## 2 Method
### BFP Ellipsometry Principle  
In a BFP ellipsometry system, the state of polarization of the incident light changes with the azimuthal angle on a BFP due to the high-\(NA\) objective lens. This incident plane rotating effect plays the role of a rotating polarizer, just like a typical rotating polarizer ellipsometer. Therefore, the annular intensity for any radius of the back focal plane is a function of the azimuthal angle. The ellipsometric parameters of the measured sample can thus be calculated by fitting the measured annular intensity to the optical model.  
As shown in Figure 1b,c, every radius of the BFP \(R\) is mapped to a unique incident angle \(\theta\). The angular range is determined by the \(NA\) of the objective lens, from \(\sin^{-1}NA\) to \(\sin^{-1}NA\). The measurement system can be modeled by the Jones matrix method, e.g.,  
\[E_{out}=R_{A}(-A)J_{A}R(A)R(-(\varphi+\frac{\pi}{2}))J_{M}J_{S}(\theta)R( \varphi)R_{P}(-P)J_{P}E_{in} \tag{1}\]  
where \(E_{in}\) and \(E_{out}\) are the input and output complex electric fields, respectively; \(J_{S}\), \(J_{A}\), and \(J_{P}\) are the Jones matrices of the sample, analyzer, and polarizer, respectively; and \(R\) is the coordinate rotation matrix. \(J_{M}\) is the reflection matrix redirected to the reference \(X\)-axis, and \(\varphi\) is the azimuth angle. \(P\) and \(A\) represent the angle of the transmittance axis of the polarizer and analyzer concerning the \(X\)-axis. By setting the polarizer and analyzer at 0 and 45 degrees, respectively, the received intensity of the captured BFP image can be expressed as follows:  
\[I_{out}(\varphi,\theta)=\alpha_{1}(\theta)+\alpha_{2}(\theta)\sin(2\varphi+ \frac{\pi}{4})+\alpha_{3}(\theta)\sin(4\varphi) \tag{2}\]  
where the three coefficients \(\alpha_{1},\alpha_{2},\text{ and }\alpha_{3}\) are the zero-, second-, and fourth-order Fourier coefficients of \(I_{out}(\varphi)\), respectively, and are the intermediate important quantities for solving subsequent parameters. For these three important intermediate Fourier coefficients, they can be easily extracted using the traditional Fourier transform. Further, the angle-resolved  
Figure 1: The measurement principle of BFP ellipsometry: (**a**) hardware configuration; (**b**) polarization change of incident and reflected beams denoted in blue and red, respectively; (**c**) the relationship between the BFP radius \(R\) and incident angle \(\theta\); (**d**) the BFP image of a SiO\({}_{2}\) film.  
ellipsometric parameters (\(\Psi(\theta)\) and \(\Delta(\theta)\)) can be calculated from the three coefficients \(\alpha_{1},\alpha_{2},\text{ and }\alpha_{3}\) using Equations (3) and (4):  
\[\tan^{2}\Psi(\theta)=\frac{\sqrt{2}\alpha_{1}(\theta)+\alpha_{2}(\theta)}{\alpha _{2}(\theta)-\sqrt{2}\alpha_{1}(\theta)}, \tag{3}\]  
\[\cos\Delta(\theta)=\frac{(\alpha_{3}(\theta)-\alpha_{1}(\theta))(\tan^{2}\Psi( \theta)+1)}{4\alpha_{1}(\theta)\tan\Psi(\theta)}. \tag{4}\]  
Subsequently, the film parameters can be determined by fitting the measured angle-resolved spectra to the theoretical model with nonlinear multivariate regression using the Levenberg-Marquardt (LM) algorithm [31]. The merit function is as follows:  
\[\text{LSE}=\frac{1}{N\!-\!1}\!\sum_{i=1}^{N}\!\left(\left|\Psi_{\text{exp}}( \theta_{i})-\Psi_{\text{the}}(\theta_{i},d)\right|^{2}+\left|\Delta_{\text{exp} }(\theta_{i})-\Delta_{\text{the}}(\theta_{i},d)\right|^{2}\right) \tag{5}\]  
where the subscripts \"exp\" and \"the\" represent the measured and theoretical values, respectively.