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

Optical Waveguide Theory

Vector modes are spatial modes that have spatially inhomogeneous states of polarization, such as, radial and azimuthal polarization. In this work, the spatially inhomogeneous states of polarization of vector modes are used to increase the transmission data rate of free-space optical communication via mode division multiplexing. A mode (de)multiplexer for vector modes based on a liquid crystal q-plate is introduced. As a proof of principle, four vector modes each carrying a 20-Gbit/s quadrature phase shift keying signal (aggregate 80 Gbit/s) on a single wavelength channel (λ∼1550  nm) were transmitted ∼1  m over the lab table with <−16.4  dB mode crosstalk. Bit error rates for all vector modes were measured at the 7% forward error correction threshold with power penalties <3.41  dB.

4 × 20 Gbit/s mode division multiplexing over free space using vector modes and a q-plate mode (de)multiplexer

We examine the spin-orbit coupling effects that appear when a wave carrying intrinsic angular momentum interacts with a medium. The Berry phase is shown to be a manifestation of the Coriolis effect in a noninertial reference frame attached to the wave. In the most general case, when both the direction of propagation and the state of the wave are varied, the phase is given by a simple expression that unifies the spin redirection Berry phase and the Pancharatnam-Berry phase. The theory is supported by the experiment demonstrating the spin-orbit coupling of electromagnetic waves via a surface plasmon nanostructure. The measurements verify the unified geometric phase, demonstrated by the observed polarization-dependent shift (spin-Hall effect) of the waves.

/pdf/coriolis-effect-in-optics-unified-geometric-phase-and-spin-41iypj0wkc.pdf

Coriolis effect in optics: unified geometric phase and spin-Hall effect.

We consider semiclassical higher-order wave packet solutions of the Schrodinger equation with phase vortices. The vortex line is aligned with the propagation direction, and the wave packet carries a well-defined orbital angular momentum (OAM) variant Planck's over 2pil (l is the vortex strength) along its main linear momentum. The probability current coils around the momentum in such OAM states of electrons. In an electric field, these states evolve like massless particles with spin l. The magnetic-monopole Berry curvature appears in momentum space, which results in a spin-orbit-type interaction and a Berry/Magnus transverse force acting on the wave packet. This brings about the OAM Hall effect. In a magnetic field, there is a Zeeman interaction, which, can lead to more complicated dynamics.

/pdf/semiclassical-dynamics-of-electron-wave-packet-states-with-1u7jcrpmbb.pdf

Semiclassical dynamics of electron wave packet states with phase vortices

We have studied the effect of the spin-orbit interaction on generation and conversion of optical vortices in multihelicoidal fibers, that is, the fibers possessing a multihelical refractive index profile. On the basis of a fully analytical approach we have obtained the spectra of coupled modes and their structure. Specifically, we have established selection rules, under which the spin-orbit interaction mediates the conversion of optical vortices into vortices with the topological charge changed by $\ifmmode\pm\else\textpm\fi{}(\ensuremath{\ell}\ifmmode\pm\else\textpm\fi{}2)$, $\ensuremath{\ell}$ being the number of helical branches in refractive index distribution. Also, we have shown that the spin-orbit interaction can lead to generation of radially and azimuthally polarized TE and TM modes from optical vortices. We have also demonstrated that if such generation is mediated by a scalar-type perturbation of the fiber's form, it is possible only for weakly deformed fibers. For strongly deformed fibers such perturbation can result only in generation of vortices with zero total angular momentum. Additionally, we have studied the possibility of polarization control over the orbital angular momentum of the generated state in such a system.

Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers

We theoretically study generation and conversion of optical vortices (OVs) in long-period helical core fibers (HCFs) produced by drawing from a perform with an eccentric core. Solving scalar waveguide equation in the helical coordinates we demonstrate that in such fibers topologically induced corrections to scalar propagation constants lead to convergence of spectral curves. At their intersection points a resonance coupling takes place between the fundamental and the vortex modes, as well as the coupling between OVs, whose topological charges differ by unity. This coupling leads to energy exchange between the coupled modes, which is manifested either in conversion of the input fundamental mode into an OV or transformation of an OV into the OV with higher by 1 (or lower, depending on core's helicity) value of the topological charge. This effect can be used for creation of all-fiber generators of singular beams from regular input beams. We also study effect of ellipticity of the core's form on transformation properties of HCFs and show that this leads to splitting of resonance fiber's parameters, at which conversion of left and right circularly polarized input regular beams into OVs occurs. We prove that in long-period HCFs one can neglect this effect due to reduction of polarization mode dispersion in twisted fibers.

Generation and conversion of optical vortices in long-period helical core optical fibers

We investigate the structure of the higher order modes in uniformly twisted weakly guiding strongly elliptical optical fibres. An analytical solution of the vector wave equation for this problem is presented based on the effective reduction of a twisted fibre equation to a straight fibre one. On the basis of the perturbation theory the structure of the higher order elliptical screw modes and polarization corrections to the scalar propagation constant are found as functions of the pitch value and the fibre parameters. It is demonstrated that at a certain rate of twisting the l = 1 modes of such fibres are presented by two pairs of almost pure arbitrary polarized optical vortices. The orbital angular momentum of the l = 1 modes is studied as a function of pitch.

Optical vortices and the higher order modes of twisted strongly elliptical optical fibres

We study the light propagation in the twisted anisotropic optical fibers endowed with torsional mechanical stress by obtaining the analytical solution of the vector wave equation. We show that at certain interplay between fiber parameters optical vortex beams of topological charge $\ensuremath{\ell}=0,\ifmmode\pm\else\textpm\fi{}1,\ifmmode\pm\else\textpm\fi{}2,...$ become the modes of the fibers in question. To explain the splitting of the optical vortex propagation constants we introduce the notions of orbital birefringence and optical Zeeman effect. Moreover, we unveil that induced by torsional stress circular birefringence makes the vortex beams with the well-defined orbital angular momentum robust against small perturbations characterized by both constant and spatially varying orientation of a director. We believe that such fibers can be successfully utilized for the long-range robust transmission of information encoded in the light's orbital degrees of freedom.

Twisted anisotropic fibers for robust orbital-angular-momentum-based information transmission

We study the field generated in the outer space by the superposition of modes of a regular circular monomode fiber array. It is shown that a supermode of the fiber array generates a discrete optical vortex; the formula for the topological charge of the vortex is obtained depending on the order of the supermode and the number of fibers in the array. The orbital angular momentum carried by an arbitrary superposition of supermodes is shown to equal the weighted sum of partial angular momenta of supermodes. It is shown that for certain combinations of supermodes the angular momentum comprises along with its intrinsic part also the extrinsic constituent. For such combinations precession of the angular momentum about the propagation axis is demonstrated. It is demonstrated that by combining supermodes one can generate in the array stable regularly rotating linear azimuthons. By creating a phased excitation of certain groups of fibers in the array one can control the global soliton-like motion of the excited domain.

Maxim A. Yavorsky

Papers

Spin-orbit-interaction-induced generation of optical vortices in multihelicoidal fibers

Generation and conversion of optical vortices in long-period helical core optical fibers

Optical vortices and the higher order modes of twisted strongly elliptical optical fibres

Twisted anisotropic fibers for robust orbital-angular-momentum-based information transmission

Linear azimuthons in circular fiber arrays and optical angular momentum of discrete optical vortices