The Generalized Partial Coherence of a Radiation Source and Its Far-field

A vector Kirchoff, aperture-field analysis is employed to derive theorems relating (1) the far-field coherency matrix and the source mutual coherence tensor and (2) the far-field mutual coherence tensor and the coherency matrix elements. These theorems are derived for monochromatic sources and thermal radiation sources. The correspondence of these relations with previous results is noted. Sufficient requirements relating the components of the source generalized mutual coherence tensor are derived for propagation of completely polarized and unpolarized light. It is demonstrated that if the radiation source is cross-spectrally pure, the far-field intensity results from a convolution of spectral densities of the source-field's spatial and temporal fluctuations. The tensor analogue of the far-field van Cittert-Zernike theorem is found to be functionally identical with this classical result; however, strict incoherence of the source is not required for the validity of this theorem.