Diffraction of plane waves by finite-radius uniform-phase disks and spiral phase plates

Scalar Fresnel and Fraunhofer diffraction by phase objects has been widely studied in recent years. However, to the best of my knowledge, no analysis of diffraction by finite-radius phase objects with variable transmittances has been done. In this work, the Fresnel diffraction of plane waves by two different kinds of circular phase objects is studied. First, a finite-radius disk with uniform phase shift is analyzed, including the effects that varying the disk and outer-plane transmittances has on the diffraction pattern. Comparisons with the exact results for a circular aperture, presented by Dubra and Ferrari, and with Fourier propagation are done for the sake of validation. Next, the Fresnel diffraction of plane waves by spiral phase plates of finite radius is considered. The integer-step spiral phase plate is studied as a particular case to study the vortex distribution and behavior within the diffraction pattern. Finally, the fractional-step spiral phase plate is considered as a more general case, and the behavior of the optical vortices in the diffracted field is analyzed.