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\begin{document}

\title{Omnidirectional dark-field imaging on a table-top X-ray source}



\author[1]{Matteo Abis}%
\affil[1]{ETH Zurich}%


\vspace{-1em}



  
  \date{May 05, 2024}


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We demonstrate the design of an omnidirectional interferometric setup suitable for a table-top source that also allows the retrieval of quantitative dark-field information without the strict constraints on the spatial coherence of the source. An analogous approach to Talbot-Lau grating interferometry is used, where a source grating $G_0$ is installed right in front of the incoherent source. A two-dimensional array of individually coherent but mutually incoherent pinholes is designed so that each source has enough coherence to generate a circular interference pattern on the detector plane, and that interference patterns from adjacent sources superimpose constructively.

To achieve these requirements, the diameter of the openings in $G_0$ has to satisfy [coherence formula for $g_0$, according to $p_1$, $L$ and $D$, and the distance between the openings $p_0$ is constrained by $p_0/p_2 = L/D$, where $p_2$ is the period of the magnified pattern at the detector plane, and $L$ is the $G_0$ to $G_1$ distance (see Figure~\ref{fig:schematic}).\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/schematic-with-arrows/schematic-with-arrows}
\caption{{Schematic of the omnidirectional interferometric setup for a
conventional table-top source.
{\label{934080}}%
}}
\end{center}
\end{figure}

The setup has a $G_0$ to $G_1$ distance of $L = 57 \text{ cm}$ and a $G_1$ to detector distance $D = 110 \text{ cm}$. The source is a COMET MXR-225/26 X-ray tube, with a 1 mm source,operated at a voltage of 60 kV [or 70? which one do we use?] and a current of 10 mA. The image is an average of 20 radiographs with 100 s exposure time each. The detector is a prototype with a CdTe sensor based on the SENTIS by Dectris Ltd. The pixel size is 75 \selectlanguage{greek}µ\selectlanguage{english}m, with a sensor thickness of 750 \selectlanguage{greek}µ\selectlanguage{english}m. The quantum efficiency of the cadmium telluride sensor is greater than 90\% at 60 keV, which ensures reasonable exposure times. As seen in Figure 2, the directional information on the fibers on a subpixel scale can be retrieved in a single shot image on a conventional X-ray source.\selectlanguage{english}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/Loop-60keV/Loop-60keV}
\caption{{Color-coded image of the principal fiber orientation measured with the
table-top interferometer.
{\label{728876}}%
}}
\end{center}
\end{figure}

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