## 3 Simulation
### Simulation Procedure  
The complete simulation was separated into three parts. In the first part, the interaction between the light wave and target was calculated with the FDTD method. In this part, the scattering matrix \(S_{i,j,k,m}\) depicted in Equation (16) is expected to be derived. The scattering matrix is dependent only on the simulated structure and wavelength of light, and it describes the complete interaction with two illumination polarizations. In the second part, an in-focus image and a set of defocused images were  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Target**} & \multirow{2}{*}{**Structure Parameter**} & \multicolumn{4}{c}{**Imaging Parameter**} \\ \cline{3-6}  & & \multicolumn{2}{c}{**Illumination**} & \multicolumn{2}{c}{**Collection**} & \multicolumn{2}{c}{**Wavelength**} \\  & & \multicolumn{2}{c}{**Numerical Aperture**} & \multicolumn{2}{c}{**Numerical Aperture**} & \multicolumn{2}{c}{**(nm)**} \\ \hline Nano-pit & Radius & 100 nm & 0.4 & 0.8 & 546 \\ \hline \multirow{3}{*}{Nano-dot} & Length & 121 nm & & & \\  & Width & 121 nm & 0.4 & 0.8 & 546 \\  & Height & 71 nm & & & \\ \hline Nano-bump & Radius & 100 nm & 0.4 & 0.8 & 546 \\ \hline \multirow{3}{*}{Double silicon lines} & Width & 30 nm & & & \\  & Height & 50 nm & & & \\  & Pitch & 20 nm & 0.4 & 0.8 & 546 \\  & Sidewall angle & \(90^{\circ}\) & & & \\ \hline \multirow{3}{*}{Dense line array} & Width & 30 nm & & & \\  & Height & 50 nm & & & \\  & Pitch & 60 nm & 0.4 & 0.8 & 546 \\  & Sidewall angle & \(90^{\circ}\) & & & \\ \hline \multirow{3}{*}{Infinite line grating} & Width & 30 nm & & & \\  & Height & 50 nm & & & \\ \cline{1-1}  & Pitch & 600 nm & & & \\ \cline{1-1}  & Sidewall angle & \(90^{\circ}\) & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Structure and imaging parameters used in the simulation.  
Figure 2: Computer-aided design (CAD) models for (**a**) nano-pit, (**b**) nano-dot, (**c**) nano-bump, (**d**) double silicon lines, (**e**) dense line array, and (**f**) infinite line grating.  
simulated by changing the value of \(D\) in Equation (14). In the last part, a TSOM image was obtained using a particular method.  
[PERSON] et al. proposed the details of obtaining both the simulated and experimental TSOM images [10], and the procedure is restated below for convenience.  
* Obtain an in-focus image and a series of defocused images of the target and a smooth background surface, respectively, as they are scanned along the axis direction.
* Normalize each through-focus image of the target and background as follows: \[Normalized\ Image=\frac{Unnormalized\ Image}{Mean\ Value}-1.\] (18)
* For the in-focus or each defocused position, directly subtract the normalized background image from the normalized target image.
* Select the inspection region with a strip window on the subtracted normalized image at each focus height, and extract the intensity profile by averaging the image intensity along the width of the window.
* Stack the averaged intensity profile of each subtracted normalized through-focus image according to its corresponding focus height to construct a raw TSOM image, which contains three-dimensional information. \(X\)- (horizontal), \(Y\)- (vertical), and \(Z\)-axes represent the spatial position across the target, the focus position, and the optical intensity, respectively.
* Interpolate and smooth the raw TSOM image and add pseudo color. The result is the final TSOM image needed.