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J. Eur. Opt. Society-Rapid Publ. 21, 27( 2025)
To furthermore maximize design robustness, axial symmetry can be demanded for the respective z D. Itisthissecond primary goal that is unique among the typical design process of chromatic systems. While achromats and apochromats only try to realize demand( I) and similar([ 47 ], p. 313ff), chromatic confocal systems only try maximize demand( II) by their request for large axial chromatic aberration [ 23, 48, 49 ].
Other secondary goals shall be the use of off-the-shelf optics, the minimization of optical elements, the minimization of the total length and the minimization of all three lateral spot-sizes for tightly localized exposure and measurement.
To design the required beam-path, the typical design methodology for optical systems is followed [ 50 ]. At first, analytical calculations with reduced complexity are desired to find a possibly working start system. In a second major design step, this start system is considered in full complexity and optimized in a commercial ray-tracing software. Here, we in particular focus on the first step and propose a new paraxial design procedure to find the important start system.
2.1 Paraxial chromatic theory
Paraxial theory shall be employed to estimate a start system for the optical beam-path of the chromatic differential confocal probe. Thereby, the space of free design parameters is reduced to the number of lenses, their respective focal length and their chosen lens materials. As shown in Appendix A. 1, for an arrangement of k lenses as illustrated in Figure 4, the power of the system from the first to the kth lens can be described recursively as:
U 1; k ¼ U 1; k�1 þ U k � ðs k � t k�1 ÞU 1; k�1 U k: ð5Þ
Thereby, t is the distance from the image-side principal plane of the previous sub-system from the first to the( k � 1) th lens to its( k � 1) th lens:
t k�1 ¼�s k�1
U 1; k�2
U 1; k�1
: ð6Þ
For k = 2 follows here the known equation for the power of two lenses U 1, 2([ 47 ], p. 154). For direct calculation, this recursive equation( 5) can also be expressed as a series:
U 1; k ¼ U k þ Xk�1
Y k
U i i¼1 j¼iþ1
� 1 �ðs j � t j�1 ÞU j: ð7Þ
This series can reproduce the given example in([ 47 ], p. 158) for a system of three lenses. Together with equation( 6) it can be used for the convenient estimation of the optical poweroflargersystems.
To find measures for the axial chromatic aberration, the material properties must be integrated. Widely used are the normalized Abbe number and the partial dispersion. The Abbe number is defined as:
m ¼ n d � 1 ð8Þ n F � n C
Figure 4. Schematic of paraxial ray crossing a system of k thin lenses characterized by their principal planes P i, P 0 i. Turquoise colored principal planes are resulting from sub-systems.
where the indices [ F, d, C ] define the evaluated wavelengths for the design. These may deviate from the original or official definition to meet the given design task. F represents the shortest wavelength of the targeted application spectrum, here the exposure wavelength. d represents a center wavelength of the spectrum. It should correspond with the design wavelength of the respective optical element. C is the longest wavelength of the targeted application spectrum. Here it represents the effective or median measurement wavelength. A higher Abbe number corresponds to lower dispersion. Therefore, it is relevant for achieving goal( I).
The second important normalized material parameter is the partial dispersion. It is defined as:
. ¼ n d � n C n F � n C
:
The higher. is, the higher the dispersion of the longer wavelength. Therefore,. is important for achieving a high axial chromatic dispersion for goal( II).
First, the achromatism( I) shall be achieved by observation of the change DU FC = U F � U C over the spectrum. For a system of k lenses, the achromatic demand must be satisfied by the power difference of the whole system
U FC 1; k ¼ UF 1; k � UC 1; k
. Viaequation( 5), this can be expressed in series representation after several rearrangement steps. Inserting equation( 8) and DU FC = U d / m yields the complete chromatic power shift of a system of k spherical lenses:
U FC 1; k ¼ Ud k m k þ Pk�1 U d i m i ð ½m i �. i þ 1ŠA i �½m i �. i ŠB i Þ; i¼1
A i ¼ Qk j¼iþ1
B i ¼ Qk j¼iþ1
1 �ðs j � t j�1 ÞU d j
1 �ðs j � t j�1 ÞU d j
m j �. j þ1 m j
m j �. j m j
:
; ð9Þ ð10Þ Note, that the t k�1 follow from equation( 6). Withthis equation for U FC
1; k
, the deviation from the primary design goal( I) can be calculated from a set of axial distances between a number of k spherical lenses with a nominal power and characterized by a certain material. Especially