## 3 Experiments
### Experimental Results on 2D Grating  
The first investigated sample was a 2D Si grating whose SEM cross-section image is shown in Figure 2. The optical properties of Si were taken from Reference [37]. As depicted in Figure 2, the cross section of the Si grating could be characterized by a symmetrical trapezoidal model with top width (\(W\)), grating height (\(H\)), sidewall angle (\(SWA\)), and period (\(P\)), whose dimensions obtained from Figure 2 were \(W\) = 350 nm, \(H\) = 472 nm, \(SWA\) = 87.63 deg, and \(P\) = 800 nm. In this case, the profile parameters under study were \(W\), \(H\), and \(SWA\), while the grating period \(P\) was fixed at its nominal dimension, namely, \(m\) = 3. The measured signature contained 15 Mueller matrix elements normalized to the first element, namely, \(l\) = 15. As mentioned above, the measurement configuration \(\mathbf{a}\) = [\(\varphi\), \(\theta\), \(\Lambda\)] was defined as a combination of fixed azimuthal angle \(\varphi\), fixed incident angle \(\theta\), but a wide waveband \(\Lambda\) for spectroscopic scatterometry. As a simple example to demonstrate our data refinement method, here, the incident and azimuthal angles were fixed at \(\theta\) = 65 deg and \(\varphi\) = 30 deg, the wavelength covered a fixed range of 300-900 nm with a step of 5 nm. We assumed that the dimensions of the three profile parameters under investigation had a deviation of about 1% from their nominal values. The differences of the 15 Mueller matrix elements between the actual and nominal profiles were calculated by our forward RCWA model under each wavelength point according to Equation (5). After considering all the 15 Mueller matrix elements and the wavelengths used, the reanalysis and eigen-analysis procedures described above were performed on the differences. According to Equations (6)-(9), the wavelength points with redundant information were identified and eliminated from the set of measured signatures in a repetitive manner. Since the investigated  
Figure 1: Principle and instrument of the dual rotating-compensator Mueller matrix ellipsometer. Figure 2: Scanning electron microscopy (SEM) cross-section image of the investigated 2D Si grating sample. sample shown in Figure 2 is a simple structure, and the model output Mueller matrix had 15 elements, we achieved the \(3\times 15\) optimized data sets by traversing the three profile parameters and the 15 Mueller matrix elements. Through merging the same wavelengths into one for the \(3\times 15\) optimized data sets, the remaining unions \(S_{\text{opt}}\), containing 19 points, were obtained and shown by the black circles in Figure 3, which are supposed to contain enough information needed for the profile reconstruction of the 2D Si grating. To examine the validity of the remaining information, regression analysis was conducted to reconstruct the three profile parameters from the full spectral range of 300-900 nm, the optimal spectrum containing 19 points, and the spectrum of 19 random points. The non-linear LM algorithm was applied to fit the measured 15 Mueller matrix elements with the modeled ones, which converged very quickly to a minimum when a suitable initial condition was chosen. During the fitting procedure, the root-mean-squared error (RMSE) was used to quantify the goodness of fit, which summed over all the measurement wavelengths the differences between the measured data and model-generated data. The lower the RMSE value, the better the fit or agreement between the measured and model-generated data. Figure 3 presents the fitting result of the model-calculated best-fit Mueller matrix elements (black circles) and the measured spectrum including (red lines) 19 optimal points. Table 1 shows the extracted profile parameter values for the fits and the SEM-measured values. The uncertainties associated with the SEM-measured values were estimated by manually measuring the SEM micrographs. The uncertainties appended to the scatterometry-determined values were estimated from 30 repeated measurements.