JEOS RP ISSN03 | Page 314

J. Eur. Opt. Society-Rapid Publ. 22, 30( 2026) 307

Table 1. The quasi-power surface contributions for spherical aberration( QPS) and coma( QPC) differ significantly from the corresponding traditional surface contributions for spherical aberration( Trad. S) and coma( Trad. C). The constant terms( Const.), which are absent(–) in the traditional approach, are also listed in the QPS and QPC columns. The values in the columns Trad. S and Trad. C are identical with the corresponding Seidel coefficients SPHA S1 and COMA S2 listed by Zemax.

5.3 Aplanatic correction

In the special case of Fulcher-like thin-lens systems it can be easily seen that the 3rd-order coma formula( equation( 42)) is also consistent with traditional aberration theory. For equal quasi-powers( as in equation( 44)) the alternating sum of squares in the coma formula vanishes. Coma itself then vanishes for a ¼ 1 = 2, which corresponds to the case of equal conjugates( i. e. transverse magnification M T ¼�1 in equation( 29)). However, if the stop is at the lens, the system is symmetric with respect to the stop, and the zero-coma value can also be derived from the traditional symmetry principle [ 9 ].

While in the traditional approach the total values of the Seidel aberrations result only from sums over surfaces, in the present approach the corresponding totals in e. g. equation( 43) include constant terms in addition to the sums of quasi-power terms over the surfaces. In equations( 16) for spherical aberration and( 41) for coma we can consider the terms c S1 ðb � aÞ 3 z 3 k and ð�1Þk�1 c C1 ðb � aÞ 2 z 2 k to be the“ quasi-power � surface contributions �”. The constant terms are then c S2 a 3 � b 3 and cC2 a 2 � b 2, respectively. The example below shows that, numerically, the quasi-power surface contributions can differ significantly from the corresponding traditional ones. Using the Fulcher approach to annul spherical aberration and symmetry to annul coma, the thin triplet with equal conjugates shown in Figure 4 can achieve 3rdorder aplanatic correction in the infrared region. For spherical aberration, the equivalent of equation( 47) for L = 3and

a ¼ 1 2( which corresponds to M T ¼�1) is

S min ¼� h4 K 3 ðn � 4Þnð5n � 2Þ; ð48Þ

108ðn � 1Þ 2 ðn þ 2Þ

which becomes zero for n = 4( germanium in infrared). For the triplet shown in Figure 4 the quasi-powers and the corresponding surface radii R k ¼ 1 c k resulting from equations( 26) and( 27) are listed in Table 1, togetherwithaberration coefficients computed using an entrance pupil diameter of 1, and a field angle of 10 degrees. We then have a ¼ 1 = 4; b ¼�1 = 4. While for spherical aberration the traditional surface contributions vary significantly( note for instance that surfaces 1 and 6 are aplanatic), all quasipower surface contributions are identical( because the z-values are identical, their cubes are also identical). The zero total spherical aberration is achieved due to the constant term, which has the opposite sign and six times the magnitude of the surface contributions. For coma, all quasi-power surface contributions have the same magnitude, but their total vanishes because of their alternating signs.

In the system shown in Figure 4 three lenses have been used to correct 3 rd-order spherical aberration and coma. It is well-known that in fact only two thin lenses are sufficient for annulling, not only these two aberrations, but axial colour as well, while keeping the desired value of the focal length [ 2 ].( Four solutions can be found, and it will be shown in a future publication that the quasi-power approach can explain the reason why the number of possible solutions is precisely four.) However, the approach used above, called by Shafer the“ relaxation design method” [ 4 ], achieves, as in the Fulcher case, more than just annulling spherical aberration for appropriate values of a, b, and n. The fact that spherical aberration given by equation( 48) is also an extremum with respect to small changes of quasi-power leads to a flat design landscape around the solution in Figure 4. Pioneered by Glatzel, the“ relaxation design method” is especially useful for systems having a high numerical aperture, due to better tolerances and reduced high-order aberrations, but often at the cost of an increased element count [ 4 ]. It is therefore unsurprising that Fulcherlike groups are often encountered as building blocks in lithographic objectives like the one shown in Figure 2.( There, a 2nd Fulcher-like building block can be found in the first wide group of lenses.)

Conclusion

This paper introduces a novel framework for analysing the aberrations of thin lenses, based on the concept of surface quasi-power. The equation( 43) shows that in this framework all Seidel aberrations follow the same remarkably simple polynomial pattern. Apart from a constant