# Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam
## 3 Numerical Results
As is seen from Figure 4, the rotation speed is nonlinear and closely depends on the value of the twist factor. However, when they reach the focal plane, they all undergo a rotation of \(\pi/2\), regardless of the value of the twist phase. Furthermore, it is of interest to see that the spectral DOC spot becomes larger as the twist factor increases due to the beam divergence caused by the twist phase. Figure 3: Modulus of the spectral degree of coherence (DOC) and the corresponding cross line of a focused twisted CCC radially polarized beam at different propagation distances for different values of \(n\). For (**a1–a4**), (**b1–b4**), (**c1–c4**) \(\mu=0.001\,\mathrm{mm}^{-1}\); (**a5,b5,c5**) \(\mu=0\,\mathrm{mm}^{-1}\). Figure 2: Normalized spectral intensity distribution, polarized spectral intensity components \(W_{xx}\left(\dot{\mathbf{r}},\dot{\mathbf{r}}^{\prime}\right)\) and \(W_{yy}\left(\dot{\mathbf{r}},\dot{\mathbf{r}}^{\prime}\right)\), and the corresponding cross line of a focused twisted CGC radially polarized beam at different propagating distances with \(n=2\). For (**a1–a3**), (**b1–b3**), (**c1–c3**) \(\mu=0.001\,\mathrm{mm}^{-1}\); (**a4,b4,c4**) \(\mu=0\,\mathrm{mm}^{-1}\). In order to learn more about the vector properties, in Figure 5 the behaviors of the DOP of a focused twisted CGC radially polarized beam at different propagation distances for different values of \(n\) is plotted. It has been shown that the on-axis DOP singularity of an SM radially polarized beam is propagation invariant [51]. However, for a twisted SM radially polarized beam, it is interesting to see from Figure 5a1-a3 that a similar on-axis DOP singularity disappears due to the existence of a twisted phase. In addition, quite different from the spectral density and the spectral DOC, a striking feature can be seen in which the twist phase does not lead to a rotation of the DOP. Figure 6 illustrates the DOP and the corresponding cross line of a focused twisted CGC radially polarized beam at different propagation distances for different values of the twist phase. It is seen that the complex DOP structure gradually degenerates and becomes more uniform. This is because the astigmatism gradually increases with the increase of twist factor. Figure 4: Modulus of the spectral DOC and the corresponding cross line of focused twisted CGC radially polarized beam at different propagation distances for different values of the twist phase. For (**a1–a4**) \(\mu=0.0005\,\)mm\({}^{-1}\); (**b1–b4**) \(\mu=0.001\,\)mm\({}^{-1}\); (**c1–c4**) \(\mu=0.002\,\)mm\({}^{-1}\); (**a5,b5,c5**) \(\mu=0\,\)mm\({}^{-1}\). Figure 5: Density plot of the DOP 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**) \(\mu=0.001\,\)mm\({}^{-1}\); (**a4,b4,c4**) \(\mu=0\,\)mm\({}^{-1}\). Next, let us further evaluate the effects of the twist phase on the state of polarization (SOP). It is well known that the CSD matrix can be represented as a sum of a completely polarized beam and a completely unpolarized beam [2]. For a partially coherent beam, the polarization ellipse is a parameter characterizing the fully polarized portion of the beam. By using the CSD matrix, one can conveniently determine the polarization ellipse, including the major and minor semiaxes of the polarization ellipse, and the orientation angle [8].