J. Eur. Opt. Society-Rapid Publ. 21, 19( 2025) 191
Figure 4. Simulation of position-, shape-, and size changes in the interference patch of a specimen near the Cat’ s Eye reference position when translated.( a) Position of the interference’ s patches center-of-gravity when moved perpendicular to the optical axis for three different axial distances.( b) Size of the interference patch when moved along the optical axis.
corresponding to the area of the interference patch over its axial position is shown. The plateau in the center corresponds to the area, were the interference patch has a circular shape. This pre-alignment method can also be adapted for form measuring interferometers other than the TWI with different beam stop apertures as long as the aperture clipping is recognizable and related to the specimen position.
When the interferogram of the interference patch is evaluated using phase shifting interferometry, one can reconstruct the phase and from there the height values of the images. These values can then be compared to the values in the reference position. To investigate the influence of misalignment on these height value differences, several simulations of different positions close to the reference positions are performed and the height value differences are analyzed. To get insight into the relation between the form of the height value differences and the position of the specimen, the optical aberrations are expressed in terms of Zernike polynomials and the numbering of the coefficients follows the ISO / ANSI numbering convention [ 26, 27 ].
In Figure 5, the Zernike coefficients fitted to the height differences in dependence of the specimen’ spositioninrela- tion to the Cat’ s Eye reference position is shown. For this purpose, the height values resulting from a specimen at the specific position and the height values resulting from a specimen with it’ sapexintheCat’ s Eye position are simulated and the results are subtracted from each other. The resulting difference wavefront is then developed into Zernike coefficients. In Figure 5a, the evolution of the Zernike coefficients from Z 0 to Z 20( Z 0
0 to Z 5 5 in the double indexing scheme) are shown for an asphere that is moved along the optical axis. It can be seen that, apart from some changes in Z 12 and Z 14, the sweep is mainly affecting the so-called Defocus-aberration( Z 4). The sweep along the optical axis was repeated with different specimens including the asphere, two spherical specimens with different radii, and a toroid. The Defocus over the axial position z is shown for them in Figure 5b. Thelinearregionfrom jz � z Cat’ sEye | 300 lm corresponds to the region, where the interference patch of the specimen has its maximum area and a round shape. The zero-crossing of the Defocus is at z Cat’ sEye, whichmakestheDefocus agoodcriterion for the axial adjustment of the specimen into the Cat’ s Eye reference position. It has to be noted that it is important to subtract the reference value simulated for the Cat’ s Eye position to align the specimen position to the position used in the model. Otherwise, the adjustment by Defocus zero-crossing is less accurate and can have an offset. Figure 5c shows the Zernike coefficients of an asphere that is translated along lateral x-axis. Here, the coefficients of the primary x-Coma( Z 8) appears to be affected the most. Therefore, in Figure 5d the change in Z 8 over the lateral position is shown for the selected specimens. Again, the zero-crossing of the Coma appears at x = x Cat’ sEye and is therefore a good criterion for a coarse alignment along the x-axis. Simulations have further shown that the same can be achieved by primary y-Coma( Z 7) for a coarse alignment along the y-axis. Note that the fine lateral alignment together with the alignment of tilt is performed in the final measurement position.
2.3
Position optimization algorithm
Following the simulations of the relation between the specimen’ s position and the resulting interferogram, an algorithm for positioning the specimen in the Cat’ s Eye reference position is introduced. The algorithm consists of three steps: First, a coarse positioning of the interference pattern to align the specimen’ s apex with the optical axis is performed. Second, a coarse positioning along the optical axis is performed to the region, where the interferograms have a round shape. Finally, the positioning along the optical axis is optimized using the reconstruction of the height values and the minimization of the Zernike coefficients of the difference topography. All three steps are described in the following in detail.
For the coarse alignment of the specimen, the specimen is first manually brought into a position, where a rectangular interferogram of sufficient size is seen on the camera image. For the alignment, the center-of-gravity of the rectangular patch is calculated. First, to get the area of the interferogram, the image is blurred. This assures that the interference fringes in the rectangular patch are evened out and do not interfere with the estimation of the patch border. Second, the camera image is binarized