What should be done to the measured Zernike coefficients when conjugating the pupil and wavefront sensor planes with...

The aim of this paper is to describe and demonstrate what should be done to the measured Zernike coefficients when conjugating the pupil and wavefront sensor planes with a 4f system. I provide theoretical and experimental evidence. The experimental setup consisted of two 4f systems of magnifications 1 and 1/3 with their corresponding wavefront sensors at their ends. Spherical and cylindrical trial lenses were measured. In addition, I measured a phase plate with high-order aberrations. I show that the Zernike coefficients of the wavefront expansion at two planes conjugated by a 4f system are related independently of the magnification of the 4f system by the following equation: b_i=(-1)a_i, with n being the order of the radial Zernike polynomial.