Conservation of electromagnetic coherent-mode structure on propagation

The coherent-mode decomposition is an invaluable tool in assessing the effects of partial optical coherence. In this work we consider paraxial and nonparaxial propagation of the coherent modes associated with partially polarized, partially coherent electromagnetic fields so as to determine whether the modes advanced from the source plane to an observation plane also constitute the coherent modes at the observation plane. We show that within the paraxial approximation the coherent-mode structure, including the expansion coefficients, is always conserved on propagation. We further demonstrate, through advancing each of the three orthogonal electric-field components separately, that in nonparaxial propagation the mode structure is preserved if the field is spatially bandlimited to include only homogeneous plane waves, analogously to a similar result for scalar fields. The longitudinal components and nonparaxial propagation of the coherent modes corresponding to the 2×2 (transverse) source coherence matrix may as well be determined by use of the Luneburg method. The three-component electric fields so obtained are however shown, in general, to differ in the source plane from the coherent modes of the full 3×3 coherence matrix. Hence the coherent modes of the transverse coherence matrix are generally not related to those of the nonparaxially propagated electric field.