304
If in equation( 36) we substitute equation( 8) and expand the square, the resulting expression must be equal to the expanded form of the first of equation( 34). Subtracting these two expressions gives for m = 1 the zero polynomial
0 ¼ t 0 ðu 1 � u 2 Þðnu 1 � u 2 Þ �c C1 ðqu 2 � u 1 Þ 2 �c C2 u 2
1 � c C3q 2 u 2 2
� ¼ u 2
2 �c �
C1q 2 � c C3 q 2 þ t 0 � u
2
1 c
ð C1 þ c C2 � nt 0 Þ þu 2 u 1 ð2c C1 q � ðn þ 1Þt 0 Þ ð37Þ
After substituting q and t’ using equations( 12) and( 34) we obtain by annulling the three coefficients of the quadratic terms the system of equations
c C1 þ c C2 ¼ hnu n�1
2c C1 ðnþ2Þ hu
¼ ð nþ1 Þ 2nþ1 n�1
hu ¼ ðnþ2Þ2 ðc C1 þc C3 Þ n�1 ð2nþ1Þ 2
J. Eur. Opt. Society-Rapid Publ. 22, 30( 2026)
ð38Þ
T j ¼ a j ðb � aÞ j
X 2L
k¼1 ð�1 Þ ðjþ1Þðk�1Þ z j k
!
� þ b j a j � b j: ð43Þ
The exponent of �1 was chosen such that for odd indices j all terms z j k have the same sign for all values of k, and that for even j the signs of z j k are alternating. For spherical aberration and coma we have j = 3 and j = 2, respectively, with T 3 ¼ S given by equation( 16) and T 2 ¼ C given by equation( 41), and the coefficients are a 3 ¼ c S1; b 3 ¼ c S2; a 2 ¼ c C1; b 2 ¼ c C2. The aberrations that have simple expressions also fit into this pattern. For astigmatism and Petzval sum( equations( 31) and( 32)) we have j = 1. The sum in equation( 43) is then 1 because it becomes the constraint( 22), therefore both aberrations are constant. According to equation( 17), forj = 1 both factors ðb � aÞ j and a j � b j are proportional to the power K, a property that is in agreement with equation( 31). The distortion T 0 is zero as expected, because for j = 0the sum with alternating terms in equation( 43) is L � L ¼ 0and we have a 0 � b 0 ¼ 1 � 1 ¼ 0. that gives for c C1 and c C2( c C3 will not be needed) c C1 ¼ hu ðnþ1
c C2 ¼
Þð2nþ1Þ; 2ðn�1Þðnþ2Þ
hu: ð39Þ 2ðnþ2Þ
Because of the factor ð�1Þ k�1 in equation( 35), fortheeven surfaces the three coefficients are exactly the opposite of those in equation( 36) and we have
� 2 C 2m ¼�c C1 l 2mþ1 � l 2m � c C2 l 2
2mþ1 � c C3l 2
2m: ð40Þ
When we sum up the surface contributions( 36) and( 40) over all lenses, all c C3 terms cancel each other out, as well as the c C2 terms, excepting those with l 2
1 ¼ u2 1 ¼ a2, and l 2 2Lþ1 ¼ u2 2Lþ1 ¼ b2.
4.3 Polynomial pattern
With the new variables defined by equation( 15), thequad- ratic terms appear in the coma expression C with alternating signs,
C ¼ c C1 ðb � aÞ 2 X2L ð�1Þ k�1 z 2 k þ c �
C2 a 2 � b 2; ð41Þ k¼1
where c C1 and c C2 are given by equation( 39). Alternatively, the coma contribution of the thin lens group, with the stop at the lens group, can be written as
" #
C ¼ �uh3 K 2 2ðn þ 2Þ ðn þ 1Þð2n þ 1Þ n � 1
X 2L
k¼1 ð�1Þ k z 2 k � 2a þ 1
: ð42Þ
For an arbitrary stop position, the contribution of the thin |
lens group to the primary aberrations can be computed by |
using the well-known stop-shift formulas [ 1, 2 ]. |
Note that, when all surfaces are spherical, all Seidel |
( monochromatic) |
aberration |
formulas |
have |
the |
same |
structure |
|
|
|
|
|
5 Examples
The thin-lens formulas for primary aberrations derived in this paper have been verified using the lens design programs CODE V and Zemax OpticStudio. For lens systems where distances between surfaces have been set to zero, the quasipowers are computed using paraxial ray-tracing data and equation( 15). Then, as shown in the supplementary data, implementing the new aberration formulas in the macro languages leads to numerical values that are identical with the corresponding coefficients listed by these programs( see the link in the Data availability statement).
5.1 Equal quasi-powers
In the examples below we consider only systems having spherical surfaces. We first focus on the spherical aberration S. We denote the sum of cubes that appears in equations
( 16) and( 18) by s ¼ P2L z 3 k. If for a system consisting of k¼1
L lenses the quasi-powers z k are considered to be variables that satisfy the constraint( 22), then it can be seen that, because of the perfect symmetry, a system having equal quasi-powers, i. e. z k ¼ 1 = ð2LÞ; ð44Þ
for all k-values, must be an extremum of s. By slightly perturbing this system by a small quantity e, z 1 ¼ 1 þ e and
ð2LÞ
( to satisfy the constraint) z 2 ¼ 1 � e; z 2L k ¼ 1 for k > 2, we
2L obtain sðeÞ ¼1 = ð4L 2 Þþ3e 2 = L which is always larger than
s min ¼ � 1: ð45Þ
4L 2
Several known results can be easily derived by starting from systems with equal quasi-powers, that are minima of s, with the minimum value s ¼ s min.