The inverse source problem for arbitrary, three-dimensional, wide-sense stationary, spatially and temporally partially coherent sources is formulated. The formulation is based on an extension of the Porter–Bojarski inverse source integral equation (ISIE) for deterministic fields and is developed in the space–frequency domain. The known data are assumed to be the cross-spectral density of the field generated by the source, along with its normal derivative, elevated on a surface enclosing the support volume of the source, whereas the unknown is the cross-spectral density of the source itself. The solution of the partially coherent ISIE is shown to be nonunique, the nonuniqueness being intimately related to the nonuniqueness of the corresponding deterministic equation. Special cases of source coherence are shown to lead to unique inversions; however, the results do not completely parallel the deterministic case. An exact solution for the radiating portion of the source cross-spectral density is presented in the form of an eigenfunction expansion, and this is compared with the space-frequency expansion developed by Wolf [ J. Opt. Soc. Am. 72, 343 ( 1982)] for arbitrary source cross-spectral densities.
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