Theory of diffraction of vortex beams from structured apertures

In this work, a comprehensive analytic study of the diffraction of vortex beams from structured apertures is presented. We formulate the near- and far-field diffraction of a vortex beam from an aperture having an arbitrary functionality in the Cartesian coordinates by two general and different approaches. We show that each of the resulting diffraction patterns can be determined by a number of successive derivatives of 2D Fourier transform of the corresponding hypothetical aperture function or equally can be obtained by a summation of 2D Fourier transforms of the corresponding modified aperture function. We implement both introduced analytic approaches in predicting of the diffraction of a vortex beam from an elliptic Gaussian aperture, an elliptic Gaussian phase mask, and a hyperbolic Gaussian phase mask in the far-field regime. It is shown that the predicted diffraction patterns by both of approaches are exactly the same. It is shown that the diffraction of a vortex beam from an elliptic Gaussian aperture at the far-field regime forms a light beam that belongs to a family of light beams we call as elegant elliptical vortex Hermite–Gaussian (vHG) beams. Our general analytic formula can be used for a large variety of apertures.