J. Eur. Opt. Society-Rapid Publ. 21, 8( 2025) 77
Fig
. 1. Four coordinate frames are frequently used in geometrical optics:( xyz) 0,( xyz),( xyz) e, and( xyz) n. These frames are referred to as the world coordinate frame, coordinate frame attached to the entrance pupil, coordinate frame attached to the exit pupil, and coordinate frame attached to the image plane, respectively. It is also shown that the entrance and exit pupils, image plane are located at( 0, 0, z),( 0, 0, z e), and( 0, 0, z n), respectively, defined by( xyz) 0.
Fig. 2. Axially symmetrical system with nine boundaries [ 18 ]. The ray path HðP 0; l 0 Þ is the path between points P 0 and P n.
To address this problem, this study converts equation( 2) from the image space to the object space using the chain rule. The numerical results show that the ray aberrations obtained by this converted equation are in very close agreement with the Zemax results.
2 Relationship between wavefront and ray aberrations in object space
This work distinguishes between a“ ray” and a“ ray path.” Referring to Figure 2, arayR i ¼ðP i; l i Þ includes the incidence point P i and its unit directional vector l i. Whena source ray R 0 ¼ðP 0; l 0 Þ travels through an axially symmetrical optical system, multiple rays( R i, i = 0ton) are distributed along the ray path HðP 0; l 0 Þ( Fig. 2).
Equation( 2) must be mathematically converted in order to operate on the polar coordinate frame in object space. One of the most useful coordinate frames in object space is( xyz), attached to the entrance pupil. Generally, the relationship of the intercepted points of a general ray path on the entrance pupil and exit pupil, P ¼ðq; / Þ and P e ¼ðq e; / e Þ, are given by q e = q e( q, /) and / e = / e
( q, /). Thus, equation( 2) can be converted using the following chain rule:
R
s sin / e @ W n s cos / e @ q þ 1
cos / e @ W q e
� sin / e @ q
P nx
P ny
@ q þ @ W @/ e @/
@/ @ q e
@/
: ð3Þ
@/ e
@ q þ @ W @ q e @/