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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 adoi:10.1364/oe.24.006063 pmid:27136800 fatcat:wxhrumjovfc2fbtydxj5wcpjsi