## 2 Operating Principle of Our OV Characterizer  
First, we briefly describe the operating principle used in characterizing OVs with arbitrary SoPs. This system consists of an SPG, two PPGs, and an imaging sensor (Figure 1). The SPG and PPG are Pancharatnum-Berry phase elements that imprint space-variant anisotropic patterns on elements [17; 18; 19; 20; 21]. The PPG is divided into two parts PPG(+) and PPG(\(-\)). The slow axes of the PPG(+) and PPG(\(-\)) are both rotated counter-clockwise along the vector direction of the grating, whereas the parabolic lines are inverted (Figure 2). Using the [PERSON] matrix formula, the functions of the SPG and the two PPGs are written as  
\[\mathbf{T}_{S}=\cos\Gamma \begin{pmatrix}1&0\\ 0&1\end{pmatrix} +\frac{1}{2}\sin\frac{\Gamma}{2}e^{i\frac{2\pi\alpha}{\Lambda} } \begin{pmatrix}i&1\\ 1&-i\end{pmatrix} +\frac{1}{2}\sin\frac{\Gamma}{2}e^{-i\frac{2\pi\alpha}{\Lambda }} \begin{pmatrix}i&-1\\ -1&-i\end{pmatrix} , \tag{1}\]  
\[\mathbf{T}_{P1}=\cos\Gamma \begin{pmatrix}1&0\\ 0&1\end{pmatrix} +\frac{1}{2}\sin\frac{\Gamma}{2}e^{i[\frac{2\pi\alpha}{\Lambda }+\phi(y)]} \begin{pmatrix}i&1\\ 1&-i\end{pmatrix} +\frac{1}{2}\sin\frac{\Gamma}{2}e^{-i[\frac{2\pi\alpha}{ \Lambda}+\phi(y)]} \begin{pmatrix}i&-1\\ -1&-i\end{pmatrix} , \tag{2}\]  
\[\mathbf{T}_{P2}=\cos\Gamma \begin{pmatrix}1&0\\ 0&1\end{pmatrix} +\frac{1}{2}\sin\frac{\Gamma}{2}e^{i[\frac{2\pi\alpha}{\Lambda }-\phi(y)]} \begin{pmatrix}i&1\\ 1&-i\end{pmatrix} +\frac{1}{2}\sin\frac{\Gamma}{2}e^{i[\frac{2\pi\alpha}{ \Lambda}-\phi(y)]} \begin{pmatrix}i&-1\\ -1&-i\end{pmatrix} , \tag{3}\]  
where \(\Gamma=2\pi\Delta nd/\lambda\) denotes the retardation of the SPG and PPGs, \(\Delta n\), \(d\), \(\Lambda\), and \(\lambda\) denote the birefringence, thickness of the polarization grating, grating pitch, and wavelength, respectively. The phase angle \(\phi(y)=\gamma y^{2}\) describes the parabolic phase pattern in the same manner as the curvature of a cylindrical lens. Here, \(\gamma\) is a curvature coefficient of cylindrical phase term. With Jones vectors \(|L\rangle=(1,\,i)/\sqrt{2}\) and \(|R\rangle=(1,-i)/\sqrt{2}\) for the left- and right-handed circular polarizations (LCP and RCP), the beam output from the straight-line and PPGs are expressed as  
\[\mathbf{E}_{S}^{\mathrm{L}}=\mathbf{T}_{S}|L\rangle=\cos\Gamma|L\rangle+\frac {1}{2}\sin\frac{\Gamma}{2}e^{i[\frac{2\pi\alpha}{\Lambda}]}|R\rangle, \tag{4}\]  
\[\mathbf{E}_{S}^{R}=\mathbf{T}_{S}|R\rangle=\cos\Gamma|R\rangle+\frac{1}{2} \sin\frac{\Gamma}{2}e^{-i[\frac{2\pi\alpha}{\Lambda}]}|L\rangle, \tag{5}\]  
\[\mathbf{E}_{P(+)}^{\mathrm{L}}=\mathbf{T}_{P1}|L\rangle=\coswhere \(\mathbf{E}_{5}^{\text{L}}\), \(\mathbf{E}_{5}^{\text{R}}\), \(\mathbf{E}_{P(+)}^{\text{L}}\), \(\mathbf{E}_{P(-)}^{\text{R}}\), \(\mathbf{E}_{P(-)}^{\text{L}}\), and \(\mathbf{E}_{P(-)}^{\text{R}}\) represent the Jones vectors of the output beams from the SPG, \(\text{PPG}_{(+)}\), and \(\text{PPG}_{(-)}\) for LCP and RCP incident beams, respectively. From these equations, the SPG and PPG both show 100% diffraction efficiency for which the retardation is \(\Gamma=\pi\). According to Equations (4) and (5), when LCP/RCP light is incident on the SPG, the transmitted light diffracts toward the direction of the \(+1\)st\(/-1\)st order mode, depending on the handedness of the circular polarization [see Figure 2a]. Similarly, from Equations (6) and (7) for the \(\text{PPG}_{(+)}\), the transmitted light is converted to cylindrically spreading/focusing light and diffracts toward the \(+1\)st\(/-1\)st order mode direction, depending on the handedness of the circular polarization [Figure 2b]. From Equations (8) and (9) for the \(\text{PPG}_{(-)}\), the transmitted light is also converted to cylindrically focusing/spreading light but diffracts toward the \(+1\)st\(/-1\)st order mode direction, depending on the handedness of the circular polarization [Figure 2c]. Given the operations of the SPG and PPGs, we now describe the characterization principle of the TC.