Zernike coefficients of a circular Gaussian pupil

We present a derivation of a formula for Zernike-Gauss circle coefficients expressed in terms of Zernike circle coefficients for a circular Gaussian pupil by revisiting a method in which theorthonormal Zernike-Gauss circle polynomials are not derived first. This is achieved by utilizing a new result, based on the extended Nijboer-Zernike diffraction theory, in which an analytical expression for the autocorrelation of any two Zernike circle polynomials in the circular Gaussian pupils is derived. We use the result to investigate the Strehl ratio of a nearly diffraction-limited optical system that is characterized using Zernike circle coefficients or classical peak-valley coefficients. The results show that aberration correction of the system with a circular Gaussian pupil is most effective if the aberrations are expressed in terms of orthonormal Zernike-Gauss circle coefficients.