Rays, waves, SU(2) symmetry and geometry: toolkits for structured light

Abstract

Structured light refers to the ability to tailor optical patterns in all its degrees of freedom, from conventional 2D transverse patterns to exotic forms of 3D, 4D, and even higher-dimensional modes of light, which break fundamental paradigms and open new and exciting applications for both classical and quantum scenarios. The description of diverse degrees of freedom of light can be based on different interpretations, e.g. rays, waves, and quantum states, that are based on different assumptions and approximations. In particular, recent advances highlighted the exploiting of geometric transformation under general symmetry to reveal the ‘hidden’ degrees of freedom of light, allowing access to higher dimensional control of light. In this tutorial, I outline the basics of symmetry and geometry to describe light, starting from the basic mathematics and physics of SU(2) symmetry group, and then to the generation of complex states of light, leading to a deeper understanding of structured light with connections between rays and waves, quantum and classical. The recent explosion of related applications are reviewed, including advances in multi-particle optical tweezing, novel forms of topological photonics, high-capacity classical and quantum communications, and many others, that, finally, outline what the future might hold for this rapidly evolving field.

Towards higher-dimensional structured light

“…This serves to highlight that the challenge is not only more DoFs and dimensions but rather those that can be practically controlled. To deal with this, an effective tool of SU(2) symmetry was exploited to design ray-wave duality structured in paraxial beams 20 , where the wave patterns can be geometrically coupled to a set of caustic rays so as to open new DoFs to be controlled 21 than prior vortex beams, for example, the number of rays, their directions and positions, and so on. We can also involve polarisation control into the ray-wave coupled states to access exotic ray-wave vector beams, see Fig.…”

Nonseparable states of light: From quantum to classical

“…The eigenfuncation of this result refers to a Hermite-Gaussian (HG) modes, the conventional laser mode solved in Cartesian coordinate system. Furthermore, the Laguerre-Gaussian (LG) modes in circular coordinate and general Hermite-Laguerre-Gaussian (HLG) modes can be easily obtained by applying additional SU(2) transformation to the ladder operators [42][43][44].…”

Nonlinear optics with structured light

“…Indeed, the very essence of structured light is the notion of superpositions, where interference (not necessarily in intensity) gives rise to the desired structure. Today one can formulate all of structured light as a linear superposition principle 1 , giving rise to geometric representations of the superpositions, from the orbital angular momentum (OAM) 23 , to the total angular momentum 24 of light, and more recently a generalised framework for multiple DoFs 25 . For example, even simple plane waves hold the potential for structure: one plane wave may have a phase gradient, two plane waves will give rise to an intensity structure (as done by Young more than 200 years ago), three plane waves can produce an optical phase singularity, while multiple plane waves can give rise to exotic families of structured light, for instance, planes waves travelling on a cone give rise to Bessel beams 26 .…”