Lissajous Singularities in Young’s Interference Experiment

Singularities in wavefields have become important objects of study, both for their unusual physical properties and the use of those properties to improve optical systems. In scalar wavefields, optical vortices, characterized by helical phase structures and lines of zero intensity, have found use in free-space optical communications, super-resolved imaging, and coronagraphy. In vector electromagnetic fields, we typically encounter polarization singularities, which for paraxial fields include lines of circular polarization (on which the orientation of the

polarization ellipse is undefined) and surfaces of linear polarization (on which the handedness of the polarization ellipse is undefined). These polarization singularities have practical applications in imaging and light-matter manipulation. Both optical vortices and polarization singularities are typically studied in monochromatic fields.

It is possible to generalize them further, and consider the types of singularities that appear in bichromatic fields where the higher frequency is a harmonic of the lower. The electric field vector then traces out a Lissajous figure instead of an ellipse; singularities of the generalized orientation of this figure are called Lissajous singularities. These singularities have potential to be used in imaging and communications, and recently a class of beams containing a single Lissajous singularity at their core was formulated. Though the topology of Lissajous singularities has been well-formulated, the conditions under which Lissajous singularities can be formed, for example through interference, are still unclear. Young’s experiment provides a unique platform for exploring a rich variety of phenomena in both classical optics and quantum optics.

In this study, we use Young’s interference experiment to investigate the superposition of two vector beams, each possessing two frequency components, and we derive sufficiency conditions under which the Lissajous-type polarization singularities are formed on the observation screen. We give examples of Lissajous patterns and singularities under these conditions and present additional cases of singularity creation.