# Besinc Pseudo-Shell Model Sources with Circular Coherence
## 3 Paraxial Propagation  
In the following, it will be assumed that the light radiated from the planar source located at the plane \(z=0\) propagates under paraxial conditions along the \(z\)-axis of a suitable reference frame. The extended Huygens-Fresnel diffraction integral will be used to obtain the CSD of the field at any plane \(z=\) constant after propagation through an ABCD optical system as [10]  
\[ \begin{array}{rcl}W\left(\mathbf{R}_{1},\mathbf{R}_{2},z\right)&=&\frac{k^{2 }}{4\pi^{2}B^{2}}\iint W\left(\mathbf{r}_{1},\mathbf{r}_{2},0\right)\exp\left[ -\frac{\mathrm{i}kA}{2B}\left(r_{1}^{2}-r_{2}^{2}\right)-\frac{\mathrm{i}kD}{ 2B}\left(R_{1}^{2}-R_{2}^{2}\right)\right]\\ &\times&\exp\left[\frac{\mathrm{i}k}{2B}r_{1}R_{1}\cos\left(\phi_{1}-\theta_{ 1}\right)-\frac{\mathrm{i}k}{2B}r_{2}R_{2}\cos\left(\phi_{2}-\theta_{2}\right) \right]\mathrm{d}\mathbf{r}_{1}\mathrm{d}\mathbf{r}_{2},\end{array}  \tag{16}\]  
where \(\mathbf{R}_{j}=(R_{j},\phi_{j})\) (with \(j=1,2\)) are two typical position vectors across the transverse plane \(z=\)constant. Figure 4: Pseudo-eigenvalues, \(\Lambda_{m,n}=n^{2}\beta_{m,n}^{2}\), for the CSD in Equation (10) with \(m=1\) as functions of the ratio \(\delta_{c}/w_{0}\). The values of \(\beta_{m,n}\) are evaluated from Equation (15). For the case of the CSD in Equation (10), the angular integration of Equation (16) gives  
\[ \begin{array}{ll}W_{m}\left(\mathbf{R}_{1},\mathbf{R}_{2},z\right)=& \frac{I_{0}\,k^{2}}{2\pi B^{2}}\exp\left[\frac{\mathrm{i}kD}{2B} \left(R_{2}{}^{2}-R_{1}{}^{2}\right)\right]\\ &\times\int\limits_{0}^{\infty}\int\limits_{0}^{\infty}\left(\frac{2\,r_{1}\,r _{2}}{w_{0}^{2}}\right)^{m}\exp\left(-\frac{r_{1}^{2}+r_{2}^{2}}{w_{0}^{2}} \right)\frac{2J_{1}\left[\zeta\left(r_{2}-r_{1}\right)\,/\delta_{c}\right]}{ \zeta\left(r_{2}-r_{1}\right)\,/\delta_{c}}\\ &\times\exp\left[\frac{\mathrm{i}kA}{2B}\left(r_{2}^{2}-r_{1}^{2} \right)\right]J_{0}\left(\frac{k}{B}R_{1}\,r_{1}\right)J_{0}\left(\frac{k}{B} R_{2}\,r_{2}\right)r_{1}\,r_{2}\mathrm{d}r_{1}\mathrm{d}r_{2},\end{array}  \tag{17}\]  
which allows us to evaluate the irradiance profile and the degree of coherence of the propagated field across any transverse plane. By choosing the ABCD parameters corresponding to a free space propagation (\(A=1\), \(B=z\), \(C=0\), \(D=1\)), the evolution of the field CSD can be obtained. Results of the numerical evaluation of Equation (17) are shown in Figures 5 and 6 for \(m=0\) and \(m=1\), respectively. Both cases present a peculiar behavior of the intensity profile during propagation. Figure 5 shows the intensity profile for different values of the ratio \(\delta_{c}/w_{0}\) for the case \(m=0\). The intensity profile shows a bell-like shape with its peak on the beam axis. It can be observed that, for sufficiently small values of the ratio \(\delta_{c}/w_{0}\), the peak of the intensity profile grows up to a maximum value that is reached at a given propagation distance from the beam waist, which coincides with the source plane. The maximum value of the intensity and the plane at which this maximum is reached depend on the selected value of \(\delta_{c}/w_{0}\). Moreover, the beam shows a sharper intensity profile after a certain propagation distance than at the beam waist, as it can be easily seen from the insets. Figure 5: Evolution of the intensity profile with propagation distance for a beam with \(m=0\) and different values of the ratio \(\delta_{c}/w_{0}\) [numerical evaluation of Equation (17) for free space propagation]. The insets show the intensity profile across the source plane and across the plane where the maximum intensity is reached. Figure 6 shows the evolution of the intensity profile with the propagation distance for the case \(m=1\). At the source plane, the intensity profile shows a donut shape with a zero value at its center (see blue lines in the insets). However, when the beam propagates, the intensity at the beam axis grows and reaches a maximum value at a given propagation distance that depends on the value of the coherence parameter.