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IEEE Journal of Quantum Electronics

Isolas in four-wave mixing optical bistability

A. E. Kaplan, Chiu Tai Law

Published Sep 1, 1985• 71 citations
Publisher

Abstract

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.

DOI
10.1109/jqe.1985.1072812

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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.

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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.
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.
Instabilities in the interaction of counterpropagating laser beams in a nonlinear medium can have important consequences for nonlinear optical processes such as optical bistability and phase conjugation. We have performed a theoretical study of the polarization characteristics of two counterpropagating waves in an isotropic Kerr medium in order to determine the stability of the multivalued solutions found by previous workers.1 Our method involves numerical integration of the coupled-amplitude equations describing the spatial and temporal evolution of the counterpropagating vector waves.2 Our results show that the multiply stable solutions for the output polarizations of the two vector fields show hysteresis, and that the order in which the two fields are turned on determines which solution is reached. For even higher input intensities we find that the output polarizations fluctuate in time.
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.
This thesis is divided into two portions. The first part explores the cross-interaction of two counter-propagating plane waves in a Kerr-like medium. It is shown analytically and numerically that due to cross-induced anisotropy, the "input-output" characteristics of such a system have not only multivalued hysteretic response but also multiple isolated branches (the so called isolas). We demonstrate also that the system may exhibit dispersion-related temporal instability above certain threshold intensity of pumping waves. Most of these effects may find applications in optical bistable and switching devices and optical signal processing. In the second part, the generation of x-ray transition radiation by low energy electron beams traversing a periodic solid-state layered medium is considered. We investigate the influence of photoabsorption and electron scattering on the phenomenon and develop a optimization procedure for selecting parameters of the system to maximize the radiation. We show that a combination of materials with high and low atomic numbers can produce an intense x-ray radiation with very narrow spectral peaks at the vicinity of K, L, …\dots… atomic absorption edges of each of the materials. Our results demonstrate that the proposed inexpensive x-ray source with relatively low electron beam energy can be used to generate narrow-line x-ray radiation.
We consider the problem of the wave propagation through a nonlinear medium. We derive a dynamic system that governs the behavior of standing (or solitary) waves. The form of this system alone suffices to understand the qualitative dependence of solutions of the original equation on the intensity of the incident wave. We solve this dynamical system in the leading order in the nonlinearity strength. We find multiple solutions of the original problem for a given incoming wave and turning points of these solutions as a function of the wave intensity. We briefly investigate stability of different branches. Our results yield an analytic description of the optical bistability phenomenon.