# Metrology of Nanostructures by Tomographic Mueller-Matrix Scatterometry
## 2 Experimental Setup  
Figure 1 presents the scheme of the developed TMS, which is a combination of a Mueller matrix ellipsometer (MME) [22] and a reflection microscope. As shown in Figure 1, a broadband white light from a light source (LS; LDLS\({}^{\text{TM}}\), EQ-99 FC, Energetiq, Woburn, MA, USA) is collimated by a flat convex lens (CL) and then filtered to be a monochromatic light by a bandpass filter (BF; FL632.8-1, Thorlabs, Inc., Newton, NJ, USA). After reflected by a rotatable flat mirror (M), which functions as a scanner of illumination directions, the light beam passes through a polarization state generator (PSG) made up of a polarizer (P; PGT5012, Union Optic, Inc., Wuhan, China) and an optimally designed zero-order quartz crystal waveplate (C\({}_{\text{r1}}\)). Then, a non-polarizing beam splitter (BS) and an achromatic doublet (L\({}_{1}\)) guides light rays to be focused on the back focal plane (BFP) of a high numerical-aperture (NA) objective lens (OB; EC Epiplan-Apochromat 50\(\times\)/0.95 HD DIC, Zeiss, Inc., Oberkochen, Germany). The emerging light from the BFP of the OB is then parallelly incident upon the sample. Afterwards, the scattered light from the sample successively traverses the OB, L\({}_{1}\), BS, a polarization state analyzer that consists of the same polarization components as a PSG but in a reverse order, another achromatic doublet (L\({}_{2}\)), and is finally collected by a detector (D; PCO.edge 5.5, PCO, Inc., Kelheim, Germany). As presented in the inset of Figure 1, the position of the focal point of the light beam on the BFP of the OB can be changed by rotating the flat mirror M, which further leads to the change of illumination direction on the sample. An epi-illumination setup is designed to collect the scattered-field distribution associated with each illumination direction by imaging the BFP of the OB. Thanks to the dual rotating-compensator configuration, a 4 \(\times\) 4 Mueller matrix associated with each point on the BFP of the OB can be obtained. Since the 16 elements of a Mueller matrix contain all polarization information that one can extract from a linear polarization scattering process, the full polarization properties of the scattered field are thus achieved. As for the Mueller matrix measurement, the Stokes vector \(S_{\text{out}}\) of the light beam detected by the detector can be expressed as the following Mueller matrix product with incident Stokes vector \(S_{\text{in}}\)[8; 22]:  
\[ \begin{split} S_{\text{out}}&=[\mathbf{M}_{\text{A}} \mathbf{R}(A_{0})][\mathbf{R}(-C_{20})\mathbf{M}_{\text{C2}}(\delta_{2}) \mathbf{R}(C_{20})][\mathbf{M}_{\text{t}}^{\text{BS}}\mathbf{M}_{\text{b}}^{ \text{OB}}\mathbf{M}_{\text{S}}\mathbf{M}_{\text{f}}^{\text{OB}}\mathbf{M}_{ \text{r}}^{\text{BS}}]\\ &\quad\times[\mathbf{R}(-C_{10})\mathbf{M}_{\text{C1}}(\delta_{1}) \mathbf{R}(C_{10})][\mathbf{R}(-\text{P}_{0})\mathbf{M}_{\text{P}}]S_{\text{in} },\end{split}  \tag{1}\]  
where \(\mathbf{M}_{\text{P}}\), \(\mathbf{M}_{\text{C1}}(\delta_{1})\), \(\mathbf{M}_{\text{C2}}(\delta_{2})\), and \(\mathbf{M}_{\text{A}}\) are the Mueller matrices of the polarizer \(\text{P}\), the first compensator \(\text{C}_{\text{r1}}\), the second compensator \(\text{C}_{\text{r2}}\), and the analyzer A; \(M_{f}^{BS}\) and \(M_{t}^{BS}\) are reflection and transmission matrices of the BS, respectively. \(M_{f}^{OB}\) and \(M_{b}^{OB}\) represent Mueller matrices of the OB in the illumination and collection directions, respectively. \(\mathbf{M}_{\text{S}}\) is the sample Mueller matrix; \(\mathbf{R}(\cdot)\) is the Mueller rotation transformation matrix for rotation by an angle, which can be the initial transmission-axis orientations of the polarizer and analyzer, \(P_{0}\) and \(A_{0}\), the initial fast-axis orientations of the 1 st and 2 nd rotating compensators, \(C_{10}\) and \(C_{20}\). \(\delta_{1}\) and \(\delta_{2}\) are the phase retardances of 1 st and 2 nd compensators, respectively.