## 3 Parameter Analysis and Theoretical Calculation  
When the incident beam is perpendicular to the optical axis, the simplified schematic diagram from Figure 1 becomes that shown in Figure 2, where the radially polarized beam transmits through a BPPF and is subsequently focused by a high NA lens. According to [PERSON]'s vector diffraction theory, the transverse and longitudinal electric fields near the focal region in Figure 2 can be written as Equation (1) and Equation (2), respectively. \[E_{r}(r,z)=A\int\limits_{0}^{\alpha}\cos^{1/2}\theta\sin(2\theta)l(\theta)T^{ \prime}(\theta)J_{1}(kr\sin\theta)e^{i\delta z\cos\theta}d\theta, \tag{1}\]  
Figure 1: (**a**) Schematic of the optical setup. The insets (**b**) and (**c**) are enlarged views of the 2D galvanometer system and the binary phase pupil filter, respectively. \[E_{z}(r,z)=2 iA\int\limits_{0}^{\alpha}\cos^{1/2}\theta\sin^{2}\theta l(\theta)T^{ \prime}(\theta)J_{0}(kr\sin\theta)e^{ikz\cos\theta}d\theta. \tag{2}\]  
In Equations (1) and (2), \(A\) is the coefficient of power (\(A=1\)), \(k\) is the wavenumber (\(k=2\pi/\lambda\)), \(J_{0}\) and \(J_{1}\) denote Bessel functions, \(l(\theta)\) describes the amplitude distribution of the incident beam, and \(\alpha\) is the maximum angle of the focus given by \(\arcsin(NA/n_{0})\), where \(NA\) is the numerical aperture. In this manuscript, \(NA\) is set to 0.95, the refractive index \(n_{0}=1\), and the wavelength \(\lambda=405\ nm\). The transmission function of the BPPF is \(T(\beta)=\exp[i\varphi(\beta)]\), where \(\varphi(\beta)\) is the phase distribution corresponding to each ring. According to the principles of constructive and destructive interference, the simulated annealing algorithm [18; 19] is used to optimize the filter structure. Using a five-ring BPPF, the angle \(\beta_{i}\)(i = 1\(\sim\)4) corresponds to the four radial positions \(r_{i}=(\sin\beta_{i})/NA\). For different ranges of the angle \(\beta\), \(\varphi(\beta)=0\) or \(\varphi(\beta)=\pi\). Therefore, the transmittance function can be expressed as  
\[T(\beta)=\left\{ \begin{array}{ll}1,&for\ 0\leq\beta<\beta_{1},\ \beta_{2}\leq \beta<\beta_{3},\ \beta_{4}\leq\beta<\alpha\\ -1,&for\ \beta_{1}\leq\beta<\beta_{2},\ \beta_{3}\leq\beta<\beta_{4}\end{array}  \right.. \tag{3}\]  
Essentially, the BPPF modulates the optical field through phase difference. When the focused beam is incident parallel to the optical axis, the oblique incidence angle is \(0^{\circ}\), and the optical path difference of the phase modulation is completely derived from the BPPF, as displayed in Figure 3a. In the case shown in Figure 1a, oblique incidence adds additional phase difference. To simplify the calculation, the oblique incidence of a radially polarized beam can be regarded as the normal incidence of a radially polarized beam, while the BPPF is placed at a certain angle to the optical axis, as shown in Figure 3b. In Figure 3, the wider step (\(h1\)) represents a phase of size \(\pi\), and the narrower step (\(h2\)) represents a phase of size 0. Figure 2: Schematic of focusing of a radially polarized beam with a binary phase pupil filter (BPPF) and high numerical aperture (NA) lens. When the BPPF is placed perpendicular to the optical axis, the optical path difference is  
\[\Delta L_{1}=h_{1}n-(\Delta hn_{0}+h_{2}n)=\Delta h(n-n_{0}). \tag{4}\]  
The corresponding phase difference is  
\[\Delta\varphi_{1}=\varphi(\theta)=\frac{2\pi}{\lambda}\Delta h(n-n_{0}), \tag{5}\]  
where \(n\) is the refractive index of the BPPF with wavelength \(\lambda\). When the BPPF is placed obliquely, the optical path difference is  
\[\Delta L_{2}=(n_{0}l_{2}+\frac{nh_{1}}{\cos\theta_{1}})-(n_{0}l_{1}+\frac{n_{ 0}\Delta h}{\cos\theta}+\frac{nh_{2}}{\cos\theta_{1}}). \tag{6}\]  
The angle between the BPPF and the optical axis is \(\theta\), and the refraction angle is \(\theta_{1}\). When \(\theta\) is small, the increased optical path of the incident beam (\(l_{1}\)) is considered equal to the increased optical path of the ejected beam (\(l_{2}\)).