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J. Eur. Opt. Society-Rapid Publ. 22, 30( 2026)
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3. Eight local minima in the vicinity of the system in Figure 2 are shown in the red boxes. Only the last six lenses are shown, which include the four lenses of interest. The lenses with the most significant change compared to Figure 2 are marked with an arrow. For the four lenses of interest the blue bar charts show the z-values that result from theory, one negative z-value and seven equal positive z-values. When these z-values are translated into surface curvatures, the lenses in the blue boxes are obtained. For comparison, the red bar charts show the z-values obtained from data extracted from the optimized systems.
assigns one negative and seven identical positive z-values to the four lenses, as illustrated by the corresponding blue bar chart.( Theory – partly already developed in [ 8 ] and to be presented in detail in a separate work – predicts the existence of such minima, characterized by one negative and seven equal positive z-values, in the vicinity of a Fulcher-like group having eight equal z-values.) The permutation symmetry in S then implies the existence of seven additional local minima, in which the negative z-value of the first minimum appears at each of the other positions within the group. The permuted z-values are shown by the blue bar charts. The lenses enclosed by blue boxes are obtained by translating each of the eight permuted sets of z-values into surface curvatures using equations( 26) and( 27).
The systems in red boxes are candidates for the predicted local minima in the optimization landscape. The red bar charts show for these systems the z-values obtained using equation( 15) and marginal-ray data from the optimized systems. Some discrepancy between the red and blue bars is expected, because the approximate error function E neglects many aberrations and because the model assumes zero-thickness lenses, whereas the optimized systems contain lenses of finite thickness.( Also, the red bars show seemingly larger discrepancies because they show relative rather
Figure 4. Aplanatic thin triplet for the infrared region( n = 4), with an effective focal length of 1, and transverse magnification M T ¼�1. The zero distances between surfaces are drawn for clarity as finite. The stop is at the thin lens. All six quasi-powers are equal to 1 / 6. Despite of the different bendings, the three spherical lenses have the same lens power 1 / 3( see equation( 28)).
than absolute differences, and because the z-values for the four surfaces of interest are significantly smaller than those of more strongly curved surfaces elsewhere in the system.) However, for the systems in red boxes the shapes of the four lenses of interest agree reasonably well with the corresponding lenses in blue boxes. This agreement supports the interpretation of these systems as the eight minima resulting from permutation symmetry.