## 3 Simulation
### Simulation Results  
The simulated TSOM images of the six targets in the illumination polarizations of 90\({}^{\circ}\) and 0\({}^{\circ}\) are illustrated in Figure 3. The relative difference of OIR (OIR\({}_{rd}\)) is calculated in Equation (19).  
\[\text{OIR}_{rd}=\frac{|\text{OIR}_{0^{\circ}}-\text{OIR}_{90^{\circ}}|}{\text {OIR}_{90^{\circ}}}. \tag{19}\]  
Figure 3: \(Cont.\)  
Table 2 provides the results.  
The first three targets were centrosymmetric. Under this condition, their TSOM images appear to be qualitatively similar. The corresponding OIRs in the two illumination polarizations were not strongly different, which illustrates a weak influence of polarization on the sensitivity of TSOM with regards to the targets of centrosymmetric features.  
The others were center asymmetric, that is, the silicon lines were oriented parallel to the \(Y\)-axis, but periodic along the \(X\)-axis. Under this condition, an apparent influence of illumination polarization on the sensitivity of TSOM can be observed as the two TSOM images and corresponding OIRs were strongly different.  
The effect brought about by illumination numerical aperture (INA) was also studied by simulating the TSOM of the double silicon line upon setting INA = 0.8 and keeping other parameters in line with those in the previous simulation. Figure 4 presents the simulation results.  
By comparing the results of Figures 3 and 4, it is demonstrated that the OIR of the TSOM image with the illumination numerical aperture of 0.8 is smaller than that with the illumination numerical aperture of 0.4. However, the two corresponding OIRs in both illumination polarizations are still different. It is concluded that the effect brought about by the illumination polarization and the  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Targets** & **Relative Difference (TSOM)** \\ \hline Nano-pit & 0.74\% \\ Nano-dot & 2.3\% \\ Nano-bump & 0.25\% \\ Double silicon lines & 31.9\% \\ Dense line array & 447\% \\ Infinite line grating & 145\% \\ \hline \hline \end{tabular}
\end{table}
Table 2: Relative difference between optical intensity ranges (OIRs) with two illumination polarizations. TSOM—through-focus scanning optical microscopy.  
Figure 3: Simulated through-focus scanning optical microscopy (TSOM) images of the six targets with different structural features with illumination polarizations of 90\({}^{\circ}\) (the first column) and 0\({}^{\circ}\) (the second column); OIR—optical intensity range; INA = 0.4.  
target structural feature exists as long as the metrology is performed in the same illuminating and imaging conditions.  
To analyze this phenomenon further, the line height of the double silicon line and dense line array were increased by 10 nm, but the other parameters are kept constant. This dimensional change produced a new TSOM image. The dimensional change was highlighted by a simple subtraction of the new and previous TSOM images and illustrated in a differential TSOM (DTSOM) image. The inspection of the vertical sensitivity of the dimensional change was directly quantified by the OIR of the DTSOM image. The DTSOM images in the two illumination polarizations are shown in Figure 5. The relative difference of OIR is provided in Table 3.