## 3 Defocus Effect Calibration and Compensation  
### Defocused BFP Incident Angle Calibration  
In the proposed method, the relationship between BFP image radius \(R\) and incident angle \(\theta\) must be accurately calibrated at each defocus position. According to [PERSON]'s sine condition [32], the relationship between \(\theta\) and \(R\) can be expressed as \(\theta=\sin^{-1}\left(R\!\cdot\!NA/R_{max}\right)\), \(NA=\sin\theta_{max}\). [PERSON] [20] built the relationship assisted by the \(NA\) of the objective lens and \(R_{max}\) of the BFP image. However, \(R_{max}\) varies with the defocus position for blur and distortion, as shown in Figure 2. Therefore, the angle distortion happens at defocus positions where \(R_{max}\) no longer corresponds to \(NA\), and the mapping relationship is distorted.  
A full-field Brewster angle method is proposed to deal with the defocused calibration problem. The method stems from the fact that the position of minimum p-polarized reflectance corresponds to the Brewster angle. In our previous study [21], the ellipsometric amplitude ratio \(\Psi(\theta)\) was demonstrated to be more robust in searching for the position of the Brewster angles than the reflectance intensities. We begin by calculating the angle-resolved ellipsometric parameter \(\Psi(\theta)\) at the BFP of a boundary reflection and then searching for the Brewster angle \(\theta_{B}\) corresponded BFP radius \(R_{B}\). Here, a local quadratic polynomial fitting determines the pixel radius \(R_{B}\) of \(\Psi(\theta)_{\text{min}}\). Thus, the relationship between \(\theta\) and \(R\) is built using Equation (6):  
\[\theta=\sin^{-1}(f^{\prime}\cdot R),\ f^{\prime}=\sin(\theta_{B})/R_{B}\, \theta_{B}=\arctan(n_{1}/n_{0}) \tag{6}\]  
Figure 2: Maximum BFP image radii of a 44.34 nm SiO\({}_{2}\) film at different defocus positions and its blurring and distortion: (**a**) a backward-defocused BFP image near the objective lens; (**b**) an in-focus BFP image; (**c**) a forward-defocused BFP image far from the objective lens.  
where \(n_{0}\) is the refractive index of air, \(n_{1}\) is the refractive index of the standard sample, and \(f^{\prime}\) is a mapping coefficient in the Brewster angle calibration.  
In order to obtain the boundary reflection for searching \(\theta_{B}\), the light transmitted into the medium needs to be completely absorbed or filtered away. For this purpose, a thick transparent plate is normally used, where large-angle light of reflection from the backside can be spatially eliminated. Considering the limitation of \(NA\) (0.9) of the objective lens, \(\theta_{B}\) of a selected standard sample should be within the limited range of the objective lens. Therefore, a thick \(K9\) glass plate was chosen as the standard sample to calibrate the incident angle distortion due to the limited focal depth and because the \(NA\) of the used objective lens and the reflected light from the backside of a thick film reflects away from the detection unit. During the calibration process, the angle-resolved \(\Psi\) of each defocused BFP image was first calculated with Equation (3). Then, the pixel radius \(R_{B}\) corresponding to the Brewster angle at different defocus positions was fitted. Further, the calibration coefficient \(f^{\prime}\) was calculated at different defocus positions with Equation (6). From a set of Brewster angle mapping coefficients calibrated at different defocus positions, an \(R\)-\(\theta\) mapping function could then be built at a specific defocus position. Using this, a defocused BFP ellipsometric spectrum can be turned to be distortion-free in an incident angle domain.