Extended Data Fig. 6: Generation of OAM-carrying Bessel-Gauss or Laguerre-Gauss beams by a single or double pass through axicons in our system.

The numerical simulations assume an input Gaussian beam with 500 nm wavelength entering the setup. The phase plate imparting OAM in the far field of the axicon has a charge ℓ = 10. a, Setup schematic for OAM-carrying Bessel-Gauss beam generation. The detection plane (vertical dashed line) is slightly shifted from the 4f plane to allow for the formation of the Bessel-Gauss beam (as in Supplementary Fig. S4a). b,c, Simulated intensity and phase of the beam at the detection plane showing a Bessel-Gauss beam with ℓ = 10. d, Setup schematic for OAM-carrying Laguerre-Gauss beam generation. In this case a second negative axicon being the conjugate of the first positive axicon is placed at 4f. The detection plane is at 2f from the second axicon. e, Due to the their complementary grating shapes, the phase contributions of the two axicons cancel out in this arrangement. f,g, Simulated intensity and phase of the beam at the detection plane showing a Laguerre-Gauss beam with ℓ = 10. In a and d the 4f imaging systems are shown unfolded for clarity and using lenses, though they can be implemented with a single spherical mirror as in Fig. 2a. An amplitude mask is used to block the zero order in the far field of the axicon.