## 2 Principles  
Figure 1 illustrates the design of the optical structure for the generation of radially and azimuthally polarized beams. It consists of a pair of confocal cylindrical lenses with a focal length ratio of 1:2, a lateral displacement polarizing beamsplitter (LDPB), h-HWP and f-HWP, and an LDB. To facilitate an understanding of the principle, an orthogonal \(x\)-\(y\)-\(z\) coordinate system is introduced. The cross-section of the laser beam (viewing against the direction of light propagation) and the associated states of polarization after passing through each component are depicted in the four quadrants of \(Q_{1}\), \(Q_{2}\), \(Q_{3}\), and \(Q_{4}\), as shown in Figure 1a-g. In Figure 1, the Jones column matrix for a linearly polarized beam propagating in the \(+z\)-direction can be expressed as [14]:  
\[E(\theta)=\left[ \begin{array}{c}\cos\theta\\ \sin\theta\end{array} \right] \tag{1}\]  
where \(\theta\) indicates the direction of polarization with respect to the \(x\)-axis; and \(\cos\theta\) and \(\sin\theta\) represent the horizontal and vertical components of the polarization, respectively. In Figure 1a, a circular Gaussian beam with a \(45^{\circ}\) linear polarization (with respect to the \(x\)-axis) is incident into a pair of cylindrical lenses in the \(+z\)-direction. Consequently, its beam cross-section is transformed into an elliptical Gaussian beam, as illustrated in Figure 1b. After passing through the LDPB, the \(45^{\circ}\) linearly polarized elliptical Gaussian beam is separated into two orthogonally polarized components, as illustrated in Figure 1c. These components continue to pass through the h-HWP and the f-HWP. The optical axes of the h-HWP and f-HWP are set at \(45^{\circ}\) and \(22.5^{\circ}\) with respect to the \(x\)-axis, respectively. Therefore, as shown in Figure 1d, the states of polarization in the right-half regions (quadrants \(Q_{1}\) and \(Q_{4}\)) for these elliptical beams are rotated by \(90^{\circ}\). In Figure 1e, the states of polarizations at the four quadrants of these elliptical beams are rotated by 45\({}^{\circ}\). Accordingly, the Jones column vector \(\mathbf{E_{Qi}}\) of each polarized beam at the \(i\)-th quadrant (\(i=1\) to 4) can be expressed as:  
\[\mathbf{E_{Qi}}=\mathbf{I_{f-HWP}}(\Theta)\mathbf{I_{h-HWP}}(\Theta)E(\theta),\ (\text{for $i=1$ then $\theta=0^{\circ}$; for $i=4$ then $\theta=270^{\circ}$)} \tag{2}\]  
and  
\[\mathbf{E_{Qi}}=\mathbf{I_{f-HWP}}(\Theta)E(\theta),\ (\text{for $i=2$ then $\theta=180^{\circ}$; for $i=3$ then $\theta=90^{\circ}$)} \tag{3}\]  
where \(\mathbf{J}\) represents the Jones matrix of the half-wave plate, and \(\Theta\) represents the direction of the optical axis of the half-wave plate with respect to the \(x\)-axis. Consequently, the Jones column vectors of the polarized beams at the four quadrants can be written as  
\[\mathbf{E_{Qi}}=\left[ \begin{array}{c}1\\ 1\end{array} \right],\ \ \mathbf{E_{Q2}}=\left[ \begin{array}{c}-1\\ 1\end{array} \right],\ \ \mathbf{E_{Q3}}=\left[ \begin{array}{c}-1\\ -1\end{array} \right],\ \ \text{and}\ \ \mathbf{E_{Q4}}=\left[ \begin{array}{c}1\\ -1\end{array} \right]\]  
Finally, the two polarized beams are concentrated after passing through and reflecting off the LDB, respectively. In Figure 1f,g, the beams are two sets of 45\({}^{\circ}\) frame-rotated Hermite-Gaussian TEM\({}_{10}\) and TEM\({}_{01}\) modes. Based on the image inversion caused by a right-angle prism and the reflection of polarized light, the Jones column vectors of the polarized beam (illustrated in Figure 1g) with a new set of quadrants \(Q_{1}\)*, \(Q_{2}\)*, \(Q_{3}\)*, and \(Q_{4}\)*, can be written as  
\[\mathbf{E_{Qi}}*=\left[ \begin{array}{c}-1\\ 1\end{array} \right],\ \ \mathbf{E_{Q2}}*=\left[ \begin{array}{c}1\\ 1\end{array} \right],\ \ \mathbf{E_{Q3}}*=\left[ \begin{array}{c}1\\ -1\end{array} \right],\ \ \text{and}\ \ \mathbf{E_{Q4}}*=\left[ \begin{array}{c}-1\\ -1\end{array} \right],\]  
where the direction of polarization occurs with respect to the \(-z\)-axis.