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<p>For the curl–curl type vector equation describing a monochromatic light wave in a Kerr medium, an exact substitution has been analyzed, which leads to a system of four first-order ordinary differential equations for functions of the transverse radial coordinate. This system includes the integer multiplicity <i>m</i> of a vortex in the longitudinal electric field component. In this case, the multiplicity of a vortex in a wave with the left and right circular polarizations is <i>m</i> – 1 and <i>m</i> + 1, respectively. With |<i>m</i>| = 1, numerical solutions of this system with appropriate boundary conditions make it possible to obtain the full information on the internal structure of a strongly nonlinear circularly polarized optical beam in a focusing medium taking into account the longitudinal field component and a small fraction of the opposite polarization. For <i>m</i> = 0, a solution in the form of a left-handed vortex in a left circularly polarized wave exists for a defocusing medium, which differs qualitatively from the right-handed vortex in the left circularly polarized wave for <i>m</i> = 2.</p>

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“Exact” Solutions for Circularly Polarized Solitons and Vortices in a Kerr Medium

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  • 发表日期：

    2025年3月1日

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  • V. P. Ruban

摘要

For the curl–curl type vector equation describing a monochromatic light wave in a Kerr medium, an exact substitution has been analyzed, which leads to a system of four first-order ordinary differential equations for functions of the transverse radial coordinate. This system includes the integer multiplicity m of a vortex in the longitudinal electric field component. In this case, the multiplicity of a vortex in a wave with the left and right circular polarizations is m – 1 and m + 1, respectively. With |m| = 1, numerical solutions of this system with appropriate boundary conditions make it possible to obtain the full information on the internal structure of a strongly nonlinear circularly polarized optical beam in a focusing medium taking into account the longitudinal field component and a small fraction of the opposite polarization. For m = 0, a solution in the form of a left-handed vortex in a left circularly polarized wave exists for a defocusing medium, which differs qualitatively from the right-handed vortex in the left circularly polarized wave for m = 2.