# Propagation Characteristics of a Twisted Cosine-Gaussian Correlated Radially Polarized Beam
## 2 Theory  
It is known that the vector electric field of a radially polarized beam can be described as the coherent superposition of TEM\({}_{01}\) Laguerre-Gaussian modes oriented along the \(x\) axis and a TEM\({}_{10}\) oriented along the \(y\) axis at \(z=0\)[43; 44; 45; 46; 47; 48]. \[\mathbf{E}(\mathbf{r})=E_{x}(\mathbf{r})\mathbf{e}_{x}+E_{y}(\mathbf{r}) \mathbf{e}_{y}=\frac{x}{w_{0}}\exp\left(-\frac{\mathbf{r}^{2}}{w_{0}^{2}} \right)\mathbf{e}_{x}+\frac{y}{w_{0}}\exp\left(-\frac{\mathbf{r}^{2}}{w_{0}^{2 }}\right)\mathbf{e}_{y}, \tag{1}\]  
where \(\mathbf{r}=\left(x^{2}+y^{2}\right)^{1/2}\) denotes the transversal distance from the beam center and \(w_{0}\) is the transverse beam size. For a vector partially coherent beam in space-frequency domain, the second-order correlation properties of a fluctuating light beam can be completely described by the CSD matrix \(\overline{\mathbf{W}}(\mathbf{r}_{1},\mathbf{r}_{2})\) with elements \(W(\mathbf{r}_{1},\mathbf{r}_{2})=\left\langle E_{\alpha}^{*}(\mathbf{r}_{1})E _{\beta}(\mathbf{r}_{2}),\alpha,\beta\in\{x,y\}\right\rangle\). \(\mathbf{r}_{1}\) and \(\mathbf{r}_{2}\) denote two arbitrary points with position vectors in the source plane. For brevity, we omit the explicit dependence of the considered quantities on frequency \(\omega\). The asterisk denotes the complex conjugate and the angular brackets represent average over the statistical ensemble. For a partially coherent radially polarized beam, the elements of the CSD matrix are described as [46]:  
\[W_{\alpha\beta}(\mathbf{r}_{1},\mathbf{r}_{2})=\frac{\alpha_{1}\beta_{2}}{w_{ 0}^{2}}\exp\left(-\frac{\mathbf{r}_{1}^{2}+\mathbf{r}_{2}^{2}}{w_{0}^{2}} \right)g_{\alpha\beta}(\mathbf{r}_{1},\mathbf{r}_{2}),\quad(\alpha,\beta\in\{ x,y\}), \tag{2}\]where \(g\) denotes the degree of coherence. It is important to note that the coherence structure is independent of spectral density profile. So, one can independently modulate the coherence function without affecting the spectral density of random sources [54]. Here, let us consider the coherence function of the initial beam to have a twisted cosine-Gaussian correlated (CGC) function [25; 54]:  
\[g_{\alpha\beta}(\mathbf{r}_{1},\mathbf{r}_{2})=\exp\left[-\frac{(\mathbf{r}_{1 }-\mathbf{r}_{2})^{2}}{2\delta_{0}^{2}}\right]\cos\left[\frac{n\sqrt{\pi}(x_{ 1}-x_{2})}{\delta_{0}}\right]\cos\left[\frac{n\sqrt{\pi}(y_{1}-y_{2})}{\delta_ {0}}\right]\exp[-ik\mu(x_{1}y_{2}-x_{2}y_{1})], \tag{3}\]  
where \(\delta_{0}\) denotes coherence parameter, \(n\) is a positive beam order parameter, and \(\mu\) represents the twist phase. When \(n=0\), a twisted CGC radially polarized beam reduces to a conventional Schell-model (SM) radially polarized beam [44; 45; 46; 47; 48]. Using Mercer's expansion, it is proven that the realizability condition for generation of such twisted random source coincides with SM sources. As a new class of twisted non-uniformly correlated vector beams, a possible experimental approach for generating twisted CGC radially polarized beams can be achieved by using a spatial light modulator and an astigmatic optical lens system [20; 25; 53]. In [25], we have demonstrated an experiment for generating a CGC correlated radially polarized beam. One may further produce a twisted CGC radially polarized beam by passing a CGC correlated radially polarized beam through an astigmatic optical lens system [53].