J. Eur. Opt. Society-Rapid Publ. 21, 27( 2025) 285
Figure 7. Paraxial estimation reliabilities 1, 2 for the dispersion of the two-lens subsystem. Figure 8. Dispersion of the three-lens system.
UFC 1; 2 þ UFC 3
U d s 3 ¼
1
� s 2: ð19Þ U F 1; 2 UF 3 � UC 1; 2 UC U d
3
1; 2
For a desired( reverse) magnification of the three-lens system of M > 1, a suitable focal length for the third stock lens could be f3
d ¼ 9 mm. For an desired Airy disc of
d F 3 lm for the exposure, this requires a collimated beam diameter of D 2 4 mm. Applying now equation( 19), should result in paraxial systems fulfilling primary goal( I). To maximize the second primary goal( II), Figure 8 shows the expected secondary spectra. Thereby, the calculated s 3 [ 28,32 ] mmvariedonlywithinasmallrange. Surprisingly, goal( II) will be significantly better fulfilled, if the second lens is of a medium dispersive material, here: LASF44 and E48R. Regarding the material choice of the third lens it is implied by Figure 8, that a highly dispersive glass would be good. Applying the reliability estimator 1, 3 equation( 12) again, reveals the safer estimations from the paraxial calculation. Figure 9 shows, that for the interesting collimator combination SF11 & E48R, a reliable start system can only include a third lens of E48R, BK7 or SiO2.
According to Figure 8 should BK7 and SiO 2 yield a slightly higher secondary spectrum, which should be advantageous for primary goal( II). However, the in the next design step added thickness of the lenses should be accounted for. This will enlarge the axial chromatic aberration significantly. Additionally considering the cost and weight of the system, the plastic E48R offers an attractive compromise. The paraxial design estimation therefore concludes with a start system consisting of a combination of a negative SF11 and two positive E48R lenses.
2.3 Ray-tracing optimization
Based on the designed start system, the more rigorous geometrical ray-tracing simulation and optimization of the system shall be performed. Therefore, real stock lenses of Table 1 are considered. The chosen three lenses are listed in Figure 10. Given the limited availabilities for stock optics, their focal length vary slightly from the paraxial design.
Figure 9. Paraxial estimation reliabilities 1, 3 for the dispersion of the three-lens system.
The resulting axial chromatic aberration for a lens system with lenses at the paraxially estimated distances s 1, s 2 and s 3 is shown in Figure 11 as turquoise initial curve. The start system nearly can achieve primary goal( I), but for a deviation of 20 lm. To eliminate this deviation, the multi-objective optimizer of the commercial ray-tracing software is employed [ 56 ]. With a heavier weight on the objective to eliminate axial color between k F and k C, solely the axial distances between the three lenses has been optimized. The optimized axial chromatic aberration in Figure 11 shows its resulting curve, that now achieves primary goal( I).
Regarding primary goal( II), the inset in Figure 11 shows the magnified relevant range for the two focal planes of the measurement wavelengths. The two resulting defocii are z D1 6 lm andz D2 �4lm. These exceed the paraxially estimated maximized defocii in Figure 8 by far. The thereby effective NA C 3 is estimated from the resulting lateralspotasNA C 3
0: 24. Using equation( 1), this results in estimated normalized axial defocii of u D1 = 6.76and u D1 = �4.51. Their average of u D ¼ 5:63 thereby lies very close to the optimal value of u D, opt = 5.61 for ideal signals. The asymmetric distribution around the achromatic plane will take effect as a shifted working point on the differential curve.