Two-dimensional signal representation on the angular Poincaré sphere

Abstract

It is common to use the Poincaré sphere for the representation of the polarization1 (or, in other words, the spin components of the angular momentum) of a radiation beam. This approach has recently been generalized2–4 to the case of Gaussian-type modes which possess orbital angular momentum. In the present paper we propose to use this formalism for the description of an arbitrary scalar twodimensional signal, which might be deterministic or stochastic (coherent or partially coherent).\nA point on the Poincar´e sphere represents – as we may call it – a certain ‘direction’ of the orbital angular momentum, whose expectation value can be found from the ten second-order moments of the signal’s Wigner distribution,5, 6 arranged in a 4 × 4 matrix M. Transforming the signal into its canonical form, corresponding to a diagonal moment matrix,7 we associate this state with the intersection of the main meridian with the equator, where Cartesian coordinates are more appropriate for signal description. Applying orthosymplectic transformations8 (which, in particular, include the antisymmetric fractional Fourier transformation, the rotation transformation, and their cascades) to this state, the entire sphere can be populated, leading to a generalized canonical form. At the poles, we thus have the states with a possible z-component of the orbital angular momentum, where polar coordinates are the best choice for signal analysis. The way in which the signal is transformed into its canonical form defines principal axes (including relative coordinate scaling) for signal representation in phase space. The proposed approach is useful for different applications: beam characterization, adaptive filtering, signal analysis and synthesis, etc.