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.

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The optical system is (almost) diffraction-limited. \( \Phi \) is identically zero, and the measured point-spread function (R) is close to an Airy disk.