Geometric phases in 2D and 3D polarized fields: geometrical, dynamical, and topological aspects

Abstract

Geometric phases are a universal concept that underpins numerous phenomena involving multi-component wave fields. These polarization-dependent phases are inherent in interference effects, spin-orbit interaction phenomena, and topological properties of vector wave fields. Geometric phases have been thoroughly studied in two-component fields, such as two-level quantum systems or paraxial optical waves. However, their description for fields with three or more components, such as generic nonparaxial optical fields routinely used in modern nano-optics, constitutes a nontrivial problem. Here we describe geometric, dynamical, and total phases calculated along a closed spatial contour in a multi-component complex field, with particular emphasis on 2D (paraxial) and 3D (nonparaxial) optical fields. We present several equivalent approaches: (i) an algebraic formalism, universal for any multi-component field; (ii) a dynamical approach using the Coriolis coupling between the spin angular momentum and reference-frame rotations; and (iii) a geometric representation, which unifies the Pancharatnam-Berry phase for the 2D polarization on the Poincaré sphere and the Majorana-sphere representation for the 3D polarized fields. Most importantly, we reveal close connections between geometric phases, angular-momentum properties of the field, and topological properties of polarization singularities in 2D and 3D fields, such as C-points and polarization Möbius strips.

21 pages, 11 figures, to appear in Rep. Prog. Phys

References in corpus (21)

• Geometrodynamics of Spinning Light
• Coriolis Effect in Optics: Unified Geometric Phase and Spin-Hall Effect
• Observation of optical vortices in momentum space
• Angular Momenta and Spin-Orbit Interaction of Nonparaxial Light in Free Space
• Geometric phase from Aharonov-Bohm to Pancharatnam-Berry and beyond
• Geometrodynamics of polarized light: Berry phase and spin Hall effect in a gradient-index medium
• Optical Momentum, Spin, and Angular Momentum in Dispersive Media
• Optical momentum and angular momentum in complex media: From the Abraham-Minkowski debate to unusual properties of surface plasmon-polaritons
• Modified geometrical optics of a smoothly inhomogeneous isotropic medium: the anisotropy, Berry phase, and the optical Magnus effect
• Electromagnetic helicity in complex media
• Angular momenta, helicity, and other properties of dielectric-fiber and metallic-wire modes
• Singular knot bundle in light
• Non-Abelian evolution of electromagnetic waves in a weakly anisotropic inhomogeneous medium
• Multi-twist polarization ribbon topologies in highly-confined optical fields
• Singular polarimetry: Evolution of polarization singularities in electromagnetic waves propagating through a weakly anisotropic medium
• Mass-polariton theory of sharing the total angular momentum of light between the field and matter
• Modal Majorana sphere and hidden symmetries of structured-Gaussian beams
• Generalized Gaussian beams in terms of Jones vectors
• The optical Möbius strip cavity: Tailoring geometric phases and far fields
• An intuitive approach to structuring the three polarization components of light
• Weak measurement of elliptical dipole moments by C point splitting

Cited by in corpus (43)

• Particle-like topologies in light
• Roadmap on structured waves
• Topological Polarization Singularities in Metaphotonics
• Topological unidirectional guided resonances emerged from interband coupling
• Water-Wave Vortices and Skyrmions
• Observation of polarization singularities and topological textures in sound waves
• Topological water-wave structures manipulating particles
• Geometric phases of light: insights from fibre bundle theory
• Geometric descriptions for the polarization for nonparaxial optical fields: a tutorial
• Evolution and global charge conservation for polarization singularities emerging from nonhermitian degeneracies
• Optical excitations of Skyrmions, knotted solitons, and defects in atoms
• Chiral topological light for detecting robust enantio-sensitive observables
• Elastic Spin and Orbital Angular Momenta
• Knotted Polarizations and Spin in 3D Polychromatic Waves
• Topological near fields generated by topological structures
• Critical coupling vortex with grating-induced high quality optical Tamm states
• Polarization singularities and Möbius strips in sound and water-surface waves
• Topological elasticity of non-orientable ribbons
• Polarization Singularities in Light Scattering by Small Particles
• Topological transitions of the generalized Pancharatnam-Berry phase
• Spin angular momentum and optical chirality of Poincaré vector vortex beams
• Optical Singularity Dynamics and Spin-Orbit Interaction due to a Normal-Incident Optical Beam Reflected at a Plane Dielectric Interface
• Vertical routing of spinning dipoles radiation from a chiral metamembrane
• Optical force and torque on small particles induced by polarization singularities
• Generic Optical Singularities in Brewster-reflected Post-paraxial Beam-fields
• Spin-orbit interaction of light in three-dimensional microcavities
• Optical polarization skyrmionic fields in free space
• Spectral Bounds on Entropy and Ergotropy via Statistical Effective Temperature in Classical Polarization and Quantum Thermal States
• Photonic Shankar skyrmion
• Manifestation of the Berry connection in chiral lattice systems
• Coupled quantum vortex kinematics and Berry curvature in real space
• Complex far fields and optical singularities due to propagation beyond tight focusing: combined effects of wavefront curvature and aperture diffraction
• Wave manipulation via delay-engineered periodic potentials
• Experimental measurement of the geometric phase of non-geodesic circles
• Topological invariants and topological charges in photonic systems
• Breakdown of Bulk-Radiation Correspondence in Radiative Photonic Lattices
• Dynamic and Geometric Shifts in Wave Scattering
• Exploring weak value arguments and Bargmann invariants in $N$-level quantum systems through the Majorana symmetric representation
• Geometric phase and a nonreciprocal spin wave circular polarizer
• Topological photonics in nanoscaled systems with far field radiation and polarization singularities
• Electric-Magnetic Geometric Phase
• Two geometric phases can dramatically differ from each other even if their evolution paths are sufficiently close in a pointwise manner
• Observation of geometric phase for unpolarized and partially polarized light fields