JEOS RP ISSN01 | Seite 84

J. Eur. Opt. Society-Rapid Publ. 21, 8( 2025) 79
Table 1. Parameters of the optical system shown in Figure 2. The thickness of the Gaussian image plane from the
i = 8th boundary are V 8 = 92.0885 and V 8 = 36.3632 for point objects placed at P 0 ¼
½ 0
17
�200
Š T and infinity,
respectively.
Boundary i
Radius
Thickness
Refractive index
1
38.2219
15.8496
1.65
2
�56.0857
5.9690
1.71736
3
�590.6820
3.0226
1
4( aperture)
0.0000
14.0208
1
5
�41.7957
2.5146
1.52583
6
29.3446
7.9248
1
7
63.5635
6.0960
1.65
8
�56.8655
V 8
1
9( Gaussian image plane)
0
1
Table 2. Transformed ray aberration coefficients and percentage errors for the system shown in Figure 2 with an object at P 0 ¼ ½ 0 17 �200 Š T.
Spherical
Coma
Astigmatism
Field curvature
Distortion
No. 1
�0.112840 10 �5
�0.102638 10 �6
0.915896 10 �8
�0.377611 10 �7
�0.157428 10 �7
No. 2
�0.165551 10 �3
�0.150483 10 �4
0.134526 10 �5
�0.553941 10 �5
�0.230920 10 �5
No. 3
�0.165562 10 �3
�0.150594 10 �4
0.134383 10 �5
�0.554041 10 �5
�0.230983 10 �5
No. 4
�0.165551 10 �3
�0.150483 10 �4
0.134528 10 �5
�0.553946 10 �5
�0.230918 10 �5
No. 5
0.007 %
0.074 %
�0.106 %
0.018 %
0.027 %
No. 6
�0.007 %
0.074 %
�0.108 %
0.017 %
0.028 %
No. 7
�1.533271
�0.112900
0.008156
�0.033625
�0.011348
No. 8
�1.533165
�0.112817
0.008164
�0.033619
�0.011345
No. 9
�1.533165
�0.112817
0.008164
�0.033619
�0.011345
No. 10
0.007 %
0.074 %
�0.106 %
0.018 %
0.027 %
No. 11
�0.014 %
0.110 %
�0.072 %
0.002 %
0.028 %
No. 12
�0.235751 10 �3
�0.190474 10 �4
0.151351 10 �5
�0.623220 10 �5
�0.230918 10 �5
No. 13
�29.777 %
�20.995 %
�11.116 %
�11.116 %
0.000 %
No. 1: wavefront aberration coefficients in( xyz) from [ 15 ]. No. 2: ray aberration coefficients in( xyz) from [ 14 ]. No. 3:
transferred ray aberration coefficients in( xyz) from equations( 11)–( 15). No. 4: ray aberration coefficients B j in( xyz)
transferred from Zemax( Eqs.( A1)–( A5)). No. 5: % error =( No. 3 – No. 2) 100 / No. 2. No. 6: % error =( No. 3 – No.
4) 100 / No. 4. No. 7: peak values of ray aberrations from No. 3. No. 8: peak values of ray aberrations from [ 14 ]. No. 9:
peak values of ray aberrations from Zemax. No. 10: % error =( No. 7 – No. 8) 100 / No. 8. No. 11: % error =( No. 7 – No.
9) 100 / No. 9. No. 12: ray aberration coefficients B e / j
in( xyz) e
from Zemax( Eqs.( A1)–( A5)). No. 13: %
difference =( No. 4 – No. 12) 100 / No. 12.
computed by Zemax based on( xyz) and( xyz) e are given in columns No. 4 and No. 12 of Table 2, respectively( see Appendix). It can be seen from column No. 13, which shows the percentage difference between them, that the spherical aberration coefficient B 1 and coma coefficient B 2 have percentage differences of �29.777 % and �20.995 % between frames( xyz) and( xyz) e, respectively. The percentage differences of the astigmatism coefficient B 3 and field curvature coefficient B 4 in the two coordinate frames are both �11.116 %. The distortion coefficients in( xyz) and( xyz) e are identical since the distortion is a function of the object height h 0 only.
Equations( 8)–( 15) are also valid for systems with an object at infinity, provided that the object height h 0 is replaced by the field angle b 0. Table 3 lists the numerical results for the system in Figure 2 with an object at infinity and b 0 = 4.377 °. The percentage errors compared with the values listed in column No. 2 and those obtained from Zemax( No. 4), respectively, are very small. Thus, the results confirm that equation( 8) can transform wavefront aberrations into ray aberrations with high precision when the image plane is Gaussian.
Similarly, equation( 1) can be revised to transform the ray aberration coefficients based on the Cartesian