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J. Eur. Opt. Society-Rapid Publ. 21, 27( 2025)
Figure
13. Measured axial response curves from a plane mirror.
Figure 15. Fitted polynomials and residues within the working range of the differential signal.
Figure 14. 30 repeated measurement of the resulting differential curve.
signals in Figure 13 can be evaluated via equation( 1). For ideal sinc 2() confocal signals [ 26 ], the normalized axial peak half-width is u HM =± 5.5662. Fromthis, theeffectiveNA may be estimated as NA F 0.22 and an average NA C 0.24. The former thereby results in an estimated Airy disk of r F 1.24 and thereby does not achieve the result from the ray-tracing resolution. The latter NA C exactly fits to the result from the ray-tracing optimization and therefore may result in the anticipated lateral spots. Given the manual adjustment procedure, this result therefore is considered as a sufficient precise adjustment.
Figure 14 shows the resulting differential curve D of the measurement signals. An asymmetry between negative and positive side can be observed. It stems from the already existing asymmetry of the confocal peaks from Figure 14 due to negative spherical aberration. It is caused by the first lens. The differential curve in Figure 14 had been derived from double-sided 30 repetitions over half an hour. From a moving window of N = 200, the repeatability at the
Figure 16. Demonstration of axial single-surface resolution of 3 nm of the differential signal( top) in comparison to the z- interferometer( bottom).
working point of the exposure focus is u D = 82.8nmwith an expansion coefficient k = 1.
To reconstruct the actual depth, the relationship between the coaxial length interferometer and the nonambiguous steep part of the differential curve D is approximated by analytical functions. Figure 15 shows the fitresult of three point-symmetric polynomials for the nonambiguous region of D [ �15 10 3 dig, 15 10 3 dig ]. Clearly, the linearity on this range is limited. The maximum of the deviation e max lin ¼ 73: 23 nm lies around the working point for the exposure laser. Therefore, higher polynomials are implemented into the control DSP of the NPMM-5D. For a fifth order polynomial, the maximum deviation is kept below e max quint 2 nm. This is assumed to be sufficient, as the remaining oscillation is caused by a stick and slip behavior of the z-stage of the NPMM-5D.
Mitigating this dynamic effect from the stage, Figure 16 shows step-movements in order to determine the axial