A coherence approach to phase-contrast microscopy II: Experiment B.D. Arhatari a,Ã , A.G. Peele a , K. Hannah a , P. Kappen a , K.A. Nugent b , G.J. Williams b , G.C. Yin c , Y.M. Chen c , J.H. Chen c , Y.F. Song c a Department of Physics, LaTrobe University, Victoria 3086, Australia b School of Physics, University of Melbourne, Victoria 3010, Australia c National Synchrotron Radiation Research Centre, Hsinchu 30076, Taiwan article info Article history: Received 21 September 2007 Received in revised form 12 November 2008 Accepted 25 November 2008 Keywords: Defocus Partial coherence Contrast transfer X-ray microscopy abstract We report an experimental investigation of the optical transfer functions for an X-ray microscope operated in defocus phase-contrast mode. The results are compared with a theoretical model of partially coherent image formation and are found to be in excellent agreement. & 2008 Elsevier B.V. All rights reserved. 1. Introduction X-ray tomography allows the non-destructive observation of the three-dimensional inner structure of an object. High resolu- tion X-ray tomography has been developed at third generation synchrotron sources. However the spatial resolution has often been limited to around 1 mm [1] due to the finite spatial resolution of the X-ray detector. One way to achieve higher resolution is to form a magnified image of the sample using X-ray microscopy. It is well established that the diffraction-limited resolution of full-field, as opposed to scanning, microscopy is improved by the use of lower-coherence illumination. This conclusion is, of course, also true of X-ray microscopy and so there are considerable potential benefits from the use of a bending magnet source, as opposed to a more coherent undulator source, at a synchrotron [2]. The ultimate diffraction-limited spatial resolution is also determined by the numerical aperture of the objective lens; so a very-high-quality objective lens is required. X-ray microscopy generally employs a Fresnel zone plate as the objective lens and this technology has achieved a spatial resolution of about 15 nm in the soft X-ray region [3]. Harder X-rays require the fabrication of thicker zone plates, limiting the feasibility of manufacturing the small zones required for very high spatial resolution, though resolutions of better than 100 nm have been achieved [4]. The use of an X-ray microscope for tomographic imaging has achieved resolution on the order of tens of nanometers [5,16]. X-ray microscopy has for many years relied on absorption contrast for image formation. More recently, the importance of phase contrast has been realised and a number of phase-contrast methodologies have been developed. These include Zernike phase contrast [2,20], differential interference contrast [21] and propa- gation-based phase contrast resulting from defocus of the image [19]. Also, in scanning microscopy, the use of a segmented detector has yielded some excellent phase-contrast results [6]. The defocus-based methods have also been shown to be able to create quantitative phase images in a number of contexts [7,8]. In this paper, we examine phase contrast resulting from application of a defocus to the standard absorption-imaging configuration. In optical microscopy, this effect is well known [7] but it has not been particularly exploited for full-field X-ray microscopy. Accordingly, we undertake a detailed study of the effect of defocus on image contrast as a function of feature size. Several studies have been reported for the imaging process in a microscope where a partially coherent formulation [9–11] has been used. While methods have been developed for electron microscopy [11] these often implicitly assume a very-high degree of spatial coherence and so do not properly describe partially coherent diffraction-limited imaging [12]. X-ray microscopy using zone plates usually operates at the diffraction limit and so the more complete descriptions developed for optical microscopy should be used. A number of these exist. Streibl [13] published a three-dimensional optical transfer function formalism that deals with partial coherence in a ARTICLE IN PRESS Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/ultramic Ultramicroscopy 0304-3991/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ultramic.2008.11.015 Ã Corresponding author. Tel.: +613 9479 1957; fax: +613 94791552. E-mail address: b.arhatari@latrobe.edu.au (B.D. Arhatari). Ultramicroscopy 109 (2009) 280–286