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Production sequence of Si-spheres and interferometrical determination of the sphere volume

Characterization of the maximum measurable slope of optical topography measuring instruments

25.01.2024

The maximum slope that can be measured with a microscope objective is an important parameter that characterizes the measurement capability of 3D optical microscopes. Due to hardware configurations or software filters, the maximum measurable slope of an objective may differ from that determined by the numerical aperture (NA) of the objective. Additionally, investigation shows that slope transfer function within the field of view (FOV) are often not homogeneous. Therefore, the actual maximum slope that can be measured using 3D optical microscopes for surface texture measurements needs to be investigated experimentally.

A proof-of-principle of a method for the characterization of the maximum measurable slopes of optical topography measuring instruments using optically smooth spheres is proposed. The sphere’s diameter is selected according to the FOV and the NA of the objective used. Figure 1 illustrates the measured sphere segment (in green) of a sphere (in blue) with radius R using an objective with a numerical aperture of NA. A sphere with a diameter in the range of 1/3 ·  χFOV ≤ 2R · NA < χFOV is recommended to ensure the measured sphere segment is large enough to characterize the slope transfer function of the objective and at the same time the whole measured sphere segment is within the FOV. Profiles of the measured sphere segment at meridian angles from α = 0° to 360° with a step size of 15° are extracted and ΦMS can be calculated by ΦMS = arcsin(RMS/R), in which the RMS is determined by using Breakpoint2 at the positions of the captured sphere’s rim.

Fig. 2 (a) shows a topography image of the sphere of 120 µm measured by a 100x objective (NA = 0.95). Profile lines from the center of the topography image along different directions are extracted and a typical profile at meridian angle of α is shown in (b).

Figure 3 shows the maximum measurable local slopes versus meridian angles α measured by the 100x objective at orientations of 0° and 90°. The measurement results show the inhomogeneity of the slope transfer function within the field of view (FOV), i.e., the maximum measurable slopes of the objective whose theoretical maximum measurable slope is 72° varies from 67° to 84°.

    
(a) Top view of the measured cap of a sphere    (b) Side view of the measured cap of a sphere
Abb. 1: Schematic illustration of the measured cap of a sphere within the FOV

    
(a) Measured topography, profile line at meridian angle α    (b) Cross section profile with two break points
Abb. 2: The measured sphere topography image, profile section at the angle of α and break points


Abb.3: Maximum measurable local slope as function of the meridian angle α measured at orientations of 0° and 90°

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