# 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.

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