J. Eur. Opt. Society-Rapid Publ. 22, 30( 2026) 305
For L = 1, we recover familiar results of traditional thinlens theory. The system with z 1 ¼ z 2 ¼ 1 = 2 corresponds then to the well-known singlet with optimal bending that has minimal spherical aberration. Traditional thin-lens theory uses the magnification variable C ¼ ða þ bÞ = ða � bÞ. Inserting in equation( 18) s min ¼ 1 = 4anda ¼ð1 þ CÞ = 2 leads to the well-known minimal spherical aberration formula [ 1 ]
S min ¼ 1 4 h4 K 3 n n ðn � 1Þ �!
C 2
: ð46Þ 2 n þ 2
For larger L, an interesting result that has a rather complex derivation in the literature follows easily from the present model. Fulcher has shown that 3rd-order spherical aberration can be corrected with thin lenses having the same power, but different bendings. In his telescope objective, four lenses with a refractive index close to n = 1.5 are used to achieve this goal [ 3 ]. For L = 4 we have for all kz k ¼ 1 = 8 and s min ¼ 1 = 64. With a ¼ 0( object at infinity) equation( 18) leads to
S min ¼� h4 K 3 nð2n � 3Þð10n � 7Þ: ð47Þ
64ðn � 1Þ 2 ðn þ 2Þ
Spherical aberration vanishes for n = 1.5 because of the first parenthesis in the numerator. It follows from equation( 28) that all four lens powers are equal, P m ¼ K = 4, despite of thefactthatthefourlenseshavedifferentcurvatures( the surface powers resulting from equation( 26) are the same as those listed in Table 1 of Ref. [ 3 ]). As shown by Shafer, Fulcher systems are good starting points for further design and lead to relaxed designs that have an axial imaging of excellent quality even at large apertures [ 4 ]. For L = 2, converting the four equal quasi-powers in equation( 44) into curvatures using equations( 26) and( 27) leads to a doublet configuration that is also given as a typical example of a relaxed design( see Fig. 1 of Ref. [ 4 ]). Figure 2 shows a Fulcher quartet appearing as a lens group in a lithographic objective having only spherical surfaces and a numerical aperture of 0.56. This system is closely related to a system in [ 5, 6 ]. The similarity with the lens shapes in the blue box supports the interpretation of the lenses in the red box as essentially a Fulcher group.
5.2 Permutation symmetry for spherical aberration
In many imaging systems, including the one shown in Figure 2, we encounter groups of lenses having reasonably small thicknesses and air spaces between them. Simplified models, including the thin-lens approximation used here, rarely yield accurate quantitative results, but the deliberate neglect of distracting complexities can reveal qualitative properties that are otherwise obscured. The principal motivation behind deriving the thin-lens formulas was to provide a simplified framework for gaining insight into the properties of the lens design landscape. Because of the extensive derivations involved, detailed examples will be presented in a separate paper. Here we show an example that helps answering a fundamental question in optical
Figure 2. Red box: Fulcher group in an optimized design in which all lenses have the same material. Blue box: the shapes of the same four lenses resulting from the present thin-lens model using z k ¼ 1 = 8 and the marginal ray angles a and b before and after the four-lens group, extracted from the optimized design. In the drawing, the lens thicknesses, which are assumed to be zero in the calculations, are kept the same as in the red box.
system optimization: why are there so many local minima in the design landscape?
The existence of certain local minima in the optimization landscape can already be explained using 3rd-order aberrations. If the surrounding landscape is not flat, higher-order aberrations only determine how deep these minima are. In an optimization landscape with specifications that make spherical aberration the most significant aberration, consider for simplicity a rough approximation of the error function, E = S tot 2. BecauseinS given by equation( 18) the quasi-powers appear in the sum of cubes s, the mathematical property of commutativity leads to permutation symmetry: if a certain set of variables z k corresponds to a local minimum, then any permutation of these variables will have the same values of s, S, S tot( given by equation( 20)) and E. Any such permutation will then correspond to a different minimum, a property that increases the number of existing minima in the landscape significantly. This permutation symmetry was not visible in earlier formalisms, because of the sequential character of ray propagation( rays pass first through surface 1, then through surface 2 etc.). However, the quasi-power formalism reveals this symmetry because the sequential character of ray propagation is now absorbed in equation( 26) and is therefore separated from the more important aberration properties resulting from equation( 18), or, more generally, from equation( 43).
Figure 3 shows an example of the effect of the permutation symmetry in S on the number of local minima. As shown previously [ 7 ], local minima in the optimization landscape surrounding the system in Figure 2 generally have localized changes in the corresponding system drawings. The red boxes in Figure 3 contain local minima for which the most significant changes occur within the group of four lenses considered in Figure 2. These systems have been obtained with CODE V, with lens curvatures as optimization variables, and distortion control added to the default error function. For this study, telecentricity was not controlled and edge thickness control inside this lens group was disabled.
In Figure 3 we consider the same group of four lenses as in Figure 2. For the upper left local minimum, the model