A theoretical analysis of the in-line method of x-ray phase-contrast imaging with a small (microfocus) source is presented. By applying the stationary-phase formula to the Fresnel integral, we derive a new variant of the transport of intensity equation (TIE) that explicitly takes image magnification into account and does not require the distance between the two observation planes to be infinitesimally small. The new derivation method provides a uniform technique for obtaining the TIE's with incident waves of different types and also leads to new practically convenient validity conditions. A new boundary problem for the TIE, which is particularly suitable for the in-line imaging, is defined and solved analytically. Numerical examples of phase retrieval using simulated data in the in-line case with a point source are also given. The present approach should be useful in quantitative phase-contrast microscopy and tomography using microfocus sources. The question of partial coherence is briefly discussed but is essentially outside the scope of the present work. #1998 Optical Society of America [S0740-3232(98)01203-4] 0CIS codes: 340.0340, 340.7480, 100.5070, 180.0180, 100.0100.
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