Bastiaans, Martin J.Alieva Krasheninnikova, Tatiana2023-06-202023-06-202010-041084-752910.1364/JOSAA.27.000918https://hdl.handle.net/20.500.14352/44417© 2010 Optical Society of America. The financial support of the Spanish Ministry of Science and Innovation under project TEC2008-04105 and the Santander-Complutense project PR-34/07-15914 is acknowledged.Based on the analysis of second-order moments, a generalized canonical representation of a two-dimensional optical signal is proposed, which is associated with the angular Poincare sphere. Vortex-free ( or zero-twist) optical beams arise on the equator of this sphere, while beams with a maximum vorticity ( or maximum twist) are located at the poles. An easy way is shown how the latitude on the sphere, which is a measure for the degree of vorticity, can be derived from the second-order moments. The latitude is invariant when the beam propagates through a first-order optical system between conjugate planes. To change the vorticity of a beam, a system that does not operate between conjugate planes is needed, with the gyrator as the prime representative of such a system. A direct way is derived to find an optical system ( consisting of a lens, a magnifier, a rotator, and a gyrator) that transforms a beam with an arbitrary moment matrix into its canonical form.engjournal articlehttp://dx.doi.org/10.1364/JOSAA.27.000918http://www.opticsinfobase.org/open access535Schell-model beamsWigner distribution functionPartially coherent beams1st-order optical-systemsLight-beamsDecompositionVortexTransformationPropagationSpectrumÓptica (Física)2209.19 Óptica Física