A uniform formulation for the self-imaging of gratings with any kind of partially coherent illumination is developed in terms of the cross mutual spectral density of the partial coherence theory. The formulation includes the time diffractive intensity distribution and the averaged diffractive intensity distribution at self-imaging distances and can be applied to both continuous and temporal illuminations with any kind of spectra. It is found that the averaged intensity distribution is related only to the intensity spectrum of illumination. The continuous polychromatic illumination and the ultrashort laser pulses with or without frequency chirp are then studied by a numerical stimulation. It is shown that the ultrashort laser pulse and the continuous polychromatic illuminations have similar averaged self-image distributions. Thus the Talbot effect may help in the study of the temporal and spectral characteristics of ultrashort laser pulses. An experiment with an LED is given, as well.
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