1. Introduction

Rapid non-destructive characterization of extended (mm sized) objects is an essential requirement for material science [1–4], security screening [5,6] and medicine [7–9]. At the mm length scale non-destructive structural probes are limited mostly to highly penetrating radiation, such as hard X-ray. Volumetric X-ray imaging techniques such as CT (computerised tomography), X-ray tomosynthesis [7] and phase contrast imaging [4,8] have developed into rapid screening tools where mass attenuation coefficients and refractive indices provide the contrast mechanisms, respectively. Many current applications would benefit from material composition information that can be directly attributed to spatially distributed components within a volume. While angular dispersive X-ray diffraction (ADXRD) is the ‘gold standard’ for conducting definitive crystalline materials structural discrimination [10], its uptake as the basis for an imaging modality has been slow.

A primary limitation is that the intensity of the diffracted X-rays is orders of magnitude weaker in comparison to the interrogating or primary X-ray beam [11,12] and therefore, relatively long data acquisition times are required. This situation is compounded by the tightly collimated sources required by high spatial resolution scanning implementations of ADXRD [13,14]. The resultant narrow beam reduces the specimen volume and number of crystallites contributing to each Debye cone. For this reason both brighter X-ray sources and more sensitive detectors are required. The integration of X-ray diffraction into a rapid screening modality is therefore problematic.

X-ray diffraction has been successfully employed to map the position, volume, orientation and stress state in grain boundaries [13,15] both statically and dynamically [16–18]. It has also been combined with conventional transmission tomography [19]. However, these methods typically employ bright (synchrotron) sources, highly collimated primary beams (to provide spatial resolution) and have been applied to samples an order of magnitude smaller than those presented here (though the work presented here is scalable). These previous works also require typically some a priori knowledge of the Bragg peaks within the sample [20] (i.e. material composition). Coherent, lensless imaging has recently been employed to produce images with very high spatial resolution [21] (nm). This concept has also been modified for reflection geometries [22] and implemented using broadband sources [23], while the goal has often been to produce imagery with high spatial resolution [24] material specific mapping has also been demonstrated [25].

Perhaps the most highly developed X-ray diffraction imaging system is TEDDI [2,3]. This energy dispersive X-ray diffraction (EDXRD) tomography approach employs an energy resolving pixelated detector, which is optically coupled to a series of high aspect ratio collimators. The sample is illuminated by a broadband or ‘white’ primary beam to enable diffraction patterns to be measured. This approach is able to identify material phases within an inspection volume with no a priori knowledge of the phase distribution. Ultimately, the materials discrimination performance is limited by the energy resolving capability of the detector. Also, the collimators restrict the solid angle subtended by the detector, which therefore limits the probability of detecting scattered photons.

Enhanced optical throughput is a primary motivation for the development of coded aperture X-ray tomography [26–29]. This approach requires a precise configuration of apertures to modulate the X-ray beam or the scattered X-rays in a known way so that the three-dimensional object structure is encoded in a series of two-dimensional intensity images. The application of an inversion algorithm solves for the origin of the scatter signals along the X-ray beam path. Material phase information can be obtained by calculating the scatter angles.

In this paper we demonstrate depth-resolved materials characterization by scanning a sample through an annular beam of X-rays. Bragg diffraction, from the sample, produces caustics in the X-ray intensity recorded by a planar detector located centrally in a circular dark region defined by the annular beam. A detailed explanation of the caustics is given in Section 2.1. The location of a crystalline phase along the beam changes the position and shape of the caustic pattern. To identify the location of diffracting objects we apply tomosynthesis. Unlike conventional absorption tomosynthesis, each perspective image in our approach is produced by rays at initially unknown angles of incidence i.e. the diffraction angle. These images are collected during a two axis translational scan by compositing the pixel streams from individual detector elements. Each “scanning” element is able potentially to record an oblique perspective view. A sequence of such images may be collected along circular paths, centred on the principal axis of the annular beam, to record depth dependent circular parallax. The application of tomosynthesis enables Bragg maxima i.e. diffraction angle specific, optical sections to be extracted at corresponding axial focal plane positions. This information may be used to calculate the associated Bragg diffraction angles and material lattice spacings for each optical section.

Unlike previous technology [2,3,26–29] post sample/diffraction collimation or complex aperture encoding is not required. Also, by not using a collimator the detector subtends larger solid angles to increase the probability of detecting a photon.

The organization of the paper is as follows. Section 2 presents the theoretical background, our new imaging technique and describes the experiment setup and conditions. Section 3 presents our empirical results and discussion. Section 4 summarizes our conclusions, discusses the broader implications of our findings and the future direction of the work.

2. Methods

2.1 Theory background

X-ray diffraction is routinely used for high fidelity materials characterization. Conventional angular dispersive X-ray diffraction (ADXRD) techniques employ a pencil or linear X-ray beam to illuminate a sample at a known location within the instrument. Typically, the data collection and analysis is designed to optimise the structural resolution and is not primarily concerned with the time interval required for this process. For example, state of the art diffractometers use highly sensitive large-area detectors with high quantum efficiency and low noise operating over relatively long integration periods. Also, they are not intended to solve for Bragg diffraction angles when the location of a crystalline phase along the X-ray beam is unknown. Therefore, conventional ADXRD techniques are not ideal as the basis for developing a scanning technique.

We have previously demonstrated that a divergent hollow cone or annular beam of X-rays incident upon a planar sample generates a continuum of coherently scattered Debye cones [11,12]. The increase in the diffracted beam intensity was ~20 times where the diffracted rays form a spot ‘focus’. The computation of lattice or d-spacings required a priori knowledge of the sample position. Also reported were extended bright patterns that may contain orders of magnitude increase in the amount of diffraction signal in comparison to the equivalent pencil beam case. For the purpose of our current work we describe these relatively bright patterns as caustics. In geometric terms, we define a caustic as the envelope of a family of curves, formed by overlapping Debye rings, in the plane of the detector. Therefore, the shape of the caustic is a curve, which is tangent to each member of a family of Debye rings at some point.

To compute accurate d-spacings requires that the depth or Z location of the scatter sources within the inspection volume be known. We have demonstrated a pencil X-ray beam technique [10] employing four different detector locations, along and normal to the primary X-ray beam axis, to help resolve the position of a sample along the beam. The reconstruction fidelity of this approach is robust with respect to non-ideal polycrystalline microstructure when, for example, the scattering distributions are adversely affected by preferred orientation or large grain size [11,30]. However, the use of multiple detection planes is not an ideal approach for a scanning system. For example, a single detector moved into different positions requires an increased time period for data collection, whereas the use of multiple detectors are relatively expensive and bulky. Many variants of 3D X-ray diffraction have been described and successfully implemented [5,10,13–20,24–27]. These techniques use either an angular-dispersive or energy-dispersive approach and may or may not require some a priori knowledge of the sample material(s) but they do all require exhaustive diffracted beam collimation. This fundamental requirement results in the loss of valuable diffracted flux. We achieve a similar result but our detector is able to subtend larger solids angles to increase the probability of detecting diffracted photons.

2.2 New imaging technique

Our tomographic technique employs the discretized scanning of an annular beam of X-rays along two orthogonal axes, X and Y, to form a coplanar sampling grid, as shown in Fig. 1. In practice, the scanning may be accomplished by translating the beam across a stationary object/sample or equivalently by translating the object relative to a fixed beam. At each discrete location during the scan an image comprising diffracted caustics is collected from the dark circular region defined by the annular beam. We refer to these images as caustic images. The detector surface is normal to the symmetry or principal axis of the annular beam. During the scan the detector occupies different overlapping coplanar positions. Therefore, an overlapping patchwork of caustic images is recovered from the detection plane shown in Fig. 1.

 

Fig. 1 The translational scan of the annular X-ray beam is represented with the aid of the coplanar grid formed by the relative positions of the point X-ray source, the detection plane formed by coplanar overlapping detector positions (at each corresponding point source location) and a potential focal plane position. All three planes are parallel.

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Focal plane positions along the Z-axis are parallel to the detection plane and also the point source plane. Reconstructed images termed optical sections may be recovered at each focal plane position.

A diagram of the annular beam is shown in Fig. 2. The diffraction pattern produced by a family of Debye cones i.e. at a constant diffraction angle of 2θ, along a specimen path at range Z produces a continuum of elliptical Debye rings; each ring is tangent to the resultant caustic as shown in Fig. 2.

 

Fig. 2 The annular X-ray beam is defined by a half opening angle $\varphi$. The separation between the point X-ray source and the detection plane is $L$. The mean diameter of the annular beam at the detection plane is 2R. The distance along the Z-axis to a circular specimen path is given by $Z$. The position of a Debye cone, of half opening angle$2\theta$, along a caustic is given by polar coordinates $\left(r,\gamma \right)$. The pole is the piercing point of the principal axis on the detection plane.

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In practice, the local polar coordinate system depicted in Fig. 2 describes the position of an element on a discretised detector, which may take the form of a rectilinear or polar sampling grid. The former is employed in the empirical work presented in this paper. The annular beam and detector are configured as a rigid body, which is sequentially translated over a matrix of (caustic) image collection points recorded by a global Cartesian $\left(\text{X,Y,Z}\right)$coordinate system described in Fig. 3.

 

Fig. 3 The local “snapshot detector” polar coordinate system is aligned at the $(0,0,0)$ “point source” plane, see Fig. 1, in the global Cartesian coordinate system. Therefore, the pole is coincident at the point source. The pole position for each successive caustic snapshot image $M_{m,n}(i_{p,q})$ along the X-axis and the Y-axis is given by $n\delta x$ and $m\delta y$, respectively. Where $n$ and $m$ are the total number of scan steps and $\delta x$ and $\delta y$ are the step sizes, respectively. $M_{m,n}(i_{p,q})$ is comprised of $\left(pq\right)$ pixel intensity values $i$, which is determined by the native “staring” resolution of the detector. The reconstructed perspective image $M_{r,\gamma }\left(i_{m,n}\right)$ is comprised of $\left(mn\right)$ pixel vales $i$ obtained from the detector element at $\left(r,\gamma \right)$.

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The global coordinate position of the pixels composing each perspective view $M_{r,\gamma }$ over a range of $0\le \gamma \le 2\pi$ is given by:

 

Fig. 4 A sequence of perspective images $M_r\left(i_{m,n}\right)$ along a caustic path $\left(r,\gamma \right)$.

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The perspective views are optically encoded with depth or Z dependent parallax $\left(P_{-r}-P_r\right)$ as illustrated in Fig. 5.

 

Fig. 5 A reconstructed caustic is formed by considering all annular beam rays (originating from different point source locations during the scan) incident on a sample at Z. The separation between conjugate points along the caustic yields depth dependent parallax $\left(P_{-r}-P_r\right)$ where$P_{-r}$ and $P_r$ are measured with respect to reference locations at $\mp r$, respectively.

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It can be appreciated from rotational symmetry that the parallax $\left(P_{-r}-P_r\right)$ is conserved along polar directions$0\le \gamma \le 2\pi$. The Z-axis position of the sample is given by

To apply Eq. (3) requires employing shift-and-add tomosynthesis to recover the corresponding optical section, $T_Z$. Initially, a reference axial focal plane position at the point source plane is formed by aligning each image at a common origin $(0,0,0)$ in the global Cartesian coordinate system. The shift distance $S_{r,\gamma (2\theta )}=\pm \left|P_{-r}-P_r\right|/2$ for each perspective image along their respective$\gamma$directions is given by

The sign of the shift accommodates $\pm r$ directions, respectively. The total number, $N$, of perspective images on a caustic are shifted sequentially before being summed and averaged to identify an optimal focus condition according to

The thickness, measured along the Z-axis, of the resultant optical section is a function of the minimum detectable increment in parallax $\delta P_r$ is given by

The value of $2\theta$ may be derived with the aid of Fig. 5 and is given by

The corresponding d-spacing for an optical section may be derived by substituting for $\theta$ from Eq. (7) in Bragg’s condition; $\lambda =2d\sin \theta$ to give

The sign of $r$ is established by the shift direction employed in Eq. (4) to recover a focused optical section via Eq. (5). It can be appreciated from Eq. (8) that many different optical sections $T_{Z(d)}$ can be present at the same axial plane position.

2.3 Experiment conditions

To demonstrate the technique, three objects of differing material phase (cyclotetramethylene - tetranitramine (HMX), copper and nickel) are distributed in a low density medium (Fig. 6(a)) to form a heterogeneous sample. These objects also vary in crystallographic structure. The copper object produces near ideal Debye cones due to small randomly oriented crystallites formed from electro-deposition. In contrast, the nickel object (subject to an annealing stage above its critical temperature) and HMX possessed a relatively large grain size. These large grains give rise to diffraction spot distributions (Fig. 6(b)) known to be problematic for EDXRD experiments [31].

 

Fig. 6 (a) Schematic depicting the full optical path of the experiment arrangement including the X-ray generator/point source, annular collimator, heterogeneous object under inspection and a CCD camera fitted with a ‘Gadox’ conversion screen. (b) Example of the diffracted caustic imagery.

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X-rays were generated by a sealed, long line focus X-ray tube with a Zr filtered molybdenum target (Mo Kα~17.4 keV). The accelerating voltage and current were 40 kV and 30 mA, respectively. The annular primary radiation was produced by a 4 mm thick bespoke brass annular collimator supporting a 17.5 mm inner and 18 mm outer annular aperture placed 150 mm from the X-ray source. The resultant annular beam had an opening angle of$\varphi \approx {3.34}^{\circ }$. A 1024x1024 (13 μm) 16 bit cooled CCD PIXIS camera with a terbium doped gadolinium oxysulphide (Gd2O2S:Tb) phosphor screen was employed to collect the scattered photons. The sample was translated in x, y by a set of Thorlab motion stages. The sample was raster scanned through 201 positions along X and Y axes with step sizes; $\delta x=\delta y=\text{0}\text{.3}\text{mm}$ (and $\delta P_r\text{=}\text{0}\text{.3}\text{mm}$). This step size was chosen to provide sufficient axial sampling in each perspective image, see Fig. 3 and Eqs. (1) and (2), to resolve the minimum feature size of approx. $\text{1}\text{mm}$on the copper and nickel samples. The thickness of an optical section along the Z-axis $\delta T_Z\approx \text{2}\text{.6}\text{mm}$was calculated using Eq. (6). Decreasing the step size or increasing $\varphi$ would reduce the thickness of the optical sections and will be considered for future experimentation. The PIXIS camera detector ‘Gadox’ phosphor screen was placed at 272 mm from the X-ray source. Each frame was exposed for 7 seconds.

3. Results and discussion

The sample volume was raster scanned along the X and Y axes according to the experiment conditions described in Section 2.3. The resultant caustic snapshot images (Fig. 7(a)) were collected and stored in digital memory according to the coordinate system presented in Fig. 3. Perspective views $M_{r,\gamma }\left(i_{m,n}\right)$ or equivalently $M_{p,q}\left(i_{m,n}\right)$ were reconstructed on radius $r$ (Fig. 7(b)). Shift-and-add tomosynthesis was applied according to Eq. (4) and Eq. (5) to identify optical sections, $T_Z$. The axial focal plane positions were calculated using Eq. (3). An example of defocussed optical sections of the HMX sample at hypothetical axial focal plane positions coincident with the point source plane in Fig. 7(c) and beyond the detector are shown and Fig. 7(e). The optical section shown in Fig. 7(d) was the best focus condition archived at $\text{Z}\text{=}\text{169}\text{.6}\text{mm}$for a true position of $Z=170mm$.

 

Fig. 7 Successive stages in the diffraction tomosynthesis of an optical section comprising the $\left(\overline{1}02\right)$ HMX Bragg maxima. (a) Stack of snapshot caustic images $M_{m,n}(i_{p,q})$ collected over a corresponding $\left(mn\right)$ grid of point source locations. (b) Pixels collected at $\left(r,\gamma \right)$ are transformed into $\left(pq\right)$ perspective images $M_{p,q}\left(i_{m,n}\right)$. Optical sections $T_Z$for different $S_{r,\gamma (2\theta )}$distances (indicated by the green color circles) corresponding to axial focal planes positioned at; (c) the X-ray point source plane at $\text{Z}\text{=}\text{0}$, (d) the HMX plane at $\text{Z}\text{=}\text{169}\text{.6}\text{mm}$and, (e) a hypothetical plane at $\text{Z}\text{=}\text{339}\text{.3}\text{mm}$ i.e. beyond the detector plane at $\text{Z}\text{=}\text{272}\text{mm}$.

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Material phase identification was demonstrated by the colocation of optical sections produced at different $2\theta$ values. A fixed shift distance $S_{r,\gamma (2\theta )}$ combined with caustics at three different radii $r$ enables synthesis from three different Bragg maxima indicative of HMX as shown in Fig. 8. The application of Eq. (7) and Eq. (8) enabled the computation of $2\theta$values and the unequivocal computation of d-spacings, respectively.

 

Fig. 8 A triplet of optical sections each for different HMX Bragg maxima with corresponding $r$ and $2\theta$ values. The corresponding Bragg plane for each optical section is; (a) $\left(020\right)$ plane, (b) $\left(\overline{1}02\right)$ plane and, (c) $\left(120\right)$ plane. A total number of $N=360$ perspective images were focused via tomosynthesis.

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While the HMX component has been examined in detail (Fig. 8 and Fig. 9(a)) other objects distributed within the sample are also readily identifiable. Diffraction originating from shaped metal objects is identified on axial focal planes at $\text{Z}\text{=}\text{215}\text{.9}\text{mm}$ and $\text{Z}\text{=}\text{239}\text{.0}\text{mm}$ (Fig. 9(b)-9(c)), which correspond to the diffraction from the $\left(111\right)$ planes of copper and nickel, respectively.

 

Fig. 9 Optical sections showing different material phases with corresponding $r$, $S$ and $Z$ values. The corresponding Bragg plane and (true $Z$value) for each optical section are (a) $\left(\overline{1}02\right)$ plane of HMX ($\text{Z=170}\text{mm}$), (b) $\left(111\right)$ plane of copper ($\text{Z=216}\text{mm}$) (c) $\left(111\right)$ plane of nickel ($\text{Z=239}\text{mm}$). A total number of $N=360$ perspective images were focused via tomosynthesis. Calculated linear distances are indicated on each image and listed here with their (true) values (a) $\text{12}\text{.0}\text{mm}\text{(12}\text{.0}\text{mm)}$, (b) $\text{21}\text{.0}\text{mm}\text{(21}\text{.0}\text{mm)}$,$\text{22}\text{.5}\text{mm}\text{(22}\text{.5}\text{mm)}$and, (c) $\text{27}\text{.9}\text{mm}\text{(28}\text{.0}\text{mm)}$,$\text{22}\text{.5}\text{mm}\text{(22}\text{.5}\text{mm)}$.

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The morphology, scale and orientation of the sample features recorded within each optical section accord well with our theory. For example, the linear distances shown in Fig. 9 were calculated using Eq. (1) and Eq. (2) and agreed well with the true values. Therefore, the scale of the optical sections $T_{Z(d)}$in the $(X,Y)$ plane were determined by the scan step sizes; $\delta x=0.3\text{mm}$, $\delta y=0.3\text{mm}$ respectively and independently of $2\theta$ and the axial focal plane position. We also show that distances measured along the Z-axis from Eq. (3) and the thickness of optical sections,$\delta T_Z$, from Eq. (6) are a function of the annular beam half opening angel, $\varphi$. For example, the total number of axial focal plane positions sandwiched between the nickel and the HMX samples can be calculated from the difference,$\Delta S$, between their respective “focused” shift values as$2\Delta S/\delta P_r\text{=}\text{2}\text{(13}\text{.95}\text{mm}\text{-}\text{9}\text{.9}\text{mm)}/\text{0}\text{.3}\text{mm}\text{=}\text{27}$ (from consideration of Eq. (4)). The corresponding axial separation may then be expressed as$\Delta Z=27\delta T_Z=69.4\text{mm}$. Alternatively the absolute distances from the “point source plane” $(0,0,0)$ to the nickel and HMX optical sections can be calculated from Eq. (3) as $\Delta Z=\text{239}\text{.6}\text{mm}\text{-}\text{169}\text{.9}\text{mm}\text{=}\text{69}\text{.7}\text{mm}$(see Fig. 9). This empirical analysis demonstrates that the reconstructed X, Y and Z parameters are independent of $2\theta$as predicted by theory. This is an interesting and counterintuitive aspect of our technique especially given that the reconstruction process employs oblique parallel projections whose angle of incidence upon the detection plane is either $\left(\varphi -2\theta \right)$ or $\left(2\theta -\varphi \right)$.

The results presented here serve to illustrate new opportunities offered by our approach as a tomographic, non-destructive structural probe for the characterization of heterogeneous samples. It also enables the same order of intensity gains observed previously for spot caustics [11,12] to be maintained for objects, which are significantly smaller than the annular beam footprint. This implementation makes the use of ADXRD as a 3D imaging tool more feasible. Another counterintuitive aspect of our method is that linear orthogonal scanning applied to an annular beam is able to produce diffraction images, which are geometrically equivalent to those collected from a sample rotating in an oblique beam. Therefore, the optical section reconstruction process is robust with respect to grains having large size, which can be problematic for other diffraction based technologies. Also, the fact that many perspective diffraction images are summed to form a single optical section helps to mitigate sample absorption. An example of this effect is illustrated in Fig. 7(c). No individual perspective view shows a complete image of the HMX object as some of the diffracted rays were absorbed by other sample components, however this does not have a significant impact upon the subsequent optical section. This result suggests that it should also be possible to reorganize the data to provide information regarding the crystallite orientation and stress states to complement the material phase identification. Additional information could also be acquired by sequentially considering each perspective image. It is also possible to combine this technique with conventional transmission tomography to provide supplementary information regarding mass attenuation coefficient by employing a larger detector area (and higher dynamic range) than employed in this paper. Also, increasing the detector area would increase the amount of reciprocal space available for collection and reconstruction.

4. Conclusions

We have demonstrated the benefits of a new method of ADXRD tomography to identify unknown crystalline phases distributed at unknown positions within an inspection volume. Our imaging technique employs a scanning annular beam to generate relatively high intensity diffraction caustics incident upon a planar detector. We show for the first time that the Bragg maxima, which form the caustics, are optically encoded with the shape and location of the crystalline features along the annular beam. We recover the spatial distribution of the Bragg maxima via “caustic” tomosynthesis. Unlike conventional absorption tomosynthesis the collocation of optical sections, each comprised of a different Bragg maximum, enables the computation of a material’s lattice spacings. The image reconstruction fidelity is enhanced by summing diffraction signals from around a circular path to mitigate intensity variation due to non-ideal polycrystalline microstructure such as large grain size.

The broader implication of our work is to challenge energy dispersive XRD as the de facto method for high speed applications. While both ADXRD and EDXRD exploit Bragg’s condition to extract structural information their physical requirements and implementation is quite different. In particular, EDXRD generally yields lower quality structural information being inexorably linked to the energy resolution of the detector. We have demonstrated the potential for translating ADXRD’s relatively enhanced specificity and sensitivity to scanning applications. Also, our method does not require complex and bulky collimation of the coherently scattered photons and is scalable with respect to both scan size and X-ray energy. Our on-going work is pursuing multiple emitter X-ray sources and higher energy spectra for enhanced penetration and scanning speed. We believe that there is wide scope for high speed analytical imaging in security screening, diagnostics and non-destructive testing.