Investigations of structured laser beams formation are one of the attractive areas of laser research in recent years. Compared to solid-state lasers, dye lasers were rarely in demand in these studies. Here we present several transverse mode selection experiments in a pulsed Rhodamine 6G dye laser ( \(\lambda\) = 580 nm) with a plano-spherical resonator under hollow beam pumping by 20 ns pulses of 532 nm radiation from the Nd:YAG laser 2-nd harmonic. Along with high-order vortex and petal Laguerre–Gaussian modes, a whole family of non-planar geometric modes of different structures was obtained at the frequency degenerate states of this dye laser resonator. The ensemble of geometric modes in the related degenerate resonator configurations is illustrated by fractal frequency spectrum calculations. The experiments performed show that a dye laser represents a fairly simple and convenient object for testing various mode selection techniques. Dye lasers operating in the visible and near-infrared spectral regions can complement the range of solid-state laser sources currently used for obtaining and studying structured laser beams, which have many applications in scientific research and applied problems.
