Scalar diffraction in terms of coherence

In scalar diffraction theory, an optical instrument can be treated as a linear system for the two limiting cases of coherent and incoherent illumination of the object, these treatments being in terms of complex amplitude and intensity, respectively. But when the illumination of the object is partially coherent, the system is no longer linear in either of these quantities and a two-stage treatment involving both quantities has been customary. Wolf has indicated the advantages of formulating diffraction theory in terms of an observable correlation function, here called the ‘coherence’, rather than in quantities such as amplitude which are not observable at optical frequencies. A Fourier theory of diffraction is developed here based on the coherence between radiation at pairs of points. As in general the coherence across a plane is a function of four spatial co-ordinates, the Fourier transforms used are in four dimensions for monochromatic light and in five for light of a finite spectral bandwidth. This diffraction theory is linear for all optical systems with illumination of any degree of coherence and leads to the concept of a ‘coherence transfer function’ to describe the performance of the instrument. In special cases, this reduces to the well-known ‘contrast transfer function' for incoherent illumination and to the transfer factors used in Hopkins’s treatment of partially coherent illumination. The theory also gives the transfer properties and the compensations required for two-beam interferometers and shows how the wave-shearing interferometer serves as an instrument for measuring coherence.