Focal-plane field when lighting double-ring phase elements

The focal-plane field amplitude is calculated when lighting double-ring phase elements by flat and Gaussian beams. Emerging conditions in the minimum or maximum centers, including flat-top maxima, are given. For the field amplitude, we obtain equations that define the radius of the first zero-intensity ring based on the deduced expressions. The root values are listed for several parameters of optical elements and incident beams due to the lack of analytical solutions. Numerical simulation results are given for flat incident beams; they are fully consistent with the theoretical calculations.