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.
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.
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.
The theory of probe beam amplification via degenerate optical wave mixing in a Kerr medium is presented. This theory includes consideration of the self-phase modulation, two-wave mixing, and multiwave mixing terms in the coupled-wave equations. Both analytical solutions, under the assumptions of a nondepleted pump beam and a transparent Kerr medium, and numerical solutions with these assumptions removed are presented. Our solutions show that under certain conditions, phase matching can be satisfied and a high probe beam gain is possible, even for a zero phase shift between the refractive index and the intensity gratings.
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.
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.
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.
We derive an exact solution for both amplitude and phase of a signal that is back-reflected by collinear nonlinear degenerate mixing of four circularly polarized waves of arbitrary intensity in a Kerr-like medium. Our analysis suggests that taking pump depletion into account may spoil the ideal phase-conjugation condition, even in the limit of weak nonlinear coupling, and indicates that substantial deterioration of the probe phase-reversal property may occur for strongly interacting waves in a manner that is highly sensitive to the anisotropy of the nonlinear susceptibility.