The one-dimensional case of the image of a sinusoidal transmittance distribution in partially coherent illumination (with the quasimonochromatic approximation) is described analytically, and shown generally to bear a nonlinear relation to the object. It is shown that the significant parameter is the ratio of coherence interval to the diameter of the Airy disk (or diffraction spot) of the imaging lens. It is further shown that since the spatial frequency of the object is related to coherence interval, typical nonlinear effects can take place at low frequencies. Since the transfer function is defined only for the incoherent limit without ambiguity, an apparent transfer function, dealing only with the image component which existed in the object, is used for comparison. The harmonics generated by the nonlinear behavior are ignored, and the variation of transfer function is observed to be a function of coherence and input modulation. It becomes apparent that the transfer function, as currently defined and measured, is inadequate to describe optical-system performance under all conditions of illumination.
RICHARD E. SWING and JAMES R. CLAY, "Ambiguity of the Transfer Function with Partially Coherent Illumination," J. Opt. Soc. Am. 57, 1180-1189 (1967)