302
J. Eur. Opt. Society-Rapid Publ. 22, 30( 2026)
Because for S 2m we have in equation( 9) the factor ð�1Þ 2m�1 ¼�1, the coefficients c S1; c S2; c S3 appear with a sign opposite to that in S 2m�1. Using equations( 7) and( 9) we can then write
S 2m�1 ¼ c S1 ðl 2m � l 2m�1 Þ 3 þ c S2 l 3
2m�1 þ c S3l 3 2m
� 3 S 2m ¼ c S1 l 2mþ1 � l 2m � c S2 l 3
2mþ1 � c S3l 3 2m ð14Þ
Note that in the first term of S 2m we have changed the order of l 2mþ1 and l 2m compared to equation( 9), therefore c S1 appears here with the plus sign.
We now replace the temporary variables l k by the new variables z k ¼ðl kþ1 � l k Þ = ðb � aÞ. The total spherical aberration S is then obtained in terms of the new variables z k by summing up the odd and even surface contributions in equation( 14) over all lenses, with m = 1... L. Notethatin this sum all terms with coefficients c S3 cancel each other out, as well as the terms with coefficients c S2, excepting those with l 3
1 ¼ u3 1 ¼ a3; and l 3
2Lþ1 ¼ u3 2Lþ1 ¼ b3. 2.3 The simple spherical aberration expression
By using equations( 8) and( 12) the new variables can be rewritten as
z k ¼ 1 n kþ1 þ 2 b � a 2n kþ1 þ 1 u kþ1 � n
k þ 2 2n k þ 1 u k ð15Þ
and, in the absence of aspheres, spherical aberration becomes
S ¼ c S1 ðb � aÞ 3 X2L z 3 k þ c �
S2 a 3 � b 3; ð16Þ k¼1
where c S1 and c S2 are given by equation( 13).
The marginal ray angles a and b, before and after the thin-lens group, are related to the total power K of the group via b ¼ a � hK: ð17Þ
Including in equation( 16) the aspheric contributions appearing in equation( 1) is straightforward because, apart from an alternating sign, the refractive index difference is the same for all surfaces. Using equation( 17), weobtain for the spherical aberration of the thin-lens group the final expression
" # S ¼ h4 K 3 n ð2n þ 1Þ 2 X 2L z 3
3ðn þ 2Þ ðn � 1Þ 2 k � 3a 2 þ 3a � 1
þ 8h 4 ðn � 1Þ X2L k¼1 k¼1
ð�1Þ k�1 G k; ð18Þ
where we have used the abbreviation a ¼ a hK: ð19Þ
If the group of L thin lenses is part of a larger system, then the 3rd-order spherical aberration of the entire system is
S tot ¼ S þ S; ð20Þ
Figure 1. The paraxially traced marginal ray( thick line) has before the first surface of the group of L lenses the angle u 1 ¼ a with the optical axis and after the last surface the angle u 2Lþ1 ¼ b. The refractive index before surface k is n k, after the surface it is n kþ1. The surface numbering for the marginal ray angles u k is the same. Inside each lens( i. e. for even k values) the refractive index is n, as shown here for the first lens( m = 1) with surfaces 1 and 2. Outside the lenses we have n 1 ¼ n 3 ¼::: ¼ n 2Lþ1 ¼ 1. In this figure, the L lenses of interest form the entire optical system, but the same notation is used when these lenses are part of a larger system. The planes of the object, image, entrance pupil and exit pupil are denoted by O, I, EP and XP respectively. In the thin-lens approximation, all axial distances between surfaces 1 and 2L will be set equal to zero in the aberration formulas.
where S * denotes the contribution of the other lenses in the larger system.
3 Quasi-powers and surface powers
The new variables z k defined by equation( 15) are essential for simplifying the entire thin-lens formalism and provide a new framework for analysing aberrations. In this section we discuss their properties as well as their relationship with the surface powers and curvatures.
When the angles u k inside the lens group are known, the corresponding z k values can be determined using equation( 15). However, as shown in the Examples section, it is sometimes possible to determine the z k values first. Then, the surface curvatures c k result from the z k values as follows. We find from equations( 15) and( 17)
l kþ1 ¼ l k þ ðb � aÞz k ¼ l k � hKz k
¼ l k�1 � hKðz k þ z k�1 Þ ¼::: ¼ a � hK Xk z i: ð21Þ i¼1
For k = 2L we expect to have, because of equation( 17), u 2Lþ1 ¼ b ¼ a � hK. The denominator in equation( 15) was therefore chosen such that the sum of all z-variables is normalized to unity,
X 2L
k¼1 z k ¼ 1:
The surface powers result from equations( 3) and( 8): ð22Þ