Zero-order and higher-order Bessel beams are well-known nondiffracting beams. Namely, they propagate with invariant profile (intensity) and carry a fixed orbital angular momentum. Here, we propose and experimentally study an anomalous Bessel vortex beam. Unlike the traditional Bessel beams, the anomalous Bessel vortex beam carries decreasing orbital angular momentum along the propagation axis in free space. In other words, the local topological charge is inversely proportional to the propagation distance. Both the intensity and phase patterns of the generated beams are measured experimentally, and the experimental results agree well with the simulations. We demonstrate an easy way to modulate the beam’s topological charge to be an arbitrary value, both integer and fractional, within a continuous range. The simplicity of this geometry encourages its applications in optical trapping and quantum information, and the like.

As an indispensable complement to an integer vortex beam, the fractional vortex beam has unique physical properties such as radially notched intensity distribution, complex phase structure consisting of alternating charge vortex chains, and more sophisticated orbital angular momentum modulation dimension. In recent years, we have noticed that the fractional vortex beam was widely used for complex micro-particle manipulation in optical tweezers, improving communication capacity, controllable edge enhancement of image and quantum entanglement. Moreover, this has stimulated extensive research interest, including the deep digging of the phenomenon and physics based on different advanced beam sources and has led to a new research boom in micro/nano-optical devices. Here, we review the recent advances leading to theoretical models, propagation, generation, measurement, and applications of fractional vortex beams and consider the possible directions and challenges in the future.

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Topological magnetic monopoles, also known as hedgehogs or Bloch points, are threedimensional (3D) nonlocal spin textures that are robust to thermal and quantum fluctuations due to their topology 1-4 . Understanding their properties is of both fundamental interest and practical applications 1-9 . However, it has been difficult to experimentally produce topological magnetic monopoles in a controlled manner and directly observe their 3D magnetization vector field and interactions at the nanoscale.Here, we report the creation of 138 stable topological magnetic monopoles at the specific sites of a ferromagnetic meta-lattice at room temperature. We further develop 3D soft xray vector ptychography to determine the magnetization vector and emergent magnetic field of the topological monopoles with a 3D spatial resolution of 10 nm. This spatial resolution is comparable to the magnetic exchange length of transition metals 10 , enabling us to probe monopole-monopole interactions. We find that the topological monopole pairs with positive and negative charges are separated by 18.3±1.6 nm, while the positively and negatively charged pairs are stabilized at comparatively longer distances of 36.1±2.4 nm and 43.1±2.0 nm, respectively. We also observe virtual topological monopoles created by magnetic voids in the meta-lattice. This work demonstrates that ferromagnetic metalattices could be used as a new platform to create and investigate the interactions and dynamics of topological magnetic monopoles. Furthermore, we expect that soft x-ray vector ptychography can be broadly applied to quantitatively image 3D vector fields in magnetic and anisotropic materials at the nanoscale.