# Besinc Pseudo-Shell Model Sources with Circular Coherence
## 2 Source Model  
The CSD function \(W\left(\mathbf{r}_{1},\mathbf{r}_{2},0\right)\), defined as the field correlation at two points \(\mathbf{r}_{j}=\left(r_{j},\theta_{j}\right)\) with \(j=1,2\) across the source plane, can be used to properly describe the spatial coherence properties of a light source [10]. Some years ago, a superposition rule was presented [26; 27] to establish a sufficient and necessary condition that any function must fulfill to represent a physically realizable CSD. According to it, \(W\left(\mathbf{r}_{1},\mathbf{r}_{2},0\right)\) is a _bona fide_ CSD provided that it can be written as  
\[W\left(\mathbf{r}_{1},\mathbf{r}_{2},0\right)=\int H^{*}\left(\mathbf{r}_{1}, \boldsymbol{v}\right)H\left(\mathbf{r}_{2},\boldsymbol{v}\right)\mathrm{d} \boldsymbol{v}, \tag{1}\]  
where \(H\left(\mathbf{r}_{2},\boldsymbol{v}\right)\) is a kernel that, in this work, has been chosen as  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]  
\(k\) being the wavenumber. Substitution of Equation (2) into Equation (1) results in the following CSD  
\[W\left(\mathbf{r}_{1},\mathbf{r}_{2},0\right)=\tau^{*}\left(\mathbf{r}_{1} \right)\tau\left(\mathbf{r}_{2}\right)g\left(r_{2}-r_{1}\right), \tag{3}\]  
where  
\[g\left(r\right)=2\pi\int_{0}^{\infty}\left|G(v)\right|^{2}J_{0}\left(krv \right)v\,dv \tag{4}\]  
and \(J_{n}(\cdot)\) is the Bessel function of first kind and order \(n\)[28]. Furthermore, by suitably normalizing the function \(G(v)\), it can be obtained that \(g(0)=1\). It can be observed that the irradiance of the source, \(I(\boldsymbol{\rho},0)=W\left(\boldsymbol{\rho},\boldsymbol{\rho},0\right)\), depends on \(\tau\left(\boldsymbol{\rho}\right)\) only, while its degree of coherence, defined as [10]  
\[\mu\left(\mathbf{r}_{1},\mathbf{r}_{2},0\right)=\frac{W\left(\mathbf{r}_{1}, \mathbf{r}_{2},0\right)}{\sqrt{W\left(\mathbf{r}_{1},\mathbf{r}_{1},0\right) W\left(\mathbf{r}_{2},\mathbf{r}_{2},0\right)}}=\frac{\tau^{*}\left(\mathbf{r}_{1} \right)\tau\left(\mathbf{r}_{2}\right)}{\left|\tau\left(\mathbf{r}_{1}\right) \tau\left(\mathbf{r}_{2}\right)\right|}g(r_{2}-r_{1}), \tag{5}\]  
is determined by the function \(g\). Moreover, the absolute value of the degree of coherence depends only on the difference between the radial distances of the considered points from the source center. This means that the coherence characteristics of such sources are shift invariant along the radial coordinate and, therefore, they belong to the class of the pseudo-Schell model sources [20; 24]. Different families of pseudo-Schell sources are generated for each selection of the functions \(G(v)\) and \(\tau\left(\mathbf{r}_{2}\right)\). In this work, we take \(G(v)\) as  
\[G(v)=\frac{k\delta_{c}}{\xi\sqrt{\pi}}\text{ circ}\left(\frac{k\delta_{c}}{\xi}v \right), \tag{6}\]corresponding to (from Equation (4))  
\[g(r)=2\frac{I_{1}(\zeta r/\delta_{c})}{\zeta r/\delta_{c}}. \tag{7}\]  
Here, \(\zeta\) is the lowest positive value for which \(J_{1}\) is zero, \(\delta_{c}\) is a distance related with the coherence area and \(\text{circ}(t)=1\) for \(|t|\leq 1\) and \(0\), otherwise. The function in Equation (7) is also known as besinc. The absolute value of the degree of coherence at the source plane turns out to be  
\[|\mu(\mathbf{r_{1}},\mathbf{r_{2}},0)|=2\ \left|\frac{I_{1}\left[\zeta(r_{2}-r_{1} )/\delta_{c}\right]}{\zeta(r_{2}-r_{1})/\delta_{c}}\right|. \tag{8}\]  
The degree of coherence at the source plane for sources described by Equation (3) is shift invariant along the radial coordinate. Hence, two points lying on the same circle concentric to the source center are completely coherent, while the coherence decreases for points belonging to different circles. In other terms, they present circular coherence [15; 16]. These results are shown in Figure 1, where the absolute value of the degree of coherence between points is plotted as a function of \(\mathbf{r_{1}}\) for fixed values of \(r_{2}\).