Necessary and sufficient conditions (termed the generalized Verdet–Stokes conditions) for light to be natural are derived. The general Verdet–Stokes conditions are requirements placed on all the moments and product moments of the two intensity projections in the Stokes–Verdet approach. The generalized Verdet–Stokes conditions are shown to be necessary and sufficient to guarantee that light be natural in that (1) all moments of the two projected intensities are independent of the rotation of the axes of the reference coordinate system, the phase retardation introduced into one of the components, and the time; (2) the product moments of the two projected intensities always decompose into products of their respective moments; (3) the probability-density functions of the projected intensities are negative exponential with the same variance. Additionally, the product moments of the Stokes parameters are obtained in order to study the behavior of the two intensity projections when the light is partially correlated. An equation governing the transfer of the covariance matrix of the Stokes parameters through a scattering medium is derived and studied. Finally a discussion of other versions of natural light in the context of the generalized Verdet–Stokes conditions is given.
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