We report on the design and realization of an x-ray imaging system for quantitative phase-contrast microscopy at high x-ray energy with laboratory-scale instrumentation. Phase and amplitude were separated quantitatively at x-ray energies up to 80 keV with micrometric spatial resolution. The accuracy of the results was tested against numerical simulations, and the spatial resolution was experimentally quantified by measuring a Siemens star phase object. This simple setup should find broad application in those areas of x-ray imaging where high energy and spatial resolution are simultaneously required and in those difficult cases where the sample contains materials with similar x-ray absorption.
© 2014 Optical Society of America
Hard x-ray radiation has a high penetration power in matter, thus enabling nondestructive inspection of the inner structure of samples. At the same time, this power is also a weakness because the contrast arising from differences in the absorption of radiation tends to vanish when partially transparent samples are illuminated. X-ray phase-contrast imaging (XPCi)  overcomes this problem because it is sensitive to the phase shifts imparted to the x-ray wave when traversing the sample. Several methods [2–15] have been developed for performing XPCi. Approaches have also been proposed for high-resolution XPCi, including free-space propagation, Zernike phase contrast, and grating-based methods [16–22]. Here we show the design, modelling, and realization of a laboratory system based on the edge-illumination [10,15] principle and implemented through its area-imaging counterpart, sometimes referred to as the coded-aperture  method. These are noninterferometric methods that do not use the Talbot self-imaging effect or Moire patterns ; contrast is generated by fine angular selection, in analogy with analyzer-based imaging which uses the rocking curve of a crystal . We aim to push the current resolution limits of a few tens of micrometers  toward microscopic resolution while still performing quantitative phase-contrast imaging at high x-ray energies. A magnified projection geometry was used in order to achieve high spatial resolution while being able to efficiently detect the radiation. The ability of the method to be quantitative and its spatial resolution are experimentally demonstrated and numerically simulated while the potential of the technique in terms of image quality is illustrated through images of a complex wood sample.
The experimental setup consists of a microfocus transmission target x-ray tube, two apertured masks, and a detector (Fig. 1). The tungsten target x-ray tube is operated at 80 kVp and has a focus of 3.5 μm. The first mask is placed at 13 cm from the focus and the source-to-detector distance is . The sample is positioned at about 14 cm from the focus, with a geometrical magnification factor of about 9. The first mask has a pitch and apertures , while and are used for the second mask . They are made of gold on a graphite () and silicon () substrate and were manufactured by Creatv Microtech (Potomac, Maryland) and Microworks GmbH (Karlsruhe, Germany), respectively. The detector is a passive pixel complementary metal-oxide semiconductor flat panel sensor (Hamamatsu Photonics C9732DK), with pixel pitch . The signal degradation due to cross talk between neighboring pixels is limited by the use of a line-skipped mask design . The main limiting factor for the field of view is , which defines the angular acceptance of the transmitted radiation , where is the thickness of the mask, about 200 μm in this case. For our setup the field of view was in the and directions, respectively. Bent masks could be used for obtaining a larger field of view with an even more compact setup . Provided that the angular spread of the beam is limited to , the pixels can be considered independent from one another, which allows us to model the image formation for a single aperture only.
Let us consider the Fourier transform of the intensity on the detector :30]. The following form for is obtained: 3) is required only to obtain a formulation easier to interpret, without the need to carry out the full integration numerically. In order to visualize this, the signal was calculated for a sinusoidal phase object, and its maximum value was plotted against the period in Fig. 2. It is interesting to note that the modulation is peaked at 3 μm, which corresponds to , and also that signals with period smaller than the aperture are detected; for example, is reduced by about 50% at 1.5 μm.
The direct comparison between the measured and the theoretical signal from a star pattern test object is carried out by means of a numerical simulation of the whole imaging system . Each monochromatic component was weighted according to the x-ray source spectrum and the detector response as function of energy . In order to perform quantitative retrieval of the absorption and refraction of the sample, we followed the approach based on the knowledge of the translation curve (TC) of the system . The TC describes how the detected intensity changes as a function of the displacement between the two masks. Images were acquired by setting the displacement between and as equal to . The resulting images can be expressed as 33], is the sample-to-detector distance, is the partial derivative of at the effective energy of the system, and is the value of the translation curve. We also introduce the following images, and . It is worth noting that the quantity depends only upon and the operational parameters of the setup ; we will refer to this quantity as the differential phase image. Moreover, depends only on the absorption properties of the sample. The following samples were imaged experimentally: a star pattern, a cylindrical two-material sample, and a bamboo wood sample. The star pattern was etched into a few hundred micrometers of crystalline silicon and provides a good approximation of a pure phase object at the x-ray energies we used. The image was acquired by using 100 sample scan steps of 0.2 μm, with an exposure time of 70 s per step. The cylindrical sample was composed of two materials, 220 μm diameter of boron with a 14 μm diameter tungsten core, and it was scanned with 16 steps of 1.2 μm, 100 s exposure time each. The bamboo wood was sliced to a thickness of about 500 μm; and were acquired by scanning the sample with 44 steps of 0.5 μm, with an exposure time of 100 s per step. For comparison, an image of the bamboo sample was also acquired in free-space propagation XPCi, with monochromatic synchrotron radiation and a high-resolution detector. The image was acquired at the I13 beamline of the diamond light source by using 9.7 keV x-rays and a detector featuring 0.8 μm size pixels. The sample was placed at about 200 m from the undulator source and the detector 30 cm downstream of the sample.
The spatial resolution of the laboratory-based system is evaluated by means of the star pattern. The image of the star is shown in Fig. 3(a) from which the intensity profile shown in Fig. 3(b) was extracted (dashed line). The experimental data are in good agreement with the numerical simulation, shown as a solid line in the plot in Fig. 3(b). As the absorption is negligible in this case, we note from Eq. (4) that depends only on and on the operational parameters of the setup. The changing visibility of the phase-contrast features around the star pattern are a property of the geometry of the setup. Let us consider the phase shift of the star pattern as a replica at various angles of a step-like feature; the detected signal is modulated by a cosine function, , where are the coordinates rotated by the angle . The smallest resolved separation between a dark and a bright fringe is 1.5 μm; the sample used hardly allowed testing finer resolutions. The quantitativeness of the method is demonstrated by retrieving differential phase and amplitude of the boron fiber with a tungsten core and comparing the extracted (dashed line) values to the expected ones (solid line) in Fig. 4(a). The theoretical was calculated assuming a cylindrical shape of the boron and tungsten fiber. The analytical profile was convolved with a Gaussian function, representing the spatial resolution of the system, of width 1.5 μm. The refractive indices of the materials were calculated by using the Xraylib library . The profile was then obtained according to  by numerically differentiating . The amplitude and differential phase images are also shown in Figs. 4(c) and 4(b), respectively. The horizontal line noise that can be noticed in Figs. 3(a), 4(b), and 4(c) is a result of the brick-like structure used to manufacture . The laboratory images of the bamboo sample are shown in Fig. 5, where the contributions from phase [Fig. 5(a)] and absorption [Fig. 5(b)] are separated. For a better appreciation of the image quality, the image acquired under nearly ideal conditions (with monochromatic synchrotron radiation and high-resolution detector) is also shown for comparison in Fig. 5(d). We note that the higher contrast of this image is expected due to the much lower x-ray energy (9.7 keV). The detail shown in the picture is similar to but not exactly the same as in the laboratory image, due to difficulty in aligning the sample to the same region of interest.
In summary, we described a hard x-ray, laboratory-based, phase-contrast microscope, obtained through the appropriate design of a high-magnification edge-illumination XPCi system. The experimental setup was built using commercially available, off-the-shelf instrumentation, and it is currently producing high-quality amplitude and differential phase-contrast images. Micrometric spatial resolution was experimentally measured, theoretically described, and numerically simulated by using a star pattern test object, with good agreement between simulation and experiment. By acquiring two separate images at different configurations, amplitude and differential phase can be quantitatively retrieved. The quantitativeness of the method was tested against theory on a two-material sample of known shape and composition. The instrument uses broadband and hard radiation, merging high resolution with the high penetration power of x-rays in a laboratory setup.
The authors are grateful to Nikon Corporation and X-Tek, part of Nikon Metrology, for their support with the experimental setup. This work was supported by the UK Engineering and Physical Sciences Research Council (Grants EP/I022562/1 and EP/I021884/1). M. Endrizzi and P. C. Diemoz are supported by Marie Curie Career Integration Grants within the Seventh Framework Programme of the European Union, PCIG12-GA-2012-334056 and PCIG12-GA-2012-333990.
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