## 2 Method  
The inverse problem in optical scatterometry is described as an objective to minimize a least-square function, which can be generally expressed as:  
\[\chi^{2}=\sum_{i=1}^{N}\bigl{[}[\mathbf{y}_{i}-\mathbf{f}(\mathbf{x},\ \mathbf{a}_{i})]^{\mathsf{T}}\mathbf{w}_{i}[\mathbf{y}_{i}-\mathbf{f}( \mathbf{x},\ \mathbf{a}_{i})]\bigr{]}, \tag{1}\]  
where, \(\mathbf{x}=[x_{1},\,x_{2},\,\ldots,\,x_{m}]^{\mathsf{T}}\) represents a column vector of \(m\) profile parameters (e.g., CD, sidewall angle, and height) and \(\mathbf{f}(\mathbf{x},\ \mathbf{a}_{i})\) is a column vector containing the calculated signature according to the forward optical model under the \(i\)th measurement configuration \(\mathbf{a}_{i}\). The measurement configuration \(\mathbf{a}_{i}\) is a combination of fixed azimuthal angle \(\varphi\), fixed incident angle \(\theta\), and the \(i\)th measurement wavelength \(\lambda_{i}\) for spectroscopic scatterometry, namely \(\mathbf{a}_{i}=[\varphi,\,\theta,\,\lambda_{i}]^{\mathsf{T}}\), and is a combination of fixed azimuthal angle \(\varphi\), fixed measurement wavelength \(\lambda\), and the \(i\)th incident angle \(\theta_{i}\) for angle-resolved scatterometry, namely, \(\mathbf{a}_{i}=[\varphi,\,\theta_{i},\,\lambda]^{\mathsf{T}}\). Correspondingly, the \(N\) in Equation (1) denotes the number of measurement configurations, which can be the number of measurement wavelengths for spectroscopic scatterometry or the number of incidence angles for angle-resolved scatterometry. The column vector \(\mathbf{y}_{i}=[y_{i1},\,y_{i2},\,\ldots,y_{i}]^{\mathsf{T}}\) consists of the corresponding measured signature under the same measurement configuration \(\mathbf{a}_{i}\). Here, the value of \(l\) depends on the specific types of the measured signature. If the measured signature is reflectance, \(l=1\), while if the measured signature is a Mueller matrix, \(l=16\) (or \(l=15\) for the normalized Mueller matrix). The weighting matrix \(\mathbf{w}_{i}\) is a \(l\times l\) positive definite matrix, which is usually chosen to be the inverse of the covariance matrix of the measured signature under the \(i\)th measurement configuration \(\mathbf{a}_{i}\). In this case, Equation (1) relates to the commonly used multivariate chi-square statistics \(\chi^{2}\). Writing the right side of Equation (1) in a matrix expression, the solution \(\mathbf{\hat{x}}\) of the inverse problem can be obtained by:  
\[\mathbf{\hat{x}}=\underset{\mathbf{x}\in\Omega}{\arg\min}\bigl{\{}[\mathbf{y} -\mathbf{f}(\mathbf{x},\ \mathbf{a})]^{\mathsf{T}}\mathbf{w}[\mathbf{y}-\mathbf{f}(\mathbf{x},\ \mathbf{a})]\bigr{\}}, \tag{2}\]  
here, \(\Omega\) is the associated profile parameter domain. The use of \(\chi^{2}\) is built on top of the belief that the measurement errors are normally distributed with zero mean, namely, the measured signature \(\mathbf{y}\) can be expressed as:  
\[\mathbf{y}=\mathbf{f}(\mathbf{\hat{x}},\ \mathbf{a})+\mathbf{\varepsilon}, \tag{3}\]  
where \(\mathbf{\varepsilon}\) is a vector of multiple random and independent variables with each element subjected to a normal distribution with zero mean. Assuming a profile parameter vector \(\mathbf{x}\) is close enough to the true or nominal parameter vector \(\mathbf{x}^{\star}\) under a measurement configuration \(\mathbf{a}_{i}\), and the function \(\mathbf{f}(\mathbf{x},\ \mathbf{a}_{i})\) is sufficiently smooth, then the function value \(\mathbf{f}(\mathbf{x},\ \mathbf{a}_{i})\) can be expanded in the vicinity of \(\mathbf{x}^{\star}\) using the first-order Taylor expansion formulation [30]:  
\[\mathbf{f}(\mathbf{x},\ \mathbf{a}_{i})\approx\mathbf{f}(\mathbf{x}\star,\ \mathbf{a}_{i})+\mathbf{J}(\mathbf{x}\star,\ \mathbf{a}_{i})\cdot\Delta\mathbf{x}, \tag{4}\]where \(\mathbf{J}(\mathbf{x}^{\star},\mathbf{a}_{i})\) is the Jacobian matrix with respect to \(\mathbf{x}\) at \(\mathbf{x}=\mathbf{x}^{\star}\) and \(\Delta\mathbf{x}\) represents the error in \(\mathbf{x}\) and is given by \(\Delta\mathbf{x}=\mathbf{x}-\mathbf{x}^{\star}\).