# Besinc Pseudo-Shell Model Sources with Circular Coherence
## 2 Source Model
It can be noted that the region where the degree of coherence is significant is a circle for \(r_{2}=0\), but, on increasing \(r_{2}\), becomes a donut (with radius \(r_{2}\)). It should be noted that, for a point at a fixed distance \(r_{2}\) from the source center, there is a set of points, located in concentric circles of radius \(r_{1}\), where the field is completely uncorrelated to the field in \(r_{2}\). The radius of these circles satisfies the condition \(J_{1}\left[\zeta(r_{2}-r_{1})/\delta_{c}\right]=0\) with \(r_{1}\
eq r_{2}\). Figure 2 shows the profile of the absolute value of the degree of coherence along a radius \(r_{1}\) for several values of \(r_{2}\). It can be noted that, for \(r_{2}>2\delta_{c}\), the thickness of the donut-shaped area of high coherence is constant, so that the coherence area grows linearly with increasing \(r_{2}\). Figure 1: Absolute value of the degree of coherence relative to a point located at the following distances \(r_{2}\) from the source center: \(r_{2}=0\) (**left**); \(r_{2}=0.5\delta_{c}\) (**center**), and \(r_{2}=\delta_{c}\) (**right**). Figure 2: Absolute value of the degree of coherence as a function of the radius \(r_{1}\) for different distances \(r_{2}\) from the source center. Once the function \(g(r)\) has been selected, different sources are obtained on changing the modulating function \(\tau\left(\mathbf{r}\right)\). Let us choose \(\tau\) as a Laguerre-Gaussian function [29] with Laguerre polynomials of order zero, that is,  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]0  
where \(I_{0}\) is a parameter having dimensions of an irradiance, \(m\) is an integer, and \(w_{0}\) is a positive quantity related to the source width. It reduces to a Gaussian function when \(m=0\) while, for \(m>0\), presents a donut-like shape.The resulting CSD will be denoted by \(W_{m}\left(\mathbf{r}_{1},\mathbf{r}_{2},0\right)\) and takes the form  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]1  
A pseudo-modal expansion in terms of coherent pseudo-modes [27; 30] can be written for the whole class of sources described by the CSD of Equation (10) in the following form:  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]2  
where \(\Lambda_{m,n}\) are the pseudo-eigenvalues and the corresponding coherent pseudo-modes are given by  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]3  
In the derivation of Equation (11), the following relation for the Bessel functions has been taken into account [30; 31]:  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]4  
The parameters \(\beta_{m,n}\) are normalization factors defined as  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]5  
in such a way that the pseudo-modes \(\Phi_{m,n}\left(\mathbf{r},0\right)\) are normalized. However, they are not orthogonal and for this reason we refer to them as pseudo-modes. By comparing Equations (10)-(14), the pseudo-eigenvalues \(\Lambda_{m,n}\) can be obtained as \(\Lambda_{m,n}=n^{2}\beta_{m,n}^{2}\) that are explicitly nonnnegative, which confirms the nonnegativeness of the proposed CSD. The normalization factors can be expressed in terms of the generalized hypergeometric functions \({}_{p}F_{q}\left(a_{1},a_{2},\ldots,a_{p};b_{1},b_{2},\ldots,b_{q};c\right)\)[31] as  
\[H\left(\mathbf{r},\boldsymbol{v}\right)=\tau\left(\mathbf{r}\right)G\left(v \right)\exp\left(ikrv\cos\phi\right), \tag{2}\]6  
Figures 3 and 4 show the dependence of the first eight pseudo-eigenvalues as functions of the ratio \(\delta_{c}/w_{0}\) for the cases \(m=0\) and \(m=1\), respectively.