The Poincaré sphere together with the Stokes parameters is widely used for the representation of the beam polarization state. A similar approach has been generalized [M. J. Padgett and J. Courtial, Opt. Lett. 24, 430 (1999); G. S. Agarwal, J. Opt. Soc. Am. A 16, 2914 (1999); T. Alieva and M. J. Bastiaans, Opt. Lett. 34, 410 (2009)] to the characterization of intrinsically anisotropic Gaussian-type modes. Recently [M. J. Bastiaans and T. Alieva, "Signal representation on the angular Poincaré sphere, based on second-order moments," J. Opt. Soc. Am. A (2010), accepted for publication] we have proposed to use this formalism for the description of an arbitrary, scalar, two-dimensional signal, which might be completely or partially coherent. In this paper, based on the analysis of the beamís second-order moments expressed as a series of Hermite-Gaussian (HG) modes, we analyze the movements on the angular Poincaré sphere during beam propagation through an isotropic ABCD system.
Two second-order moments invariants allow to divide two-dimensional signals into two classes: isotropic and anisotropic. Using the Iwasawa decomposition of the ray transformation matrix and bringing the second-order moments matrix to its diagonalized form, we are able to associate the signal with a certain point on the sphere similar to the one applied for Gaussian mode representation. The latitude of this point describes the vorticity of the signal, while its longitude corresponds to the orientation of the beam's principal axes. The propagation through the isotropic ABCD system modifies the diagonalizing matrix; consequently, the point on the Poincaré sphere is moving, which corresponds to a change of the vorticity state. We analyze these movements using the HG modes composition of the signal. In particular we confirm - as expected - that stable modes do not change their position on the Poincaré sphere during beam propagation through an isotropic system.
The proposed approach is useful for beam characterization, analysis and synthesis.
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