Design of Lissajous beams

Singularities in optical wavefields, such as optical vortices in scalar fields and polarization singularities in

vector fields, have become an area of great theoretical and practical interest, becoming the field known as singular

optics. Such singularities have been used for diverse applications such as imaging, trapping and rotation of

particles, and free-space communications.

Most studies of singular optics have focused on fields which are monochromatic or quasi-monochromatic. In

2003, however, Kessler and Freund noted that a new class of singularities could be defined for fields which possess

two harmonic frequencies, with a fundamental frequency w and a first harmonic 2w. The singularities in this

case are lines in three-dimensional space where a generalized orientation of the polarization figure, now a Lissajous

figure, is undefined. Freund elaborated on the properties of these Lissajous singularities in detail, focusing on the

case where the fields are produced by second harmonic generation.

In recent years, Lissajous singularities have been observed and investigated in a number of projects. These

include a study of the dynamical evolution of Lissajous singularities in free-space propagation, control of

the polarization of a Lissajous fields in high-harmonic generation, and the topological features of Lissajous

fields.

Though researchers have suggested a number of potential applications for such polarized bichromatic beams,

there has been relatively little work investigating the properties of beams possessing Lissajous singularities. For

example, though fundamental classes of beams containing a single optical vortex or polarization singularity on the

propagation axis have been derived, no comparable class of Lissajous beams has been introduced.

In this paper, we will introduce a class of Lissajous beams, discuss the various types of beams that are possible,

and investigate their basic propagation properties.