Аңдатпа
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$\epsilon$and the distance$d$between the focal planes in two transverse directions,$\chi$and$y$, which we refer to as the astigmatism parameter. The plane wave is controlled by the relative amplitude$P$and relative phase$\phi$.