Photonic Wheels and Polarization Möbius Strips in Highly-Confined Trigonometric Beams

A photonic wheel describes a special structure in which the electric vector spins (with time) in a plane containing the propagation direction, while around a C-point (a point with circular polarization) the polarization Möbius strip can be formed topologically in three-dimensional (3D) space. These two interesting but very different structures are both unique to 3D structured fields, and in this article they are studied together and also their connections are analyzed. By highly confining the trigonometric beams, the three-dimensional (3D) fields with a wide area of photonic wheels are constructed, and the characteristics of the photonic wheels in different regions are analyzed. The expression for the C-lines of the photonic wheels is derived, which provides a simple way to get the exact position of the C-lines and also is a basis for observing polarization Möbius strips. Along a C-line, the behaviors of the polarization Möbius strips, including two typical (generic) Möbius strips with opposite indices, and two special topological events seemingly violating the topological law are examined. Our results show that the trick and the paradox in the two special events actually are the manifestations of the trace-dependent property and the observer-dependent property of the polarization Möbius strips respectively. The finding also suggests that even in a short segment of a C-line rich topological events related to polarization Möbius strips can be observed. Our research provides a way to observe the photonic wheels, the polarization Möbius strips and their connection, also supplies a theoretical foundation for special structures in 3D fields.