It is a sufficient condition only, not a necessary and sufficient condition, for decomposing wavefront aberrations

Abstract

The classic equation for decomposing the wavefront aberrations of axis-symmetrical optical systems has the form, $$ W({h}_0,\rho,\phi )=\sum_{j=0}^{\propto } \sum_{p=0}^{\propto } \sum_{m=0}^{\propto } {C}_{\left(2j+m\right)\left(2p+m\right)m}({h}_0{)}^{2j+m}(\rho {)}^{2p+m}(\mathrm{cos}\phi {)}^m $$ where j, p and m are non-negative integers, ρ and ϕ are the polar coordinates of the pupil, and h0 is the object height. However, one non-zero component of the aberrations (i.e., C133h0ρ3cos3ϕ) is missing from this equation when the image plane is not the Gaussian image plane. This implies that the equation is a sufficient condition only, rather than a necessary and sufficient condition, since it cannot guarantee that all of the components of the aberrations can be found. Accordingly, this paper presents a new method for determining all the components of aberrations of any order. The results show that three and six components of the secondary and tertiary aberrations, respectively, are missing in the existing literature.