## Materials and Methods
### A Model for the Uncertainty Analysis of the Measurements
#### 3.3.2 Uncertainty Analysis for LTM Measurement Setup  
In this case, \(\delta_{A}\) still represents Type A uncertainty, whereas \(\delta_{B}\) are the Type B uncertainty. The main sources of type B uncertainty considered are:  
* Power Supply uncertainty has been evaluated through the data sheet and a Gaussian distribution has been assumed;
* LTM system uncertainty has been evaluated considering its maximal lateral resolution, limited by diffraction of the light; a uniform distribution with 0.6 \(\upmu\)m semi-amplitude has been assumed. * Digital image uncertainty has been evaluated considering the digital resolution of the acquired images (1.359 px/\(\upmu\)m). A uniform distribution with 2 pixels semi-amplitude has been assumed. * Uncertainty due to the plane planarity, the focusing plane variation, and the vibration of the DUT have been considered negligible, thanks to optical table with pneumatic vibration isolators that increase the stability of the device and reduce vibrations that could cause an incorrect characterization. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Uncertainty Source** &
\begin{tabular}{c} **Probability Density** \\ **Function (PDF)** \\ \end{tabular} & **Type (\({}^{2}\))** & **m** & \(\delta\) **(OPS)** & \(\delta\) **(LTM)** & **Unit** \\ \hline Power Supply: voltage accuracy & \(N\left(m,\sigma\right)\) & B & 0.06 & 0.04 & V \\ \hline Optical System: maximal lateral resolution due to diffraction & \(U\left(m,\sigma\right)\) & B & 0 & 0.57 & 0.57 & \(\upmu\)m \\ \hline Digital Image: digital conversion & \(U\left(m,\sigma\right)\) & B & 0 & 3.14 & 1.43 & \(\upmu\)m \\ \hline Image Processing Software: uncertainty in point identification & \(U\left(m,\sigma\right)\) & A & \(\pm\)1.9 & \(\pm\)1.9 & pixel \\ \hline Uncertainty in ROI position & & & \(\pm\)1.9 & \(\pm\)1.9 & pixel \\ (template-matching [3]) & & & \(\pm\)1.9 & \(\pm\)1.9 & pixel \\ \hline Uncertainty in ROI size & & & \(\pm\)1.9 & \(\pm\)1.9 & pixel \\ (template-matching [3]) & & & & \(\pm\)1.9 & \(\pm\)1.9 & \\ \hline Uncertainty in the & Monte Carlo template-matching algorithm & B & 0.02 & 0.02 & \({}^{\circ}\) \\ \hline \hline \end{tabular}  
* Notes: (\({}^{1}\)) \(N\left(m,\sigma\right)\) is a Gaussian \(PDF\) with mean \(m\) and standard deviation \(\sigma\); (\({}^{2}\)) Type A and type B uncertainty as in standard [47]; (\({}^{3}\)) the uncertainty value \(\delta\) for OPS and LTM is referred to 95% of confidence level.  
\end{table} 
Table 3: OPS and LTM measurement setup uncertainty sources. To evaluate the total uncertainty of the two measurement systems, according to Equation (2), two contributions have been determined: the first is related to the uncertainty contribution evaluable with a statistic analysis of the measurements dispersion obtained with the measurement system (\(\delta_{A}\)), while the other is due to the overall uncertainty from the main error sources related to the experimental setup (\(\delta_{B}\)). In order to evaluate the \(\delta_{B}\) uncertainty, two main contributions have been determined: the first is associated to the derivative of the function that expresses the angular displacement \(\theta\) of the comb-drive with respect to the applied voltage \(\frac{\partial\theta}{\partial V}\), multiplied by the uncertainty due to the power supply \(\delta_{V}\), while the other is related to the measurement of the comb-drive angle \(\delta\alpha_{t}\)[47]  
\[\delta_{B}=\delta_{\alpha}=\sqrt{\left(\frac{\partial\theta}{\partial V} \delta_{V}\right)^{2}+\left(\delta\alpha_{t}\right)^{2}} \tag{3}\]  
In particular, a second order polynomial function  
\[\theta=a\cdot V^{2}+b\cdot V+c\, \tag{4}\]  
where \(a\), \(b\), and \(c\) are obtained experimentally, approximates the trend of the angular displacements.