Zernike polynomiales for optical systems with rectangular and square apertures of area equal to

Abstract

The property of orthogonality of Zernike circle polynomials and their representation of balanced spherical aberrations made them in a widespread use for wavefront analysis. In the present paper, we derived closed form polynomials that are orthonormal over horizontal and vertical rectangular pupils of an area equal to just the same as the area of Zernike circular pupil. Then the polynomials suitable for a square aperture with the same area are found, using the circular polynomials as the basis functions. The polynomials are given in cartesian and polar coordinates. The values of standard deviation for balanced and unbalanced primary aberration are also calculated for the concerned apertures and compared with that of circular aperture.