# Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam
## 3 Numerical Results  
Now, we numerically analyze the focusing properties of a twisted CGC radially polarized beam with the help of the theoretical results derived above. Let us consider the source beam passes through a thin lens with focal length\(f\) and then arrives at the receiver plane. The ray matrix of such optical system reads as:  
\[\left( \begin{array}{cc}A&B\\ C&D\end{array} \right)=\left( \begin{array}{cc}1&z\\ 0&1\end{array} \right)\left( \begin{array}{cc}1&0\\ -1/f&1\end{array} \right)\left( \begin{array}{cc}1&f\\ 0&1\end{array} \right)=\left( \begin{array}{cc}1-z/f&f\\ -1/f&0\end{array} \right). \tag{12}\]  
The global parameters used in the following calculations are set as \(\lambda=632.8\,\mathrm{nm},w_{0}=1\,\mathrm{mm}\), \(\delta_{0}=0.2\,\mathrm{mm},\mu=0.001\,\mathrm{mm}^{-1}\), and \(f=150\,\mathrm{mm}\), unless different values are specified. In Figure 1, we depict the normalized spectral density distribution and the corresponding cross line of a twisted CGC radially polarized beam focused by a thin lens at several propagation distances for different values of \(n\). For the case of \(n\
eq 0\), it is shown that the spectral density gradually decomposes into a four-beamlets array distribution, and the distance between each beamlet grows as the beam order \(n\) increases. Moreover, one clearly sees that the spectral density distribution of a twisted CGC radially polarized beam in the focal plane is significantly different from that in the absence of the twist phase. As expected, for a CGC radially polarized beam without the twist phase, there is a Fourier-like relation between the spectral density in the focal plane and the initial correlation function [25]. However, for a twisted CGC radially polarized beam, this genuine reciprocal relationship disappears due to the existence of the twist phase, see Figure 1a3,b3,c3. The reason for this is that the twist phase leads to a beam astigmatism during diffraction, which causes a focal shift. In order to study the spectral density distribution of polarization components \(W_{xx}\left(\mathbf{r},\mathbf{r}^{{}^{\prime}}\right)\) and \(W_{yy}\left(\mathbf{r}^{{}^{\prime}},\mathbf{r}^{{}^{\prime}}\right)\), Figure 2 plots the normalized polarized spectral density components and the corresponding cross lines at several propagation distances with \(n=2\). It is clearly seen that the spectral density not only splits during spreading, but also each spectral density component undergoes a rotation of \(\pi/2\) within the focal length. In addition, for a CGC radially polarized beam, see Figure 2b4,c4, one finds that the two orthogonal spectral density components have the same distribution in the focal plane. Figure 3 shows the modulus of the spectral DOC and the corresponding cross line at different propagating distances for different values of \(n\). One finds that the array structure of the spectral DOC becomes more complex with the growth of the order parameter \(n\). Similar to the spectral density, it is clearly seen that the spectral DOC rotates during the spreading and also experience a rotation of \(\pi/2\) when they reach the focal plane. At the same time, it is found that the rotation speed depends only on the twist phase, see Figure 3b1-b3,c1-c3. In addition, due to the existence of the twist phase, the Fourier reciprocal relationship between the spectral DOC in the focal plane and the initial spectral intensity no longer exists. Thus, the spectral DOC of a twisted CGC radially polarized beam in the focal plane is quite different from that without the twist phase [25]. Figure 1: Normalized spectral density distribution and the corresponding cross line of a focused twisted cosine-Gaussian correlated (CGC) radially polarized beam at different propagating distances for different values of \(n\). For (**a1**–**a3**), (**b1**–**b3**), (**c1**–**c3**) \(\mu=0.001\,\mathrm{mm}^{-1}\); (**a4,b4,c4**) \(\mu=0\,\mathrm{mm}^{-1}\). Figure 4 illustrates the effect of the twist phase on the spectral DOC.