Multi-twist polarization ribbon topologies in highly-confined optical fields

Abstract

Electromagnetic plane waves, solutions to Maxwell's equations, are said to be 'transverse' in vacuum. Namely, the waves' oscillatory electric and magnetic fields are confined within a plane transverse to the waves' propagation direction. Under tight-focusing conditions however, the field can exhibit longitudinal electric or magnetic components, transverse spin angular momentum, or non-trivial topologies such as Möbius strips. Here, we show that when a suitably spatially structured beam is tightly focused, a three-dimensional polarization topology in the form of a ribbon with two full twists appears in the focal volume. We study experimentally the stability and dynamics of the observed polarization ribbon by exploring its topological structure for various radii upon focusing and for different propagation planes.

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

“…The corresponding spatial distribution of the major semiaxis of the polarization ellipse has a discontinuity A → −A highlighted in yellow. This forms the polarization Möbius strip [33][34][35][36][37][38][39]. The transition between the two cases occurs each time that the spatial contour crosses a non-degenerate C-line.…”

“…Case 2: Upon continuous cyclic evolution, the polarization-ellipse vectors return to the values opposite to the initial ones: (A, B) in = (−A, −B) fin . Each of these vectors, traced along the spatial contour of evolution, forms a 3D structure similar to a Möbius strip with a half-integer number of turns around the contour [33][34][35][36][37][38][39]. In this case, the vectorsū 1 andū 2 swap after the cyclic evolution, (ū 1 ,ū 2 ) in = (ū 2 ,ū 1 ) fin , and form a single closed loop over the Poincarana sphere, Fig.…”

“…However, with the rapid development of nano-optics and photonics, more attention has been placed on inhomogeneous 3D fields whose polarization may be nonparaxial. In particular, structured fields can give rise to knotted 3D singularities [28][29][30][31][32] and polarization Möbius strips [33][34][35][36][37][38][39], as well as to the coupling of the spin and orbital AM [11,[40][41][42][43][44][45] where geometric phases play a key role.…”

Optical framed knots as information carriers

“…Such discontinuities, which can be present in features such as optical phase 4 or polarization 5 , are known as optical singularities and can be employed to produce optical beams of varying complexity from those carrying a single singularity 6 to more exotic wavefields forming structures such as topological bands and knots. The latter include Möbius strips [7][8][9][10] , multi-twist ribbons 11 , knots within scalar optical fields [12][13][14] , knotted topologies within bichromatic fields 15 , and knots in polarization fields, which include both knotted electromagnetic field lines [16][17][18][19][20] and knotted polarization singularities 21 . These structured optical fields carrying topological features have found numerous applications in modern science.…”

Atoms in axially shifted tightly focused counter-propagating beams: the role of the Gouy and curvature phases

“…Most work on laser cooling and trapping of atoms has dealt with fairly broad beams but recent developments have led to the generation and the application of optical beams that are focused to sub-wavelength sizes [14][15][16]. Dorn et al [17] demonstrated experimentally that a radially polarized field can be focused to a spot size significantly smaller (0.16λ 2 ) than for a linearly polarized field (0.26λ 2 ).…”