# Metrology of Nanostructures by Tomographic Mueller-Matrix Scatterometry
## 3 Methods  
### Inverse Modeling  
The inverse modeling includes the solution of a direct problem and an indirect problem. The direct problem involves the establishment of a model to simulate the theoretical Mueller matrices of a nanostructure. The indirect problem involves the reconstruction of the nanostructure profile with the objective of finding an optimal input to the above established model whose simulated Mueller matrices can best match the measured values. As for the direction problem here, the rigorous coupled-wave analysis (RCWA) [23; 24] is used to simulate theoretical Mueller matrices of a periodic nanostructure. Figure 2 shows the case of polarized light incidence for a one-dimensional grating structure. The grating is denoted by vector  
Figure 1: Scheme of the tomographic Mueller-matrix scatterometer (TMS). LS, light source; AS, aperture stop; CL, collimated lens; BF, bandpass filter; M, mirror; P and A, polarizer and analyzer; \(\text{C}_{\text{r1}}\) and \(\text{C}_{\text{r2}}\), 1 st and 2 nd rotating compensators; FS, field stop; BS, Beam splitter; \(\text{L}_{1}\) and \(\text{L}_{2}\), imaging lenses; OB, objective lens; S, sample; D, detector. The inset on the right side illustrates the principle of illumination direction scanning. \(\mathbf{x}=[x_{1},x_{2},\ldots,x_{n}]^{T}\) where \(x_{1}\), \(x_{2}\),..., \(x_{n}\) represent structural parameters such as top CD, height, sidewall angle, and so on. As shown in Figure 2b, the grating is divided by several different layers with boundaries joined together. Inverse modeling includes three steps: (1) derive electromagnetic expression from [PERSON]'s equations; (2) expand the permittivity function and electromagnetic fields into Fourier series; (3) and match the tangential field components at boundaries between different layers, reducing the boundary-value problem to an algebraic problem. According to the reflection coefficients calculated by solving the algebraic problem, a 2 \(\times\) 2 Jones matrix \(\mathbf{J}\) associated with the diffracted light can be formulated by:  
\[\left[ \begin{array}{c}E_{rp}\\ E_{rs}\end{array} \right]=\mathbf{J}(\mathbf{x})\left[ \begin{array}{c}E_{ip }\\ E_{is}\end{array} \right]=\left[ \begin{array}{cc}r_{pp}&r_{ps}\\ r_{sp}&r_{ss}\end{array} \right]\left[ \begin{array}{c}E_{ip}\\ E_{is}\end{array} \right], \tag{2}\]  
where \(E_{\text{s,p}}\) refers to the electric field component perpendicular and parallel to the plane of incidence, respectively. When depolarization could be ignored, the Mueller matrix of the sample can be obtained by:  
\[\mathbf{M}=\mathbf{A}(\mathbf{J}\otimes\mathbf{J}^{*})\mathbf{A}^{-1}, \tag{3}\]  
where the symbol \(\otimes\) denotes the Kronecker product. \(\mathbf{J}^{*}\) is the complex conjugate of \(\mathbf{J}\), and the matrix \(\mathbf{A}\) is given by:  
\[\mathbf{A}=\left[ \begin{array}{cccc}1&0&0&1\\ 1&0&0&-1\\ 0&1&1&0\\ 0&\mathbf{i}&-\mathbf{i}&0\end{array} \right]. \tag{4}\]  
when depolarization could not be ignored, the sample Mueller matrix can be represented as the sum of multiple non-depolarizing Mueller matrices \(\mathbf{M}=\sum\limits_{i}M_{i}\), and each non-depolarizing Mueller matrices can be calculated as mentioned above [25]. In practice, the Mueller matrix \(\mathbf{M}\) is usually normalized to the (1, 1)th element \(M_{11}\), with the */normalized Mueller matrix elements being \(m_{ij}=M_{ij}/M_{11}\). As for the indirect problem, a weighted least-squares regression analysis (Levenberg-Marquardt algorithm [26]) is performed, during which the structural parameters under measurement are varied until the calculated and measured data match as close as possible.