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  • Phenomenological description of the scattering signal for polychromatic grating interferometry in the Compton regime

    Matteo Abis\(^{1,2}\), Pablo Villanueva Perez\(^{1,2}\), Spyros Gkoumas\(^{1,2}\), Zhentian Wang\(^{1,2}\), Marco Stampanoni\(^{1,2}\)
    1 Swiss Federal Institute of Technology, Rämistrasse 101, 8092 Zurich, Switzerland
    2 Paul Scherrer Institute, 5232 Villigen, Switzerland

    Abstract

    A description of dark-field or scattering contrast in grating interferometry is presented, by extending earlier theoretical derivations from monochromatic synchrotron light to a polychromatic X-ray tungsten anode tube in an energy range dominated by Compton scattering at 160 kVp, henceforth called the Compton regime. We introduce and validate a model that allows to quantitatively describe the ratio of the scattering signal and absorption signal for table-top sources. Two experiments, one on homogeneous bulk samples and one on microspheres, with design energies of 120 keV and 100 keV respectively, have been performed to show that this description allows to recover a signal that depends only on the size of unresolved structures in the sample, an order of magnitude smaller than the detector pixel size, independent on the sample thickness and material.

    Introduction

    Grating interferometry

    Talbot-Lau interferometers, as depicted in Fig. \ref{fig:schematic}, employ the self-imaging property of a periodic structure, the grating \({G_{1}}\), in order to produce a fine pattern of bright and dark lines. A sample placed close to \({G_{1}}\) will absorb, refract and scatter the X-rays, resulting in a lower photon count, a lateral displacement of the fringes and a reduction of their amplitude respectively (David 2002).

    To achieve sufficient phase sensitivity, the pitch of these fringes has to be in the micron range. Micrometer resolution is not achievable for large scale applications, therefore a second grating \({G_{2}}\) is placed in front of the detector with the same pitch as the interference pattern produced by \({G_{1}}\). The grating \({G_{2}}\) then slides across one period of the interference pattern in small steps. This procedure is known as phase stepping (Momose 2003, Weitkamp 2005). The convolution between the pattern and the transmission function of \({G_{2}}\) is recorded by the detector, producing a phase stepping curve. The following notation will be employed, where \(a_{n,s}\) and \(a_{n,f}\) are the moduli of the \(n\)-th coefficients of the discrete fourier transform of the phase stepping curve with and without sample respectively. The absorption and scattering signals are \[\begin{aligned} A &= \frac{a_{0,s}}{a_{0,f}}\\ B &= \frac{a_{1,s}}{a_{1,f}}\frac{a_{0,f}}{a_{0,s}}. \label{eqn:define-ab}\end{aligned}\] The signal \(B\) is also known in literature as dark-field signal or visibility reduction signal. We also introduce the ratio of the logarithms \(R\) \[R = \frac{\log B}{\log A}.\] Finally, in the Talbot-Lau configuration, a third grating \({G_{0}}\) is installed right after the source to produce an array of individually coherent sources (Pfeiffer 2006).

    Grating interferometry at 160 kVp

    This study employs the technique of edge-on illumination (David 2014, Thüring 2014) to achieve a sufficiently large aspect ratio of the absorption gratings. The absorbing lines have to be thicker to block higher energy X-rays while the period needs to be shorter to achieve sufficient sensitivity on shorter wavelenghts. The current fabrication techniques cannot achieve such extreme aspect ratios on large areas. The gratings are here illuminated from the side, so that the length of the grating is used an the absorption thickness. This results in a one dimensional setup, where the samples are scanned across a collimated fan beam.