J. Eur. Opt. Society-Rapid Publ. 2025, 21, 8 Ó The Author( s), published by EDP Sciences, 2025 https:// doi. org / 10.1051 / jeos / 2025003 Available online at: https:// jeos. edpsciences. org
Journal of the European Optical Society-Rapid Publications
RESEARCH ARTICLE
Transformation equation from wavefront aberrations to ray aberrations based on coordinate frames in object space
Psang Dain Lin * Department of Mechanical Engineering, National Cheng Kung University, Tainan 70101, Taiwan
Received 21 October 2024 / Accepted 15 January 2025
Abstract. Object and image spaces are widely used in geometrical optics to describe the functions of optical systems. However, the mapping between these two spaces is non-linear, meaning that a function expressed in coordinate frames in the image space cannot be directly applied in the object space, and vice versa. For example, the relationship between the wavefront and ray aberrations given by Rayces J.( 1964) Opt. Acta 11, 85 – 88. https:// doi. org / 10.1080 / 713817854 is valid for coordinate frames in the image space but fails for coordinate frames in the object space. To overcome this limitation, this study converts the wavefront-ray aberration relationship using the chain rule so that it can also be applied to coordinate frames in the object space. The numerical results obtained for the primary ray aberrations using the proposed converted relationship are shown to be in close agreement with the Zemax simulation results.
Keywords: Wavefront aberrations, Ray aberrations, Axially symmetrical systems, Object sapce.
1 Introduction
Many approaches for deriving the primary wavefront aberrations of an axially symmetrical system have been proposed( e. g., [ 1 – 5 ]). One of the most widely used methods is that developed by Buchdahl [ 1 ]. In this method, the marginal and chief paraxial rays at a single refractive boundary are traced to compute the Buchdahl aberration coefficients, and these coefficients are then converted to the image space by applying the Gaussian imaging equation sequentially, surface by surface. The final wavefront aberrations are expressed in terms of the object height h 0 and coordinates( x e, y e) in the image space as W( h 0, x e, y e). When the values of the wavefront aberrations are much smaller than the reference sphere radius R s, the ray aberration( DP nx, DP ny) can be estimated as([ 2, 6 ], p. 13 of [ 3 ], [ 7 – 10 ])
P nx
P ny
R s n s
@ W ðh 0; x e; y e Þ =@ x e @ W ðh 0; x e; y e Þ =@ y e
; ð1Þ
where n s is the refractive index of the reference sphere. Notably, the partial derivatives in equation( 1) must be performed in coordinate frames in the image space. One frequently used coordinate frame is( xyz) e attached to the exit pupil( Fig. 1). The need to utilize coordinate frames in the image space arises from the fact that, by doing so, the normal vector of the wavefront can be
* Corresponding author: pdlin @ mail. ncku. edu. tw determined directly by the gradient of the wavefront expression to determine ray aberrations( see Eq.( 1.11) of [ 3 ], [ 11, 12 ]). Equation( 1) is given in a Cartesian coordinate frame( xyz) e. If the wavefront aberration is expressed in terms of polar coordinates as W( h 0, q e, / e), its transformation equation to( DP nx, DP ny) can be obtained from equation( 1) simply by using( x e, y e)=( q e sin / e, q e cos / e)( see Fig. 1), to yield
P nx
P ny
R s n s
sin / e @ W þ 1
cos / e @ W cos / e @ q e q e � sin / e
@/ e
: ð2Þ
The errors of the transferred ray aberrations from equations( 1) or( 2) are not quantified analytically, as stated in the Abstract of [ 13 ].
However, wavefront aberrations can also be determined in terms of the object space parameters without using the Gaussian imaging equation [ 14 – 17 ]. For example, the unit directional vector l 0 originating from a point object P 0 ¼ð0; h 0; P 0z Þ is expressed in terms of the intercepted
in-plane point P ¼ðx; yÞ of the entrance pupil( Fig. 1). The Taylor series expansion can then be employed to obtain the wavefront aberration function as W( h 0, x, y), where( xyz) are coordinate frame in the object space attached to the entrance pupil. Nonetheless, transforming the wavefront aberrations into ray aberrations based on a coordinate frame in the object space is still challenging.
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