The suggested measurement algorithm has three main beneﬁts compared to the matrix-based scanning methods:
1. It can save the amount of necessary measured information while measuring with very high resolution, thus shortening the measurement time and minimizing the probe wear while preserving the necessary resolution on critical surface structures.
2. It allows the user to perform further local reﬁnements easily while the sample is still in the microscope, not necessarily based on the algorithm criteria but also on the user's requirements.
3. It can automatically measure the data allowing the user to perform zooms offline, e. g. in the data processing phase after the measurement. This can be useful namely for automated image processing or inspection purposes.
In order to document the effect of the reduction of the necessary scanned path, we compare the path length saved by the presented algorithm to the standard matrix-based approach for the surface shown in Figure . The resulting total tip path reductions for two diﬀerent ﬁnal accuracies are summarized in Table . It can be noted that the more reﬁnement levels are performed the more data are saved. This is however not valid generally but only on regular structures like the one presented here. As stated above, for completely random data, we cannot save anything using the reﬁnement algorithm.
Performance of the measurement algorithm
In order to display the real performance of the algorithm if implemented in the microscope, we compare the results of the matrix algorithm and the presented algorithm on real data of a calibration grating. The same ﬁnal resolution was requested. In Figure , we can see the result of the subtraction of images obtained using the two algorithms. We can see that the local diﬀerences are caused only by relatively high noise of the presently used long-range system z scanner and a slight mismatch between the two images (due to a slight shift caused by retraction and approach between the two measurements, the images had to be cropped and aligned manually).
Figure 5 Comparison of the matrix algorithm and the presented algorithm. This ﬁgure shows the result of a subtraction of images obtained using the two algorithms. Note that the local diﬀerences are caused only by relatively high noise of present (more ...)
It should be noted that the obtained data have diﬀerent statistical properties than a typical matrix-based SPM image. There is no diﬀerence between the fast and slow axes as both x and y directions are used for the measurement. It has no sense therefore to distinguish between directions while evaluating direct or statistical quantities. All parasitic eﬀects, such as drift or noise, inﬂuence data almost isotropically. This could be understood as a drawback in some sense; on the other side, it can simplify the treatment of uncertainties while obtaining proﬁles of different orientation with respect to the main axes or 2D statistical properties evaluation.
An important issue is the inﬂuence of drift on the measurement process. Drift, namely of thermal origin, can be observed in many SPM systems [8
]. In a typical matrix-based SPM image, drift is mainly seen in the slow scanning axis, leading to image distortion as seen in Figure . Note that the coordinate system origin in this simulation is in the top left corner of the image. Based on the analysis of the known surface structures or based on repetitive scans, we can determine the drift rate, which is usually a time-dependent decaying function, with a maximum right after the instrument start-up or a sample exchange [10
]. If we use the scanning approach presented in this work, the data in the image are not measured successively, so the straightforward drift determination from AFM image is not possible as the drift inﬂuences interleaved values signiﬁcantly (see Figure ). However, drift can still be evaluated. As in each reﬁnement level, we measure on the same area as was already measured in the previous iteration, and we even obtain data at exactly the same points (at row/column crossings); we can determine the drift rate already during reﬁnement process for areas that are being reﬁned. We can do this using the following approach:
Figure 6 Drift inﬂuence. Microchip surface (A) and inﬂuence of general xyz drift to standard matrix-based measurement process (B), iterative reﬁnement (C) and measurement process and iterative reﬁnement process after drift guess (more ...)
• Create an interpolated image from one reﬁnement level (using the standard procedure from the previous section);
• Create an interpolated image from the next level skipping the data measured in the previous level;
• Use cross-correlation to determine the shift between the two data sets (in all three axes), which are the x and y drift values; and
• Shift one data set according to the cross-correlation result and subtract the two data sets, which gives us the z drift value.
As an illustration of the process, we have simulated data measurement with a constant drift vector of (3, 3, 0.3) nanometres per second in Figure . We have used the part of the microchip surface seen in Figure , using already measured data (without observable drift) and adding the drift during the simulated measurement (all movements were performed with the same velocity). Drift was evaluated from levels 1 and 2 of the reﬁnement process where, for this sample, we still measure on nearly the whole area (the local reﬁnement criterion still holds everywhere). Using the above-mentioned process, we have evaluated the drift vector as (3.3 ± 0.5, 2.7 ± 0.3, 0.29 ± 0.15) nanometres per second which is a good estimate of the drift rate. In Figure , a simulated measurement with correction based on this estimated drift rate is shown (for drift estimation, the ﬁrst two iterations were used), obviously leading to signiﬁcant correction of the image. Of course, if the drift rate is not constant, the above-mentioned approach would not be an optimal one; however, the user could repeat it after several reﬁnement iterations to correct the drift rate estimate. Generally, it can be seen that the drift leads to a much more evident image distortion when our adaptive reﬁnement algorithm is used. On the other side, even the data obtained in a regular matrix approach are inﬂuenced by the drift, and if we do not know the measured structure properties, we need to use a similar correlation technique to determine the drift. Data are therefore 'wrong' in both cases, but in the regular matrix case, they look better and allow the user to ignore the systematic errors caused by the drift. The large inﬂuence of the drift on the presented algorithm could therefore be seen even as a beneﬁt for a metrologist - if the systematic error is quantitatively the same in both cases anyway, the adaptive approach prevents the fact that it would rest hidden in the data.