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J. Eur. Opt. Society-Rapid Publ. 21, 27( 2025)
Figure A1. Secondary spectra of two-lens achromats.
Figure A2. Chromatic focal shift for a two-lens Achromat with a first lens of E48R. h iþ1 h i þ s i c iþ1: ðA: 1bÞ
As illustrated by Figure 4, c 1 is the angle of incidence of a single ray on the firstthinlensattheheighth 1. U is the power of said thin lens, the inverse of its focal length:
UðkÞ ¼ 1 f ðkÞ: ðA: 2Þ
It depends on the refractive index of the lens and therefore on the wavelength. This depends on the specific optical element. For example, a spherical lens can be described by the famous lensmaker equation.
For a sequential combination for the first two lenses in Figure 4, theopticalpowerU 1, 2 of their combination shall be derived. For the first lens follows from equation( A. 1a):
c 2 ¼�U 1 h 1:
Inserting this into equation( A. 1b) it yields: h 2 ¼ h 1 � s 1 U 1 h 1:
Adapting again equation( A. 1a) to find c 3, itbecomes: c 3 ¼�U 1 h 1 � U 2 ðh 1 � s 1 U 1 h 1 Þ:
Dividing this by �h 1 yields the the power U 1, 2 of the two-lens combination via equation( A. 1a) and( A. 1b) as also derived in([ 47 ], p. 151):
U 1; 2 ¼� c 3 h 1
¼ U 1 þ U 2 � s 1 U 1 U 2:
This system’ s image-side principal plane P 0 1; 2 is at an axial distance t 1 from the second lens: t 1 ¼�s 1
U 1
U 1; 2
:
From these statements it can be deduced, that a sequence of k lenses can be always divided into a system of two lenses where the first lens is actually system of thin lenses characterized by P 0 1; k and its optical power U 1, k.
Therefore, for an arrangement of k lenses as indicated by Figure 4, the power of the system from the first to the kth lens can be described in analogue manner to equations( A. 1a) and recursively as:
U 1; k ¼ U 1; k�1 þ U k �ðs k � t k�1 ÞU 1; k�1 U k;
which is used in this article as equation( 5). ðA: 3Þ
A. 2 False-positives for two-lens achromats
A first benchmark for the suggested equations( 10) and( 11) shall be the design of a two-lens achromatic system. From equation( 10) follows for the realization of the desired achromatism between k F and k C for the distance between the thin lenses:
s 2 ¼ Ud 1 þ Ud 2 m 1 m 2
m 1 m 2 U d 1 Ud 2ð m 1 �. 1 þ m 2 �. 2 þ 1Þ
¼ f2 dm 2 þ f1 dm 1
: ðA: 4Þ ðm 1 �. 1 þ m 2 �. 2 þ 1Þ
Note, that this equation deviates from the textbook version, e. g. found in([ 47 ], p. 316), because the difference has not been approximated by a differential. Obeying this equation for s 2 satisfying demand( I), the evaluation for suitable material combinations can be judged by the second primary demand( II). For two lenses, equation( 11) becomes:
U dC 1; 2 ¼. 2 Ud 2 þ U d. 1 U d 1 1
� s Ud 2 2 ðm 1. m 2 m 1 m 1 m 2 þ m 2. 1 �. 1. 2 Þ:
2 ðA: 5Þ
Regarding the optical tool head, the exposure beam emitted by the common fiber must be collimated and focused onto the substrate. For two lenses, such a system could be a combination of f 1 = 50mmandf 2 = 12.5 mm. Theoretically, this would achieve afourtimesminified image of the single mode fiber core. Figure A. 1 shows the from equation( A. 5) calculated secondary power shifts. As this should be a measure for demand( II), the shown values should correspond to the local slope at k C.
For verification of this, the graph for a firstlensmadeofE48R shall be analyzed in detail. Figure A. 2 is showing the paraxially calculated focal shifts for every wavelength for the combination of an E48R collimator with an objective made of the materials of Table 1. Indeed, the estimation of the slope at k C generally