Scalar analytical expressions for the field dependence of Zernike polynomials in asymmetric optical systems with circular symmetric surfaces

Alessandro Grosso,Toralf Scharf

doi:10.1364/osac.396288

Abstract

In this paper we derive scalar analytical expressions describing the full field dependence of Zernike polynomials in optical systems without symmetries. We consider the general case of optical systems constituted by arbitrarily tilted and decentered circular symmetric surfaces. The resulting analytical formulae are inferred from a modified version of the full field dependent wavefront aberration function proposed in the Nodal Aberration Theory (NAT). Such formula is modified with the scope of solving few critical points arising when primary and higher order aberrations are both present in an optical system. It is shown that when secondary aberrations are taken into account in the wavefront aberration function, the final effect is a perturbation to the symmetry of the field dependence of the Zernike polynomials. In particular, the centers of symmetry of the Zernike polynomial field dependences are shifted with respect to the locations predicted using the NAT equations as a consequence of the presence of higher order aberrations. The retrieved analytical expressions are verified through surface fitting to real ray-trace data obtained for a simple optical system.

Highlights

The Nodal Aberration Theory (NAT) proposed by Thompson [1,2] constitutes a fundamental step forward to the development of the wave theory of aberrations for non-circular symmetric optical systems characterized by tilted and decentered circular symmetric surfaces
The terms in Eq (18) correspond to the first nine Zernike polynomials using the Fringe indexing scheme. These terms are at most of 4th order in their pupil coordinates dependence. Considering instead their field coordinates dependence, Eq (18) explicitly shows that in an asymmetric optical system the full field behavior of Zernike polynomials is described by a superposition of polynomial surfaces whose respective displacements in the field of view (FOV) plane are defined by the centers of symmetry of the inherent aberration types contributing to the description of the Zernike terms in question
We have presented scalar analytical formulae describing the full field dependence of Zernike polynomials deriving from the field behavior of their inherent coefficients in asymmetric optical systems characterized by tilted and decentered circular symmetric surfaces

Summary

Introduction

The Nodal Aberration Theory (NAT) proposed by Thompson [1,2] constitutes a fundamental step forward to the development of the wave theory of aberrations for non-circular symmetric optical systems characterized by tilted and decentered circular symmetric surfaces. Developing Eq (2) up to the 6th order, we obtain the following Eq (4) describing the wavefront aberration function constituted by primary and secondary aberrations for asymmetric optical systems with tilted and decentered circular symmetric surfaces: W(Hì , ρì) = W000 + W020(ρì · ρì) + W111[(Hì − ìa111) · ρì]. The complete scalar full field wavefront aberration function W(Hx, Hy, ρ, θ) for asymmetric optical systems constituted by tilted and decentered circular symmetric surfaces is given by the sum of terms exposed in equations from Eq (7) to Eq (15). The scalar wavefront aberration function W(Hx, Hy, ρ, θ) is used in the following part to retrieve the field behavior of Zernike coefficients in asymmetric optical systems constituted by tilted and decentered surfaces

Full field dependence of Zernike polynomials in asymmetric systems

Validation of results

Conclusions