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X-ray imaging using a Talbot-Lau interferometer, consisting of three binary gratings, is a well-established approach to acquire x-ray phase-contrast and dark-field images with a polychromatic source. However, challenges in the production of high aspect ratio gratings limit the construction of a compact setup for high x-ray energies. In this study we consider the use of phase gratings with triangular-shaped structures in an x-ray interferometer and show that such gratings can yield high visibilities for significantly shorter propagation distances than conventional gratings with binary structures. The findings are supported by simulation and experimental results for both cases of a monochromatic and a polychromatic source.
© 2014 Optical Society of America
1. Introduction
Conventional, absorption-based x-ray imaging suffers from a relatively low contrast in soft tissue. X-ray phase-contrast and dark-field imaging are modalities that can significantly increase the soft tissue contrast [1–5] and add complementary imaging information, previously not accessible [6–8]. Recently, a method has been reported that showed the feasibility of x-ray phase-contrast and dark-field image acquisition at a compact setup, operated with a polychromatic laboratory source [9–12]. The approach is based on the introduction of a Talbot grating interferometer into the x-ray beam. The interferometer consists of a binary phase grating and a binary analyzer grating. In case of a non-microfocus laboratory source, a third grating is placed close to the source emission point, to introduce sufficient spatial coherence, resulting in a so-called Talbot-Lau interferometer. Grating interferometry provides conventional absorption, phase-contrast and dark-field contrast simultaneously. The signals are obtained by moving one of the gratings laterally in a number of steps over one grating period and acquiring an image for each step. The ratio in each pixel of the amplitude of a thus obtained stepping curve and its offset is referred to as the visibility: V = (Imax − Imin)/(Imax + Imin), where Imax and Imin are the maximum and minimum measured intensities, respectively. The signal-to-noise ratio in the phase-contrast and dark-field images is strongly dependent on the visibility.
One of the practical limitations of this imaging approach is the size of the setup at clinically relevant x-ray energies in the range of 60–120 keV. The analyzer grating is placed typically at a Lohmann distance [13], also referred to as fractional Talbot distance [14], behind the phase grating to obtain the highest visibility. A typical binary phase grating has π/2 [15] or π [16,17] phase shifting structures with a 0.5 duty cycle (see Fig. 1(a)). Binary phase gratings with higher phase-shifts: (m+1/2)π, m ∈ ℕ are equivalent in the visibility behavior to the π/2 grating and gratings with a phase shift (2m + 1)π are equivalent to a π shifting grating. For a binary phase gratings with π/2 or π phase shifting structures the shortest Lohmann distance is given by [14,18] and [14,18], respectively, where Zt = 2p2/λ is the full Talbot distance, determined by the x-ray wavelength λ and the phase grating period p. However, a π phase grating introduces an interference pattern with a doubled spatial frequency and requires, therefore, the use of an analyzer grating with half the period of the phase grating. The availability of high aspect-ratio analyzer gratings is limited by manufacturing possibilities. Construction of a compact, high energy imaging setup with a grating interferometer using a binary phase grating is currently only feasible, if very small, 1–2 μm period gratings were available with structure heights of several hundred μm.
Fig. 1 Simulated monochromatic intensity carpets for ideal (a) binary, π/2 phase grating and triangular gratings with phase shift (b) π/2, (c) 3π/2 and (d) 5π/2. (e) Simulated visibility for binary π/2, 3π/2 and 5π/2 gratings (black, dashed) and triangular π/2 (mangenta, dashed), 3π/2 (blue, bold solid) and 5π/2 (red, solid) gratings.
In the present letter, the specific case of phase gratings with triangular-shaped structures instead of binary ones in a Talbot-Lau interferometer is discussed. Simulated visibilities are studied for the case of a monochromatic and for a polychromatic source and are compared with experimental measurements. Finally, we show that the size of the setup can be significantly reduced, substituting binary phase gratings with triangular-shaped gratings.
2. Simulation
The use of triangular-shaped phase gratings for x-ray phase-contrast imaging with a Talbot-Lau interferometer was evaluated using a simulation. The two cases of a monochromatic and a polychromatic source were considered separately.
2.1. Monochromatic source
In the case of a monochromatic source it was assumed that a phase grating is illuminated with a coherent, monochromatic plane wave with an amplitude set to unity. Furthermore, the phase grating is ideal and shows no absorption. The propagation of the wave behind the phase grating was achieved in Fourier space, evaluating the field using the Fresnel approximation (based on paraxial approximation) [14, 19], which is applicable to near-field effects. The wavelength was set to 0.12 Å (100 keV) and the period of the phase grating to 4.0 μm. The interference pattern behind the phase grating was calculated up to the first full Talbot distance Zt = 2.58 m using 800 equidistant propagation steps. The cases of a triangular and a binary phase grating with π/2, 3π/2 and 5π/2 phase shifting structures were simulated and are shown in Fig. 1. The interference carpet for a binary grating with a 5π/2 phase shift is identical to the case of π/2, and the interference carpet for a binary grating with a 3π/2 phase shift differs from the carpet shown in Fig. 1(a) only by a lateral shift of the pattern by half a grating period. Thus, the visibility for ideal binary gratings with a phase shift π/2, 3π/2 and 5π/2 is identical. The given phase shift for a triangular-shaped grating refers to the phase shift the wave acquires, passing through the thickest part of the grating relative to the thinnest part. From the simulated intensity carpets, theoretical visibility was calculated using an ideal analyzer grating stepping approach [10, 12]. The stepping was performed for all distances with an analyzer grating with the same period as the phase grating with a duty cycle of 0.5. The results are shown in Fig. 1(e) for both triangular and binary phase gratings.
The simulation results reveal that, contrary to binary gratings, triangular-shaped phase gratings do not yield equivalent interference patterns for the phase shift of π/2, 3π/2 and 5π/2. All three simulated triangular-shaped gratings give visibilities high enough to be used for x-ray phase-contrast and dark-field imaging with a Talbot-Lau interferometer, but the highest visibility was obtained for the 5π/2 grating. We empirically observed that the first visibility peak behind the phase grating has always the highest visibility. The propagation distances for all visibility peaks have been estimated empirically from the simulation and are summarized in Table 1.
Table 1. Propagation distances with visibility peaks for triangular-shaped gratings for a monochromatic source.
2.2. Polychromatic source
The use of triangular-shaped gratings was also simulated for the case of a tungsten-target poly-chromatic source. The x-ray spectrum used for the simulation [20] is shown in Fig. 2(a). The source peak voltage was set to 60 kV, with an average photon energy of Emean = 45 keV. Five specific cases of 8,16, 24, 32 and 40 μm high non-absorbing nickel phase gratings - corresponding to a phase shift of π/2, π, 3π/2, 2π and 5π/2 for Emean - were studied for a binary and a triangular grating. The pitch was set to 5 μm, resulting in a full Talbot distance of Zt = 1.8 m for Emean. From the simulated intensity carpets, assuming an ideal analyzer grating, theoretical visibilities were calculated and are presented in Fig. 2. Triangular-shaped phase gratings yield similarly high visibilities as binary gratings. For 3π/2 and 5π/2 phase shifts the first visibility peak behind the triangular-shaped grating lies at a significantly shorter distance than for the binary grating. These results can be especially interesting for design of compact x-ray interferometers. For a polychromatic spectrum, binary phase gratings with π and 2π phase shift yield low peak visibilities. On the contrary triangular-shaped gratings produce visibilities high enough for imaging purposes.
Fig. 2 (a) 60 kVp normalized tungsten source spectrum used in the simulation. Simulated (polychromatic) visibility for ideal binary and triangular gratings with a phase shift (b) π/2, (c) π, (d) 3π/2, (e) 2π and (f) 5π/2 for the mean energy. Visibilities obtained with an analyzer grating with the same period as the phase grating.
3. Experimental validation
The presented simulation results were validated experimentally. In a parallel beam triangular-shaped phase gratings can be simply achieved by tilting a binary grating. For a cone beam, constant magnification over the grating area is necessary. In this case a triangular-shaped phase grating can be produced by writing binary structures on a tilted wafer. Figure 3 illustrates how different degrees of skewing lead to diverse phase structures under illumination perpendicular to the substrate.
3.1. Monochromatic source
The simulated results for a monochromatic source were validated experimentally by direct intensity carpet measurements at the coherence beamline P10 at the Petra III synchrotron radiation facility in Hamburg, Germany. A monoenergetic 19.2 keV beam (bandwidth ∼ 0.01%) was used to illuminate a binary nickel phase grating with h = 10 μm and period p = 5 μm, introducing a phase shift of 3π/2 to the beam. The source size was 35 × 6 μm2. The phase grating was positioned approximately 95 m away from the source resulting in a transverse coherence length of 175 μm. The field of view was 700 μm in vertical direction and several mm in horizontal direction. A YAG scintillator was combined with a factor 25 magnification optics and a PCO (Kehlheim, Germany) CCD camera with an effective pixel size of 360 nm. To acquire the intensity carpet the detector system was mounted on a motorized stage and was moved downstream in the beam behind the binary phase grating for one full Talbot distance (0.77 m), acquiring intensity at 154 equidistant positions with an exposure time of 0.5 seconds. Subsequently, the phase grating was tilted by 14 degrees, so that a triangular shape was achieved and the intensity measurements were repeated. The measurement results are presented in Fig. 4. In order to compare the simulated visibilities with experimental results, the experimental carpets were convoluted with the transmission function of an ideal analyzer grating with a duty cycle of 0.5 and the same period as the phase grating. The visibility was averaged over 10 periods of the phase grating. Thus obtained results are plotted in Fig. 4 along with simulated visibilities. In the simulation the absorption from the nickel phase grating was taken into account. The simulated and experimental results show a good agreement. However, the simulated visibility is higher than the measured one, which can be explained by grating imperfections (e.g. bridges, duty cycle deviations), the efficiency of the detector and the mechanical stability of the stage.
Fig. 4 Top row: directly measured intensity carpets for (a) a binary grating and (b) a binary grating, tilted to a triangular shape. Bottom row: visibilities calculated from the measured carpet for (c) binary and (d) triangular phase grating (solid, blue) and the simulated values (dashed, red).
3.2. Polychromatic source
The simulation results for a polychromatic source were also validated experimentally. The setup used for the measurements consisted of a tungsten MXR-160HP/11 (Comet, Flamatt, Switzerland) x-ray source, a three grating Talbot-Lau interferometer and a PaxScan 2520D (Varian, Palo Alto, USA) scintillator detector (pixel size 127 × 127 μm2). The source grating and the analyzer grating had the same height and period: h=150 μm, p = 10 μm. During the experiment three phase gratings were tested. Two were binary nickel phase gratings with period p = 5 μm and height h = 16 μm and h = 8 μm, respectively. The third phase grating was a nickel triangular-shaped grating with p = 5 μm and maximum height h = 14 μm and the tilt angle 8.9 degrees. The source grating was fixed approximately 10 cm behind the source and the analyzer grating was placed 7 cm in front of the detector.
During the measurements, the distance between the source and the detector was varied, keeping the distance from the source grating to the phase grating as well as from the phase to the analyzer grating equal. Thus, the setup was symmetric and had a magnification factor of two. A stepping curve [10, 12] was acquired for each distance between the source and the detector and the visibility was calculated. The source current was adapted in such a way that the photon flux on the detector for each distance was approximately the same. Due to geometrical setup restrictions, it was only possible to acquire measurements for distances from 19 cm to 125 cm between the source and the phase grating. Experimentally determined visibilities are presented in Fig. 5. Due to the divergent beam the visibility achieved at the edges of the wafer was significantly lower than in the middle. The presented values are visibilities measured in the central part of the wafer, averaged over an area of 100 × 100 pixels. The visibilities for the h = 14 μm triangular and h = 16 μm binary phase grating were determined at 40 kVp. For this source voltage the phase shift for the photons with mean energy corresponds approximately to 3π/2 for both gratings. Subsequently, the visibilities for the h = 14 μm triangular and h = 8 μm binary phase grating were determined at 60 kVp. For this source voltage the triangular grating introduced a phase shift of approximately π and the binary grating a phase shift of π/2 to the photons with mean energy. From Fig. 5 one can see that for both measurements similar peak visibilities were achieved with triangular and binary phase gratings. However, the distances for the maximum visibilities for triangular-shaped structures are shorter. As predicted by the simulation (see Fig. 2(c)), π phase grating with triangular-shaped structures, yields visibilities high enough for imaging purposes with an analyzer grating with the same period as the phase grating. Furthermore, comparison of the experimental results, presented in Fig. 5 with simulations in Fig. 2 shows a good agreement, even though the measured visibilities are significantly lower than predicted by the simulation. The reasons for this could be absorption and imperfections in the phase grating, non-ideal analyzer grating and the detector efficiency. Furthermore, the simulation was based on the assumption that the grating is illuminated by a fully coherent plane wave.
Fig. 5 Measured visibilities with a polychromatic source for different distances between the phase grating G1 and the analyzer grating G2 at 40 and 60 kVp source voltage.
4. Phantom imaging
A phantom was used to demonstrate the actual feasibility of phase-contrast and dark-field imaging with a compact Talbot-Lau interferometer with a triangular phase grating. The phantom consisted of three small plastic tubes, filled with sand (right), water (middle) and a paper roll (left). The phantom was imaged using a compact prototype phase-contrast and dark-field small-animal CT scanner [15,21,22]. The x-ray source is a fixed anode tungsten-target MCBM 65B-50 W tube: (RTW, Neuenhagen, Germany) with a focal spot size of approximately 50 μm in diameter. The detector is a flat-panel C9312SK-06 Hamamatsu (Hamamatsu, Japan), GOS scintillator, with 50 μm pixel size. The distances between the source (period 15 μm, gold height 50 μm), the triangular phase grating (period 5 μm, nickel height 14 μm) and the absorption grating (period 7.5 μm, gold height 45 μm) are 300 and 145 mm respectively. The triangular phase grating is the same grating as used for visibility measurements with a polychromatic source. The images were acquired at 50 kVp source voltage with a visibility of approximately 15%. Images for 8 source grating phase steps [12] were acquired with an exposure time of 1 second per step. Transmission, differential phase-contrast and dark-field images, obtained subsequently with Fourier signal processing [10], are presented in Fig. 6. The complementarity of the three imaging modalities can be clearly seen e.g. water and paper show approximately the same transmission values. However, x-ray scattering on sub-pixel sized structures in paper is registered in the strong dark-field signal, whereas water does not show any scattering.
Fig. 6 (A) Transmission, (B) differential phase-contrast and (C) dark-field images of a phantom, consisting of plastic tubes with sand (right), water (center) and paper (left), acquired with a triangular phase grating.
This is a proof-of-principle result, showing that imaging with a Talbot-Lau interferometer with triangular-shaped phase grating is feasible.
5. Conclusion
In conclusion, this proof-of-principle study has shown that triangular-shaped phase gratings can be used for x-ray phase-contrast and dark-field imaging with a Talbot-Lau interferometer instead of binary gratings. Indeed, the use of triangular-shaped gratings offers some advantages. First, propagation distances for peak visibilities for triangular gratings are shorter than for the binary ones. This implies that with triangular-shaped phase gratings a compact high energy x-ray phase-contrast imaging setup can be realized. Second, triangular gratings do not have the property that gratings with (2m + 1)π with m ∈ ℕ phase shift produce an interference pattern with double spatial frequency. Analyzer gratings with the same period as the phase grating yield visibilities high enough to be used for imaging. This relaxes the requirements for analyzer grating production. Table 2 gives example of the periods for the source (P0), the phase (P1) and the analyzer (P2) gratings, needed to realize a compact setup at 100 keV. In the setup, the distance from the source to the phase grating is set to 90 cm and the distance between the phase and the analyzer grating is 10 cm.
Table 2. Periods of the source (P0), the phase (P1) and the analyzer (P2) gratings, needed to realize a compact (total length 1 meter) setup at 100 keV.
Using triangular phase gratings, the requirements for the period of the analyzer and the source gratings can be significantly reduced. The needed aspect-ratio for a gold analyzer grating decreases significantly from 274 (transmission 0.1), needed for a binary π/2 phase grating, to 119 for a 5π/2 triangular grating. This means that a compact Talbot-Lau interferometer for high energies becomes much more feasible.
Triangular-shaped phase gratings have also some limitations. Plane gratings in a cone-beam geometry suffer from visibility drop towards the wafer edges. However, this can be solved by bending the gratings. Furthermore, the triangular-shaped structures must have the exact height, otherwise resulting in a different grating shape (see Fig. 3).
Future studies should focus on implementation of triangular-shaped gratings into high energy setups for e.g. biomedical applications. Other non-binary grating shapes may also offer some advantages for x-ray phase-contrast and dark-field imaging. Consequently, future studies should also consider the use of e.g. sinusoidal and parabolic phase gratings, even if their experimental realization is more challenging than the triangular shape.
Acknowledgments
We acknowledge Michael Sprung and Sergej Bondarenko from Petra III for the kind and productive support during the intensity carpet measurements. We acknowledge financial support through the DFG Cluster of Excellence: Munich-Centre for Advanced Photonics (MAP), the DFG Gottfried Wilhelm Leibniz program and the European Research Council (ERC, FP7, StG 240142). This work was carried out with the support of the Karlsruhe Nano Micro Facility (KNMF, www.kit.edu/knmf), a Helmholtz Research Infrastructure at Karlsruhe Institute of Technology (KIT). A.Y., A.M., T.B., M.S. and A.T. would like to thank the Technische Universität München Graduate School for the support of their studies. J.W. would like to thank the GSISH Graduate School for the support of his studies.
References and links
1. F. Pfeiffer, O. Bunk, C. David, M. Bech, G. L. Duc, A. Bravin, and P. Cloetens, “High-resolution brain tumor visualization using three-dimensional X-ray phase contrast tomography,” Phys. Med. Biol. 52, 6923–6930 (2007). [CrossRef]
2. M. Stampanoni, Z. Wang, T. Thüring, C. David, E. Roessl, M. Trippel, R. A. Kubik-Huch, G. Singer, M. K. Hohl, and N. Hauser, “The first analysis and clinical evaluation of native breast tissue using differential phase-contrast mammography,” Invest. Radiol. 46, 801–806 (2011). [CrossRef]
3. D. W. Parsons, K. Morgan, M. Donnelley, A. Fouras, J. Crosbie, I. Williams, R. C. Boucher, K. Uesugi, N. Yagi, and K. W. Siu, “High-resolution visualization of airspace structures in intact mice via synchrotron phase-contrast X-ray imaging (PCXI),” J. Anat. 213, 217–227 (2008). [CrossRef]
4. T. Donath, F. Pfeiffer, O. Bunk, C. Grünzweig, E. Hempel, S. Popescu, P. Vock, and C. David, “Toward clinical X-ray phase-contrast CT: Demonstration of enhanced soft-tissue contrast in human specimen,” Invest. Radiol. 45, 445–452 (2010).
5. A. Momose, W. Yashiro, S. Harasse, and H. Kuwabara, “Four-dimensional X-ray phase tomography with Talbot interferometry and white synchrotron radiation: dynamic observation of a living worm,” Opt. Express 19, 8423–8432 (2011). [CrossRef]
6. H. Wen, E. E. Bennett, M. M. Hegedus, and S. Rapacchi, “Fourier X-ray scattering radiography yields bone structural information,” Radiology 251, 910–918 (2009). [CrossRef]
7. S. Schleede, F. G. Meinel, M. Bech, J. Herzen, K. Achterhold, G. Potdevin, A. Malecki, S. Adam-Neumair, S. F. Thieme, F. Bamberg, K. Nikolaou, A. Bohla, A. Yildirim, R. Loewen, M. Gifford, R. Ruth, O. Eickelberg, M. Reiser, and F. Pfeiffer, “Emphysema diagnosis using X-ray dark-field imaging at a laser-driven compact synchrotron light source,” Proc. Natl. Acad. Sci. U. S. A. 109, 17880–17885 (2012). [CrossRef]
8. A. Velroyen, M. Bech, A. Malecki, A. Tapfer, A. Yaroshenko, M. Ingrisch, C. C. Cyran, S. D. Auweter, K. Nikolaou, M. Reiser, and F. Pfeiffer, “Microbubbles as a scattering contrast agent for grating-based X-ray dark-field imaging,” Phys. Med. Biol. 58, N37 (2013). [CrossRef]
9. A. Momose, “Phase-sensitive imaging and phase tomography using X-ray interferometers,” Opt. Express 11, 2303–2314 (2003). [CrossRef]
10. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express 13, 6296–6304 (2005). [CrossRef]
11. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low-brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006). [CrossRef]
12. F. Pfeiffer, M. Bech, O. Bunk, P. Kraft, E. F. Eikenberry, C. Bronnimann, C. Grünzweig, and C. David, “Hard-X-ray dark-field imaging using a grating interferometer,” Nat. Mater. 7, 134–137 (2008). [CrossRef]
13. A. W. Lohmann and J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990). [CrossRef]
14. T. J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36, 4686–4691 (1997). [CrossRef]
15. A. Tapfer, M. Bech, A. Velroyen, J. Meiser, J. Mohr, M. Walter, J. Schulz, B. Pauwels, P. Bruyndonckx, X. Liu, A. Sasov, and F. Pfeiffer, “Experimental results from a preclinical X-ray phase-contrast CT scanner,” Proc. Natl. Acad. Sci. U. S. A. 109, 15691–15696 (2012). [CrossRef]
16. T. Thüring, R. Guggenberger, H. Alkadhi, J. Hodler, M. Vich, Z. Wang, C. David, and M. Stampanoni, “Human hand radiography using X-ray differential phase contrast combined with dark-field imaging,” Skeletal Radiol. 42, 827–835 (2013). [CrossRef]
17. G.-H. Chen, N. Bevins, J. Zambelli, and Z. Qi, “Small-angle scattering computed tomography (SAS-CT) using a Talbot-Lau interferometer and a rotating anode X-ray tube: theory and experiments,” Opt. Express 18, 12960–12970 (2010). [CrossRef]
18. V. Arrizón and E. López-Olazagasti, “Binary phase grating for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995). [CrossRef]
19. J. Goodman, Introduction to Fourier Optics, 3rd ed. (McGraw-Hill, 2005).
20. J. H. Siewerdsen, A. M. Waese, D. J. Moseley, S. Richard, and D. A. Jaffray, “Spektr: A computational tool for X-ray spectral analysis and imaging system optimization,” Med. Phys. 31, 3057–3067 (2004). [CrossRef]
21. A. Tapfer, M. Bech, B. Pauwels, X. Liu, P. Bruyndonckx, A. Sasov, J. Kenntner, J. Mohr, M. Walter, J. Schulz, and F. Pfeiffer, “Development of a prototype gantry system for preclinical X-ray phase-contrast computed tomography,” Med. Phys. 38, 5910–5915 (2011). [CrossRef]
22. A. Yaroshenko, F. G. Meinel, M. Bech, A. Tapfer, A. Velroyen, S. Schleede, S. Auweter, A. Bohla, A. O. Yildirim, K. Nikolaou, F. Bamberg, O. Eickelberg, M. F. Reiser, and F. Pfeiffer, “Pulmonary emphysema diagnosis with a preclinical small-animal X-ray dark-field scatter-contrast scanner,” Radiology 269, 427–433 (2013).
