# Metrology of Nanostructures by Tomographic Mueller-Matrix Scatterometry
## 3 Methods
### Pitch Measurement  
In scatterometric data analysis, the grating pitch is commonly fixed to extract other profile parameters. However, inaccurate pitch values would lead to a poor result. Moreover, it is difficult to set the pitch dimension \(\Lambda\) as a parameter, because it is coupled with other parameters which may make measurements imprecise. Next, we show how to measure grating pitch by TMS.  
For grating periods \(\Lambda\) larger than \(\lambda/2\), where \(\lambda\) is the wavelength of the incident light, the first-order diffracted light can be easily observed at an emergent angle \(\theta\). Assuming \(\theta\) is the difference between the diffraction angles of the zeroth- and first-order diffracted light, according to the grating equation the grating pitch can be written as:  
\[\Lambda=\frac{\lambda}{\sin\theta\cos\theta+(1-cos\theta)sin\theta}. \tag{6}\]  
It is easy to see the variable \(\theta\) become the only parameter to measure pitch. There are three ways to obtain the value of \(\theta\). The first one is a so-called Littrow configuration [27]. The schematic drawing of the method is presented in Figure 3. The main point is to rotate the grating to ensure the incidence light coincides with the first-order diffracted light. Assumed rotating angle of the grating is \(\alpha\), then it is simple to know \(\theta\) = 2\(\alpha\) and \(\theta\) = 90\({}^{\circ}\) - \(\alpha\). Equation (6) could be rewritten as:  
\[\Lambda=\frac{\lambda}{2\sin\alpha}. \tag{7}\]  
The second method is to use a grating whose pitch is known as a reference. Because a camera is used as a detector, the number of pixels can be acquired between the zeroth- and first-order diffracted light spot projected on the camera, set as \(d_{1}\) and \(d_{2}\) for reference and sample grating, respectively. Next, we can calculate the difference of diffraction angles \(\theta_{1}\) and angle of zeroth-order diffracted light \(\beta_{0}\), which is equal to the incidence angle \(\theta\), using the grating equation. When the sample grating  
Figure 3: Littrow configuration for grating pitch measurement.  
is illuminated by the same incidence light, \(\beta_{0}\) is same to that of the reference one. Afterwards, the difference of diffraction angles \(\vartheta_{2}\) for simple grating can be obtained through the geometric relation:  
\[\frac{\tan\vartheta_{2}-\tan\beta_{0}}{\tan\vartheta_{1}-\tan\beta_{0}}=\frac{ d_{2}}{d_{1}}. \tag{8}\]  
In case of the influence of aberrations of the optical system, test targets such as concentric circle and crosshair grid targets are used to calibrate and compensate errors induced by the system. If there is no reference grating available, as a third method we could use two different lights whose wavelengths (\(\lambda_{1}\) and \(\lambda_{2}\)) are known to illuminate sample grating at the same incidence angle. It is similar to the second method, thus there is no more detailed description. For this case, another geometric relation is:  
\[\frac{\sin(\theta^{\prime}{}_{2}+\beta_{0})+\sin\beta_{0}}{\sin(\theta^{\prime }{}_{1}+\beta_{0})+\sin\beta_{0}}=\frac{\lambda_{2}}{\lambda_{1}}. \tag{9}\]  
The intensity of diffracted light is recorded by the camera. However, for a real light beam, the diameter could not be extremely small, so on the camera there usually is a spot with several pixels. When we deal with the signal, the data from a pixel is chosen for analysis. For a practical optical system, there are inevitably aberrations and misplacements of optical components. Due to the asymmetrical intensity distribution, the pixel chosen would influence the result. Therefore, a pixel picked from a spot shouldn't be random. The approach is based on calculating the center of mass (COM),  
\[p=\frac{\sum_{i=1}^{N}\sum_{j=1}^{M}iI_{ij}}{\sum_{i=1}^{N}\sum_{j=1}^{M}I_{ij} },\hskip 28.452756 ptq=\frac{\sum_{i=1}^{N}\sum_{j=1}^{M}jI_{ij}}{\sum_{i=1}^{N} \sum_{j=1}^{M}I_{ij}}, \tag{10}\]  
where \(I_{ij}\) is the intensity of pixel \((i,j)\) in a \(N\times M\) image, and \((p,q)\) is the COM of light spot.