78
J. Eur. Opt. Society-Rapid Publ. 21, 8( 2025)
From the numerical simulation results, it is found that o // o / e 1, o // oq e 0, and oq / o / e 0holdovertheentire entrance pupil domain. Thus, equation( 3) can be further simplified as
P nx
P ny
R s n s
sin / e cos / e
@ W @ q
@ q @ q e þ 1 q e
cos / e � sin / e
@ W @/
Since equation( 4) is an approximation equation, it can be further simplified without degrading its effectiveness by setting cos / e = cos /, sin / e = sin /, andq / q e = oq / o q e to obtain
P nx
P ny
R s n s
@ q sin / @ q e cos /
@ W @ q þ 1 q
cos / � sin /
@ W @/
: ð4Þ
: ð5Þ
The numerical value of oq / oq e depends on q, theradial coordinate of the entrance pupil intercepted by the ray path HðP 0; l 0 Þ. To simplify equation( 5), oq / oq e can be approximated either by @ q @ q e
¼ q max q e = max
; ð6Þ
where q max and q e / max are the maximum entrance and exit pupil radii, respectively, or
@ q
¼ @ q; ð7Þ
@ q e @ q e axis
where( oq / oq e) axis is the pupil magnification evaluated by paraxial optics. The numerical results show that, compared with equation( 7), equation( 6) may introduce additional errors into equation( 5). It is thus discarded here. By substituting equation( 7) into equation( 5), the following relationship between the wavefront and ray aberrations applicable in the object space is obtained:
P nx
P ny
R s n s
@ q @ q e axis
sin / cos /
@ W @ q þ cos / 1
� sin / q
@ W @/
Consider the illustrative optical system with a Gaussian image plane shown in Figure 2. Assume that the point object is located at P 0 ¼ð0; h 0; P 0z Þ¼ð0; 17; �200Þ. The lengths of this paper are specified in millimeters. The parameters of the optical system are presented in Table 1. The primary wavefront aberrations W( h 0, q, /) andthe ray aberrations( DP nx, DP ny), based on the polar coordinate frame built at the entrance pupil, are given respectively by( Eq.( 1) of [ 15 ] and Eqs.( 2a) and( 2b) of [ 14 ]) as follows:
Wðh 0; q; / Þ¼C 040 q 4 þ C 131 h 0 q 3 cos /
: ð8Þ þ C 222 h 2
0 q2 ðcos / Þ 2 þ C 220 h 2 0 q2 þ C 311 h 3
0q cos /; ð9Þ
P nx
P ny
sin / ¼ B 1 q 3 cos /
þ B 4 h 2 þ B 3 h 2
0 q sin / 3cos /
þ B 2 h 0 q 2
0 q sin / cos /
|
sinð2 / Þ 2 þ cosð2 / Þ |
|
|
þ B 5 h 3
0
0
1
|
: ð10Þ
It is noted that the second-order wavefront aberration( i. e., defocus wavefront aberration C 020 q 2, magnification wavefront aberration C 111 h 0 qcos /) andfirst-order defocus ray aberration( i. e., A 1 q( sin /, cos /)) are not included in equations( 9) and( 10), respectively, since their coefficients, C 020, C 111, andA 1, are zero-valued when the image plane is Gaussian. By substituting equations( 9) and( 10) into equation( 8), the following transformed ray aberration coefficients are obtained:
B 1 ¼ 4C 040
R s n s
B 2 ¼ C 131
R s n s
B 3 ¼ C 222
R s n s
@ q @ q e
@ q @ q e
@ q @ q e
B 4 ¼ ð2C 220 � C 222 Þ R s n s
B 5 ¼ C 311
R s n s
@ q @ q e
; ð11Þ axis
; ð12Þ axis
; ð13Þ axis
@ q @ q e
; ð14Þ axis
; ð15Þ axis
where( oq / oq e) axis = 1.11521263, and the radius and refractive index of the reference sphere are R s = 131.564842 and n s = 1, respectively.
The numerical values of the transformed ray aberration coefficients obtained from equations( 11) to( 15) are given in column No. 3 of Table 2. The ray aberration coefficients from Zemax simulation are listed in No. 4( see Appendix). The percentage errors of these transformed ray aberration coefficients relative to the values in columns No. 2 and No. 4 are listed in columns No. 5 and No. 6, respectively. The results presented in columns No. 5 and No. 6 show that equations( 11)–( 15) transfer the wavefront aberration coefficients to ray aberration coefficients with significant accuracy. The peak values of the transferred ray aberrations( i. e., B 1 q 3 max, B 2q 2 max, B 3h 2
0 q max, B 4 h 2 0 q max, andB 5 h 3
0, where object height h 0 = 17 mm and maximum opening radius of the entrance pupil q max = 21 mm) and those obtained from [ 14 ] and Zemax simulation are listed in columns No. 7, No. 8, and No. 9, respectively. The percentage errors of No. 7 with respect to No. 8 and No. 9 are shown in columns No. 10 and No. 11, respectively.
It should be noted that the numerical values of the ray aberration coefficients based on( xyz) are different from those based on( xyz) e due to the non-linearities of q e = q e( q,/) and / e = / e( q,/). The ray aberration coefficients