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: $\overrightarrow{\text{E}}=\left[{\overrightarrow{\text{E}}}_1\exp (\text{ikz})+{\overrightarrow{\text{E}}}_2\exp (-\text{ikz})\right]{\text{e}}^{-\text{i}\omega t}$ where ${\overrightarrow{\text{E}}}_1$ and ${\overrightarrow{\text{E}}}_2$ are envelope vectors of forward and backward waves respectively. Based on ², the nonlinear component of electric displacement ${\overrightarrow{\text{D}}}^{\text{NL}}$ in a Kerr-like medium can be expressed as:${\overrightarrow{\text{D}}}^{\text{NL}}={ϵ}_{\text{o}}\chi \left[\text{A}\overrightarrow{\text{E}}\left(\overrightarrow{\text{E}}\cdot {\overrightarrow{\text{E}}}^{*}\right)+{\text{B}\overrightarrow{\text{E}}}^{*}(\overrightarrow{\text{E}}\cdot \overrightarrow{\text{E}})\right]$ where ϵ_o 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 ${\overrightarrow{\text{E}}}_1$ and ${\overrightarrow{\text{E}}}_2$ into a sum of two circularly polarized components: ${\overrightarrow{E}}_{\text{j}}=\frac{1}{\sqrt{2}}\left[{\text{u}}_{\text{j}}\left({\widehat{e}}_{\text{x}}+\text{i}{\widehat{e}}_{\text{y}}\right)+{\text{v}}_{\text{j}}\left({\widehat{\text{e}}}_{\text{x}}-\text{i}{\widehat{\text{e}}}_{\text{y}}\right)\right],(\text{j}=1,2)$ and using the vectorial nonlinear wave equation for ${\overrightarrow{\text{E}}}_1$ and ${\overrightarrow{\text{E}}}_2$ (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.

© 1985 Optical Society of America