions of all the beams parallel to one another. The two pump beams, the probe beam, and the phase-conjugate beam are denoted by the subscripts 1,2,4 and 3, respectively. The nonlinear medium is placed between z = 0 and z = 1; the values of the intensities of the beams at z = 0 and 1 are denoted by Ii0 andIi1, where i = 1,2,3,4. Beams 2 and 4 are incident at z = 1; and beam 1 is incident at z = 0. We have determined the dependence of the phase-conjugate reflectivity R = I31/I41 and the pump transmission T = I11/I10 on the following parameters: the probe ratio, q=I41/(I10 + I21); the pump ratio, r = I21/I 1 0; the absorption coefficient–path length product, al; and the coupling constant of the medium, q. In the absence of nonlinear refraction, pump depletion, and linear absorption in the medium, R = tan2q. When all of these effects are taken into account, the maximum value of R is given by Rmax = exp (-2α1)/(q(1 + r)). The minimum value that T can reach is proportional to q for q < 1. Even for small values of q, multistable solutions for R and T are obtained when q is greater than a threshold value, which we denote by qth. We have determined the dependence of qth on the parameters q and r.
Optical phase conjugation by four-wave mixing in a Kerr medium is considered. Two strong counterpropagating lasers pump a nonlinear medium in which they excite the third-order polarization. Mixing with a weak incident field then generates the phase-conjugated image of that field. It is shown how the polarization of the pumps and the tensorial nature of the nonlinear interaction can be accounted for by a geometrical polarization tensor. The electric field is shown to obey two coupled wave equations, which couple a positive- and negative-frequency component of the field. These wave equations allow plane-wave solutions, and we derive the dispersion relation for the wave vectors of these modes. By matching the field of such a mode across the boundaries of the medium to an external incident field, we are able to obtain analytically the Fresnel coefficients for reflection and transmission. Our expressions reduce to earlier results in the appropriate limits.