The generalized Jones vectors (GJVs) (t indicates the transpose) that describe the linear, circular, and elliptical polarization states of an arbitrary three-dimensional (3-D) monochromatic light field are determined in terms of the geometrical parameters of the 3-D vibration of the time-harmonic electric field. In three dimensions, there are as many distinct linear polarization states as there are points on the surface of a hemisphere, and the number of distinct 3-D circular polarization states equals that of all two-dimensional (2-D) polarization states on the Poincaré sphere, of which only two are circular states. The subset of 3-D polarization states that results from the superposition of three mutually orthogonal x, y, and z field components of equal amplitude is considered as a function of their relative phases. Interesting contours of equal ellipticity and equal inclination of the normal to the polarization ellipse with respect to the x axis are obtained in 2-D phase space. Finally, the generalized Jones calculus, in which elastic scattering (e.g., by a nano-object in the near field) is characterized by the 3-D linear transformation , is briefly introduced. In such a matrix transformation, and are the GJVs of the incident and scattered waves and T is the generalized Jones matrix of the scatterer at a given frequency and for given directions of incidence and scattering.
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