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Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 8 — Apr. 22, 2013
  • pp: 9308–9314
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X-ray phase imaging with a laboratory source using selective reflection from a mirror

Daniele Pelliccia and David M. Paganin  »View Author Affiliations


Optics Express, Vol. 21, Issue 8, pp. 9308-9314 (2013)
http://dx.doi.org/10.1364/OE.21.009308


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Abstract

A novel approach for hard x-ray phase contrast imaging with a laboratory source is reported. The technique is based on total external reflection from the edge of a mirror, aligned to intercept only half of the incident beam. The mirror edge thus produces two beams. The refraction x-rays undergo when interacting with a sample placed before the mirror, causes relative intensity variations between direct and reflected beams. Quantitative phase contrast and pure absorption imaging are demonstrated using this method.

© 2013 OSA

1. Introduction

Many imaging methods have been developed to date which allow quantitative phase contrast imaging. Some rely on free space propagation to render visible as intensity variations the transverse optical phase gradients imparted on a beam upon traversing a sample, with other techniques relying on the use of optical elements to analyze the beam after the sample.

Free space propagation methods are sensitive to the Laplacian of the phase shift (see e.g [1

1. D. M. Paganin, Coherent X-Ray Optics (Oxford University, New York, 2006).

].), and whilst the absence of any optical elements between the sample and the detector makes them very simple to implement, they always produce a mixed absorption/phase image. Their main advantage is that the second transverse derivative of the optical phase changes very quickly at the interface between different materials and this greatly enhances the edge contrast. On the other hand quantitative phase imaging may not be straightforward even though several strategies have been devised in the context of the Transport-of-Intensity equation [2

2. K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996). [CrossRef] [PubMed]

4

4. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002). [CrossRef] [PubMed]

].

When analyzer optics are inserted between the sample and the detector, it usually becomes possible to discriminate between an absorption image and first order differential phase contrast. Two widely used techniques are Diffraction Enhanced Imaging (DEI) [5

5. V. A. Somenkov, A. K. Tkalich, and S. Shil’stein, “Refraction contrast in X-ray microscopy,” Sov. Phys. Tech. Phys. 3, 1309–1311 (1991).

7

7. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]

] – using a crystal – and the grating interferometer (GI) [8

8. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]

,9

9. 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(16), 6296–6304 (2005). [CrossRef] [PubMed]

], which makes use (in its simplest form) of a pair of gratings. Both techniques are sensitive to the refraction shift that an x-ray beam undergoes when interacting with a sample.

A variation of the analyzer-based methods has been introduced by Olivo and associates [10

10. A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001). [CrossRef] [PubMed]

]. In this case the measurement of the refraction shift is accomplished by enhancing the detection sensitivity by aligning an absorbing mask to partially cover the active surface of a given row of pixels. In this way the effective angular aperture of the pixel is decreased. The refraction shift is then detected as an intensity change in any given row of pixels when the sample is scanned across the x-ray beam. This method has been demonstrated with a synchrotron source [10

10. A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001). [CrossRef] [PubMed]

], laboratory source [11

11. A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50(12), 1765–1769 (2011). [CrossRef] [PubMed]

] and applied to multi-keV x-ray imaging [12

12. A. Olivo, P. C. Diemoz, and A. Bravin, “Amplification of the phase contrast signal at very high x-ray energies,” Opt. Lett. 37(5), 915–917 (2012). [CrossRef] [PubMed]

]. Recently a quantitative method based on this principle has been reported [13

13. P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013). [CrossRef] [PubMed]

].

In this paper we present an alternative technique based on a similar principle. We introduce a mirror with a sharp edge between the sample and the detector. The mirror edge is aligned to partially cover the active surface of a given row of pixels. In this way the presence of the edge decreases the angular aperture of such row – thus enhancing the sensitivity – and the mirror plays the same role of the absorbing mask used in [10

10. A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001). [CrossRef] [PubMed]

13

13. P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013). [CrossRef] [PubMed]

]. On the other hand the mirror is not actually blocking part of the incoming beam, as is the case when an absorbing mask is used. The reflected beam contains complementary information that can be recorded on another row of pixels. Therefore two images are recorded at once, by scanning the sample in the direction perpendicular to the rows of pixels, in the direct and reflected beam respectively. Importantly, the contrast variations displayed by the two beams are opposite, thus the two images contain complementary information that can be used to retrieve a quantitative differential phase contrast map of the sample.

This paper is organized as follows: in Section 2 we will introduce the theoretical model and explain the measurement principle underpinning our method for laboratory-source quantitative x-ray phase imaging using selective reflection from a mirror. In Section 3, we report on the experimental results obtained using a rotating anode hard x-ray source. The measurement of a known nylon fiber is used to calibrate the setup. The calibration is used for quantitative phase imaging of an unknown sample. The results are discussed in Section 4, and the conclusions are drawn in Section 5.

2. Theoretical model

Let us consider the schematic diagram in Fig. 1
Fig. 1 Schematic drawing of the experimental setup for x-ray phase imaging with a laboratory source using selective reflection from a mirror (not to scale).
. For clarity, the beam at the detector position is drawn smaller than the pixel, while in reality the beam size extends over several pixels. For the model described below only the pixel rows partially covered by the mirror shadow (in both direct and reflected beam) are actually involved in the data analysis. When the sample is scanned across the beam, the refraction shift can be detected as intensity variations Id(x) and Ir(x) on the pixels intercepting the edge of the direct and reflected beam respectively. Therefore, for any given sample-detector distance, the sensitivity to the refraction shift is increased by reducing the effective pixel size and therefore a tradeoff is found for the optimal sensitivity-to-intensity ratio.

On the other hand, by measuring a reference mono-material sample, the value of s can be obtained by the direct measure of the projected thickness, without having to scan the mirror across. This is the approach used here, since our reference sample was composed of a single material. In this case the refractive index n (and hence μ) is constant. Therefore the projected thickness t(x) can be quantitatively determinedy by the sole absorption measurement using Eq. (4a). Moreover the differential phase becomes proportional to the first derivative of the projected thickness, via the real part δ of the refractive index:
Φx=kδtx.
(5)
Therefore by obtaining t(x) from Eq. (4a), the value of s can be determined from Eq. (4b).

3. Experimental phase contrast results

Data were taken at the x-ray laboratory located at the School of Physics, Monash University, Australia. The x-ray source was a rotating anode (Rigaku FrE + Superbright) with Cu target and accelerating voltage set to 45 kV. A pair of parabolic multilayer mirrors (AXO Dresden) was placed after the source to collimate the beam and also provide an effective monochromatization at a wavelength λ = 0.154 nm. The use of the collimating optics is crucial not only by reducing the divergence, but also by strongly attenuating the high energy component of the beam which would increase the measurement background. No other monochromator was used in the setup, to preserve flux. Two sets of slits were placed after the mirror to define the beam size and reduce the scattering background. Their distance from the mirrors was 520 mm and 970 mm respectively. The FWHM value for the beam divergence, measured after the slits, was α = (0.18 ± 0.01) mrad.

The relevant experimental quantities are specified in Fig. 1. We already introduced z1 as the distance from the sample to the mirror edge. Moreover z2 is the distance between the mirror edge and the detector plane. In our case z1 = 100 mm and z2 = 210 mm. A Si wafer was used as the x-ray mirror. The wafer was carefully cleaved to obtain a precise edge. The total length of the mirror was 80 mm, sufficient to intercept the whole x-ray beam size at the chosen angle of incidence. In our case we set ϕ = 3.3 mrad, slightly less than the critical angle for Si at the given energy: ϕc = 3.9 mrad.

The lateral position of the mirror was carefully aligned in order to halve the intensity in a line of pixels located roughly at the beam center. This value, after normalization to the counts measured in the absence of the sample, corresponds to the intensity Id(ξ0). The same procedure was repeated for the reflected intensity Ir(ξ0), in this case acting on the mirror angle ϕ. The rotation axis of the mirror was set on the mirror edge, to be able to adjust ϕ without changing the lateral position of the mirror. The detector was a Medipix with silicon chip, thickness of the chip 500 μm and pixel size p = 55 μm.

The reference object was a nylon fiber, 0.52 mm diameter, placed immediately after the second slit. The fiber was scanned across the beam with step of 11 μm and exposure time of 10 s for each image. The results are reported in Fig. 2
Fig. 2 Measurement of the nylon fiber. (a) Normalized intensity Id(x). (b) Normalized intensity Ir(x). (c) Retrieved projected thickness (d) Refraction image. (e) Comparison of the projected thickness profile (open diamonds) with a theoretical curve (solid line).
. Figures 2(a) and 2(b) show the 2D maps obtained by merging the 1D intensity profiles Id(x) and Ir(x) respectively, while scanning the sample across the beam. The obtained images have been subsequently rescaled to have equal resolution in both directions. In the maps x is the horizontal coordinate, corresponding to the scan direction. They clearly show inverted contrast, as expected.

Figure 2(c) is the projected thickness obtained by Eq. (4a) assuming δ = 4.016 × 10−6 and β = 7.28 × 10−9 as the real and imaginary part of the index of refraction of nylon at the given wavelength. Consequently from Eq. (4b) the refraction image can be obtained, shown in Fig. 2(d). By comparison the value of the slope is estimated to be s = 2.29 mm−1. The accuracy of the measure of the projected thickness is demonstrated by the plot in Fig. 2(e). Here the profile along the central pixel of the projected thickness map of Fig. 2(c) is compared with a theoretical profile calculated for the nylon fiber with good agreement between the two profiles.

The measured value of the slope has been used for a subsequent measurement of an unknown sample, in this case a fly (Little House Fly, Fannia Canicularis). The step size was set to 11 μm and exposure time to 10 s. In Fig. 3(a)
Fig. 3 (a) Absorption, (b) refraction and (c) phase images of a fly.
the absorption image, obtained with Eq. (4a), is shown. The periodic pattern of vertical stripes in the background is due to small mechanical deflections caused by temperature variations. The refraction image is shown in Fig. 3(b). In this case the intensity oscillation is not visible, being normalized out by the difference in Eq. (4b). The horizontal strokes mostly visible in the refraction image are due to a change in the count rate of certain pixels during the scan. Finally, Fig. 3(c) shows the phase map of the fly, obtained by integration from the differential phase contrast map. Numerical integration typically amplifies the input noise in the resulting image. To reduce such noise, the value of the phase has been obtained by applying the Fourier derivative theorem (see e.g [14

14. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, Boston, 2000).

].) implemented via fast Fourier transform (FFT):
Φ(x)=F1[1iqF[Φx]]
(6)
In Eq. (6) F and F1 represent the forward and inverse Fourier transform respectively, q is the conjugate variable to x and i is the imaginary unit. To avoid the numerical instability in q = 0, the argument of the inverse Fourier transform in Eq. (6) has been replaced with the FFT of the phase obtained by direct integration for q = 0.

4. Discussion

The most important advantage in using a mirror edge instead of an absorbing mask is the possibility of acquiring both direct and reflected beam information in parallel, thus allowing for quantitative phase retrieval in a single scan. On the other hand using the mirror implies introducing the additional distance z2, between the mirror edge and the detector. Such a distance primarily depends on the detector resolution, as it must be chosen to distinctly resolve direct and reflected beam. Thus the divergence of the primary beam, namely the position and size of the slit with respect to the mirror edge, limits the sensitivity of the setup, i.e. the ability to distinguish small changes of the refraction angle. In Fig. 3(c), the wing of the fly is at the limit of detection. From it we can estimate the sensitivity to be about 5 μrad.

Denoting by α the beam divergence, the penumbral blurring of the mirror edge at the detector position is ΔxD = αz2. In this case the experiment must be designed in order to keep the width of the blurred mirror edge well within the size of a single pixel, not to degrade the resolution. In our experimental situation ΔxD≈40 μm, well below the pixel size.

Finally it is worth remarking that in our model we neglect the effects of Fresnel diffraction, i.e. non-geometrical wave propagation effects. This seems to be the case of the measurements presented here, in which the spatial resolution is moderate. It is expected that free-space propagation must be included if higher resolution is needed. The formalism has been already developed in a rather general context and applied to the specific case of DEI [15

15. K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004). [CrossRef]

17

17. P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005). [CrossRef] [PubMed]

]. The application of the complete formalism to our system is currently under study and will be the subject of a future work.

5. Conclusions

We have demonstrated a novel technique to obtain quantitative x-ray phase contrast imaging. The phase sensitivity is obtained by detecting the refraction shift caused by the sample by inserting the edge of an x-ray mirror in the beam path, after the sample. The refraction causes complementary intensity variations in the direct and reflected beam respectively. This information is used to quantitatively retrieve the phase shift produced by an object. In the case of a mono-material sample, the method is able to reconstruct the projected thickness of the sample. The approximations used in the model have been explained and discussed.

Acknowledgment

The authors acknowledge funding by the Australian Research Council.

References and links

1.

D. M. Paganin, Coherent X-Ray Optics (Oxford University, New York, 2006).

2.

K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett. 77(14), 2961–2964 (1996). [CrossRef] [PubMed]

3.

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80(12), 2586–2589 (1998). [CrossRef]

4.

D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc. 206(1), 33–40 (2002). [CrossRef] [PubMed]

5.

V. A. Somenkov, A. K. Tkalich, and S. Shil’stein, “Refraction contrast in X-ray microscopy,” Sov. Phys. Tech. Phys. 3, 1309–1311 (1991).

6.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard X-rays,” Nature 373(6515), 595–598 (1995). [CrossRef]

7.

D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol. 42(11), 2015–2025 (1997). [CrossRef] [PubMed]

8.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]

9.

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(16), 6296–6304 (2005). [CrossRef] [PubMed]

10.

A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys. 28(8), 1610–1619 (2001). [CrossRef] [PubMed]

11.

A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt. 50(12), 1765–1769 (2011). [CrossRef] [PubMed]

12.

A. Olivo, P. C. Diemoz, and A. Bravin, “Amplification of the phase contrast signal at very high x-ray energies,” Opt. Lett. 37(5), 915–917 (2012). [CrossRef] [PubMed]

13.

P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express 21(1), 647–661 (2013). [CrossRef] [PubMed]

14.

R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, Boston, 2000).

15.

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys. 37(19), 2746–2750 (2004). [CrossRef]

16.

K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Kitchen, K. K. W. Siu, J. E. Gillam, K. Uesugi, N. Yagi, M. J. Morgan, and R. A. Lewis, “Unification of analyser-based and propagation-based X-ray phase-contrast imaging,” Nucl. Instrum. Methods Phys. Res. A 548(1-2), 163–168 (2005). [CrossRef]

17.

P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat. 12(2), 241–245 (2005). [CrossRef] [PubMed]

OCIS Codes
(100.5070) Image processing : Phase retrieval
(340.7440) X-ray optics : X-ray imaging
(340.7470) X-ray optics : X-ray mirrors

ToC Category:
X-ray Optics

History
Original Manuscript: February 13, 2013
Revised Manuscript: March 26, 2013
Manuscript Accepted: March 30, 2013
Published: April 8, 2013

Virtual Issues
April 30, 2013 Spotlight on Optics

Citation
Daniele Pelliccia and David M. Paganin, "X-ray phase imaging with a laboratory source using selective reflection from a mirror," Opt. Express 21, 9308-9314 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-8-9308


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References

  1. D. M. Paganin, Coherent X-Ray Optics (Oxford University, New York, 2006).
  2. K. A. Nugent, T. E. Gureyev, D. J. Cookson, D. Paganin, and Z. Barnea, “Quantitative phase imaging using hard X rays,” Phys. Rev. Lett.77(14), 2961–2964 (1996). [CrossRef] [PubMed]
  3. D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett.80(12), 2586–2589 (1998). [CrossRef]
  4. D. Paganin, S. C. Mayo, T. E. Gureyev, P. R. Miller, and S. W. Wilkins, “Simultaneous phase and amplitude extraction from a single defocused image of a homogeneous object,” J. Microsc.206(1), 33–40 (2002). [CrossRef] [PubMed]
  5. V. A. Somenkov, A. K. Tkalich, and S. Shil’stein, “Refraction contrast in X-ray microscopy,” Sov. Phys. Tech. Phys.3, 1309–1311 (1991).
  6. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, “Phase contrast imaging of weakly absorbing materials using hard X-rays,” Nature373(6515), 595–598 (1995). [CrossRef]
  7. D. Chapman, W. Thomlinson, R. E. Johnston, D. Washburn, E. Pisano, N. Gmür, Z. Zhong, R. Menk, F. Arfelli, and D. Sayers, “Diffraction enhanced x-ray imaging,” Phys. Med. Biol.42(11), 2015–2025 (1997). [CrossRef] [PubMed]
  8. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys.42(Part 2, No. 7B), L866–L868 (2003). [CrossRef]
  9. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, “X-ray phase imaging with a grating interferometer,” Opt. Express13(16), 6296–6304 (2005). [CrossRef] [PubMed]
  10. A. Olivo, F. Arfelli, G. Cantatore, R. Longo, R. H. Menk, S. Pani, M. Prest, P. Poropat, L. Rigon, G. Tromba, E. Vallazza, and E. Castelli, “An innovative digital imaging set-up allowing a low-dose approach to phase contrast applications in the medical field,” Med. Phys.28(8), 1610–1619 (2001). [CrossRef] [PubMed]
  11. A. Olivo, K. Ignatyev, P. R. T. Munro, and R. D. Speller, “Noninterferometric phase-contrast images obtained with incoherent x-ray sources,” Appl. Opt.50(12), 1765–1769 (2011). [CrossRef] [PubMed]
  12. A. Olivo, P. C. Diemoz, and A. Bravin, “Amplification of the phase contrast signal at very high x-ray energies,” Opt. Lett.37(5), 915–917 (2012). [CrossRef] [PubMed]
  13. P. R. T. Munro, L. Rigon, K. Ignatyev, F. C. M. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, “A quantitative, non-interferometric X-ray phase contrast imaging technique,” Opt. Express21(1), 647–661 (2013). [CrossRef] [PubMed]
  14. R. N. Bracewell, The Fourier Transform and its Applications, 3rd ed. (McGraw-Hill, Boston, 2000).
  15. K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Morgan, and R. A. Lewis, “Linear systems with slowly varying transfer functions and their application to x-ray phase-contrast imaging,” J. Phys. D Appl. Phys.37(19), 2746–2750 (2004). [CrossRef]
  16. K. M. Pavlov, T. E. Gureyev, D. Paganin, Y. I. Nesterets, M. J. Kitchen, K. K. W. Siu, J. E. Gillam, K. Uesugi, N. Yagi, M. J. Morgan, and R. A. Lewis, “Unification of analyser-based and propagation-based X-ray phase-contrast imaging,” Nucl. Instrum. Methods Phys. Res. A548(1-2), 163–168 (2005). [CrossRef]
  17. P. Coan, E. Pagot, S. Fiedler, P. Cloetens, J. Baruchel, and A. Bravin, “Phase-contrast X-ray imaging combining free space propagation and Bragg diffraction,” J. Synchrotron Radiat.12(2), 241–245 (2005). [CrossRef] [PubMed]

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