tokens: spread function of the system. Its energy corresponds to the incoming flux and is normalized to 1. At this point, it is important to note that the sensitivity is bounded. Since the mask transmission, | m |, cannot be greater than 1 (| m | ≤ 1), Eq. ( 7 ) implies that TF| f ≤ 2. Hence, given Eq. ( 8 ), we conclude that, in the frame of our normalizations, the sensitivity cannot be greater than 2: This is an important result as it defines the theoretical limit for the sensitivity of an FFWFS. 2.3. Application to the ZWFS The convolutional formalism introduced in the previous section was then applied to the ZWFS in order to find a simple formula of its sensitivity according to the mask parameters. As previously mentioned, the ZWFS mask is defined by two free parameters. The first is the depth (phase shift) of the dot δ ; for the classical ZWFS, δ = π /2. The second is its diameter p ; for the classical ZWFS, p = 1.06 λ / D . As described in N’Diaye et al. (2013) , the dot diameter value was chosen in order to get an equivalent flux inside and outside the focal plane dot. This configuration with p = 1.06 λ / D also allows a uniform reference intensity distribution to be obtained. The purpose of this section is to demonstrate that this choice of p = 1.06 λ / D is actually not optimal, and that the sensitivity can be improved with a larger dot. For the sake of clarity, we carried out this study in one dimension. As such, the spatial frequency vector f becomes the scalar frequency f . We further assumed that the weighting function ω , which corresponds to the PSF, can be described as a top-hat function of diameter radius a . This simplified geometry is summarized by Fig. 1 a. From this simplified geometry, we then computed TF| f and plot this quantity in Fig. 1 b. The full derivation can be found in Appendix B . Fig. 1. Simplified description of the ZWFS. ( a ): simplified one-dimensional framework for convolutional derivations; in black, the dot diameter is equal to p and the phase shift is equal to δ . In green, the PSF is approximated by a normalized top-hat function with a diameter a . ( b ): transfer function of the ZWFS for two cases; in red the dot size is smaller than the PSF, and in blue the dot is larger than the PSF. The optimal case appears for p = a and not p = a /2, as done for the classical ZWFS. From Fig. 1 b one can distinguish two cases. The first is p ≥ a (i.e., the dot diameter is larger than the PSF characteristic size). Frequencies above ( p + a )/2 reach the sensitivity 2sin( δ ). Frequencies below ( p − a )/2 are at 0. The second is p a : the dot diameter is smaller than the PSF size. Frequencies over p have a value of 2 p / a × sin( δ ), which is smaller than the theoretical limit of 2. From this simplified model one can first conclude that a phase shift of δ = π /2 will maximize the sensitivity, as expected. But surprisingly, it shows that for this phase shift a dot radius of p = a offers a sensitivity of the optimal
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