J. Eur. Opt. Society-Rapid Publ. 21, 19( 2025) 195
Figure 9. Measurement results of the repeated specimen adjustment into the Cat’ s Eye reference position for four different specimens:( a) Axial position measured by DMI over measurement number with mean value subtracted.( b) Residual Defocuscoefficient over measurement number.
function. The main adjustment into the reference position uses the hexapod for the positioning. This algorithm can be performed with or without readjusting the lateral position of the specimen. For setting the reference position for the repeated measurements, an initial iteration of the positioning algorithm is performed. With the specimen in the Cat’ s Eye position, the retroreflector is adjusted for optimal beam backreflection into the DMI. The specimen is then brought back to a starting position for the repetitions and the internal counter of the DMI is set to zero for marking this position as a reference position. From there, repeated adjustments into the Cat’ s Eye position are performed with going back to the starting reference position after every iteration. To make sure that the algorithm performs reliably under different starting conditions, the starting position can be altered by a random offset in every iteration of the loop. The DMI value and the Zernike coefficients of the height values are recorded for every time that the Cat’ s Eye position is reached.
3 Results and discussion
Repeated measurements with various test specimens were made following the process described above( Sect. 2.4). The utilized test specimens encompass an asphere [ 25 ], a toroidal surface [ 17 ] and two spherical test specimens( with radii of curvature R = 40mmandR = 15 mm). Measurements both with and without lateral repositioning in the Cat’ s Eye plane were performed and the DMI-value, the hexapod encoder values, and the Zernike coefficients from
Z 0
3
0 up to Z
3 were recorded. Since the DMI is prone to occasional glitches of the internal counter, a sufficiently long section of the data without glitch is selected to show the short term repeatability of the system. The reasons of the glitches and methods to reduce them will be investigated in future work. In Figure 9, repeated measurements for the Cat’ s Eye adjustment along the optical axis without lateral repositioning can be seen for the four different test specimens.
The data in Figure 9a show a good short-term repeatability of the axial position measured with the DMI for the investigated specimens. The standard deviations are below
30 nm for all tested specimens. The individual values of the standard deviation r z are given in Table 1. In addition, the residual Defocus-values of the measurements shown in Figure 9b display standard deviations below 2 nm, hinting a good convergence of the positioning algorithm.
In addition to the repeatability of the axial position adjustment, the optional lateral position adjustment subsequent to the axial positioning was investigated with the test specimens described above in order to investigate, if the above described sensitivity of the Zernike coefficients primary x- and y-Coma to the lateral misalignment in the reference position is already large enough to achieve highly accurate lateral specimen positioning in this step. Since the DMI can only measure along the optical axis, the lateral position of the specimen is tracked by the axis encoders of the positioning hexapod. They don’ t have the accuracy as the DMI, but are still good enough to estimate the lateral repeatability of this coarse alignment procedure. After each individual positioning, the axial position is resetted into a position showing a rectangular interference patch. In addition, the lateral position is changed by a random value between �50 m and 50 m for each axis to ensure that the centering works independent of the starting condition in a reasonable region. From the recorded positions for the repeated measurements the mean position is subtracted and the results for the four different test specimens are shown in Figure 10.
For the asphere and the sphere with the radius of 15 mm, the position measurements are densely packed around the mean position. However, for the larger sphere with 40 mm radius and for the toroid, the adjusted positions are more widely distributed. Especially for the larger sphere the positioning is rather unstable, leading to larger outliers. Here, the movement range during the positioning could be limited to prevent coarse misalignment. The standard deviation r x and r y of this positioning step are shown in Table 1. It can be seen that the positioning deviations are not equal for both axes, which may be caused by additional tilts of the specimens in the holder. This can be fixed by a subsequent fine alignment of the specimen in the measurement position. For estimating the resulting axial displacement from this lateral displacement, the