Comparison of material measures for the determination of transfer characteristics of surface topography measuring instruments

The star-shaped groove material measure was evaluated with the following routine: the inner 80% of the functional area were extracted and a plane-fit was applied for the alignment of the extracted surface. The subsequent evaluation of the lateral period limit was performed as described by Giusca and Leach by extracting profiles through neighbouring upper and lower petals [20]. The entire transfer functions of the different instrument setups were determined by assigning the transferred amplitude coefficients of the various spatial frequencies that were measured. In doing so, it is possible to assign every position on the Siemens star surface a spatial frequency f that is directly based on the distance to the centre r (see [20]):

$$\begin{eqnarray}&&f=\displaystyle \frac{N}{r\cdot 2\pi },\end{eqnarray} \tag{ 1 }$$

where N is the number of petals. The height difference ${\rm{\Delta }}d$ between the two extracted profiles through the lower and upper petals is divided by the nominal amplitude A of the petals and leads to the transfer function TF that represents the relative amplitude ${A}_{rel}$ as a function of the spatial frequency:

$$\begin{eqnarray}&&{A}_{rel}=\displaystyle \frac{{\rm{\Delta }}d}{A}\cdot 100 \% \to TF={A}_{rel}\left(f\right),\end{eqnarray} \tag{ 2 }$$

Corresponding to the discretization of the measured topography dataset, an almost continuous determination of the transfer function is possible. The difference profile between the two initial profiles represents directly the transferred height difference. Based on the transfer function, the lateral period limit is subsequently determined. Its definition is similarly as in the definition of linear filters the spatial frequency which is transferred with 50% of the original amplitude. For the different instruments and microscope configurations, multiple repetitive measurements were obtained.

The results for the CM and WLI measuring instruments are summarized for different magnifications in figure 3. For every setup, four repetitions of the measurement were performed whose results are imaged in the figure. With the CM 50× setup, eight measurement results are imaged as both samples featuring a size of 200 μm × 200 μm and 400 μm × 400 μm were measured four times each. The results show that with increasing magnification, the lateral period limit decreases for both instruments. Additionally, the repeatability improves due to the increasing resolution capabilities. Similar values for both examined surface topography measuring instruments can be calculated. The lateral period limit values are all smaller than 11 μm which means that geometric structures down to this size can be adequately resolved. This shows that both instruments feature a high precision. Because one pair of points in the topography represents every spatial frequency in the measured datasets, there is some scattering of the results and a significant effect of single outliers on the transfer functions. However, with the ASG geometry, a large range of spatial frequencies can be imaged and used to determine the instruments transfer characteristics.

Zoom In Zoom Out Reset image size

The chirped standard that was manufactured by DLW was also only measured and evaluated for the areal surface topography measuring instruments. The pre-processing of the topography dataset by applying an F-operator for alignment was identical to the previously described evaluation of the ASG material measure. After this, a central profile of the 3D-topography was used to apply the small scale fidelity evaluation (see [22]). An example for this evaluation is shown in figure 4. Multiple sinusodial waves are fitted into the measured dataset and the amplitudes and wavelengths of the individual fits are calculated. The transfer function of the instrument can be calculated when all transferred amplitudes were divided by the nominal amplitude in order to obtain the transfer characteristics for varying spatial frequencies. As the chirped standard features a limited number of sinusoidal waves, there is a finite number of spatial test frequencies for this kind of transfer function determination. For example, the PTB chirped standard features 24 wavelengths ranging from 91 μm to 10 μm [6]. Additionally to the transfer function, for each measurement the small scale fidelity limit was determined which is defined as the shortest wavelength which is transferred within a certain threshold value for the deviation of the amplitude transmission. In the example of figure 4, the nominal amplitude of the chirped standard equals 400 nm. When a maximum amplitude deviation of ±25% is applied, the smallest wavelength which features a fitted amplitude between 300 nm and 500 nm is considered as the small scale fidelity limit. In the given example, the small scale fidelity limit equals 29.87 μm.

Zoom In Zoom Out Reset image size

For the specific evaluation the threshold for the small scale fidelity limit was chosen to ±50% of the original amplitude in order to ensure the best possible comparability to the results of the type ASG sample. Also, the amplitude transmission for all fitted wavelengths was calculated in order to characterize the transfer functions of the various measuring instruments and measuring configurations. The results of all chirp evaluations are shown in figure 5. There, the resulting points of the sinusoidal fits are shown for the CM and WLI measuring instruments for the varying magnification values 20×, 50× and 100×. The amplitudes of the different sinusodial fits from multiple measurements are summarized for each optical configuration. When the separate points of the various transfer functions are interpreted, the following conclusions can be drawn: just as with the results from the type ASG measurements, the transfer capabilities improve with higher magnification and the entire transfer function is shifted towards higher frequencies. Also the scattering of the transferred amplitudes is reduced with increasing magnifications—leading to more reproducible results. For the CM 50× measurements however, the determination of suitable illumination parameters was challenging due to optical artefacts that occur in regions with high frequencies when the illumination is set too large or too small. Four measurements for every setup were evaluated.

Zoom In Zoom Out Reset image size

Additionally, it should be noticed that with the CIN-DLW sample only a limited number of frequencies can be imaged within the samples that feature a lateral length between 100 μm and 800 μm. Thus, also the ultra-precision turned sample CIN-UPT was measured which represents a larger number of spatial frequencies and can additionally be measured with the profile measuring devices. The CIN-DLW material measure can fulfil its design purpose to determine reproducible transfer functions for areal surface topography measuring instruments.

The evaluation of the CIN-UPT sample was performed identical as for the CIN-DLW sample in the previous section. For all confocal setups, profiles of two areal topography measurements were evaluated, otherwise four repetitions were performed and evaluated. The results of the transfer functions of the four different measuring instruments just as well as their determined small scale fidelity limit are summarized in figure 6. Similar results to the previous sample can be observed and similar small scale fidelity limit values in the micrometre-range were determined. However, for the areal instruments CM and WLI one phenomenon can be observed: because stitching is required to measure the entire set of different wavelength the results tend to scatter more significantly already for lower frequencies. The overall sample has a length of several millimetres. The results for the cut-off frequencies of the different measuring instruments which are again represented by the small scale fidelity limit are similar to the previously described sample CIN-DLW. For the two profile sensors SI and CCM, good results with small scale fidelity limits in the micrometre-range can be identified—these measuring systems are designed to acquire larger measuring lengths and no stitching is required. Thus, it can be concluded that the chirped sample which features larger structures is more suitable for the calibration of profile measuring instruments due to the required sampling length which is larger than for the CIN-DLW sample. The CIN-DLW samples are more suitable for areal measuring instruments because all frequencies can be imaged within a typical field of view.

Zoom In Zoom Out Reset image size

The three previously described samples are specifically designed for the transfer function determination and it was shown that each of them has a specific field of application. In order to benchmark the results, they are compared with a different approach that was suggested by the authors and is based on time series modeling [23]. The approach is based on an autoregressive-moving-average (ARMA) model. When the time series $u\left[h\right],h=1,\,2,\,\mathrm{...},\,N$ is considered as input signal, the output dataset z can be determined based on an ARMA model with the coefficients $a[k]$ and $b[l]$ [32]:

$$\begin{eqnarray}&&\begin{array}{l}z[n]\,=\,-\sum _{k=1}^{p}a[k]\cdot z[n-k]\,+\,\sum _{l=0}^{q}b[l]\cdot u\left[n-l\right],\\ \,n\,=\,\{\begin{array}{l}p+1,\,\ldots ,\,N\,\mathrm{if}\,p\geqslant q\\ q+1,\,\ldots ,\,N\,\mathrm{else}\end{array}\end{array}\end{eqnarray} \tag{ 3 }$$

The mathematical term $\displaystyle {\sum }_{k=1}^{p}a[k]\cdot z[n-k]$ describes the auto-regressive part with the degree p and $\displaystyle {\sum }_{l=0}^{q}b[l]\cdot u\left[n-l\right]$ the moving-average part with the degree q. When the corresponding model is interpreted in a way that the control dataset for the ultra-precision manufacturing of a certain deterministic rough surface is the input data u of the measuring instrument and the measured data represents the output data z of the model, the ARMA-coefficients describe the transfer behavior of the measuring instrument. Thus, when equation (3) is transformed into the frequency domain, and an algebraic fit of the model coefficients $a[k]$ and $b[l]$ is executed, the frequency-dependent transfer behavior of the measuring instrument can be determined based on its input and output data. This was investigated and proven for different surfaces (see [23]). In doing so, it was shown that the Rk-standard, whose nominal geometry for manufacturing is based on an actual component surface and known precisely, can lead to the most precise determination of the transfer function for different measuring instruments. The virtual manufacturing dataset is considered as the input and the measured data of a topography measuring instrument as the output of a time series. Thus, with the fit of a model for the transfer behaviour determination, additional information for comparison can be obtained based on this fourth sample.

The Rk-standard was measured once with every areal topography measuring setup and four repetitions with the two profile measuring instruments were performed. Based on the obtained measured data, a time series model was determined based on only one profile per measuring configuration which provides a linear approximation of the instruments transfer behaviour. It is examined whether this approach is more universally applicable than the previously described samples. In doing so, the benchmark displayed in figure 7 was determined where all transfer functions that were determined with the numerous methods are displayed. This includes as well the transfer functions that were determined based on the fit of the ARMA-model.

Zoom In Zoom Out Reset image size

It can be observed that the transfer functions based on the ARMA-approach do well agree with the other transfer functions and realistic results can be generated. The cut-off frequencies of the different approaches do agree well for all instrument setups. Only for high frequencies the approach based on the ARMA-model leads to deviations if the sampling theorem is violated. Thus, the transfer functions were calculated to a maximum frequency of ${f}_{\max }\,\leqslant \,1000\,{{\rm{mm}}}^{-1}.$ Generally, it can be shown that the approach of time series modeling with the Rk-Standard is a universally applicable method for the transfer function determination of different profile and areal surface topography measuring instruments. However, one limitation of the model should be considered: as a linear model, the time series model cannot image some physical effects that have non-linear characteristics.

One particular example for physical effects that can be imaged with the ARMA-model can however be observed for the CCM measurement. It can be assumed that curvature depending effects at large spatial frequencies lead to an increase of the amplitude transmission when the CIN-UPT sample is measured. Mauch et al described this effect [33]. This effect would be also expected for the confocal microscope but can only be observed for the CCM measurement. This effect can also be imaged with the transfer function of the ARMA-model. However, all measuring instruments feature non-linear effects that cannot be considered by the model. As the agreement between the measured transfer functions and the transfer functions of the ARMA-model is high, it can be concluded that these effects do not have a major influence on the transfer behaviour.