Optical Vortex Rings: Reconnections and Catastrophes

Gaussian laser beams in free space represent surprisingly rich system supporting complex topologies of vortex lines, from reconnections [1] to vortex rings [2] and ring transformations [3]. We study theoretically the experimentally accessible optical vortex rings generated by destructive interference of an elliptic Gaussian beam and a plane wave. The shape of the Gaussian beam is governed by the ellipticity of the transverse profile <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\epsilon$</tex> and the distance <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$d$</tex> between the focal planes in two transverse directions, <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\chi$</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$y$</tex>, which we refer to as the astigmatism parameter. The plane wave is controlled by the relative amplitude <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$P$</tex> and relative phase <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\phi$</tex>.