#### Aberration correction using the extended Nijboer–Zernike theory and PhaseLift

Phase retrieval is used to correct an aberration in an optical breadboard.

At sample time $$k = 0$$, the optical system is almost diffraction-limited. An aberration is introduced at $$k = 1$$. Three measurements of the PSF are collected, namely the initial aberration $$d_1$$ at $$k = 1$$, and two defocused measurements $$d_2$$ and $$d_3$$, respectively at $$k = 2$$ and $$k = 3$$. At this point, the phase retrieval algorithm is applied to the three measurements to obtain an estimate of the aberration. Using such an estimate, the aberration correction is performed with a DM at $$k = 4$$. To assess the quality of the correction, the residual aberration $$\Phi$$ is recorded using a Shack-Hartmann wavefront sensor.

The extended Nijboer-Zernike theory is used for modelling the PSF, and the phase retrieval problem is solved using PhaseLift, a signal recovery method based on matrix rank minimisation.

More details are found here.

 select frame: [0]    1     2     3     4 The optical system is (almost) diffraction-limited. $$\Phi$$ is identically zero, and the measured point-spread function (R) is close to an Airy disk.