Zernike-basis expansion of the fractional and radial Hilbert phase masks

• Zernike functions presented in the context of Nijboer-Zernike theory.
• Extended Nijboer-Zernike (ENZ) coefficients for the fractional Hilbert mask.
• ENZ coefficients for the radial Hilbert phase mask (optical vortex).

The linear Hilbert phase mask or transform has found applications in image processing and spectroscopy. An optical version of the fractional Hilbert mask is considered here, comprising an imaging system with a circular, unobscured pupil in which a variable phase delay is introduced into one half of the pupil, split bilaterally.The radial Hilbert phase mask is also used in image processing and to produce optical vortices which have applications in optical tweezers and the detection of exoplanets.We subjected the fractional and radial Hilbert phase masks to Zernike function expansion in order to compute the image plane electromagnetic field distribution using Nijboer-Zernike theory. The Zernike functions form an orthogonal basis on the unit circle. The complex-valued Zernike expansion coefficients for these two phase masks were derived for use in the context of the Extended Nijboer-Zernike (ENZ) theory of image formation. The ENZ approach is of interest in that it allows a greater range of defocus to be dealt with, provides a simple means of taking a finite source size into account and has been adapted to high Numerical Aperture (NA) imaging applications.Our image plane results for the fractional Hilbert mask were verified against a numerical model implemented in the commercial optical design and analysis code, Zemax®. It was found that the Nijboer-Zernike result converged to the Zemax® result from below as the number of Zernike terms in the expansion was increased.