Numerical simulation illustrating the principle of Nijboer-Zernike retrieval applied on the vortex coronagraph. (Up to down), numerical simulation for each circular polarization with l_p = 2 and l_p =  −2 respectively using the first 860 Zernike polynomials (40 complete modes) and a Lyot stop of 99%. The Strehl ratio of the PSF before coronagraphic filtering is 95%. The Lyot stop remove the strong diffraction value in ρ = 1 but allows us to show all images without scaling the coefficients (). (Left to right), final monochromatic (λ = 650 nm) coronagraphic image obtained with the sum of analytical functions, the FFT of the direct sum of Zernike polynomials in the pupil plane, and finally, the classical phase function obtained by . Due to the difference of basis, the final simulation with the classical phase function presents minor discrepancies in the speckle background (I_c < 10^-7, I_c is the coronagraphic final intensity). The image scales are not linear .