# Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam
## 3 Numerical Results
Here, let us assume that the initial spectrum is a Lorentz type with \(\omega_{0}\) being the central frequency and \(\Gamma_{0}\) being the half-width at half-maximum, and \(\omega\) is the angular frequency. Then the elements of the CSD matrix of a focused polychromatic twisted CGC radially polarized beam in the output plane are given as:  
\[ \begin{array}{ll}W_{xx}\left(\mathbf{r}_{1}^{{}^{\prime}},\mathbf{r}_{2}^{{}^ {\prime}},\omega\right)&=\frac{k^{2}}{64u_{0}^{2}B^{2}N_{1}^{2}(N_{2}-\Omega)^{ 2}}\cdot\frac{\Gamma_{0}^{2}}{\left[(\omega-\omega_{0})^{2}+\Gamma_{0}^{2} \right]}\exp\Bigl{[}-\frac{ikD}{2B}\left(\mathbf{r}_{1}^{{}^{\prime}2}-\mathbf{ r}_{2}^{{}^{\prime}2}\right)\Bigr{]}\\ &\times\sum\limits_{a_{1}=\pm a}\sum\limits_{a_{2}=\pm a}\biggl{[}\xi_{i\pi 1 1}(\xi_{i\pi 21}+i\Pi_{1})+\delta_{0}^{-2}-\frac{(\vec{i}_{\pi 21}-\Pi_{1})^{2}}{2 \delta_{0}^{2}(N_{2}-\Omega)}+\frac{k\mu(\xi_{i\pi 21}+i\Pi_{1})(\vec{i}_{\xi 2 02}-\Pi_{2})}{2(N_{2}-\Omega)}\biggr{]}\\ &\times\exp\biggl{[}-\frac{\vec{\xi}_{i\pi 21}^{{}^{\prime}2}+\vec{\xi}_{i \pi 1}^{{}^{\prime}2}}{4N_{1}}+\frac{(\vec{i}_{\xi 21}-\Pi_{1})^{2}+(\vec{i}_{\xi 2 22}-\Pi_{2})^{2}}{4(N_{2}-\Omega)}\biggr{]},\end{array}  \tag{16}\]  
\[ \begin{array}{ll}W_{yy}\left(\mathbf{r}_{1}^{{}^{\prime}},\mathbf{r}_{2}^{{}^ {\prime}},\omega\right)&=\frac{k^{2}}{64u_{0}^{2}B^{2}N_{1}^{2}(N_{2}-\Omega)^{ 2}}\cdot\frac{\Gamma_{0}^{2}}{\left[(\omega-\omega_{0})^{2}+\Gamma_{0}^{2} \right]}\exp\Bigl{[}-\frac{ikD}{2B}\left(\mathbf{r}_{1}^{{}^{\prime}2}-\mathbf{ r}_{2}^{{}^{\prime}2}\right)\Bigr{]}\\ &\times\sum\limits_{a_{1}=\pm a}\sum\limits_{a_{2}=\pm a}\biggl{[}\xi_{i\pi 1 2}(\xi_{i\pi 22}+i\Pi_{2})+\delta_{0}^{-2}-\frac{(\vec{i}_{\xi 22}-\Pi_{2})^{2}}{2 \delta_{0}^{2}(N_{2}-\Omega)}-\frac{k\mu(\xi_{i\pi 22}+i\Pi_{1})(\vec{i}_{\xi 2 02}-\Pi_{1})}{2(N_{2}-\Omega)}\biggr{]}\\ &\times\exp\biggl{[}-\frac{\vec{\xi}_{i\pi 21}^{{}^{\prime}2}+\vec{\xi}_{i \pi 1}^{{}^{\prime}2}}{4N_{1}}+\frac{(\vec{i}_{\xi 22}-\Pi_{1})^{2}+(\vec{i}_{\xi 22}-\Pi_{1})^{2}}{4(N_{2}-\Omega)} \biggr{]},\end{array}  \tag{17}\]  
\[ \begin{array}{ll}W_{xy}\left(\mathbf{r}_{1}^{{}^{\prime}},\mathbf{r}_{2}^{{}^ {\prime}},\omega\right)&=\frac{k^{2}}{64u_{0}^{2}B^{2}N_{1}^{2}(N_{2}-\Omega)^ {2}}\cdot\frac{\Gamma_{0}^{2}}{\left[(\omega-\omega_{0})^{2}+\Gamma_{0}^{2} \right]}\exp\Bigl{[}-\frac{ikD}{2B}\left(\mathbf{r}_{1}^{{}^{\prime}2}-\mathbf{ r}_{2}^{{}^{\prime}2}\right)\Bigr{]}\\ &\times\sum\limits_{a_{1}=\pm a}\sum\limits_{a_{2}=\pm a}\biggl{[}\xi_{i\pi 11}(\xi_{i\pi 22}+i\Pi_{ 2})+ik\mu+ik\mu\frac{(\vec{i}_{\xi 22}-\Pi_{2})^{2}}{2(N_{2}-\Omega)}-\frac{(\vec{i}_{\xi 22}-\Pi_{2})(\vec{i}_{\xi 2 02}-\Pi_{1})}{2\delta_{0}^{2}(N_{2}-\Omega)}\biggr{]}\\ &\times\exp\biggl{[}-\frac{\vec{\xi}_{i\pi 21}^{{}^{\prime}2}+\vec{\xi}_{i \pi 1}^{{}^{\prime}2}}{4N_{1}}+\frac{(\vec{i}_{\xi 22}-\Pi_{1})^{2}+(\vec{i}_{\xi 22}-\Pi_{1})^{2}}{4(N_{2}-\Omega)}\biggr{]}.\end{array}  \tag{18}\]  
Now, we numerically investigate the relative spectral changes of a polychromatic twisted CGC radially polarized beam focused by a thin lens. The spectral shift \(\Delta\omega\) is the difference between the peak frequency \(\omega_{\max}\) of the spectrum after propagation and the central frequency \(\omega_{0}\) of the source spectrum. A positive value of \(\Delta\omega\) means a blueshift, while a negative value denotes a redshift. The relative spectral shift can be defined as:  
\[\gamma=(\omega_{\max}-\omega_{0})/\omega_{0}. \tag{19}\]  
The parameters are set as \(\omega_{0}=3.6\times 10^{15}\) and \(\Gamma_{0}=1\times 10^{14}\). Figure 9 shows the relative spectral shift \(\gamma\) of a focused polychromatic twisted CGC radially polarized beam versus the propagation distance \(z\). For a polychromatic partially coherent radially polarized beam without the twist phase, a study showed that an on-axis blueshift can be found before and after the focus [47]. However, for a polychromatic twisted radially polarized beam, the on-axis spectral shift is always redshifted, with the minimum redshift occurring in the focal plane, see Figure 9e.