Fig. 1. Poincaré sphere of electromagnetic spatial coherence. For an arbitrary partially polarized and partially coherent beam two coherence Poincaré vectors ${\mathbf{q}}_{12}$ and${\mathbf{q}}_{21}$ with the same length$P_{\mathrm{\Omega }}{\mu }^2$ but different orientations are generally needed (Sphere A). If the beam is fully polarized the lengths of these vectors display the degree of coherence,$\left|{\mathbf{q}}_{12}\right|=\left|{\mathbf{q}}_{21}\right|={\mu }^2$, and their directions show the polarization state of the beam at points${\mathbf{r}}_2$ and${\mathbf{r}}_1$, respectively (Sphere B). If the beam is uniformly polarized, these vectors coincide and a single coherence Poincaré vector$\mathbf{q}$ is sufficient to represent the beam, with its length again showing the degree of coherence and the direction specifying the polarization state (Sphere C). At a single point the formalism reduces to the traditional polarization Poincaré sphere where the distance from the origin is given by the degree of polarization$P$ [1] (Sphere D).