A novel cubic‐exp evaluation algorithm considering non‐symmetrical axial response signals of confocal microscopes

The depth discrimination in confocal microscopy is based on the digital analysis of depth response signals obtained by each camera pixel during measurement. Various signal‐processing algorithms are used for this purpose. The accuracy of these algorithms is inter alia restricted by the axial symmetry of the signals. However, in practice response signals are rather asymmetrical especially in case of measurement objects with critical surface structures such as edges or steep flanks. We present a novel signal‐processing algorithm based on an exponential function with a cubic argument to handle asymmetrical and also symmetrical depth response signals. Results obtained by this algorithm are compared to those of commonly used signal processing algorithms. It turns out that the novel algorithm is more robust, more accurate and exhibits a repeatability of a similar order compared to other algorithms.

In this work, we present a novel fitting algorithm, which is based on an exponential function with a cubic argument to handle nonsymmetrical signals more accurately compared to established signal processing algorithms. The new algorithm is validated using depth responses obtained from rectangular grating structures and a tilted plane mirror. For this purpose, simulated signals are used for validation in order to avoid effects resulting from signal noise and real surface deviations. Furthermore, repeatabilities obtained from the novel algorithm are compared with those of centroid, parabolic, and Gaussian fitting algorithms. It is investigated how well the different signal processing approaches can handle consecutive response signals of varying axial sample point locations. Finally, the algorithms are compared by use of response signals obtained from a layer thickness standard using a commercial confocal microscope. The analysis shows that the cubic fitting algorithm locates the position of the detected response maximum most reliably. An improvement in the detection of the signal maximum using a fitting function of increased order is already mentioned by Kim et al. (2006). Chen et al. (2019) present a corrected parabolic fitting algorithm to reduce systematic deviations of a parabolic fitting. This algorithm is applied to symmetrical and asymmetrical spectra of a chromatic confocal microscope. It is shown that the results of the corrected parabolic fitting algorithm exhibit a lower standard deviation compared to other algorithms. However, an improvement for detecting the maximum of non-symmetrical signals is not shown. Another approach is given by Seewig et al. (2013) where a robust signal processing of noisy response signals is performed using maximum-likelihood estimation. An asymmetrical shape of the intensity signal is significantly suppressed during the signal processing.
However, the focus is on suppressing signal noise, and an analysis of asymmetrical compared to symmetrical depth response signals is not discussed. Furthermore, signal filtering can lead to a loss of information. A further algorithm that handles asymmetrical depth response signals based on the centroid method is introduced by Chen et al. (2022). However, this algorithm can be insufficient in determining the exact peak location and thus probably may lead to systematic deviations in height determination, for example, in the case of layer thickness measurements, where signals from different interfaces may be more or less asymmetric (see Section 3.4).

| AXIAL RESPONSE SIGNAL FORMATION AND EVALUATION ALGORITHM
First, the formation of an axial response signal of a confocal microscope is briefly discussed in this section. Then, the cubic signal-processing algorithm as well as further approaches such as centroid, parabolic, and Gaussian fitting algorithms are introduced for comparison.

| Signal formation
The working principle of a confocal microscope is based on microscopic imaging. While the temporal coherence of the illumination source does not affect the basic principle of confocal microscopy, spatially coherent illumination is crucial. Spatial coherence can be achieved due to a pinhole located in the illumination arm of the microscope as schematically illustrated in Figure 1. Due to the microscope objective lens, an image of the pinhole occurs on the specimen's surface. Light scattered from the specimen is collected by the objective lens and focused to the detector pinhole. If the axial location of the surface under investigation matches the working distance of the microscope objective (visualized by the blue line), the collected light passes the detector pinhole. In this case, the maximum intensity is detected by the detector or the corresponding camera pixel. On the other hand, the detected intensity decreases with increasing or decreasing distance of the surface under investigation with respect to the axial focus position. Consequently, if the distance between objective lens and specimen is continuously changed during the depth scan, a depth response signal results. Compared to conventional optical microscopes the depth response signal of a confocal microscope is characterized by a narrow intensity peak, which falls to zero if the surface is far away from the focus position. A more detailed description of the depth response signal is given in (Corle & Kino, 1996;Pahl et al., 2021). Signals measured by a confocal microscope are exemplarily depicted in Figure 2. These signals are obtained from a rectangular surface structure of 6 μm period length using the commercial confocal microscopeÂμsurf custom (Nanofocus AG) with a numerical aperture (NA) of 0.95 using a cyan LED for illumination (Hagemeier, 2022;Hagemeier et al., 2019). For better comparability, the intensity curves are normalized and the locations of the maximum are axially shifted to zero. The signal depicted in Figure 2a is obtained from an upper plateau of the rectangular structure and nearly F I G U R E 1 Schematic illustration of a confocal microscope. The blue line illustrates the imaging beam path, if the specimen is in focus of the microscope objective. An out of focus beam path is exemplarily represented by the red dashed line. symmetrical, whereas the signal presented in Figure 2b is obtained from an edge of 190 nm height of the structure and shows some asymmetry.
Consequently, both symmetrical and asymmetrical signals occur from the rectangular grating enabling to validate signal processing algorithms for both, asymmetrical and symmetrical response signals.

| Signal analysis
In order to discriminate height information from confocal response signals several signal processing algorithms are used in practice. One of the fastest and most frequently used computation algorithm is the centroid algorithm, represented by for linear or for square relation, where the intensity of the response signal is represented by I j ð Þ at the position z j ð Þ of the depth scan in z direction. Further approaches rely on least squares approximation of the measured intensity signal by a known mathematical function such as a parabola or a Gaussian function. A polynomial function of order n can be described by where the indices l and r represent the left and right border of the intensity tuple above a certain threshold. The parameters a i (i 1,…,n) are determined by solving the equation system using QR decomposition. In case of a parabola (n ¼ 2) the axial position of the maximum results from The same procedure can be used to calculate the axial position of the maximum of a fitted Gaussian function. Here, the natural logarithm of the intensity I z ð Þ is taken in order to receive the argument of the exponential function, leading to These signal-processing algorithms are suitable for symmetrical depth response signals. However, as shown above measured depth response signals are often asymmetric, where the strength of asymmetry depends on the surface texture and the components of the confocal microscope. Even the depth response signal in Figure 2a obtained from a flat surface section is not perfectly symmetric.
Asymmetries can be considered by use of an exponential function with a cubic argument (n ¼ 3) for fitting confocal signals. The fitting procedure is similar to that of the Gaussian according to Equation (6) and the axial position of the maximum can be taken from Since a cubic function comprises two extreme values (a maximum and a minimum) and only the maximum is relevant, the correct axial position arises from Note that the presented evaluation algorithm determines the axial position of a single depth response curve and must be applied to each signal that is each camera pixel in a full-field measuring confocal microscope.

| VALIDATION OF DEPTH DISCRIMINATION ALGORITHMS
In order to validate the different evaluation algorithms, simulated depth response signals obtained from certain surface textures are investigated using a rigorous simulation model (Pahl et al., 2021  This is probably a consequence of the so-called batwing effect (Sheppard et al., 1988;Xie, 2017), which occurs through interference of light diffracted at the edges of the investigated surface structure and leads to the overshoots shown in Figure 3. Such an asymmetric depth response signal obtained at the lateral position x ¼ 1:5 μm is depicted in Figure 4 and used to demonstrate how well the different approaches fit the signal. Note that the CF as well as the other algorithms introduced in Section 2.1 are not designed to correct for systematic shifts of the signal peak caused by batwings or other systematic effects. In this work, we investigate which signal processing algorithm meets the measured intensity curves and especially the location of their maxima best. Figure 4 shows the intensity signal which affects the height values determined by PF and GF less. However, a higher threshold value has the consequence that the number of sampling points used for signal processing decreases and thus a higher measurement uncertainty is to be expected in the presence of noise. Note that in all cases the CF method fits the corresponding response signal well and thus, the determined height values are constant. Therefore, the overshoots caused by the batwing effect are best approximated using the CF approach. This is the first step of the signal processing chain and must be followed by further steps, if batwings are to be eliminated.

| Rectangular grating
Batwings may be reduced based on adequate simulation models (Pahl et al., 2021;Pahl, Hagemeier, Künne, et al., 2020), by use of filtering algorithms (for example., median filtering) applied to the measured topography data , or by adjusting the evaluation wavelength in case of other surface measurement methods such as coherence scanning interferometry . Further, physical approaches to reduce batwings are given by, for example, adjusting the polarization of the illumination (Pahl, Hagemeier, Hüser, et al., 2020;Pahl, Hagemeier, Künne, et al., 2020) and the wavelength of the illumination source  with respect to the measured surface texture.
In order to visualize the quality of the different fitting algorithms in numbers, the standard deviations of the fitted curves obtained by the different algorithms with respect to the measured intensity signal are listed in Table 1. For a threshold of 0.5, the fitting curves calculated using the PF and GF approaches show much higher deviations compared to the CF approach, as expected from  T A B L E 1 Standard deviations of the fitted curves obtained from the different signal processing algorithms (shown in Figure 4) with respect to the measured intensity signal. F I G U R E 5 Measured rectangular grating with 6 μm period length obtained from the RS-N standard using a commercial 100Â confocal microscope with a numerical aperture of 0.95. of the measured grating are higher compared to the nominal value, as shown in Figure 5. In order to analyze the overestimation of simulated and measured grating heights, simulations are performed for gratings of same height (190 nm) but with different period lengths L. Figure 6 displays simulated grating profiles obtained by GF algorithm for L ¼ 3 μm, L ¼ 6 μm and L ¼ 10 μm. It should be mentioned that a monochromatic light source is assumed in the simulation, since simulations considering broader spectral bandwidths are time consuming and the results shown here are sufficient to explain the overestimation. Hence, the result obtained for L ¼ 6 μm in Figure 6 slightly differs from the profiles presented in Figure 3.
Comparing the grating profiles in Figure 6, the grating with L ¼ 3 μm shows the highest height difference, whereas the height difference for L ¼ 10 μm almost corresponds to the nominal height.
Therefore, the overestimation of measured step heights can be explained by the overshoots, which slightly decay with distance to the edges. If the period length of the grating is too small, neighboring edges influence each other due to diffraction and the measured height differences exceed the nominal height values leading to the obtained overestimation.
A further example of interest is given a rectangular grating with a period length of 400 nm and a step height of 140 nm, taken from the same RS-N standard. Simulated profiles resulting from the different signal processing algorithms are depicted in Figure 7 for several thresholds. As expected, the resulting profiles do not correspond to the nominal rectangular grating profile, since the period length of the grating is close to the lateral optical resolution limit of the confocal microscope considered by the simulation program. While the profiles calculated by the fitting algorithms are sinusoids, the profiles determined by the centroid algorithms exhibit a stepped shape. Due to the fact that the surface's period length is close to the lateral resolution limit, only scattered light up to the first diffraction order is captured by the microscope objective lens and thus, a sinusoidal shape is expected. Hence, the results obtained by the fitting methods appear to be more correct   Note: σ Δh represents the empirical standard deviation determined for height differences of each signal processing algorithm.

| Tilted plane mirror
This section is intended to investigate how well the signal processing algorithms can handle asymmetrical response signals, whose sampling points are varied with respect to their axial position. For this purpose, response signals for a tilted plane mirror are simulated with a lateral sampling interval of Δx ¼ 160 nm. A tilted mirror is an example of practical relevance, since real measurement objects often show tilted specularly reflecting surface sections. A tilt angle θ tilt with respect to the x-axis leads to an axial shift of are considered in the simulations similarly to Corle and Kino (1996).
The total simulated intensity I x, z ð Þ is calculated by where the scattered and incident wave vectors k s θ in , φ in ð Þ , k in θ in , φ in ð Þ with corresponding z-components k s,z , k in,z as well the filter function Θ θ in , φ in ð Þ are computed as it is already described by Siebert et al. (2022). The sine and cosine functions represent a homogeneous pupil illumination (Corle & Kino, 1996;Wilson et al., 1980).

| Repeatability
In the previous paragraph noise-free depth response signals are used to compare systematic deviations of different signal processing algorithms. In this section, the repeatability of these algorithms depending on the signal-to-noise ratio (SNR) as specified by (Tereschenko, 2018) is investigated. Here, σ 2 signal and σ 2 noise represent the variance of the signal under investigation and the variance of the noise, respectively. Note that the noise amplitudes added to the asymmetrical response signals obey the normal distribu- The empirical standard deviations obtained for thresholds of 0.6 and 0.8 are depicted in Figure 10a,b, whereas the average height F I G U R E 8 Residual errors of profiles corresponding to a tilted plane mirror, depending on the signal processing algorithm. The determined profiles are vertically shifted to each other to increase visibility.
T A B L E 4 Standard deviations σ determined for the residual errors depicted in Figure 8. F I G U R E 1 1 Averaged standard deviations of the fitted curves obtained by the different signal processing algorithms with respect to the simulated intensity signal. The solid lines represent σ obtained for a threshold of 0.6 and 0.8 in case of the dashed lines.
F I G U R E 9 Simulated depth response signals with different signalto-noise ratio (SNR) values.
by the approximated signal courses obtained by the fitting approaches, the standard deviation between the real intensity signal and the approximated signal course is calculated for each of the 1000 repeated intensity signals at identical SNR and finally, their average is taken. This procedure is repeated for each SNR value and compared in Figure 11. For Both thresholds, the averaged standard deviations determined by the PF and GF approaches exhibit higher values compared to the CF approach, as expected from the results listed in Table 1.

| Layer thickness
Another example of improved accuracy achieved by the CF approach is related to layer thickness measurement. In the case of response signals obtained from a perfectly adjusted surface flat, it can be expected that the signal processing algorithms provide nearly the same relative height values independent of the lateral position even for asymmetric response signals. In contrast, response signals obtained from multiple layers show multiple peaks, which differ in their shape. This may result in offsets between the height values obtained for different layers at the same lateral position, especially if the signal processing algorithm cannot handle asymmetrical depth response signals.
For demonstration, a response signal obtained from a layer thickness standard is depicted in Figure 12. This standard comprises a 4:1 μm thick SU-8 (transparent photoresist) located on a 10 nm thick chromium (Cr) layer, which is deposited on an approximately 525 μm thick silicon layer (Brand et al., 2011). The right peak of the response signal results from the transition of air to SU-8 and exhibits a nearly symmetrical shape, whereas the left confocal peak results from the SU-8 to Cr transition and shows an asymmetrical course. While the signal processing algorithms represent the course of the right lobe well in almost the same manner, the course of the left lobe is well approximated only by the CF algorithm. Note that a threshold factor of 0.6 is used for signal processing. An increased threshold factor leads to a slightly improved representation of the signal course, as shown in (Hagemeier, 2022).
Depending on the depth discrimination algorithm, multiple measured signal courses lead to varying height values for the left confocal peak, whereas those obtained for the right peak are similar, as represented by the lower and upper lines in Figure 13 for a measured pro- According to these results, the CF approach is the most suitable F I G U R E 1 2 Response signals (blue) obtained experimentally from a layer thickness standard, including the fits of the different fitting algorithms as well as crosses to mark the position determined by centroid approaches. The curves shown in the magnified sections, which are determined by parabolic fitting (PF) and Gaussian fitting (GF) as well as the crosses representing the linear centroid (LC) and square centroid (SC) approaches, are shifted vertically by a certain offset to improve visibility.
F I G U R E 1 3 Profiles of the measured height values obtained from a layer thickness standard using different signal processing algorithms.
T A B L E 5 Averaged height differences Δh and standard deviations σ Δh of the different layer thickness profiles depicted in Figure 13.  (Brand et al., 2011;Cox & Sheppard, 2001;Kühnhold et al., 2015;Sheppard et al., 1994). However, once the correct axial locations of the maxima are known, systematic height measurement errors such as batwings can be identified and reduced in further steps, for example, by model-based approaches considering an appropriate simulation model, by appropriate surface topography filtering, and so forth.

| CONCLUSION
A disadvantage of the cubic algorithm is the higher computation time compared to the other algorithms presented in this work.
However, due to increasing computational resources, the processing time is in an acceptable range, for example, 984 Â 984 depth response signals of 6 μm length and a step size of 30 nm are computed in 4050 ms for the cubic, in 3588 ms for the Gaussian and in 2915 ms for the parabolic fitting approaches as well as in 1932 ms for the square and in 1799 ms for the linear centroid methods using a personal computer with an Intel i9-9900K CPU (Intel Corporation, 2023).
Furthermore, the computation time can be reduced by software optimization.

CONFLICT OF INTEREST STATEMENT
The authors declare no conflict of interest.

ACKNOWLEDGMENT
Open Access funding enabled and organized by Projekt DEAL.

DATA AVAILABILITY STATEMENT
The data that support the findings of this study are available from the corresponding author upon reasonable request.