X-ray tomographic imaging of the complex refractive index P. J. McMahon, A. G. Peele, D. Paterson, K. A. Nugent, A. Snigirev, T. Weitkamp, and C. Rau Citation: Applied Physics Letters 83, 1480 (2003); doi: 10.1063/1.1602155 View online: http://dx.doi.org/10.1063/1.1602155 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/83/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in High-refractive-index CuI waveguide with aligned cylindrical micropores for high-resolution X-ray imaging J. Appl. Phys. 116, 013515 (2014); 10.1063/1.4887136 Computed Tomographic Xray Velocimetry AIP Conf. Proc. 1266, 35 (2010); 10.1063/1.3478193 Transmission images and evaluation of tomographic imaging based scattered radiation from biological materials using 10, 15, 20 and 25 keV synchrotron Xrays: An analysis in terms of optimum energy AIP Conf. Proc. 705, 1364 (2004); 10.1063/1.1758055 Optimization of the number of soft x-ray arrays and detectors for the SST-1 tokamak by the tomographic method Rev. Sci. Instrum. 74, 2353 (2003); 10.1063/1.1544052 X-ray imaging microscopy using a micro capillary X-ray refractive lens AIP Conf. Proc. 507, 566 (2000); 10.1063/1.1291213
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APPLIED PHYSICS LETTERS
VOLUME 83, NUMBER 7
18 AUGUST 2003
X-ray tomographic imaging of the complex refractive index P. J. McMahon,a) A. G. Peele, D. Paterson,b) and K. A. Nugentc) School of Physics, University of Melbourne, Victoria, 3010, Australia
A. Snigirev, T. Weitkamp,d) and C. Rau European Synchrotron Radiation Facility, 6 rue Jules Horowitz, Boıˆte Postale 220, 38043 Grenoble, France
共Received 7 April 2003; accepted 17 June 2003兲 We present a quantitative three-dimensional reconstruction of the complex refractive index of boron clad tungsten fiber using 35 keV x rays. The reconstruction provides a quantitatively accurate measurement with a three-dimensional spatial resolution of approximately 2 m. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1602155兴
Research over the last few years has demonstrated that the effects of phase can be important in the propagation of x rays, and that the additional information may be used to develop modes of x-ray imaging1 such as phase-sensitive projection imaging for synchrotron1 and laboratory2 sources; and quantitative phase imaging methods based on interferometry3 and propagation.4 Phase-sensitive tomography based on both interferometry and propagation-based phase recovery methods3,5,6 has also been demonstrated. In this letter we extend this work to show that it is possible to use synchrotron radiation to quantitatively measure the complex field and thereby simultaneously determine the real and the imaginary part of the complex refractive index. Absorption x-ray tomography has undergone many advances in recent years and is based on the well known equation for the propagation of x rays through a medium
冋 冕
˜A 共 r兲 ⫽ 冑I 共 r兲 exp ⫺k
册 冋冕
 共 r,z 兲 dz exp ik
册
␦ 共 r,z 兲 dz , 共1兲
where we assume that the x rays are able to pass through the sample with negligible deviation. Here ˜A (r) is the amplitude distribution at a position r in a plane perpendicular to the propagation direction of the synchrotron beam, I is the observed intensity, k⫽2 / is the wave number for the radiation, and is its wavelength. The terms ␦ (r,z) and  (r,z) are, respectively, the spatial distributions of the real and imaginary components of the energy dependant x-ray complex refractive index n 共 r,z 兲 ⫽1⫺ ␦ 共 r,z 兲 ⫺i  共 r,z 兲
共2兲
over three-dimensional space. Absorption-contrast tomography uses measurements of 兩 ˜A (r) 兩 2 which is, according to Eq. 共1兲, only sensitive to the value of 兰  (r,z)dz. The effect of the phase distribution is ignored. Propagation-based phase recovery uses the observation that spatial gradients in the real part of the refractive index a兲
Present address: Platform Sciences Laboratory, Defense Science and Technology Organization, Fisherman’s Bend, Victoria 3001, Australia. b兲 Present address: Advanced Photon Source, Argonne National Laboratory, 9700 South Cass Ave., Argonne, IL 60439. c兲 Electronic mail:
[email protected] d兲 Present address: Paul Scherrer Institut 共PSI兲, CH-5232 Villigen PSI, Switzerland.
act to redistribute energy on propagation and so is able to produce observable effects in the intensity distribution. These intensity effects may be inverted to recover the phase distribution. There are a number of ways in which the phase may be recovered. In this work we use the so-called transport of intensity equation.7 The transport of intensity equation relates the rate of change of the intensity, I(r⬜ ), of a paraxial wave along the beam propagation direction z to the phase of the wave ⌽(r⬜ ): ⫺k
I 共 r⬜ 兲 ⫽ⵜ⬜ "关 I 共 r⬜ 兲 ⵜ⬜ ⌽ 共 r⬜ 兲兴 . z
共3兲
ⵜ⬜ is the transverse gradient operator at the detector plane and r⬜ is a position vector lying in a plane perpendicular to the direction of the z axis. In the absence of phase discontinuities,8,9 this expression specifies the phase uniquely and enables a quantitative determination independent of the intensity. This approach has been confirmed for a wide range of radiation types.10 In practice, the intensity derivative is estimated by forming the intensity difference between two downstream planes a small distance apart and placed equally around the plane of interest. The intensity data enable the reconstruction of the absorptive structure, while the intensity data combined with the intensity derivative data allow the independent reconstruction of the phase distribution. In this work, the phase will be recovered a distance downstream from the object and it is assumed that the propagation distance is sufficiently small that the field in this plane is only slightly different from that in the object. Experiments were carried out at the European Synchrotron Research Facility 共ESRF兲 at the microfluorescence and diffraction beam line ID22. An undulator was 65 m upstream on a high- section of the ESRF storage ring and produced an effective source with a horizontal full width at half maximum 共FWHM兲 of 700 m and vertical FWHM of 50 m. The beam divergence was 30⫻20 rad. The beam line itself consists of flat Pt coated silicon mirror to remove high energy x rays with final energy selection carried out with a fixed-exit vertical flat double-crystal design Si关111兴 monochromator (⌬E/E⫽10⫺4 ). The experiment was carried out at an energy of 35 keV.
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The sample examined in this experiment was a composite fiber of boron 共100 m diameter兲 with an inner core of tungsten 共15 m diameter兲. This sample was chosen as it had a component that would predominantly influence the absorption and another that would primarily influence the phase. The tomographic data were acquired by rotating the fiber around its length in the horizontal direction in order to utilize the higher vertical spatial coherence of the source. Imaging data were collected using the FReLon 2000 x-ray to visible conversion charge coupled device 共CCD兲 camera system.11 This 14 bit fast readout camera consists of a 2048⫻2048 CCD with a 14 m pixel size. The conversion of x rays to visible light was carried out with single crystal 12-m-thick doped yttrium–aluminum–garnet scintillator deposited onto a 170-m-thick lutetium–aluminum–garnet substrate. The subsequent optical images of the sample were magnified 40 times by a lens system to produce an effective FIG. 1. Tomographic reconstructions of the sample. 共a兲 Slice through the pixel size of 0.35 m on the CCD camera. reconstruction of the phase distribution. 共b兲 A slice through the reconstruction of the absorption part of the reconstruction. This image has a logarithA total of 1250 equally spaced projection images were mic gray scale and was taken at the same location as the image in 共a兲. 共c兲 collected over 180° for each sample to detector setting. In an Surface rendered image of the phase reconstruction showing the boron coateffort to reduce sampling artifacts a region of interest of ing. The cutaway allows the interior tungsten wire to be seen. 共d兲 Surface 1024⫻1024 was selected for each intensity image and the rendering of the imaginary part of the refractive index. As only the tungsten wire shows significant absorption this image only shows that component of maximum projected dimension of the object was 330 pixels the object. All of these images are able to yield quantitative results for the wide. Tomographic data sets for absorption based reconstrucrefractive index, as shown in Table I. tions were collected at a sample to scintillator distance of 4 mm. Tomographic data sets for phase based reconstructions refractive index for both boron and tungsten are consistent were taken at sample-scintillator distances of 150 and 200 with our measurements, although no reliable measurement of mm. These settings were used to estimate the intensity dethe imaginary component for boron was possible at the enrivative required by Eq. 共1兲 and were selected in order to ergy used in this experiment. minimize variations in image contrast due to subpixel strucCareful analysis of the reconstructions indicates that a ture within the sample, such as small angle scattering. conservative estimate of the voxel spatial resolution is apTo facilitate image normalization, dark current subtracproximately 1 m in each direction for the imaginary comtion and flat field division were carried out on all sample ponent of the refractive index, and ⬃2 m for the real part. images prior to any processing. The sinograms were aligned This result suggests that phase-based reconstructions display about the center of rotation and backlash correction was persimilar resolution limitations to traditional tomographic techformed in software. The tomographic reconstructions were niques. The spatial resolution of the detector system is carried out using filtered backprojection on both the logaslightly better than 1 m and so the phase recovery introrithm of the intensity data and directly on the phase sinoduced a small amount of further degradation of the spatial grams. resolution. Figure 1 depicts the results obtained for both phase and The phase recovery method is very fast12 and can easily amplitude reconstructions from the boron clad fibre. Figure be performed as quickly as the data are acquired. The prin1共a兲 shows a slice through the tomographic phase reconstruccipal limitation of this approach, then, is the need to acquire tion and so depicts a map of the real part of the refractive additional data in order to recover the phase. The significant index. Figure 1共b兲 depicts the same slice reconstructed using benefit of this technique is the addition of an additional paconventional absorption tomography and is displayed with a rameter for analysis of samples. logarithmic gray scale in order to allow visualization of the In conclusion, we have shown that it is now possible to boron cladding simultaneously with the tungsten core. The quantitatively recover the full complex x-ray refractive index high degree of spatial coherence in the x-ray beam gives rise map by using an extension to standard tomographic techto a very small amount of artifact due to phase contrast in the niques. The technique is rapid and in principle only requires absorption image and this is responsible for the visibility of the boron fibre outline. TABLE I. Tabulated and measured results for the complex refractive index Figure 1共c兲 shows a surface rendering and cut out of the of the sample. The tabulated values of the real 共␦兲 and imaginary 共兲 comvolume reconstruction of the real part of the x-ray complex ponents were interpolated from the National Institute of Standards and Technology tables. It can be seen that the results agree to within experimental refractive index, and Fig. 1共d兲 depicts the corresponding error. imaginary component. The boron can only be seen in the phase measurements. Experimental Tabulated Experimental Tabulated These data sets allow a quantitative determination of the value of ␦ value of ␦ value of  value of  complex refractive index and the results of this analysis are Boron (4.0⫾0.4)⫻10⫺7 3.7⫻10⫺7 ¯ 1.0⫻10⫺11 ⫺6 ⫺6 ⫺8 given in Table I. The table clearly shows that tabulated valTungsten (2.3⫾0.3)⫻10 2.6⫻10 (7.7⫾0.7)⫻10 7.5⫻10⫺8 to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. 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one additional projection measurement for each angle. Application of the technique is simple to implement and may be used with all existing tomographic techniques. We anticipate that the ability to acquire this extra information about samples may find application in a number of areas. The authors would like to acknowledge the technical assistance of J.-M. Rigal and D. Fernandez of the ESRF, and J.-C. Labishe of the ESRF for the development of the FReLon CCD camera. The authors also acknowledge the support of the Australian Research Council. 1
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