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| RAPID, SHORT PUBLICATIONS ON THE LATEST IN OPTICAL DISCOVERIES

  • Editor: Xi-Cheng Zhang
  • Vol. 39, Iss. 11 — Jun. 1, 2014
  • pp: 3332–3335
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Phase-contrast microscopy at high x-ray energy with a laboratory setup

Marco Endrizzi, Fabio A. Vittoria, Paul C. Diemoz, Rodolfo Lorenzo, Robert D. Speller, Ulrich H. Wagner, Christoph Rau, Ian K. Robinson, and Alessandro Olivo  »View Author Affiliations


Optics Letters, Vol. 39, Issue 11, pp. 3332-3335 (2014)
http://dx.doi.org/10.1364/OL.39.003332


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Abstract

We report on the design and realization of an x-ray imaging system for quantitative phase-contrast microscopy at high x-ray energy with laboratory-scale instrumentation. Phase and amplitude were separated quantitatively at x-ray energies up to 80 keV with micrometric spatial resolution. The accuracy of the results was tested against numerical simulations, and the spatial resolution was experimentally quantified by measuring a Siemens star phase object. This simple setup should find broad application in those areas of x-ray imaging where high energy and spatial resolution are simultaneously required and in those difficult cases where the sample contains materials with similar x-ray absorption.

© 2014 Optical Society of America

Hard x-ray radiation has a high penetration power in matter, thus enabling nondestructive inspection of the inner structure of samples. At the same time, this power is also a weakness because the contrast arising from differences in the absorption of radiation tends to vanish when partially transparent samples are illuminated. X-ray phase-contrast imaging (XPCi) [1

1. R. Fitzgerald, Phys. Today 53(7), 23 (2000). [CrossRef]

] overcomes this problem because it is sensitive to the phase shifts imparted to the x-ray wave when traversing the sample. Several methods [2

2. U. Bonse and M. Hart, Appl. Phys. Lett. 6, 155 (1965). [CrossRef]

15

15. P. R. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, Proc. Natl. Acad. Sci. USA 109, 13922 (2012). [CrossRef]

] have been developed for performing XPCi. Approaches have also been proposed for high-resolution XPCi, including free-space propagation, Zernike phase contrast, and grating-based methods [16

16. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, Opt. Express 11, 2289 (2003). [CrossRef]

22

22. J. Choi and Y.-S. Park, Appl. Phys. Express 5, 042503 (2012). [CrossRef]

]. Here we show the design, modelling, and realization of a laboratory system based on the edge-illumination [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, Med. Phys. 28, 1610 (2001). [CrossRef]

,15

15. P. R. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, Proc. Natl. Acad. Sci. USA 109, 13922 (2012). [CrossRef]

] principle and implemented through its area-imaging counterpart, sometimes referred to as the coded-aperture [23

23. A. Olivo and R. Speller, Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]

] method. These are noninterferometric methods that do not use the Talbot self-imaging effect or Moire patterns [24

24. P. R. Munro, L. Rigon, K. Ignatyev, F. C. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, Opt. Express 21, 647 (2013). [CrossRef]

]; contrast is generated by fine angular selection, in analogy with analyzer-based imaging which uses the rocking curve of a crystal [25

25. P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]

]. We aim to push the current resolution limits of a few tens of micrometers [26

26. M. Marenzana, C. K. Hagen, P. D. N. Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, Phys. Med. Biol. 57, 8173 (2012). [CrossRef]

] toward microscopic resolution while still performing quantitative phase-contrast imaging at high x-ray energies. A magnified projection geometry was used in order to achieve high spatial resolution while being able to efficiently detect the radiation. The ability of the method to be quantitative and its spatial resolution are experimentally demonstrated and numerically simulated while the potential of the technique in terms of image quality is illustrated through images of a complex wood sample.

The experimental setup consists of a microfocus transmission target x-ray tube, two apertured masks, and a detector (Fig. 1). The tungsten target x-ray tube is operated at 80 kVp and has a focus of 3.5 μm. The first mask M1 is placed at 13 cm from the focus and the source-to-detector distance is zsd=130cm. The sample is positioned at about 14 cm from the focus, with a geometrical magnification factor G=zsd/(zsdzod) of about 9. The first mask M1 has a pitch p1=20μm and apertures a1=3μm, while p2=98μm and a2=29μm are used for the second mask M2. They are made of gold on a graphite (M2) and silicon (M1) substrate and were manufactured by Creatv Microtech (Potomac, Maryland) and Microworks GmbH (Karlsruhe, Germany), respectively. The detector is a passive pixel complementary metal-oxide semiconductor flat panel sensor (Hamamatsu Photonics C9732DK), with pixel pitch p3=50μm. The signal degradation due to cross talk between neighboring pixels is limited by the use of a line-skipped mask design [27

27. K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, Rev. Sci. Instrum. 82, 073702 (2011). [CrossRef]

]. The main limiting factor for the field of view is a1, which defines the angular acceptance of the transmitted radiation θma1/tM1, where tM1 is the thickness of the mask, about 200 μm in this case. For our setup the field of view was 2mm×5mm in the x and y directions, respectively. Bent masks could be used for obtaining a larger field of view with an even more compact setup [28

28. T. Thuering, P. Modregger, T. Grund, J. Kenntner, C. David, and M. Stampanoni, Appl. Phys. Lett. 99, 041111 (2011). [CrossRef]

]. Provided that the angular spread of the beam is limited to θl<8p3/zod340μrad, the pixels can be considered independent from one another, which allows us to model the image formation for a single aperture only.

Fig. 1. Experimental setup: the x-ray beam, generated by a microfocus x-ray source S, is shaped by the first mask M1, traverses the sample, and is analyzed by the second mask M2 before being recorded by the pixels P of a digital detector.

Fig. 2. Evaluation of Eq. (3) as a function of the period of a sinusoidal phase object.

The direct comparison between the measured and the theoretical signal from a star pattern test object is carried out by means of a numerical simulation of the whole imaging system [31

31. F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, Appl. Opt. 52, 6940 (2013).

]. Each monochromatic component was weighted according to the x-ray source spectrum and the detector response as function of energy [32

32. M. Endrizzi, P. Oliva, B. Golosio, and P. Delogu, Nucl. Instrum. Methods Phys. Res. A 703, 26 (2013). [CrossRef]

]. In order to perform quantitative retrieval of the absorption and refraction of the sample, we followed the approach based on the knowledge of the translation curve (TC) of the system [25

25. P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]

]. The TC describes how the detected intensity changes as a function of the displacement Δξ between the two masks. Images were acquired by setting the displacement between M1 and M2 as equal to Δξ1=Δξ2=2μm. The resulting images can be expressed as [25

25. P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]

]
IL,R=exp[Oμ¯dz]ITC(Δξ1,2zodϕ¯x/G),
(4)
where μ¯ denotes the linear attenuation coefficient of the sample evaluated at the effective energy of the system [33

33. P. R. T. Munro and A. Olivo, Phys. Rev. A 87, 053838 (2013). [CrossRef]

], zod is the sample-to-detector distance, ϕ¯x=ϕ¯/x is the partial derivative of ϕ at the effective energy of the system, and ITC is the value of the translation curve. We also introduce the following images, IΣ=IL+IR and IΔ=ILIR [25

25. P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]

]. It is worth noting that the quantity IΔ/IΣ depends only upon ϕ¯x and the operational parameters of the setup [25

25. P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]

]; we will refer to this quantity as the differential phase image. Moreover, IΣ depends only on the absorption properties of the sample. The following samples were imaged experimentally: a star pattern, a cylindrical two-material sample, and a bamboo wood sample. The star pattern was etched into a few hundred micrometers of crystalline silicon and provides a good approximation of a pure phase object at the x-ray energies we used. The IL image was acquired by using 100 sample scan steps of 0.2 μm, with an exposure time of 70 s per step. The cylindrical sample was composed of two materials, 220 μm diameter of boron with a 14 μm diameter tungsten core, and it was scanned with 16 steps of 1.2 μm, 100 s exposure time each. The bamboo wood was sliced to a thickness of about 500 μm; IL and IR were acquired by scanning the sample with 44 steps of 0.5 μm, with an exposure time of 100 s per step. For comparison, an image of the bamboo sample was also acquired in free-space propagation XPCi, with monochromatic synchrotron radiation and a high-resolution detector. The image was acquired at the I13 beamline of the diamond light source by using 9.7 keV x-rays and a detector featuring 0.8 μm size pixels. The sample was placed at about 200 m from the undulator source and the detector 30 cm downstream of the sample.

The spatial resolution of the laboratory-based system is evaluated by means of the star pattern. The image of the star is shown in Fig. 3(a) from which the intensity profile shown in Fig. 3(b) was extracted (dashed line). The experimental data are in good agreement with the numerical simulation, shown as a solid line in the plot in Fig. 3(b). As the absorption is negligible in this case, we note from Eq. (4) that IL depends only on ϕ¯x and on the operational parameters of the setup. The changing visibility of the phase-contrast features around the star pattern are a property of the geometry of the setup. Let us consider the phase shift of the star pattern as a replica at various angles φ of a step-like feature; the detected signal is modulated by a cosine function, S2cos(φ)uϕ(u,v), where (u,v) are the (x,y) coordinates rotated by the angle φ. The smallest resolved separation between a dark and a bright fringe is 1.5 μm; the sample used hardly allowed testing finer resolutions. The quantitativeness of the method is demonstrated by retrieving differential phase and amplitude of the boron fiber with a tungsten core and comparing the extracted (dashed line) values to the expected ones (solid line) in Fig. 4(a). The theoretical ϕ¯x(x) was calculated assuming a cylindrical shape of the boron and tungsten fiber. The analytical profile was convolved with a Gaussian function, representing the spatial resolution of the system, of width 1.5 μm. The refractive indices of the materials were calculated by using the Xraylib library [34

34. T. Schoonjans, A. Brunetti, B. Golosio, M. S. del Rio, V. A. Solé, C. Ferrero, and L. Vincze, Spectrochim. Acta B 66, 776 (2011). [CrossRef]

]. The profile was then obtained according to IΔ/IΣ=(zodϕ¯x/G)/(ITC(Δξ)/ITC(Δξ)) [25

25. P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]

] by numerically differentiating ϕ¯. The amplitude and differential phase images are also shown in Figs. 4(c) and 4(b), respectively. The horizontal line noise that can be noticed in Figs. 3(a), 4(b), and 4(c) is a result of the brick-like structure used to manufacture M1. The laboratory images of the bamboo sample are shown in Fig. 5, where the contributions from phase [Fig. 5(a)] and absorption [Fig. 5(b)] are separated. For a better appreciation of the image quality, the image acquired under nearly ideal conditions (with monochromatic synchrotron radiation and high-resolution detector) is also shown for comparison in Fig. 5(d). We note that the higher contrast of this image is expected due to the much lower x-ray energy (9.7 keV). The detail shown in the picture is similar to but not exactly the same as in the laboratory image, due to difficulty in aligning the sample to the same region of interest.

Fig. 3. Star pattern phase object. (a) Differential phase-contrast experimental image (scale bar is 50 μm). (b) Simulation (solid line) compared against the experimental profile (dashed line) extracted from the highlighted region in the center of the star.
Fig. 4. Boron fiber with a tungsten core sample. (a) Comparison of the experimental intensity profile [highlighted region in (b) sector] against the theoretically expected one for a single fiber. (b) Differential phase and (c) amplitude images of two fibers at an angle.
Fig. 5. Image of the bamboo sample. (a) Differential phase and (b) amplitude images, quantitatively separated by using the laboratory setup (scale bar 100 μm). (c) Zoom of the highlighted 120μm×120μm regions of (a) and (b), top and bottom, respectively. (d) Free-space propagation image acquired with a high-resolution detector and monochromatic synchrotron radiation.

In summary, we described a hard x-ray, laboratory-based, phase-contrast microscope, obtained through the appropriate design of a high-magnification edge-illumination XPCi system. The experimental setup was built using commercially available, off-the-shelf instrumentation, and it is currently producing high-quality amplitude and differential phase-contrast images. Micrometric spatial resolution was experimentally measured, theoretically described, and numerically simulated by using a star pattern test object, with good agreement between simulation and experiment. By acquiring two separate images at different configurations, amplitude and differential phase can be quantitatively retrieved. The quantitativeness of the method was tested against theory on a two-material sample of known shape and composition. The instrument uses broadband and hard radiation, merging high resolution with the high penetration power of x-rays in a laboratory setup.

The authors are grateful to Nikon Corporation and X-Tek, part of Nikon Metrology, for their support with the experimental setup. This work was supported by the UK Engineering and Physical Sciences Research Council (Grants EP/I022562/1 and EP/I021884/1). M. Endrizzi and P. C. Diemoz are supported by Marie Curie Career Integration Grants within the Seventh Framework Programme of the European Union, PCIG12-GA-2012-334056 and PCIG12-GA-2012-333990.

References

1.

R. Fitzgerald, Phys. Today 53(7), 23 (2000). [CrossRef]

2.

U. Bonse and M. Hart, Appl. Phys. Lett. 6, 155 (1965). [CrossRef]

3.

K. Goetz, E. Foerster, P. Zaumseil, M. P. Kalashnikov, I. A. Mikhailov, G. V. Sklizkov, and S. I. Fedotov, Kvantovaia Elektronika Moscow 6, 1037 (1979).

4.

E. Foerster, K. Goetz, and P. Zaumseil, Kristall und Technik 15, 937 (1980). [CrossRef]

5.

V. N. Ingal and E. A. Beliaevskaya, J. Phys. D 28, 2314 (1995). [CrossRef]

6.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995). [CrossRef]

7.

T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature 373, 595 (1995). [CrossRef]

8.

S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature 384, 335 (1996). [CrossRef]

9.

A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996). [CrossRef]

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, Med. Phys. 28, 1610 (2001). [CrossRef]

11.

C. David, B. Nohammer, H. H. Solak, and E. Ziegler, Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]

12.

T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005). [CrossRef]

13.

F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006). [CrossRef]

14.

Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, Phys. Rev. A 79, 013815 (2009). [CrossRef]

15.

P. R. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, Proc. Natl. Acad. Sci. USA 109, 13922 (2012). [CrossRef]

16.

S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, Opt. Express 11, 2289 (2003). [CrossRef]

17.

M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, Appl. Phys. Lett. 90, 224101 (2007). [CrossRef]

18.

Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, Appl. Phys. Express 1, 117002 (2008). [CrossRef]

19.

W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, Phys. Rev. Lett. 103, 180801 (2009). [CrossRef]

20.

C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, Nat. Phys. 6, 883 (2010). [CrossRef]

21.

H. Kuwabara, W. Yashiro, S. Harasse, H. Mizutani, and A. Momose, Appl. Phys. Express 4, 062502 (2011). [CrossRef]

22.

J. Choi and Y.-S. Park, Appl. Phys. Express 5, 042503 (2012). [CrossRef]

23.

A. Olivo and R. Speller, Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]

24.

P. R. Munro, L. Rigon, K. Ignatyev, F. C. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, Opt. Express 21, 647 (2013). [CrossRef]

25.

P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]

26.

M. Marenzana, C. K. Hagen, P. D. N. Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, Phys. Med. Biol. 57, 8173 (2012). [CrossRef]

27.

K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, Rev. Sci. Instrum. 82, 073702 (2011). [CrossRef]

28.

T. Thuering, P. Modregger, T. Grund, J. Kenntner, C. David, and M. Stampanoni, Appl. Phys. Lett. 99, 041111 (2011). [CrossRef]

29.

J. P. Guigay, C. R. Acad. Sc. Paris 284B, 193 (1977).

30.

J. P. Guigay, Optik 49, 121 (1977).

31.

F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, Appl. Opt. 52, 6940 (2013).

32.

M. Endrizzi, P. Oliva, B. Golosio, and P. Delogu, Nucl. Instrum. Methods Phys. Res. A 703, 26 (2013). [CrossRef]

33.

P. R. T. Munro and A. Olivo, Phys. Rev. A 87, 053838 (2013). [CrossRef]

34.

T. Schoonjans, A. Brunetti, B. Golosio, M. S. del Rio, V. A. Solé, C. Ferrero, and L. Vincze, Spectrochim. Acta B 66, 776 (2011). [CrossRef]

OCIS Codes
(110.7440) Imaging systems : X-ray imaging
(180.7460) Microscopy : X-ray microscopy

ToC Category:
Microscopy

History
Original Manuscript: February 14, 2014
Revised Manuscript: April 10, 2014
Manuscript Accepted: April 24, 2014
Published: May 30, 2014

Virtual Issues
Vol. 9, Iss. 8 Virtual Journal for Biomedical Optics

Citation
Marco Endrizzi, Fabio A. Vittoria, Paul C. Diemoz, Rodolfo Lorenzo, Robert D. Speller, Ulrich H. Wagner, Christoph Rau, Ian K. Robinson, and Alessandro Olivo, "Phase-contrast microscopy at high x-ray energy with a laboratory setup," Opt. Lett. 39, 3332-3335 (2014)
http://www.opticsinfobase.org/ol/abstract.cfm?URI=ol-39-11-3332


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References

  1. R. Fitzgerald, Phys. Today 53(7), 23 (2000). [CrossRef]
  2. U. Bonse and M. Hart, Appl. Phys. Lett. 6, 155 (1965). [CrossRef]
  3. K. Goetz, E. Foerster, P. Zaumseil, M. P. Kalashnikov, I. A. Mikhailov, G. V. Sklizkov, and S. I. Fedotov, Kvantovaia Elektronika Moscow 6, 1037 (1979).
  4. E. Foerster, K. Goetz, and P. Zaumseil, Kristall und Technik 15, 937 (1980). [CrossRef]
  5. V. N. Ingal and E. A. Beliaevskaya, J. Phys. D 28, 2314 (1995). [CrossRef]
  6. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995). [CrossRef]
  7. T. J. Davis, D. Gao, T. E. Gureyev, A. W. Stevenson, and S. W. Wilkins, Nature 373, 595 (1995). [CrossRef]
  8. S. W. Wilkins, T. E. Gureyev, D. Gao, A. Pogany, and A. W. Stevenson, Nature 384, 335 (1996). [CrossRef]
  9. A. Momose, T. Takeda, Y. Itai, and K. Hirano, Nat. Med. 2, 473 (1996). [CrossRef]
  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, Med. Phys. 28, 1610 (2001). [CrossRef]
  11. C. David, B. Nohammer, H. H. Solak, and E. Ziegler, Appl. Phys. Lett. 81, 3287 (2002). [CrossRef]
  12. T. Weitkamp, A. Diaz, C. David, F. Pfeiffer, M. Stampanoni, P. Cloetens, and E. Ziegler, Opt. Express 13, 6296 (2005). [CrossRef]
  13. F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, Nat. Phys. 2, 258 (2006). [CrossRef]
  14. Z.-F. Huang, K.-J. Kang, L. Zhang, Z.-Q. Chen, F. Ding, Z.-T. Wang, and Q.-G. Fang, Phys. Rev. A 79, 013815 (2009). [CrossRef]
  15. P. R. Munro, K. Ignatyev, R. D. Speller, and A. Olivo, Proc. Natl. Acad. Sci. USA 109, 13922 (2012). [CrossRef]
  16. S. Mayo, T. Davis, T. Gureyev, P. Miller, D. Paganin, A. Pogany, A. Stevenson, and S. Wilkins, Opt. Express 11, 2289 (2003). [CrossRef]
  17. M. Engelhardt, J. Baumann, M. Schuster, C. Kottler, F. Pfeiffer, O. Bunk, and C. David, Appl. Phys. Lett. 90, 224101 (2007). [CrossRef]
  18. Y. Takeda, W. Yashiro, T. Hattori, A. Takeuchi, Y. Suzuki, and A. Momose, Appl. Phys. Express 1, 117002 (2008). [CrossRef]
  19. W. Yashiro, Y. Takeda, A. Takeuchi, Y. Suzuki, and A. Momose, Phys. Rev. Lett. 103, 180801 (2009). [CrossRef]
  20. C. Holzner, M. Feser, S. Vogt, B. Hornberger, S. B. Baines, and C. Jacobsen, Nat. Phys. 6, 883 (2010). [CrossRef]
  21. H. Kuwabara, W. Yashiro, S. Harasse, H. Mizutani, and A. Momose, Appl. Phys. Express 4, 062502 (2011). [CrossRef]
  22. J. Choi and Y.-S. Park, Appl. Phys. Express 5, 042503 (2012). [CrossRef]
  23. A. Olivo and R. Speller, Appl. Phys. Lett. 91, 074106 (2007). [CrossRef]
  24. P. R. Munro, L. Rigon, K. Ignatyev, F. C. Lopez, D. Dreossi, R. D. Speller, and A. Olivo, Opt. Express 21, 647 (2013). [CrossRef]
  25. P. R. Munro, C. K. Hagen, M. B. Szafraniec, and A. Olivo, Opt. Express 21, 11187 (2013). [CrossRef]
  26. M. Marenzana, C. K. Hagen, P. D. N. Borges, M. Endrizzi, M. B. Szafraniec, K. Ignatyev, and A. Olivo, Phys. Med. Biol. 57, 8173 (2012). [CrossRef]
  27. K. Ignatyev, P. R. T. Munro, R. D. Speller, and A. Olivo, Rev. Sci. Instrum. 82, 073702 (2011). [CrossRef]
  28. T. Thuering, P. Modregger, T. Grund, J. Kenntner, C. David, and M. Stampanoni, Appl. Phys. Lett. 99, 041111 (2011). [CrossRef]
  29. J. P. Guigay, C. R. Acad. Sc. Paris 284B, 193 (1977).
  30. J. P. Guigay, Optik 49, 121 (1977).
  31. F. A. Vittoria, P. C. Diemoz, M. Endrizzi, L. Rigon, F. C. Lopez, D. Dreossi, P. R. T. Munro, and A. Olivo, Appl. Opt. 52, 6940 (2013).
  32. M. Endrizzi, P. Oliva, B. Golosio, and P. Delogu, Nucl. Instrum. Methods Phys. Res. A 703, 26 (2013). [CrossRef]
  33. P. R. T. Munro and A. Olivo, Phys. Rev. A 87, 053838 (2013). [CrossRef]
  34. T. Schoonjans, A. Brunetti, B. Golosio, M. S. del Rio, V. A. Solé, C. Ferrero, and L. Vincze, Spectrochim. Acta B 66, 776 (2011). [CrossRef]

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