  The Zernike wavefront sensor (ZWFS) is a concept belonging to the wide class
Fourier-filtering wavefront sensor (FFWFS). The ZWFS is known for its extremely
high sensitivity while having a low dynamic range, which makes it a unique
sensor for second stage adaptive optics (AO) systems or quasi-static
aberrations calibration sensor. This sensor is composed of a focal plane mask
made of a phase shifting dot fully described by two parameters: its diameter
and depth. In this letter, we aim to improve the performance of this sensor by
changing the diameter of its phase shifting dot. We begin with a general
theoretical framework providing an analytical description of the FFWFS
properties, then we predict the expected ZWFS sensitivity for different
configurations of dot diameters and depths. The analytical predictions are then
validated with end-to-end simulations. From this, we propose a variation of the
classical ZWFS shape which exhibits extremely appealing properties. We show
that the ZWFS sensitivity can be optimized by modifying the dot diameter and
even reach the optimal theoretical limit, with a trade-off for low spatial
frequencies sensitivity. As an example, we show that a ZWFS with a 2{\lambda}/D
dot diameter (where {\lambda} is the sensing wavelength and D the telescope
diameter), hereafter called Z2WFS, exhibits a sensitivity twice higher than the
classical 1.06{\lambda}/D ZWFS for all the phase spatial components except for
tip-tilt modes. Furthermore, this gain in sensitivity does not impact the
dynamic range of the sensor, and the Z2WFS exhibits a similar dynamical range
as the classical 1.06{\lambda}/D ZWFS. This study opens the path to the
conception of diameter-optimized ZWFS.
