Tinkerbell beams in a non-linear ring resonator

In this article the dynamical behaviour of a beam that behaves according to a Tinkerbell map (i.e. Tinkerbell beam), within a nonlinear ring phase-conjugated resonator is modelled. The ABCD matrix of an optical device able to generate a two dimensional Tinkerbell map is found in terms of the map parameters, the state variables and the resonator parameters. For the first time to our knowledge the conditions in order to obtain the dynamics of a beam behaving according to a Tinkerbell map are found within an optical resonator. Finally some of the main technical problems to build a resonator intracavity element able to produce Tinkerbell beams are presented.