## 2 Theoretical Formulation  
We consider the RVVB propagation along the z-axis in the Cartesian coordinate system [1; 2; 3]:  
\[E(r,\theta,z=0)=A(r)[\exp(i2\pi(r/r_{0})^{2}+i\theta_{0})e_{x}+\exp(-i2\pi(r/r _{0})^{2}+i\theta_{0})e_{y}], \tag{1}\]  
where \(r=(x^{2}+y^{2})^{1/2}\) is the polar radius and \(\theta=\arctan(y/x)\) is the azimuthal angle in the polar coordinate system. \(A(r)\) denotes the amplitude of the RVVB, \(\theta_{0}\) is the initial phase, \(e_{x}\) and \(e_{y}\) are the \(x\)- and \(y\)-direction unit vectors. Figure 1 shows the initial SoP alignment in a Gaussian beam cross-section. By using the Fourier transform of the initial RVVB, the transverse components of the vector angular spectrum \(A_{x}\) (\(\rho\)cos\(\phi\), \(\rho\)sin\(\phi\)) and \(A_{y}\) (\(\rho\)cos\(\phi\), \(\rho\)sin\(\phi\)) can be given as [20; 21; 22]:  
\[\left( \begin{array}{c}A_{x}(\rho\cos\phi,\rho\sin\phi)\\ A_{y}(\rho\cos\phi,\rho\sin\phi)\end{array} \right)=\frac{k}{2\pi}\left( \begin{array} []{c}\int_{0}^{\infty}\int_{0}^{2\pi}A(r)\exp\left[i2\pi(r/r_{0})^{2}+i\theta _{0}-ikr\rho\cos(\theta-\phi)\right]rd\theta dr\\ \int_{0}^{\infty}\int_{0}^{2\pi}A(r)\exp\left[-i2\pi(r/r_{0})^{2}+i\theta_{0} -ikr\rho\cos(\theta-\phi)\right]rd\theta dr\end{array} \right), \tag{2}\]  
where \(\rho\) and \(\phi\) are in relation with the transverse Fourier-transform variables \(\mu\), \(\
u\): \(\mu=\rho\)cos\(\phi\), \(\
u=\rho\)sin\(\phi\), and \(\xi=(1-\rho^{2})^{1/2}\). \(k\) is the wavenumber. For the Gaussian beam \(A(r)=\exp(-r^{2}/w^{2})\) where \(w\) indicates the waist of Gaussian beam, the \(x\), \(y\), and \(z\) components of the angular spectrum of a RVVB (Equation (1)) are:  
Figure 1: (Color online) The states of polarization (SoP) arrangement in the cross-sections of a radially-variant vector beam (RVVB) with a Gaussian amplitude. Here, \(\lambda\) is the wavelength. \[A_{x}(\rho\cos\phi,\rho\sin\phi)=\frac{{k^{2}{r_{0}}^{2}}{w^{2}}}{4\pi({r_{0}}^{2} -2\pi{w^{2}}i)}\exp(-\frac{{k^{2}\rho^{2}{r_{0}}^{2}}{w^{2}}}{4{r_{0}}^{2}-8\pi{w ^{2}}i}+i\theta_{0}), \tag{3}\]  
\[A_{y}(\rho\cos\phi,\rho\sin\phi)=\frac{{k^{2}{r_{0}}^{2}}{w^{2}}}{4\pi({r_{0}}^{ 2}+2\pi{w^{2}}i)}\exp(-\frac{{k^{2}\rho^{2}{r_{0}}^{2}}{w^{2}}}{4{r_{0}}^{2}+8 \pi{w^{2}}i}+i\theta_{0}), \tag{4}\]  
\[A_{z}(\rho\cos\phi,\rho\sin\phi)=\cos\phi A_{x}(\rho\cos\phi,\rho\sin\phi)+ \sin\phi A_{y}(\rho\cos\phi,\rho\sin\phi). \tag{5}\]  
The RVVB in the plane of propagation distance \(z\) can be calculated as [20; 21; 22]:  
\[E(r,z)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}A(\rho\cos\phi,\rho\sin \phi)\exp[ik(ux+vy+\xi z)]dudv. \tag{6}\]  
Therefore, the \(x\)-, \(y\)- and \(z\)- components of the RVVB can be obtained by Equations (3)-(6):  
\[E_{x}(r,z)=\frac{{k^{2}{w^{2}}{r_{0}}^{2}}\exp(i\theta_{0})}{2({r_{0}}^{2}-2 \pi{w^{2}}i)}\int_{0}^{\infty}\exp[-\frac{{k^{2}\rho^{2}}{r_{0}}^{2}{w^{2}}}{4 {r_{0}}^{2}-8\pi{w^{2}}i}+ik\sqrt{1-\rho^{2}}z]J_{0}(rk\rho)\rho d\rho, \tag{7}\]  
\[E_{y}(r,z)=\frac{{k^{2}{w^{2}}{r_{0}}^{2}}\exp(i\theta_{0})}{2({r_{0}}^{2}+2 \pi{w^{2}}i)}\int_{0}^{\infty}\exp[-\frac{{k^{2}\rho^{2}}{r_{0}}^{2}{w^{2}}}{4 {r_{0}}^{2}+8\pi{w^{2}}i}+ik\sqrt{1-\rho^{2}}z]J_{0}(rk\rho)\rho d\rho, \tag{8}\]  
\[E_{z}(r,z)=\frac{{ik^{2}{w^{2}}{r_{0}}^{2}}\exp(i\theta_{0})}{2}\Big{[}\frac{ \sin\theta}{{r_{0}}^{2}-2\pi{w^{2}}i}\int_{0}^{\infty}\exp[-\frac{{k^{2}\rho^ {2}}{r_{0}}^{2}{w^{2}}}{4{r_{0}}^{2}-8\pi{w^{2}}i}+ik\sqrt{1-\rho^{2}}z]J_{0}( rk\rho)\rho d\rho \tag{9}\]  
When the integrations are in the range: \(1\leq\rho<\infty\), they denote the evanescent wave component. On the other hand, if the integrations are in the region: \(0\leq\rho<1\), they refer to the propagating wave component [20; 21; 22].