Contrast transfer functions for Zernike phase contrast in full-field transmission hard X-ray microscopy

Full-field transmission hard X-ray microscopy (TXM) has been widely applied to study morphology and structures with high spatial precision and to dynamic processes. Zernike phase contrast (ZPC) in hard X-ray TXM is often utilized to get an in-line phase contrast enhancement for weak-absorbing materials with little contrast differences. Here, following forward image formation, we derive and simplify the contrast transfer functions (CTFs) of the Zernike phase imaging system in TXM based on a linear space-shift-invariant imaging mode under certain approximations. The CTFs in ZPC in their simplified forms show a high similarity to the one in free-space propagation X-ray imaging systems.