J. Eur. Opt. Society-Rapid Publ. 21, 19( 2025) 193
Figure 6. Reconstruction process from phase shifted interference images including phase reconstruction, phase unwrapping with calculation of height values, subtraction of reference height values, and subsequent development into Zernike polynomials.
steps of p between u 2 1 = 0 and u 5 = 2p. The recorded intensity images I 1 to I 5 are then used with Hariharans phase reconstruction method [ 15 ] to retrieve the phase of the measured interferogram. In order to now get rid of the 2p-ambiguity of the phase image, phase unwrapping using the Goldstein method [ 16 ] is performed. The simulated height values of a specimen in the Cat’ s Eye position are then subtracted from the height values measured. This ensures that the reference position for the real adjustment matches the simulation of the adjustment using the model of the TWI. The resulting height value differences are developed into Zernike polynomials. The Defocus-term( Z 0
2
,[ 26 ]) of the Zernike developement is then a measure for the proximity of the specimen’ s current position in relation to the Cat’ s Eye reference position [ 20, 31 ]. The process of wavefront reconstruction and Zernike coefficient extraction is shown in Figure 6.
Since the Defocus develops linearly near the Cat’ sEye reference position and has it’ s zero-crossing directly at this position, several measurements and linear regression can be used to estimate the axial coordinate value. Therefore, the height values and the resulting Zernike-coefficients are repeatedly measured and the specimen is translated along the optical axis( z) between the measurements. In this particular case n = 5 measurements were performed and the specimen was moved by Dz = 75lm between the measurements. Then, a linear regression of the estimated Defocus values was performed and the zero-crossing of the Defocus is calculated. The specimen is moved to this position and the height values are measured again. Since the lateral position in relation to the Cat’ s Eye reference position has alinearrelationtotheprimary x- and y-Coma( Z �1
3 and
Z 1 3
) the positioning perpendicular to the optical axis can be optimized here as well to achieve a coarse lateral alignment. In this case, both lateral coordinates( x and y) are optimized simultaneously by using the secant method, which is a special case of the Regula falsi [ 32 ]. The new value of x is given by x nþ1 ¼ x n � d x f ðx n Þ with d x being the finite difference quotient defined as d x ¼ðx n � x n�1 Þ = ðf ðx n Þ�f ðx n�1 ÞÞ. Theiterationsofy follow in the same way. The first iteration from x 0 to x 1 and y 0 to y 1, respectively, use fixed steps Dx 0 and Dy 0 instead of d x f ðx n Þ and d y f ðy n Þ. Afterafixed number of iterations( here: n = 5), the position of the specimen perpendicular to the optical axis is refined sufficiently. After the lateral position optimization, the specimen might have moved and the distance between objective lens surface and the specimen’ s surface measured along the optical axis might have changed. Therefore, the adjustment along the optical axis is refined by an additional optimization step. However, even if the lateral positioning was not performed, the refinement step is still applied to account for inaccuracies in the first axial optimization step. For this refinement, the current Defocus-value and the slope calculated previously in the linear regression step are used. This step can be repeated if necessary, since the slope might have changed slightly due to the lateral repositioning. The algorithm of the specimen positioning using the Zernike coefficients is shown in Figure 7.
2.4
Repeatability of the Cat’ s Eye positioning algorithm
For measuring the repeatability of the specimen positioning in the Cat’ s Eye reference position, multiple specimens, containing spheres with different radii, aspheres, and freeform surfaces, are inserted into the TWI measurement setup and repeatedly brought into the reference position by the method described above. The final position along the optical axis is measured by a distance measuring interferometer( DMI) and the results are compared to reveal the spreading of the final position. The measurement setup is shown in Figure 8.
The setup consists of the TWI itself, from which only the objective is shown, the large z-axis with a positioning