Beam mapping on the orbital Poincaré sphere

Representation of two-dimensional optical signals on the orbital angular Poincarý sphere is useful for beam analysis, synthesis and comparison. This mapping is based on the measurement of the second-order moments, which are widely used for beam characterization. It is well known that two second-order moments invariants allow dividing two-dimensional signals into two classes: isotropic and anisotropic. Using the modified Iwasawa decomposition of the ray transformationmatrix and bringing the second-order moments matrix to its diagonalized form, we are able to associate the anisotropic signal with a certain point on the sphere. The latitude of this point describes the vorticity of the signal, while its longitude corresponds to the orientation of the beam's principal axes. Apart from that, the beam's scaling and its curvature can be defined. Before beam comparison, it is thus appropriate to perform first its normalization and mapping on the Poincarý sphere. There are many very different beams associated with the same point and therefore this procedure makes sense for fine analysis of beams whose intensity distributions have similar forms. Moreover, every point on the sphere is associated with an orthonormal set of Hermite-Laguerre-Gaussian modes, which can be used for the corresponding beam decomposition that is important for its synthesis and analysis. The developed algorithm for the beam mapping is demonstrated on several examples.