# Metrology of Nanostructures by Tomographic Mueller-Matrix Scatterometry
## 4 Results and Discussion
### Measurement of Grating Structural Parameters  
To reduce errors caused by sample nonuniformity, the measurements of grating pitch and other structural parameters were carried out on the same grating region. Figure 5 presents the scattered field of the photoresist grating collected at the wavelength of 633 nm, the incidence angles varied from 0\({}^{\circ}\) to 65\({}^{\circ}\) at interval of 1\({}^{\circ}\), azimuthal angles from 0\({}^{\circ}\) to 90\({}^{\circ}\) at interval of 5\({}^{\circ}\). The reason why the azimuthal angle was limited to the range of 0\({}^{\circ}\)\(\sim\)90\({}^{\circ}\) rather than 0\({}^{\circ}\)\(\sim\)360\({}^{\circ}\) is that the Mueller matrices stay unchanged after a 180\({}^{\circ}\) rotation, due to rotation symmetry of the photoresist grating. The grating also has reflection symmetry relative to the plane that is perpendicular to the periodic direction. Since the photoresist grating sample is a periodic structure, the collected scattered field is actually a diffracted field. Due to the limitation of the NA of the objective lens in the TMS, for each illumination direction, only the zeroth-order diffracted light was collected. As shown in Figure 5, the panel associated with each Mueller matrix element consists of three sectorial parts, of which the lower right part corresponds to the TMS-measured data. The polar coordinate system of the sectorial part is set, as shown in the inset of Figure 5, with the radial coordinate being the sine of the incidence angle sin\(\theta\) and the angular coordinate being the azimuthal angle \(\varphi\). As can be observed, as the azimuthal angles change, the grating exhibits anisotropy with the 2 \(\times\) 2 off-diagonal Mueller matrix elements deviating from about 0 at \(\varphi=0^{\circ}\), and finally approaching to about 0 when \(\varphi=90^{\circ}\). We also calculated the depolarization index distribution associated with the TMS-measured data according to \(DI=\sqrt{\left[\mathrm{Tr}(\mathbf{M}\mathbf{M}^{\mathrm{T}})-m_{11}^{2} \right]/3m_{11}^{2}}\) (0 \(\leq DI\leq 1\)) [33], with \(DI=1\) and \(DI=0\) representing a totally non-depolarizing and a totally depolarizing Mueller matrix, respectively. The calculated  
Figure 4: SEM micrograph and geometric model of the photoresist grating. depolarization indices indicated that \(|DI-1|<0.028\), so the depolarization effect could be ignored in the following data analysis. It should be noted that the angle-resolved Mueller polarimetry [34] can also acquire the diffracted field of a grating sample similar to the TMS. However, it should be pointed out that the measurement modes of the angle-resolved Mueller polarimetry and TMS are inherently different. For the angle-resolved Mueller polarimetry, polarization properties are acquired simultaneously by illuminating a sample simultaneously at all possible incidence angles and imaging the BFP of a high NA objective lens. When the sample is a periodic dense structure, both the TMS and angle-resolved Mueller polarimetry are appropriate for characterizing the sample. However, for a case in which the pitch of a periodic dense structure is greater than \(\lambda\)/(2\(\times\)NA), or an isolated or non-periodic structure is detected, an overlapping between the polarized scattering information occurs at the BFP of the objective lens in the angle-resolved Mueller polarimetry. In comparison, the TMS overcomes these limits and can distinguish information appropriately. As shown in Figure 5, the upper left one of the three sectorial parts of each panel presents the calculated best-fit data to the TMS-measured data. To make it clear, the difference between the TMS-measured data and the calculated best-fit data is presented in the lower left sectorial part of each panel. It was found that the difference of most points was less than 0.02. To make a comparison, we also provided the results measured by the commercial MME.