Discrete-Gauss states and the generation of focussing dark beams

Discrete-Gauss states are a new class of gaussian solutions of the free Schr\"odinger equation owning discrete rotational symmetry. They are obtained by acting with a discrete deformation operator onto Laguerre-Gauss modes. We present a general analytical construction of these states and show the necessary and sufficient condition for them to host embedded dark beams structures. We unveil the intimate connection between discrete rotational symmetry, orbital angular momentum, and the generation of focussing dark beams. The distinguishing features of focussing dark beams are discussed. The potential applications of Discrete-Gauss states in advanced optical trapping and quantum information processing are also briefly discussed.