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

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

At sample time \(k = 0\), the optical system is almost diffraction-limited.

An aberration is introduced at \(k = 1\). Three measurements of the point spread function 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 deformable mirror. To assess the quality of the correction, the residual aberration \(\Phi\) is recorded using a Shack-Hartmann 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.

- At time instant \(k = 0\) the optical system is (almost) diffraction-limited. \(\Phi\) is identically zero, and the measured PSF (R) is close to the Airy disk.
- At \(k = 1\), an aberration is introduced. The measured residual aberration is \(\operatorname{rms}(\Phi) \approx 1\), and the PSF is visibly aberrated (\(d_1=0\)).
- At \(k = 2\), a first defocus diversity \(d_2 < 0\) is introduced with the DM, and the PSF is measured.
- At \(k = 3\), a second defocus diversity \(d_3 > 0\) is introduced with the DM, and the PSF is measured.
- Finally, at \(k = 4\), the PR algorithm is applied to the PSF measurements \(d_1\), \(d_2\), and \(d_3\), which takes \(\Delta t_c\) seconds. After that, the aberration correction is applied, and an improved PSF is measured.