This Review article provides an overview of the fundamental origins and important applications of the main spin–orbit interaction phenomena in modern optics that play a crucial role at subwavelength scales. Light carries both spin and orbital angular momentum. These dynamical properties are determined by the polarization and spatial degrees of freedom of light. Nano-optics, photonics and plasmonics tend to explore subwavelength scales and additional degrees of freedom of structured — that is, spatially inhomogeneous — optical fields. In such fields, spin and orbital properties become strongly coupled with each other. In this Review we cover the fundamental origins and important applications of the main spin–orbit interaction phenomena in optics. These include: spin-Hall effects in inhomogeneous media and at optical interfaces, spin-dependent effects in nonparaxial (focused or scattered) fields, spin-controlled shaping of light using anisotropic structured interfaces (metasurfaces) and robust spin-directional coupling via evanescent near fields. We show that spin–orbit interactions are inherent in all basic optical processes, and that they play a crucial role in modern optics.

Spin-orbit interactions of light

Bessel-Gauss beams

A few years ago the possibility of coupling and inter-converting the spin and orbital angular momentum (SAM and OAM) of paraxial light beams in inhomogeneous anisotropic media was demonstrated. An important case is provided by waveplates having a singular transverse pattern of the birefringent optical axis, with a topological singularity of charge q at the plate center, hence named 'q-plates'. The introduction of q-plates has given rise in recent years to a number of new results and to significant progress in the field of orbital angular momentum of light. Particularly promising are the quantum photonic applications, because the polarization control of OAM allows the transfer of quantum information from the SAM qubit space to an OAM subspace of a photon and vice versa. In this paper, we review the development of the q-plate idea and some of the most significant results that have originated from it, and we will briefly touch on many other related findings concerning the interaction of the SAM and OAM of light.

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Spin-to-orbital conversion of the angular momentum of light and its classical and quantum applications

A system of two aspheric lenses is described, which efficiently converts a collimated Gaussian beam to a flattop beam. Departing from earlier designs, both aspheric surfaces were convex, simplifying their fabrication; the output beam was designed with a continuous roll-off, allowing control of the far-field diffraction pattern; and diffraction from the entrance and exit apertures was held to a negligible level. The design principles are discussed in detail, and the performance of the as-built optics is compared quantitatively with the theoretical design. Approximately 78% of the incident power is enclosed in a region with 5% rms power variation. The 8-mm-diameter beam propagates approximately 0.5 m without significant change in the intensity profile; when the beam is expanded to 32 mm in diameter, this range increases to several meters.

Design and performance of a refractive optical system that converts a Gaussian to a flattop beam.

The choice of the mathematical form of spatial correlation functions for optical fields is restricted by the constraint of nonnegative definiteness. We discuss a sufficient condition for ensuring the satisfaction of such a constraint.

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Devising genuine spatial correlation functions

An example of a Collett-Wolf source

We describe monochromatic light propagation in uniaxial crystals by means of an exact solution of Maxwell’s equations. We subsequently develop a paraxial scheme for describing a beam traveling orthogonal to the optical axis. We show that the Cartesian field components parallel and orthogonal to the optical axis are extraordinary and ordinary, respectively, and hence uncoupled. The ordinary component exhibits a standard Fresnel behavior, whereas the extraordinary one exhibits interesting anisotropic diffraction dynamics. We interpret the anisotropic diffraction as a composition of two spatial geometrical affinities and a single Fresnel propagation step. As an application, we obtain the analytical expression of the extraordinary Gaussian beam. We then derive the first nonparaxial correction to the paraxial beam, thus giving a scheme for describing slightly nonparaxial fields. We find that nonparaxiality couples the Cartesian components of the field and that the resultant longitudinal component is greater than the correction to the transverse component orthogonal to the optical axis. Finally, we derive the analytical expression for the nonparaxial correction to the paraxial Gaussian beam.

Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond.

We deduce the expressions for the two circularly polarized components of a paraxial beam propagating along the optical axis of a uniaxial crystal. We find that each of them is the sum of two contributions, the first being a free field and the second describing the interaction with the opposite component. Moreover, we expand both components as a superposition of vortices of any order, thus obtaining a complete physical picture of the interaction dynamics. Consequently, we argue that a left-hand circularly polarized incoming beam, endowed with a circular symmetric profile, gives rise, inside the crystal, to a right-hand circularly polarized vortex of order 2. The efficiency of this vortex generation is investigated by means of a power exchange analysis. The Gaussian case is fully discussed, showing the relevant features of the vortex generation.

Circularly polarized beams and vortex generation in uniaxial media.

We investigate the paraxial propagation along the optical axis of a uniaxially anisotropic crystal of a general paraxial beam whose boundary Cartesian components possess cylindrical symmetry. This property allows us to obtain expressions whose dependence on the azimuth angle ϕ (in cylindrical coordinates) is fully described and very simple. We also find that the beam loses its boundary cylindrical symmetry during propagation, as a consequence of medium anisotropy. Further, these expressions elucidate the way in which the anisotropy changes the state of polarization. As an example, we discuss the case of a Gaussian beam focused into the crystal by a thin spherical lens.

Claudio Palma

Papers

An example of a Collett-Wolf source

Optical propagation in uniaxial crystals orthogonal to the optical axis: paraxial theory and beyond.

Circularly polarized beams and vortex generation in uniaxial media.

Propagation of cylindrically symmetric fields in uniaxial crystals.

Observation of optical redshifts and blueshifts produced by source correlations