Partially coherent imaging in phase space

Conventional optical microscopes, such as brightfield, darkfield, phase contrast or differential interference contrast microscopes are partially coherent imaging systems. Imaging in a partially coherent system was first analyzed by Hopkins only in 1953. He propagated the mutual intensity through the optical system, but did not give an expression for the mutual intensity of the image itself. The mutual intensity is a four dimensional (4D) quantity that contains information about the modulus and phase of the image wave field, which depends on the object's complex refractive index in 3D. The mutual intensity is related to other representations such as the Wigner distribution function (WDF) and ambiguity function. Explicit expressions for different phase space representations of the image wave field are given. The expressions separate into system and object dependent parts. In addition, explicit relationships between the defocused partially coherent cross-coefficient and phase space representations in the image plane are derived.