Journal of Microscopy, Vol. 232, Pt 1 2008, pp. 145–157 Received 6 September 2007; accepted 19 March 2008 The fractional Talbot effect in differential x-ray phase-contrast imaging for extended and polychromatic x-ray sources M. ENGELHARDT ∗ , C. KOTTLER †, O. BUNK †, C. DAVID †, C. SCHROER ‡, J. BAUMANN §, M. SCHUSTER § & F. PFEIFFER ∗∗ ∗ Siemens AG, M ¨ unchen, Germany and Technische Universit¨ at M ¨ unchen, Germany †Paul Scherrer Institut, Villigen, Switzerland ‡Technische Universit¨ at Dresden, Germany §Siemens AG, M ¨ unchen, Germany ∗∗ Paul Scherrer Institut, Villigen, Switzerland and Ecole Polytechnique F´ ed´ erale de Lausanne, Switzerland Key words. Grating interferometry, phase retrieval, talbot effect, x-ray microscopy. Summary The influence of different physical parameters, such as the source size and the energy spectrum, on the functional capability of a grating interferometer applied for phase- contrast imaging is discussed using numerical simulations based on Fresnel diffraction theory. The presented simulation results explain why the interferometer could be well combined with polychromatic laboratory x-ray sources in recent experiments. Furthermore, it is shown that the distance between the two gratings of the interferometer is not in general limited by the width of the photon energy spectrum. This implies that interferometers that give a further improved image quality for phase measurements can be designed, because the primary measurement signal for phase measurements can be increased by enlargement of this distance. Finally, the mathematical background and practical instructions for the quantitative evaluation of measurement data acquired with a polychromatic x-ray source are given. Introduction X-ray imaging is an important technique for medical and technical applications. Normally, the absorption contrast is used for imaging. However, for imaging of weakly absorbing or thin structures, the applicability of this method is limited. Phase-sensitive x-ray imaging can overcome these shortcomings because it has the potential for a significantly increased contrast (Fitzgerald, 2000; Momose, 2003, 2005). For phase-contrast imaging, propagation-based methods are Correspondence to: M. Engelhardt. Tel: 0049 179 3934233; e-mail: martin.engelhardt@mytum.de often applied at synchrotron x-ray sources (Snigirev et al., 1995; Nugent et al., 1996; Cloetens et al., 1999; Peele et al., 2005), but can also be well combined with commercial or custom-made laboratory x-ray sources (Wilkins et al., 1996; Mayo et al., 2003) in order to determine the phase shift of x rays transmitted through matter. Using recently developed grating- based methods (David et al., 2002; Momose, 2003; Weitkamp et al., 2005a, b, 2006; Pfeiffer et al., 2006, 2007b), the differential phase contrast (first derivative of the phase shift) is measured. This is an advantage in comparison to propagation- based methods, where the data contain essentially the second derivative of the phase shift. Differential x-ray phase-contrast imaging using a grating interferometer was combined with a magnifying cone beam geometry using a conventional high-brilliance microfocus x- ray source. By contrast to propagation-based methods and previous grating-based methods, this offers the possibility to apply an efficient low-resolution detector for phase- contrast imaging with high-spatial resolution. It was shown experimentally how the actual measurement values depend on the magnification factor (Engelhardt et al., 2007a). The potential of the method was demonstrated by a phase tomography of an insect (Engelhardt et al., 2007a; Pfeiffer et al., 2007a). Furthermore, quality validation measurements of refractive x-ray lenses were performed (Engelhardt et al., 2007b). Grating-based interferometer With grating-based methods, the beam deflection angle, induced by the differential phase shift, is determined. In Fig. 1, the principle of the grating interferometer used for differential phase imaging is outlined. The phase grating g 1 consists of C  2008 Siemens AG Journal compilation C  2008 The Royal Microscopical Society