Manipulation of the orbital angular momentum of soft x-ray beams by consecutive diffractive optics

Abstract

Production and manipulation of orbital angular momentum (OAM) of coherent soft x-ray beams is demonstrated utilizing consecutive diffractive optics. OAM addition is observed upon passing the beam through consecutive fork gratings. The OAM of the beam was found to be decoupled from its spin angular momentum (SAM). Practical implementation of angular momentum control by consecutive devices in the x-ray regime opens new experimental opportunities, such as direct measurement of OAM beams without resorting to phase sensitive techniques, including holography. OAM analyzers utilizing fork gratings can be used to characterize the beams produced by synchrotron and free electron lasers sources; they can also be used in scattering experiments.

Figures (3)

• Figure 1: Schematic of the experimental set-up for the generation and manipulation of soft x-ray OAM beams using two consecutive diffraction fork gratings. The first grating ($L_1$) produces the OAM beams diffracted along the vertical $z$-direction, using the coherent gaussian beam from a synchrotron beamline collimated by an 8 $\mu$m pinhole. One of these OAM beams, $L_1$=+2$\hbar$, is then selected to impinge on the second fork grating ($L_2$) used as the sample which diffracts the beams with modified OAM along the $z'$-direction. Both diffraction patterns are observed on the area detector in straight-through configuration. In the experiment shown, the two-step diffraction setup yields the beams with total OAM given by $L_{\rm tot}$=(2$\pm$2k)$\hbar$, where $k$=0, 1, 2, 3, etc. Intersecting white lines on the detector image are added to illustrate the increasing size of the diffracted beams with increasing diffraction order away from the $L_{\rm tot}$ =0 position.
• Figure 2: (a) Detector images of the diffraction patterns containing the OAM beams generated by different combinations of consecutive fork gratings. OAM beams of different momenta ($L_1$= -4$\hbar$, 4$\hbar$, -1$\hbar$, 2$\hbar$, 3$\hbar$) are used selectively to illuminate the 2nd grating. The diffraction from the 2nd grating changes the OAM of the incident beam according to the OAM addition rule. (b) and (c) Solid lines depict the intensity profiles of the linecuts through the centers of the OAM beams generated by 1-fork grating illuminated by OAM beams with $L_1$=4$\hbar$ and -4$\hbar$, respectively. Different LG modes of the diffracted beams are indexed by the orbital angular momentum quantum number $\ell$. Gaussian beams ($\ell$=0) are shaded. Vertical solid lines indicate the positions of the transmitted beams from the second grating at $z'$=0. Dashed lines show the numerical simulations, as described in the text. (d) The OAM beam size as function of the total OAM, $L_{\rm tot}$=$L_1$+$L_2$. Vertical error bars reflect experimental resolution.
• Figure 3: (a) Detector images of the diffraction patterns produced by a 2-fork grating illuminated by $\ell$=+1 beam generated by 1-fork grating. The three panels correspond to different polarization states of the incoming beam from the synchrotron beamline, as indicated. (b) Line profiles through the centers of the diffracted beams shown in (a).