On genuine cross-spectral density matrices

For an electromagnetic stochastic beam, the choice of the mathematical structure of the cross-spectral density matrix is limited by the constraint of non-negative definiteness. We present a sufficient condition for building these matrices in such a way that this constraint is automatically satisfied. This allows us to put into evidence that electromagnetic beams can exhibit very peculiar correlation properties, some of which would not be encountered in scalar treatments. These results are illustrated by means of a number of examples.