Transport of intensity and spectrum for partially coherent fields

Transport-of-intensity and transport-of-spectrum equations are derived using the coherent mode decomposition for paraxial fields having an arbitrary state of coherence. We give a simple example that demonstrates the difference between a partially coherent and a fully coherent transport of intensity or spectrum. The results presented here may be used to estimate the intensity response in a variety of phase-contrast imaging modalities and may form the basis for improved phase-retrieval techniques.