Centroid error due to non-uniform lenslet illumination in the Shack-Hartmann wavefront sensor

Images formed by individual Shack-Hartmann wavefront sensor lenslets are displaced proportionally to the average wavefront slope over their aperture. This principle fails when the lenslet illumination is non-uniform. Here we demonstrate that the resulting error is proportional to the linear component of the illumination intensity, the quadratic wavefront component, and the lenslet size. For illustrative purposes, we compare the error due to centered Gaussian illumination decaying by 30% at the pupil edge against the error due to assuming the wavefront at the lenslet center being equal to the wavefront average across each lenslet. When testing up to ninth-order Zernike polynomial wavefronts and simulating nine lenslets across the pupil, the maximum centroid errors due to non-uniform illumination and sampling are 1.4% and 21%, respectively, and 0.5% and 6.7% when considering 25 lenslets across the pupil in the absence of other sources of error.