1 1 Ja n 20 05 Vortex Knots in Light

Optical vortices generically arise when optical beams are combined. Recently, we reported how several laser beams containing optical vortices could be combined to form optical vortex loops, links and knots embedded in a light beam (Leach et al 2004 Nature 432 165). Here, we describe in detail the experiments in which vortex loops form these structures. The experimental construction follows a theoretical model originally proposed by Berry and Dennis, and the beams are synthesised using a programmable spatial light modulator and imaged using a CCD camera. Optical vortices (phase singularities, wave dislocations) are ubiquitous in nature, occurring whenever three or more light waves interfere. They are places in the interference field where the intensity is zero, and the phase is undefined, and usually occur at points in 2-dimensional fields, and along lines in 3 dimensions (see, for example, [1, 2, 3]). Within the laboratory, vortices are commonly produced using computer generated holograms or diffractive optics. Holograms of the design originally demonstrated by [4, 5] are now widely used to create light beams with vortices embedded, and the orbital angular momentum [6, 7, 8] with which they are associated. In 2001, Berry and Dennis showed theoretically [9, 10] that specific superpositions of beams could be generated in which vortex loops could be linked together and even knotted. Here, we describe in detail our experimental generation [11] of such laser beams. An optical vortex embedded on the axis of a laser beam is usually characterised by a function, R |l| exp(ilφ) describing the local structure of the optical field around the vortex point in a transverse cross section of the beam. The exp(ilφ) factor describes the phase singularity nature of the vortex, R |l| shows that this must be accompanied by a zero of intensity. The integer l determines the number of phase cycles made in a right-handed circuit of the vortex and is called the strength of the singularity. Although l can take any integer value, positive or negative, vortices with |l| > 1 are unstable and, in the presence of astigmatism, degenerate to form multiple vortices of l = ±1 upon propagation. Many observations have been reported on the seeming interaction between two or more vortex points as light propagates from one plane to the next (see, for instance, [12] and references therein). In this view, two vortices with oppositely signed § To whom correspondence should be addressed. Vortex Knots …