Improved description of diffraction effects for phase contrast imaging with applications to magnetically confined fusion plasmas

We present an improved diffraction model for phase contrast imaging (PCI), relevant to the measurement of electron density fluctuations in magnetically confined fusion plasmas. The model applies to optical systems containing multiple special components, such as phase plates, localization masks, and apertures, connected by segments describable in terms of ray transfer matrices. Fresnel diffraction in the connecting segments may be described using linear canonical transforms, while the special components are characterized by their respective transfer functions. For a PCI system with ideal placement of the phase plate and localization mask, deviations between the Fresnel and Fraunhofer diffraction models are entirely determined by the distance from the diffraction plane to the object plane of the optical system. Application of the developed model to the synthetic PCI diagnostic at the Wendelstein 7-X (W7-X) stellarator shows minimal impact of diffraction on the DC detector power. However, significant amplitude modifications due to phase scintillation from diffraction are expected for high-wavenumber PCI sound wave calibration signals at W7-X, in qualitative agreement with the experimental observations. In contrast to previous expectations, diffraction is found to have only a small impact on the localization mask response characteristics at W7-X. Based on this observation, we show that the synthetic PCI mask response at W7-X can be reproduced by a model which only considers the signal at the beam center in the image plane without diffraction. We further use a similar model to reproduce the experimentally observed mask response from the DIII-D tokamak, indicating the general applicability of such models.