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.
Imaging in a bright field or phase contrast microscope is partially coherent. We have found that the image can be conveniently considered and modeled in terms of the Wigner distribution function (WDF) of the object transmission. The WDF of the object has a simple physical interpretation for the case of a slowly varying object. Basically, the image intensity is the spatial marginal of the spatial convolution of the object WDF with the phase space imager kernel (PSIkernel), a rotated version of the transmission cross-coefficient. The PSI-kernel can be regarded as a partially-coherent generalization of the point spread function. <p> </p>This approach can be extended to consider the partial coherence of the image itself. In particular, we can consider the mutual intensity, WDF or ambiguity function of the image. It is important to note that the spatial convolution of the object WDF with the PSI-kernel is <i>not </i>a WDF, and not the WDF of the image. The phase space representations of the image have relevance to phase reconstruction methods such as phase space tomography, or the transport of intensity equation approach, and to the three-dimensional image properties.