200 J. Eur. Opt. Society-Rapid Publ. 21, 20( 2025)
Figure 1. MTF measurement setup.
Figure 2. Focus point position z( P f) determined from the measurement data maximum, focus scan with d m = 5 lm pitch, 20 repetitions [ 6 ].
value as the focus point position z( P f). This was quite robust as it is present in almost every data set. But the maximum value is strongly influenced, for example, by noise and thus prone to focus point deviations that directly correlate with the focus scan step width Dz ðc Pf; data max
¼ 1 zÞ. 2
This is clearly visible in Figure 2 where the results obtained from repeated focus scans at the same position are shown. The focus point position determined using the maximum MTF data value results in basically two different locations that are approximately 5 lm apart( i. e. the applied focus scan step width). However, this also means that minor changes in the measurement conditions like the ambient temperature or camera temperature( cf. [ 5 ]) are not recognised as long as they do not result in the data maximum being reached at a different measurement value.
In measurements presented in [ 6 ] on obtaining the focus point more precisely polynomial fits were used and a 4th degree polynomial was proposed to be used for the determination of the focus point location z( P f)( cf. Fig. 3). In that contribution the use of higher order fits appeared to be less robust and did not result in a significant reduction of the fit residuals.
Recent investigations though show that using a 6th degree polynomial for the focus point fit is preferable, as the residuals could be further reduced and the robustness of the polynomial fit couldbesignificantly improved by more suitable boundary conditions for the fit maximum. The focus point position z( P f) isobtainedusingtheroots of the fit polynomials first derivative, if one of the roots is within the measurement range, is non-complex, and the second derivative of the fit polynomial at that position is negative.
3 Temperature dependence of the focus point P f
Another question regarding the focus point is how it is affected by temperature variations and how the temperature dependence of the focus point position influences the determined MTF value of the test object. To investigate the influence of temperature changes repeated focus scans have been performed. The orientation of a slit target imaged by the measurement system was changed from tangential to sagittal and vice versa between consecutive focus scans. Since the line spread function( LSF) of the sample under test is determined from the camera image( cf. Fig. 4), the image coordinate of the LSF maximum represents the position of the target image on the camera sensor. Deviations in the maximum position hint towards a motion of the measurement setup perpendicular to the optical axis, either in horizontal( obtained from the measurements with tangential / vertical target orientation) or vertical direction( from measurements with sagittal / horizontal target orientation).
While the influence of the camera temperature during warm-up has been shown in [ 5 ], in the investigations presented here the ambient temperature was changed during repeated through focus scans and the influence on the LSF maximum and the focus point position representing the x / y and z shift of the focus point was examined during a change of the ambient temperature by about DT �0.5 K.
3.1
LSF maximum shift
In Figure 5, the LSF curves of 130 repeated focus scans are shown“ from above”. Each horizontal line represents the