Light's capacity to carry angular momentum is integral to our knowledge of physics and ability to probe matter. In addition to spin, photons can occupy free-space orbital angular momentum (OAM) eigenstates.[1, 2] Visible light OAM is used in quantum information experiments, superresolution microscopy, optical tweezers, and angular momentum transfer to atoms in optical lattices.[3, 4] Soft x-ray OAM applications, slowed by the lack of suitable optics and rarity of coherent x-ray sources, could enable the direct alteration of atomic states through OAM exchange, and methods to study the electronic properties of quantum materials. We have made soft x-ray diffractive optics that generate single Laguerre-Gauss modes, observed carrying up to 30h, or their superpositions. We also present Hermite-Gauss diffractive optics, and a soft x-ray OAM analyzer. This set of tools could enable both the manipulation and finer characterization of topologically complex or strongly correlated electronic matter, such as magnetic skyrmions or stripes in superconductors. Efforts to generate and control x-ray and other short wavelength OAM beams is an active field of research. To create OAM in EUV (λ > ∼ 10 nm), groups have used higher-order harmonic up-conversion with optical vortex laser pumps[5-7], off-axis synchrotron undulator radiation[8], and laser-seeded free electron laser techniques[9, 10]. (See Ref. [11], also.) In the hard x-ray regime (λ < ∼ 0.1 nm), phase singular optics, such as stepped phase plates[12] and spiral Fresnel zone plates[13] have been used. Soft x-rays are ideally suited for probing the electronic structure and spin states of molecules and condensed matter. However, these wavelengths are too long to be compatible with hard x-ray optics, and too highly absorbing for use with visible/UV apparatus, making soft x-ray OAM difficult to generate. Binary amplitude diffractive optical elements[14, 15] have successfully been used to create soft x-ray beams characterized by single orbital angular momentum modes. In particular, Sakdinawat et al.[15] created and tested spiral zone plates for phase contrast enhancement in soft x-ray microscopy. We chose to fabricate and test the performance of binary amplitude diffraction gratings that produce OAMcarrying Laguerre-Gauss (LG) TEM modes. LG modes
