JEOS RP ISSN01 | Page 288

J. Eur. Opt. Society-Rapid Publ. 21, 27( 2025) 283

Table 1. In this article considered materials of stock lenses.

Material SF11 SF5 F2 LASF44 E48R BK7 UV-FS CaF

1 The availabilities only consider the major web stores in the year 2024.

for a low lens count, this enables fast minimization of axial chromatism to U FC

1; k ¼ 0.

In order to fulfill the second primary design goal( II) a second statement resembling the high axial dispersion at k C is required. In analogue manner to the partial dispersion, the change of the power between k d and k C, DU dC, shallbe used as an estimator for that dispersion. This aligns with common lens-design approaches and is called secondary spectrum of a lens [ 47, 51 ]. With DU dC =. U d / m for spherical lenses follows from equation( 5), via insertion of equations( 8) and( 9), after several transformations the series representation for the secondary spectrum U dC

1; k of a system of k spherical lenses as:

U dC 1; k ¼. k Ud k þ Xk�1 m k i¼1

�U d i m i �. i m i

Y k

j¼iþ1

Y k

U d i j¼iþ1

1 �ðs j � t j�1 ÞU d j

m

1 �ðs j � t j�1 ÞU d j �.! j j

: ð11Þ m j

With this expression, the second primary goal( II) can be estimated by maximizing this U dC

1; k from the choice of available materials, distances and focal length for a set of k spherical lenses. Together, equations( 10) and( 11) now completely describe the paraxial system in order to achieve the two primary goals( I) and( II). The complexity is further reduced, by limiting the choice to stock lenses. A short survey of the current products limits the material choice to the ones given in Table 1.

However, the application paraxial theory is an approximation and may lead to erroneous estimations about the optical system. As laid out in the Appendix A. 2, evenfor two positive lenses, equation( 10) leads to wrong achromatic systems due to the negligence of the thickness of the lenses. Nonetheless, this negligence is desirable as it reduces the complexity of the initial design problem drastically. On the other hand, many combinations of a positive and a negative lens are known to realize axial achromatic focussing. Figure 5 shows such a combination of a positive BK7 and a negative SF11 lens. Especially the combination of high-m and low-m materials can result in achromatic systems. The negative lens hereby reverses the before imprinted dispersion of the positive lens( orange box in Fig. 5). Here this reversal shall be called negative dispersion

Figure

5. Illustrative ray-tracing of a the two lens achromat consisting of a positive and negative lens fulfilling the goal for achromatism( I) that was estimated with paraxial theory.

( turquoise box in Fig. 5) inrelationtothelateralchromatic deflection order.

This observation leads to the idea, that for certain combinations of positive and negative dispersing optical elements, the derived paraxial equations are still valid and useful. Therefore, we suggest the introduction of a new discriminator:

¼ Xk i¼1 s Ref f i

. i m i

; k 2: ð12Þ

This discriminator is built on the idea to sum the optical powers to a single value. A higher optical power, positive or negative, has stronger positive or negative dispersion. It then is multiplied by the ratio of partial dispersion and the Abbe number to weight the dispersion of longer wavelengths stronger, as the reversal of their chromatic dispersion is harder within the visual spectrum. This is caused by the smaller slope of the refractive index of typically used glasses. For infrared light and diffractive optical elements this does not apply.

The factor s Ref in equation( 12) is a reference length for normalization of the optical power. It should be adjusted to the expected lens thickness. In this article s Ref = 1 mm is used, because of the given design volume. It may also become s Ref = s i for more accurate results, but then cannot be estimated a priori.

The discriminator should be kept sightly below zero 0. This ensures, that there is sufficient negative