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J. Eur. Opt. Society-Rapid Publ. 21, 27( 2025)
With this result, the optical design of the combined exposure and chromatic differential confocal measurement through a common fiber is finished. It marks a first example for the proposed design methodology for such systems.
3 Realized integrated tool head
Figure
10. Via ray-tracing optimized three lens system with turning optics.
As indicated in Figure 10, the beam-path features additional functions to enhance the usability and the ability to adjust to the local surface slope of the substrate. The main beam-path is folded by two 90 ° deflections due to space restrictions. For the first deflection a beam-splitter is inserted to integrate a camera chip. This 2D-array detector is observing the deflection of the beam to provide a feedback signal for the rotational NPMM-5D axes.
The movement of the spot on the camera is evaluated via the application of a threshold and a following centroid algorithm [ 57 ]. This provides a very fast processing time. Thereby the calculations are performed on the basis of a reference sphere. Its apex defines the center pixels x px, 0, y px, 0 on the camera. The two required surface angles can now be derived from the shift of the light spot on the camera. For an ideal sphere and square pixels follows for the azimuthal angle u:
u M ¼ arctan 2 y px; M � y px; 0 x px; M � x px; 0
: ð20Þ
For the polar angle #, the surface slope, follows for lateral positions x IM, y IM on an ideal reference sphere with the diameter D 0 as measured by the length interferometers:
# Ref ¼ arcsin 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
x 2 IM þ y2 IM: ð21Þ
D 0
However, the optical system towards the camera imposes its imaging characteristics. Therefore, the measured # M is only a rough estimate, that can be normalized by the number of total pixels N px to allow the calculation of the arcus sine:
0 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1
# M ¼ arcsin 2 ðx px; M � x px; 0 Þ 2 þðy px; M � y px; 0 Þ 2 @
A:
N px
Figure 11. Chromatic focal shift of ray-tracing simulations before and after numerical optimization.
Besides this axial chromatic behavior, this simple beampath achieves a lateral Airy radius of r F 1.1 lm according to the ray-tracing software. This is below the nominal Airy disk of the third lens for an input beam of D 2 4 mm. For the medium measurement wavelength follows r C 1.66 lm. Therefore, the simulated system is considered optimized. For the actual measurement wavelengths that are focused on the defocused planes, the Airy radii r C1 2.77 lm andr C2 3.89 lm are slightly larger. This lateral widening indicates also an axial widening of the confocal peaks. ð22Þ
Equations( 21) and( 22) can now be compared for every lateral position deviating from the reference sphere apex. Now, a position-depending correction coefficient can be derived:
�
# Ref
C # x IM; x IM; x px; M; y px; M ¼ #: ð23Þ
M
After this characterization, the polar angle # M ¼ C # # M can be reconstructed online and used for the control of the rotational axes of the NPMM-5D.
Another not negligible function is the navigation and overview of the user on the substrate. Although the fibercoupled confocal signals can be used for the reconstruction of surfaces, users often face different substrates of various