Correlation geometry and topology of structured optical beams
Correlation geometry and topology in a random, scalar beam carrying Orbital Angular Momentum (OAM) in $L$ modes are shown to be linked to the real and imaginary parts, respectively, of its orbitalization matrix (OM). The OM is obtained by filtering the cross-spectral density in the polar Fourier basis at a given cross-section and radius for each pair of OAM indices. The symmetric real part of the OM has the diagonal canonical form, specifying the ellipsoid-like correlation geometry of the beam. The skew-symmetric imaginary part of the OM has the Darboux canonical form, enabling decomposition into $N\leq L/2$ mutually orthogonal circulation states and $L-2N$ trivial circulation-free states, mutually orthogonal and $L-2N$ trivial circulation states thereby defining the $N$-dimensional orbitalization torus associated with the correlation topology of the beam. The canonical forms are then used to define the degrees of linear and circular orbital anisotropy and stability, and the degrees of linear and circular orbitalization, in analogy with corresponding quantities in 2D and 3D polarization theory. These results demonstrate the emergence of fundamentally new correlation structures in random multimode OAM-carrying light, not accessible in lower-dimensional systems and absent in the deterministic limit.