In the paraxial regime, the Wigner function and ambiguity function are two phase-space representations that can be used to describe propagation of waves. For example, Brenner, Lohmann, and Ojeda-Castañeda described how the ambiguity function can be used as a polar display of the defocused optical transfer function (OTF). However, it is well known that, although these phase-space representations are useful in the paraxial regime, for highly convergent fields their form changes upon propagation, an effect analogous to the introduction of aberrations. Wolf et al. introduced a form of Wigner function for 2D nonparaxial wavefields called the angle-impact marginal, which has the properties of being real and invariant under translation or rotation.
An alternative representation for wave propagation is based on the concepts of the generalized pupil function, introduced by McCutchen, and the generalized OTF, introduced by Mertz and Frieden. These are 2D or 3D functions for 2D or 3D wavefields, respectively. The term “generalized” is used to distinguish these functions from the ordinary defocused pupil function and OTF, which are defined for a fixed defocus. The generalized transfer functions have been investigated in the paraxial regime, and for highly convergent scalar and vector wavefields. We have found that the concept of the generalized OTF is useful in visualizing the derivation of the Wigner function.
The different representations have also found use in the phase retrieval problem, where knowledge of the intensity in the focal region can be used to reconstruct the phase variations.
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