Features of Zernike function properties affecting aberration recognition capabilities

In this paper, we study in detail the properties of both individual Zernike functions and their superpositions, which are of particular interest from the point of view of the possibilities of detecting and recognizing wavefront aberrations. The paper shows that paired Zernike functions that differ only in the sign of the angular index provide the formation of distributions whose intensity in the focal plane is invariant to rotation. In addition, we obtain an analytical expression for degenerate superpositions of the Zernike function with the same radial index, which provides the formation of distributions with a pronounced 1D dependence. Due to the noted features, some types of the superpositions considered may complicate their recognition by the intensity pattern in the focal plane using standard error reduction algorithms.