Transport of intensity phase retrieval and computational imaging for partially coherent fields: The phase space perspective

• The conventional TIE is reformulated in the phase-space using Wigner distribution functions.
• The phase-space formulation clarifies the physical meaning of the phase of partially coherent fields.
• The phase-space representation enables explicit account of partial coherence effects on phase retrieval.
• High-resolution light field can be realized in a purely computational way.

The well-known transport of intensity equation (TIE) allows the phase of a coherent field to be retrieved non-interferometrically given positive defined intensity measurements and appropriate boundary conditions. However, in many cases like the optical microscopy, the imaging systems often involve extended and polychromatic sources for which the effect of the partial coherence is not negligible. In this work, we present a phase-space formulation for the TIE for analyzing phase retrieval under partially coherent illumination. The conventional TIE is reformulated in the joint space-spatial frequency domain using Wigner distribution functions. The phase-space formulation clarifies the physical meaning of the phase of partially coherent fields, and enables explicit account of partial coherence effects on phase retrieval. The correspondence between the Wigner distribution function and the light field in geometric optics limit further enables TIE to become a simple yet effective approach to realize high-resolution light field imaging for slowly varying phase specimens, in a purely computational way.