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.