Generation of transversely oriented optical polarization M\"obius strips

We report a time-reversal method based on the Richards-Wolf vectorial diffraction theory to generate transversely oriented optical M\"obius strips that wander around an axis perpendicular to the beam propagation direction. A number of sets of dipole antennae are purposefully positioned on a prescribed trajectory in the y = 0 plane and the radiation fields are collected by one high-NA objective lens. By sending the complex conjugate of the radiation fields in a time-reversed manner, the focal fields are calculated and the optical polarization topology on the trajectory can be tailored to form prescribed M\"obius strips. The method can be extended to construct various polarization topologies on three-dimensional trajectories in the focal region. The ability to control optical polarization topologies may find applications in nanofabrication, quantum communications, and light-matter interactions.

Optical Möbius strips, featuring a one-sided surface, are formed by the major axes of threedimensional polarization ellipses. Optical Möbius strips can be created by tightly focusing optical fields emerging from a q-plate [10]. The interference of two noncoaxial circularly polarized beams of opposite handedness with different scalar topological charges also result in Möbius strips topological structures [9,18]. Möbius strips are also discovered in light scattered from high-index dielectric nanoparticles [19].
Most optical Mobius strips discovered so far are within the cross section of a beam. In the current paper, we report a time-reversal method based on the Richards-Wolf vectorial diffraction theory for generating transversely oriented Möbius strips that wander around an axis perpendicular to the beam propagation direction [20][21][22][23][24][25][26][27]. A number of dipole antennae are positioned on a circular trajectory in the x-z plane and their radiation fields are collected by one high-NA microscope objective. By sending the radiation fields in time reversed order, optimal light spot arrays are formed at the source location, forming transversely oriented ringshaped focal fields with Möbius strips polarization topologies. The method can be further extended to create other optical topological structures along arbitrary prescribed trajectories in three-dimensional space. The ability to tailor optical polarization topologies may spur various applications in nanofabrication and light-matter interactions.

Time reversal method
The time-reversal theory indicates that an optimal light spot can be formed at the source location by sending in time reversed order the radiation fields received from an infinitesimal dipole antenna. If three dipole antennae with orthogonal oscillating directions are positioned at the same location, not only can an optimal light spot be formed but also its threedimensional state of polarization can be controlled by the relative amplitude and phase of the individual dipole antennae. In this work, we demonstrate that if a number of sets of dipole antennae are positioned on a prescribed trajectory, it is even possible to tailor the optical polarization topologies along arbitrary three-dimensional curves in a time-reversed manner. Figure 1 shows the schematic of the proposed time-reversal scheme. To reconstruct the polarization topology of Möbius strips, a number of sets of dipole antennae are uniformly positioned in the y = 0 plane along a circular trajectory of radius r 0 . Each set of dipole antennae occupy one spot on the trajectory and contain three individual dipoles oscillating along the x, y, and z directions. The radiation fields generated from all dipole antennae are coherently combined in the curved surface of a high-NA objective lens.
The electric field data of Möbius strip polarization topologies is obtained from a tightly focused beam emerging from a q-plate based on the Richards-Wolf vectorial diffraction theory. The x-, y-, and z-components of the electric field data at each sampled point are connected with the relative amplitude and phase of the three orthogonally oscillating dipoles of each set of dipole antenna on the prescribed trajectory. The complex conjugate of the radiation fields received from all sets of dipole antennae are focused back with one high-NA objective lens. According to the Richards-Wolf vectorial diffraction method, the electric fields in the focal region are given by [26]: where ( ) We examine the three-dimensional electric fields and the associated three-dimensional states of polarization in the y = 0 plane in the focal region. The optical polarization topology, traced along a circular trajectory, can be deciphered by deriving the major and minor axes and the normal vector of the three-dimensional polarization ellipses [10].
where Re, Im and * E  denote the real part, imaginary part, and the complex conjugate of E  , respectively.

Numerical simulation
Two examples are considered to demonstrate that the method is capable of generating transversely oriented optical polarization Möbius strips in the y = 0 plane in the focal region. The parameters used in the numerical simulation include: the number of sets of dipole antennae N = 12, the numerical aperture of the objective lens NA = 1, and the radius of the circular trajectory 0 r λ = .
For the first example, the data of optical fields generated from a q-plate of topological charge 1/2 is evenly sampled and interpolated. The x-, y-, and z-components of the electric field data is assigned to the x-, y-, and z-dipole at each point on the circular trajectory of   To determi amplitude and Figures 3(a) t 0 plane, respe components in 3(g), that is n ellipses form in fig. 3(h). T red to indicate The second plane, that ha spatially vari distributions distributions o istributions of utions of the E total intensity d polarization e fig. 3(h) fig. 4(a). The b) to 4(c). Th espectively.

Conclusions
In summary, we report a scheme that utilize the time-reversal method and the vectorial diffraction theory to generate transversely oriented optical Möbius strips polarization topology through tight focusing. The topological structures possessing polarization topological charges of -1/2 and -3/2 are constructed along prescribed trajectories in the y = 0 plane. The method can be further extended to build various polarization topologies along arbitrary three-dimensional trajectories in the focal region. Tailoring polarization topologies empowers applications in nanofabrication and quantum communications.