Magnified x-ray phase imaging using asymmetric Bragg reflection: Experiment and theory Peter Modregger, 1, * Daniel Lübbert, 2,1 Peter Schäfer, 1 and Rolf Köhler 1 1 Institut für Physik, Humboldt-Universität zu Berlin, Newtonstr. 15, D-12489 Berlin, Germany 2 Institut für Synchrotronstrahlung, Forschungszentrum Karlsruhe, D-76344 Eggenstein-Leopoldshafen, Germany Received 14 March 2006; revised manuscript received 16 June 2006; published 22 August 2006 X-ray imaging using asymmetric Bragg reflection in the hard x-ray regime opens the way to improve the spatial resolution limit below 1 m by magnifying the image before detection, simultaneously providing a strong phase contrast. A theoretical formalism of the imaging process is established. Based on this algorithm, numerical simulations are performed and demonstrate that both Fresnel propagation and Bragg diffraction contribute to contrast formation. The achievable resolution of this technique is investigated theoretically; the results obtained can be used to improve future experimental setups. Furthermore, the minimum detectable phase gradient is estimated, for comparison with other phase sensitive imaging techniques. Results from biological objects demonstrate that the technique is viable for imaging both in two and three dimensions. Refraction contrast images are extracted from experimental projection images by an algorithm similar to diffraction-enhanced imaging DEI, and used to achieve three-dimensional tomographic reconstruction. DOI: 10.1103/PhysRevB.74.054107 PACS numbers: 87.59.-e, 87.57.Ce, 61.10.-i I. INTRODUCTION Contrast enhancement in x-ray imaging can be achieved by exploiting phase contrast in addition to absorption con- trast. This is particularly advantageous in the case of light organic, weakly absorbing objects. One of the most often used techniques for experimentally obtaining phase contrast is Fresnel propagation or in-line holography. 1–4 A principal alternative is analyzer-based imaging, also known as diffraction-enhanced imaging DEI. 5 DEI is based on the combination of transmission of an x-ray wave field through the object under study radiogra- phy with subsequent Bragg diffraction at an analyzer crys- tal. Owing to the narrow angular range of Bragg diffraction, the analyzer efficiently selects rays locally deviated at the object by specific angular amounts due to phase gradients within the sample, thus enhancing image contrast. This technique has proven useful for the investigation of laser-fusion targets, 6 for x-ray diffraction topography, 7 and biological and medical imaging. 8 Advanced analysis tech- niques have been developed 5,8–10 which combine several im- ages taken at different angular positions of the analyzer crys- tal. In this way, the effects of x-ray absorption, refraction, and small-angle scattering can be experimentally separated and a set of complementary images with specific contrast features is obtained. While the standard DEI setup is based on symmetric re- flection from the analyzer surface, asymmetric reflections make it possible to simultaneously realize image magnification. 6 By using two consecutive reflections from a pair of analyzers, magnification can be achieved in both im- age dimensions. 7,9,11–13 For a given analyzer surface orienta- tion, the magnification factor uniquely depends on the x-ray photon energy. Synchrotron radiation is advantageous, since its energy tunability gives one the flexibility to widely vary the magnification by small changes in energy. Asymmetric-reflection DEI thus opens the way to phase- contrast imaging with sub-micrometer spatial resolution. It represents a path out of a conflict encountered in direct x-ray imaging: Particularly when using state-of-the-art CCD cam- eras, spatial resolution improvements usually require reduc- ing the converter screen thickness, at the expense of detec- tion efficiency. By exploiting post-transmission image magnification, commercial CCD cameras with moderate pixel sizes but high sensitivity remain usable even for appli- cations requiring highest spatial resolution. In this way, magnified x-ray imaging allows one to simul- taneously realize submicrometer resolution and phase con- trast. While the magnification factor is seemingly unlimited, the actual resolution is determined by a complex interplay of several concurring factors. For a precise determination of resolution limits, a complete theory of the imaging process is therefore mandatory; resolution cannot simply be equated with geometrical quantities such as the projected x-ray pen- etration depth in the analyzer. In this article, we present an instrumental realization of the principle of magnified imaging—the “Bragg Magnifier”—optimized in view of high spatial resolution. In the theoretical part, a comprehensive description of the im- age formation process is developed, including wave propa- gation behind the object Fresnel diffraction and reflection at the analyzer crystals Bragg diffraction. Based on this theory, the achievable spatial resolution is estimated numeri- cally for several different scenarios and model objects. The results reveal significant differences between the limiting cases of absorption objects and phase objects. Moreover, im- plications for further instrumental optimization are deduced. The phase sensitivity of the technique is quantified. An ex- ample of an experimental measurement performed on bio- logical objects will be shown, including results of three- dimensional imaging after tomographic reconstruction. II. THEORY OF IMAGE FORMATION In this section we will discuss the image formation of the Bragg magnifier sketched in Fig. 1: A monochromatic wave is transmitted through the sample and diffracted twice by two PHYSICAL REVIEW B 74, 054107 2006 1098-0121/2006/745/05410710 ©2006 The American Physical Society 054107-1