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We present a method for x-ray imaging of objects inside the focal spot of polycapillary optics that resolves details smaller than the focal spot dimensions. This method employs coded aperture imaging, in which the micro-structure of polycapillary optics is treated as the coding pattern. Projection of the object is decoded from a magnified x-ray image of the polycapillary structure which is specifically sharpened by the object. Field of view can be extended by scanning the object across the focal spot.
© 2013 Optical Society of America
1. Introduction
Focusing polycapillary devices are reflective x-ray optical elements composed of arrays of glass micro-capillaries [1–3]. Radiation in individual capillaries propagates by means of multiple total external reflections. A special bending of capillaries enables to concentrate x-rays into an intense micro-spot, hereafter referred to as the focal spot of a polycapillary optics. It is worth to emphasize, that while polycapillary elements are frequently referred to as “lenses” they cannot be described in terms of classical imaging optics [4]. For polycapillary optics, the formation of the focal spot can be usually described using geometrical optics as a result of the overlap of beams emitted from individual capillaries [5, 6]. The lateral dimensions of the focal spot are thus of the order of ∼2fθc, where θc is the critical angle for the total external reflection (θc[mrad] ≈ 30/E[keV]) and f is the focal length determined by the bending radius of capillaries. For low x-ray energies and/or small capillary diameters, interference effects can introduce fine structures to the shape of the focal spot [7, 8].
For a recent review of polycapillary optics discussing applications in spectrometry, imaging and diffraction see [9]. Most frequently, focusing polycapillary optics is used in scanning geometry for element specific imaging [10]. The use of two crossed lenses permits a 3D confocal x-ray microscopy [11] with a table-top setup [12]. In such geometries, imaging resolution is usually defined as the size of the finest focal spot that can be formed using the optics and is typically at the level of 5–50 μm. In contact geometries the resolution is limited to several widths of the capillary channels and also has a value at the level of several microns [13].
In this work we present a method to resolve details of an object located inside the focal spot of polycapillary x-ray optics, which is based on the coded aperture principle [14–17]. The proposed approach works for hard and polychromatic x-rays. In the proof-of-principle experiment we obtained a five-fold improvement in the resolution as compared to the size of the focal spot.
2. Principle of x-ray imaging inside the focal spot of polycapillary optics
The principle of x-ray imaging inside the focal spot of polycapillary optics is presented in Fig. 1(a). For its description we introduce a simplified model based on geometrical optics. For brevity we restrict the calculation to the xz plane. Consider an extended source at z = 0 (exit surface of lens) having a brightness S(x)G(α), an object located at z = d and x-ray camera placed at z = D. S(x) describes the internal structure of the polycapillary optics [c.f. Fig. 1(b)] and the transmission of the lens. G(α) defines the angular distribution of the radiation emitted from each point of optics. Since each capillary emits radiation in a shallow cone directed towards the focal spot, we assume that G(α) = exp(−0.5α2/σ2), where σ ∼ θc, and α is the angle between vectors r − rs and f − rs, where r = (x, D) is the observation point, rs = (xs, 0) runs over the exit surface of the lens and f = (0, f) is the position of the focal spot. In the paraxial approximation α ≈ (x − Mxs)/D, where M = (f − D)/f. The intensity at the detector can be written as:
where T(x) = exp[−P(x)] describes the transmission of the object and P(x) = μ(x,z)dz is the projection of the object’s absorption coefficient. Convolution of I(x) with the point-spread function of the the detector should be also taken into account in Eq. (1).
Fig. 1 Scheme of an experiment for x-ray imaging inside the focal spot of a polycapillary optics using the coded aperture principle. (a) The internal structure of the polycapillary optics is treated as a coding aperture. Projection of the object is decoded from a magnified x-ray image of the polycapillary structure which is specifically sharpened by the object placed at the focal spot. (b) Optical microscope image of the exit surface of the lens. The marked hexagon corresponds to a single capillary bundle. (c) Transmission scan of the focal spot with a 5 μm pinhole. Points show a profile through the center and the line is a Gaussian fit with a FWHM of 40 μm.
An image recorded without an object will be denoted as I0. I0 evaluated for D = f gives the intensity distribution at the focal spot: a Gaussian with a FWHM of 2.35 fσ. While the information about the structure of S(x) is lost in the focal plane [c.f. Fig. 1(c)], an x-ray camera placed at position D > f records a magnified (by a factor of M) and blurred image of the internal structure of the polycapillary element. If an object is placed at d > f then, in a crude approximation, the focal spot can be treated as a secondary source producing a magnified point-projection of the object. The imaging resolution in such a case is limited by penumbra blur resulting from the size of the focal spot [18].
The most interesting situation arises when an object is placed inside the focal spot (d = f). In such a case Eq. (1) becomes:
From Eq. (2) it follows that the shadow of the object vanishes from the camera. Instead of this, the image of the capillary structure changes drastically. Since the object’s details introduce high frequencies to the convolution, the image I is unblurred as compared to I0 in a similar way as in a pinhole camera.What’s most important is that Eq. (2) describes coded aperture imaging [15–17], where the capillary pattern S defines the coded aperture. In fact, in our notation, S combines information about geometrical micro-structure of the polycapillary and its x-ray transmission characteristics.
3. Experimental setup
Imaging inside the focal spot of polycapillary optics was realized using polychromatic radiation from an x-ray tube with a W anode and a 75 μm spot (Oxford Instruments, XTF 5011) operating at 1 mA and 50 keV, which was focused by a polycapillary lens (IfG). The optics had an input focal length of 40 mm, output focal length of 14 mm, length of 100 mm and exit surface diameter of 2.4 mm. The acceptance cone of the optics had an opening angle of 7°. The optical microscope image of the lens exit surface [Fig. 1(b)] reveals a hexagonal mesh of individual capillaries (diameter 3.5 μm) as well as a coarser one corresponding to bundles of capillaries. The focal spot of the lens was featureless and had an approximate Gaussian shape with FWHM of (40 ± 1) μm as demonstrated in Fig. 1(c). This image was obtained by scanning the focal spot with a 5 μm pinhole and measuring the transmitted intensity. The total flux in the focal spot was approx. 108 photon/s and the mean energy of the beam was 22 keV. The intensity gain in the focal spot was at the level of 5 × 103 as compared to the intensity of the direct beam through 5 μm pinhole. X-ray images were acquired using a CMOS camera (Rad-Icon, RadEye1). The sensor had 1024×512 pixels separated by 48 μm and was optically coupled to a scintillator. The camera was placed at a distance D = 171 mm from the lens. Since θc changes with energy (a slow 1/E dependence), for polychromatic radiation the effective focal spot size depends on energetic response of the detector. Therefore, to maintain constant experimental conditions, auxiliary x-ray transmission scans were performed with the same x-ray camera but used as a point-detector just by integrating the intensity over all camera pixels.
4. Results and discussion
4.1. X-ray images of simple objects placed inside the focal spot
Let us first demonstrate experimental images obtained for simple objects placed inside the focal spot of a polycapillary lens. Figure 2(a) shows x-ray images recorded without an object (I0) and for two simple objects: a 5 μm pinhole in a tungsten foil and a 25 μm tungsten wire. For a better presentation ratios I/I0 are shown. For the pinhole, the image was corrected by a subtraction of an image recorded with the pinhole moved aside of the focal spot. This eliminated the influence of high-energy radiation passing through the foil. The upper insets in the images show projections recorded for objects placed at d = f + 20 mm. The regions in images outside the shadow of the objects are almost uniform which is due to normalization by I0.
Fig. 2 X-ray imaging of simple objects placed inside the focal spot. (a) Left: image recorded without object (I0). Middle: data for a 5 μm pinhole. Right: data for a 25 μm wire. Pinhole and wire images were divided by I0. Insets at the bottom show zooms of the data. The zoom of I0 was flattened for a better comparison. The upper insets show projection-like images obtained for objects displaced from focal spot along z axis. (b) Corresponding logarithms of the Fourier transform intensities. Numeric labels provide spatial frequencies in the focal plane.
Conversely, for x-ray images recorded with objects inside the focal spot, the hexagonal poly-capillary mesh is clearly visible despite normalization by I0. This means that the object influences this structure or, in other words, the information about the object is encoded in the entire image. Let us discuss the most important features of images from Fig. 2. First, the visible hexagonal structure corresponds to bundles of capillaries [c.f. Fig. 1(b)] and individual capillaries are not resolved. Hence, in this pilot experiment, capillaries determine the behavior of the G function, but the S function is determined by the structure of capillary bundles. Second, the object inside the focal spot produces a sharpening of the images. The degree of sharpening is higher for the smaller object. For the wire, sharpening takes place only along the horizontal direction. These observations are confirmed by the analysis of Fourier transforms presented in Fig. 2(b). Furthermore, a clear contrast reversal in images recorded for the wire and for the pinhole can be observed.
4.2. Reconstruction of the object’s projection
Next, let us describe the process of decoding or in other words reconstruction of the object’s projection. Since this procedure is based on Eq. (2), the knowledge of the S(x) function characterizing the capillary structure i.e. the coded aperture is essential. This function has to be known only up to a maximal frequency of the magnified image that can be resolved by the detector. Therefore, S(x) can be approximated by an image IP recorded for a pinhole placed inside the focal spot. In our case, x-ray camera ultimately cuts-off frequencies above ∼ (10μm)−1. Therefore IP recorded for a 5 μm pinhole, that is shown in Fig. 2, is a sufficient approximation of S(x).
The decoding was performed by a direct implementation of the convolution theorem:
For thin objects, U(x) approximates P(x)G(x). Fourier transforms from Fig. 2(b) show discrete peaks corresponding to the periodicity of the capillary bundles. The deconvolution was limited to this discrete set of values which extremely reduced noise. However, let us show that the reconstruction using such a discrete set is at all possible. Shannon sampling theorem [19] says that if the object has a finite support (−a/2, a/2) then its shape can be completely reconstructed from Fourier transform sampled at least at a critical frequency νC = 1/a. In our case, the signal from an arbitrary large object originates from the area of the focal spot. In practice it is sufficient to assume that the focal spot is confined to a diameter of 2×FWHM. The corresponding critical frequency νC is marked in Fig. 2(b) and is higher than the spatial frequency of the 1-st order peaks. This denotes a proper sampling.In order to test the decoding procedure, coded aperture images were measured for two golden grids with different meshes (600 and 1000 mesh grids). Results of the reconstruction are presented in Fig. 3(a). The mesh structure is clearly resolved inside the area of the focal spot. This proves the possibility of imaging objects inside the focal spot of a polycapillary lens. However, the quality of images is only moderate. Moreover, the increase of the acquisition time did not significantly improve the quality.
Fig. 3 Reconstruction of coded aperture images recorded for golden grids with different meshes placed inside the focal spot of a polycapillary optics. (a) Reconstruction from a single image. Left: pitch 42 μm, hole 37 μm. Right: pitch 25 μm, hole 19 μm. (b) Corresponding reconstructions from a set of images. The number of images and their acquisition times are given at the bottom of the plots.
The improvement of quality and, simultaneously, the possibility of imaging of larger objects can be realized by an extension of the reconstruction procedure. X-ray coded aperture images can be recorded for the object scanned across the focal spot in such a way that there is an overlap between neighboring decoded images i.e:
where Δx and Δy are smaller than 4fσ. This procedure is very loosely related to the x-ray ptychographic procedure [20]. This simple operation improves the data quality. It corrects the image for the shape of the focal spot and for the asymmetry of the deconvolution kernel as shown in Fig. 3(b).A reconstruction of a more complex object (number 12 from a copper electron microscopy finder grid) is presented in Fig. 4(a). The reconstruction was performed from a set of 22×18 coded aperture images (5 s acquisition each) and is compared with a standard x-ray transmission scan for d = f and a projection image for d = f + 20 mm. Data in Fig. 4 show a drastic improvement in the resolution of the coded aperture imaging as compared to conventional scanning. Features separated by 10 μm are very well resolved. Features separated by 8 μm are at the limit of resolution. The corresponding frequency of (16 μm)−1 is marked in Fig. 2(b). This denotes a five-fold improvement as compared to the transmission scan for which the resolution is limited by focal spot size to 40 μm.
Fig. 4 Comparison of coded aperture imaging with a standard x-ray transmission scan and a projection image. (a) Reconstruction from a set of coded aperture images. The line plot is a profile at the position marked with a dashed line. (b) Transmission scan for d = f. (c) Projection image (I/I0) for d = f + 20mm. (d) Optical microscope image of the object.
The resolution of the projection from Fig. 4(c) is slightly better than for the scan from Fig. 4(b). In projection geometry the penumbral blur is reduced by a factor of (D − d)/(D − f) ≈ 0.86 as compared to the focal spot dimensions. In fact, for d → D this factor tends to zero. However, the magnification of the point projection is given by (D − f)/(d − f) and tends to unity. Hence, for large d the resolution of the detector becomes the dominant factor. This is exemplified in Fig. 4(c) where individual pixels of the detector can be observed. It means that coded aperture imaging gives optimal spatial resolution for a given polycapillary and detector combination.
Finally, we will discuss the limits of the resolution. The fundamental limit is the highest spatial frequency of the S(x) function describing the coded aperture. In practice, this frequency is set by the highest resolved harmonics of the capillary structure. Commercially available lenses have capillary channels with diameter smaller than 1 μm. In order to resolve such a frequency one has to obtain a sufficiently high magnification M. For example, a standard polycapillary optics having f = 5 mm and a 1k×1k pixel detector with a resolution of 50 μm placed at D = 300 mm could provide resolution better than 1 μm. Precise measurement of S(x) with a correction for directly transmitted beam seems to be possible for pinholes with such a diameter.
5. Conclusions
We presented a method for resolving details of objects inside the focal spot of polycapillary optics which treats the micro-structure of polycapillary optics as a coded aperture. The proof-of-principle experiment was performed using a low-resolution equipment and the formalism used to model the image formation and reconstruction was largely simplified. After improvements, resolution at the level of 1 μm could be possible with commercially available optics. Further developments in x-ray optics fabrication and use of more exact theories of x-ray propagation in capillaries [3, 21] could help to obtain sub-micron resolution. The presented results can have impact on other imaging geometries using polycapillaries [13,22–24]. Presumably, the proposed approach could be used with other kinds of micro or nano-structured optics [25, 26].
Acknowledgments
This work was financially supported by the Polish National Science Center (grant no. DEC-2011/01/B/ST3/00506) and Jagiellonian University’s DSC program ( DSC/000692/2012).
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