# Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam
## 3 Numerical Results
The orientation angle \(\varphi\) is given by the following formula:  
\[\varphi\Big{(}\mathbf{r}^{{}^{\prime}}\Big{)}=\frac{1}{2}\mathrm{arctan} \left(\frac{2\mathrm{Re}\Big{[}W_{xy}\Big{(}\mathbf{r}^{{}^{\prime}},\mathbf{ r}^{{}^{\prime}}\Big{)}\Big{]}}{W_{xx}\big{(}\mathbf{r}^{{}^{\prime}},\mathbf{ r}^{{}^{\prime}}\big{)}-W_{yy}\big{(}\mathbf{r}^{{}^{\prime}},\mathbf{r}^{{}^{ \prime}}\big{)}}\right),\quad(-\pi/2\leq\varphi\leq\pi/2), \tag{13}\]  
and the major and minor semiaxes of the polarization ellipse take the following form:  
\[A_{\pm}\Big{(}\mathbf{r}^{{}^{\prime}},\mathbf{r}^{{}^{\prime}}\Big{)}=\frac {1}{\sqrt{2}}\bigg{\{}\sqrt{\big{(}W_{xx}-W_{yy}\big{)}^{2}+4\big{|}W_{xy} \big{|}^{2}}\,\pm\sqrt{\big{(}W_{xx}-W_{yy}\big{)}^{2}+4\big{(}\mathrm{Re} \big{[}W_{xy}\big{]}\big{)}^{2}}\bigg{\}}^{1/2}. \tag{14}\]  
Then, the degree of ellipticity \(\varepsilon\) characterizing the shape of the polarization ellipse of an electromagnetic beam is defined by:  
\[\varepsilon=A_{-}\Big{(}\mathbf{r}^{{}^{\prime}},\mathbf{r}^{{}^{\prime}} \Big{)}/A_{+}\Big{(}\mathbf{r}^{{}^{\prime}},\mathbf{r}^{{}^{\prime}}\Big{)}, \quad 0\leq\varepsilon\leq 1. \tag{15}\]  
On submitting from Equations (5)-(8) into Equations (13)-(15), we numerically investigated the behaviors of the SOP in Figure 7 of a focused twisted CGC radially polarized beam at different propagation distances for different values of \(n\). Owing to the twisted phase, it is of interest to find from Figure 7a1-a4 that the initial radially polarized structure gradually evolves into an azimuthally polarized structure in the focal plane. Moreover, similar to the spectral DOC, one finds that the SOP experiences a rotation of \(\pi/2\) with the focal length, regardless of the value of order \(n\). In addition,  
Figure 6: Density plot of the DOP the corresponding cross line of a focused twisted CGC radially polarized beam at different propagation distances for different values of the twist phase with \(n=1\). For (**a1**–**a3**) \(\mu=0.0005\,\mathrm{mm}^{-1}\); (**b1**–**b3**) \(\mu=0.001\,\mathrm{mm}^{-1}\); (**c1**–**c3**) \(\mu=0.002\,\mathrm{mm}^{-1}\); (**a4**,**b4**,**c4**) \(\mu=0\,\mathrm{mm}^{-1}\). for the case of \(n\
eq 0\), the SOP splits with the splitting of the spectral intensity, which is the same as that in absence of the twist phase [25; 26]. Figure 8 shows the dependence of the SOP on the twist phase. Similar to the spectral density and the spectral DOC, it is seen that although the twist speed of the SOP closely depends on the value of the twist phase, the SOP always rotates \(\pi/2\) within the focal length compared to that without twist factor. Therefore, it is worthwhile to manipulate novel complex vector beams by taking advantage of the twist phase and tailored coherence structure. Figure 8: The SOP and the corresponding cross line of focused twisted CGC radially polarized beam at different propagation distances for different values of the twist phase with \(n=2\). For (**a1–a4**) \(\mu=0.0005\,\mathrm{mm}^{-1}\); (**b1–b4**) \(\mu=0.001\,mm^{-1}\); (**c1–c4**) \(\mu=0.002\,\mathrm{mm}^{-1}\). Figure 3:0 The state of polarization (SOP) and the corresponding cross line of a focused twisted CGC radially polarized beam at different propagation distances for different values of \(n\). For (**a1–a3**), (**b1–b3**), (**c1–c3**) \(\mu=0.001\,\mathrm{mm}^{-1}\); (**a4,b4,c4**) \(\mu=0\,\mathrm{mm}^{-1}\). Finally, we concentrate our attention on the spectral properties of a focused polychromatic twisted CGC radially polarized beam. It is well known that the correlation-induced spectral change (also called the \"Wolf effect\") is an important feature of partially coherent beams. It was demonstrated that Wolf effect has important applications in optical signal processing, information encoding and exchange [5; 6; 67; 68; 69; 70]. Since the spectral change is closely related to the coherence and polarization, it is meaningful to study the spectral shift of a twisted nontrivial CGC radially polarized beam.