## 3 Experiments
### Experimental Results on 3D Grating  
The second sample was a photoresistive grating consisting of a cylinder array deposited on silicon with a bottom anti-reflective coating (BARC), which is a 3D grating whose SEM cross-section image is shown in Figure 4. The 3D grating of the cylinder array was chosen for this study due to the high computation effort of its forward optical model. As depicted in Figure 4, the cross section of the 3D grating is characterized by a model with cylinder diameter (\(D\)) and height (\(H_{1}\)), BARC height (\(H_{2}\)), and periods of arrays in two directions (\(P_{\mathrm{x}}\) and \(P_{y}\)). The dimensions obtained from Figure 4 are \(D=226.7\) nm, \(H_{1}=355.1\) nm, \(H_{2}=104.3\) nm, and \(P_{\mathrm{x}}=P_{y}=835\) nm. In our experiments, optical properties of the photoresist were modelled by a two-term Forouhi-Bloomer model [38], whose eight undetermined parameters were determined by measuring a photoresist film deposited on the silicon substrate using the above ellipsometer and were \(A_{1}=0.006029\), \(A_{2}=0.020598\), \(B_{1}=14.195303\) eV, \(B_{2}=14.196431\) eV, \(C_{1}=50.523937\) eV\({}^{2}\), \(C_{2}==50.537878\) eV\({}^{2}\), \(n\) (\(\infty\)) = 1.436087, and \(E_{g}=4.774050\) eV. Similarly, optical properties of the BARC were modelled using the Tauc-Lorentz model [39], whose five undetermined model parameters were obtained as \(\varepsilon_{1}\) (\(\infty\)) = 1.42680, \(E_{g}=3.45971\) eV, \(A=21.14955\), \(E_{0}=9.94921\), \(C=0.98767\). For more details about physical meanings of the above parameters, one can consult References [38; 39], which are omitted here for brevity. Here, the investigated profile parameters of the 3D grating included the \(D\), the \(H_{1}\), and the \(H_{2}\), namely, \(m=3\), while the two grating periods were fixed at their nominal dimensions. In contrast to the experiment settings above, the incidence and azimuthal angle were fixed at \(\theta=65^{\circ}\) and \(\varphi=0^{\circ}\), and the terms of \(N\), \(C\), and \(S\) parameters versus wavelength were used as measured signatures, which \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Measuring Mode**} & \multicolumn{2}{c}{**Dimensions (nm/\({}^{\circ}\))**} & \multirow{2}{*}{**RMSE**} & **Time** \\ \cline{2-2} \cline{4-4}  & \(W\)/nm & \(H\)/nm & \(SWA\)/\({}^{\circ}\) & **(s)** \\ \hline SEM & \(350.3\pm 4.74\) & \(472.1\pm 4.87\) & \(87.63\pm 0.611\) & — & — \\ Full spectrum & \(347.3\pm 0.17\) & \(468.9\pm 0.20\) & \(86.89\pm 0.019\) & \(9.44\) & \(548\) \\
**Optimal (19)** & \(347.4\pm 0.15\) & \(476.9\pm 0.12\) & \(87.27\pm 0.008\) & \(8.27\) & \(143\) \\ Random (19) & \(345.2\pm 0.74\) & \(486.4\pm 0.86\) & \(86.31\pm 0.103\) & \(13.59\) & \(225\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of fitting results of the 2D grating extracted from different spectrum and SEM measurements. Figure 4: SEM cross-section image of the investigated 3D grating sample. were derived from ellipsometric Psi (\(\Psi\)) and Delta (\(\Delta\)) and taken as \(N=\cos\) (2\(\Psi\)), \(C=\sin\) (2\(\Psi\)) cos (\(\Delta\)), and \(S=\sin\) (2\(\Psi\)) sin (\(\Delta\)). Considering that the investigated 3D sample deviated largely from its nominal values, its forward model was highly complex and nonlinear, and the model output was \(N\), \(C\), and \(S\), namely, \(l=3\), we assumed that the dimensions of the three profile parameters changed \(k=6\) times to ensure that the optimization procedure was stable, and the deviations were about \(\pm 1\%\), \(\pm 5\%\), or \(\pm 10\%\) from their nominal dimensions, respectively. According to our proposed method, \(3\times 3\times 6\) optimized data sets remained, then, the union \(S_{\text{opt}}\) of the remaining data sets, consisting of the 36 wavelength points shown in Figure 5, were achieved for the inverse problem solution.