With the advent of Talbot-Lau interferometers for x-ray phase-contrast imaging, oblique and grazing incidence configurations are now used in the pursuit of sub-micron grating periods and high sensitivity. Here we address the question whether interferometers having oblique incident beams behave in the same way as the well-understood normal incidence ones, particularly when the grating planes are non-parallel. We derive the normal incidence equivalence of oblique incidence geometries from wave propagation modeling. Based on the theory, we propose a practical method to correct for non-parallelism of the grating planes, and demonstrate its effectiveness with a polychromatic hard x-ray reflective interferometer.

© 2011 OSA

1. D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, “An interferometer for atoms,” Phys. Rev. Lett.66(21), 2693–2696 (1991). [CrossRef] [PubMed]
2. J. F. Clauser and S. F. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A49(4), R2213–R2216 (1994). [CrossRef] [PubMed]
3. P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett.22(14), 1059–1061 (1997). [CrossRef] [PubMed]
4. C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett.81(17), 3287–3289 (2002). [CrossRef]
5. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jap. J. Appl. Phys. Part 2-Letters42, 866–868 (2003).
6. 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]
7. H. F. Talbot, “LXXVI. Facts relating to optical science. No. IV,” Philos. Mag.9, 401–407 (1836).
8. E. Lau, “Beugungserscheinung an Dopperlrastern,” Ann. Phys. (Leipzig)6(7-8), 417–427 (1948). [CrossRef]
9. J. Jahns and A. W. Lohmann, “Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun.28(3), 263–267 (1979). [CrossRef]
10. B. J. Chang, R. Alferness, and E. N. Leith, “Space-invariant achromatic grating interferometers: theory,” Appl. Opt.14(7), 1592–1600 (1975). [CrossRef] [PubMed]
11. S. K. Lynch, C. Liu, L. Assoufid, N. Y. Morgan, D. Mazilu, E. E. Bennett, C. K. Kemble, X. Xiao, and H. H. Wen, “Multi-layer coated micro-grating array for x-ray phase-contrast imaging,” Proc. SPIE8076, 80760F.1-80760F.10 (2011).
12. J. Thibaud, “Soft x-ray emission and absorption spectra with tangential grating,” Nature121(3044), 321–322 (1928). [CrossRef]
13. L. A. Sayce and A. Franks, “N.P.L. gratings for x-ray spectroscopy,” Proc. R. Soc. London Ser. A Math. Phys. Sci.282(1390), 353–357 (1964). [CrossRef]
14. J. Filevich, K. Kanizay, M. C. Marconi, J. L. A. Chilla, and J. J. Rocca, “Dense plasma diagnostics with an amplitude-division soft-x-ray laser interferometer based on diffraction gratings,” Opt. Lett.25(5), 356–358 (2000). [CrossRef] [PubMed]
15. Y. Liu, X. Tan, Z. K. Liu, X. D. Xu, Y. L. Hong, and S. J. Fu, “Soft x-ray holographic grating beam splitter including a double frequency grating for interferometer pre-alignment,” Opt. Express16(19), 14761–14770 (2008). [CrossRef] [PubMed]
16. D. Stutman, M. Finkenthal, and N. Moldovan, “Development of microperiodic mirrors for hard x-ray phase-contrast imaging,” Appl. Opt.49(25), 4677–4686 (2010). [CrossRef] [PubMed]
17. C. K. Kemble, J. Auxier, S. K. Lynch, E. E. Bennett, N. Y. Morgan, and H. Wen, “Grazing angle Mach-Zehnder interferometer using reflective phase gratings and a polychromatic, un-collimated light source,” Opt. Express18(26), 27481–27492 (2010). [CrossRef] [PubMed]
18. J. D. Jackson, “Scalar diffraction theory” in Classical Electrodynamics (Wiley, New York 1998), pp. 478–481.
19. T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys.106(5), 054703 (2009). [CrossRef]
20. D. L. Voronov, M. Ahn, E. H. Anderson, R. Cambie, C. H. Chang, E. M. Gullikson, R. K. Heilmann, F. Salmassi, M. L. Schattenburg, T. Warwick, V. V. Yashchuk, L. Zipp, and H. A. Padmore, “High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths,” Opt. Lett.35(15), 2615–2617 (2010). [CrossRef] [PubMed]
21. J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett.36(8), 1398–1400 (2011). [CrossRef] [PubMed]
22. M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun.129(3-4), 167–172 (1996). [CrossRef]

Ahn, M.
Alferness, R.
Anderson, E. H.
Assoufid, L.
Auxier, J.
Baruchel, J.
Baumann, J.
Bennett, E. E.
Bunk, O.
Cambie, R.
Chabior, M.
Chang, B. J.
Chang, C. H.
Chilla, J. L. A.
Clauser, J. F.
Cloetens, P.
da Silva, P.
David, C.
De Martino, C.
Diaz, A.
Donath, T.
Druart, G.
Ekstrom, C. R.
Filevich, J.
Finkenthal, M.
Franks, A.
Fu, S. J.
Goncharenko, A. M.
Guérineau, N.
Guigay, J. P.
Gullikson, E. M.
Hamaishi, Y.
Heilmann, R. K.
Hempel, E.
Hoheisel, M.
Hong, Y. L.
Idir, M.
Jahns, J.
Kanizay, K.
Kawamoto, S.
Keith, D. W.
Kemble, C. K.
Khilo, N. A.
Koyama, I.
Lau, E.
Leith, E. N.
Li, S. F.
Liu, C.
Liu, Y.
Liu, Z. K.
Lohmann, A. W.
Lynch, S. K.
Marconi, M. C.
Mazilu, D.
Mercère, P.
Mohr, J.
Moldovan, N.
Momose, A.
Morgan, N. Y.
Nohammer, B.
Padmore, H. A.
Pfeiffer, F.
Popescu, S.
Primot, J.
Pritchard, D. E.
Reznikova, E.
Rizzi, J.
Rocca, J. J.
Salmassi, F.
Sayce, L. A.
Schattenburg, M. L.
Schlenker, M.
Schuster, M.
Solak, H. H.
Stampanoni, M.
Stutman, D.
Suzuki, Y.
Takai, K.
Talbot, H. F.
Tan, X.
Testorf, M.
Thibaud, J.
Turchette, Q. A.
Vincent, G.
Voronov, D. L.
Warwick, T.
Weitkamp, T.
Wen, H.
Wen, H. H.
Xiao, X.
Xu, X. D.
Yashchuk, V. V.
Ziegler, E.
Zipp, L.

Ann. Phys. (Leipzig)

• E. Lau, “Beugungserscheinung an Dopperlrastern,” Ann. Phys. (Leipzig)6(7-8), 417–427 (1948). [CrossRef]

Appl. Opt.

• B. J. Chang, R. Alferness, and E. N. Leith, “Space-invariant achromatic grating interferometers: theory,” Appl. Opt.14(7), 1592–1600 (1975). [CrossRef] [PubMed]
• D. Stutman, M. Finkenthal, and N. Moldovan, “Development of microperiodic mirrors for hard x-ray phase-contrast imaging,” Appl. Opt.49(25), 4677–4686 (2010). [CrossRef] [PubMed]

Appl. Phys. Lett.

• C. David, B. Nohammer, H. H. Solak, and E. Ziegler, “Differential x-ray phase contrast imaging using a shearing interferometer,” Appl. Phys. Lett.81(17), 3287–3289 (2002). [CrossRef]

J. Appl. Phys.

• T. Donath, M. Chabior, F. Pfeiffer, O. Bunk, E. Reznikova, J. Mohr, E. Hempel, S. Popescu, M. Hoheisel, M. Schuster, J. Baumann, and C. David, “Inverse geometry for grating-based x-ray phase-contrast imaging,” J. Appl. Phys.106(5), 054703 (2009). [CrossRef]

Jap. J. Appl. Phys. Part 2-Letters

• A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of x-ray Talbot interferometry,” Jap. J. Appl. Phys. Part 2-Letters42, 866–868 (2003).

Nature

• J. Thibaud, “Soft x-ray emission and absorption spectra with tangential grating,” Nature121(3044), 321–322 (1928). [CrossRef]

Opt. Commun.

• J. Jahns and A. W. Lohmann, “Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun.28(3), 263–267 (1979). [CrossRef]
• M. Testorf, J. Jahns, N. A. Khilo, and A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun.129(3-4), 167–172 (1996). [CrossRef]

Opt. Express

• Y. Liu, X. Tan, Z. K. Liu, X. D. Xu, Y. L. Hong, and S. J. Fu, “Soft x-ray holographic grating beam splitter including a double frequency grating for interferometer pre-alignment,” Opt. Express16(19), 14761–14770 (2008). [CrossRef] [PubMed]
• 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]
• C. K. Kemble, J. Auxier, S. K. Lynch, E. E. Bennett, N. Y. Morgan, and H. Wen, “Grazing angle Mach-Zehnder interferometer using reflective phase gratings and a polychromatic, un-collimated light source,” Opt. Express18(26), 27481–27492 (2010). [CrossRef] [PubMed]

Opt. Lett.

• P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, and M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett.22(14), 1059–1061 (1997). [CrossRef] [PubMed]
• J. Filevich, K. Kanizay, M. C. Marconi, J. L. A. Chilla, and J. J. Rocca, “Dense plasma diagnostics with an amplitude-division soft-x-ray laser interferometer based on diffraction gratings,” Opt. Lett.25(5), 356–358 (2000). [CrossRef] [PubMed]
• D. L. Voronov, M. Ahn, E. H. Anderson, R. Cambie, C. H. Chang, E. M. Gullikson, R. K. Heilmann, F. Salmassi, M. L. Schattenburg, T. Warwick, V. V. Yashchuk, L. Zipp, and H. A. Padmore, “High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths,” Opt. Lett.35(15), 2615–2617 (2010). [CrossRef] [PubMed]
• J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett.36(8), 1398–1400 (2011). [CrossRef] [PubMed]

Philos. Mag.

• H. F. Talbot, “LXXVI. Facts relating to optical science. No. IV,” Philos. Mag.9, 401–407 (1836).

Phys. Rev. A

• J. F. Clauser and S. F. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A49(4), R2213–R2216 (1994). [CrossRef] [PubMed]

Phys. Rev. Lett.

• D. W. Keith, C. R. Ekstrom, Q. A. Turchette, and D. E. Pritchard, “An interferometer for atoms,” Phys. Rev. Lett.66(21), 2693–2696 (1991). [CrossRef] [PubMed]

Proc. R. Soc. London Ser. A Math. Phys. Sci.

• L. A. Sayce and A. Franks, “N.P.L. gratings for x-ray spectroscopy,” Proc. R. Soc. London Ser. A Math. Phys. Sci.282(1390), 353–357 (1964). [CrossRef]

Proc. SPIE

• S. K. Lynch, C. Liu, L. Assoufid, N. Y. Morgan, D. Mazilu, E. E. Bennett, C. K. Kemble, X. Xiao, and H. H. Wen, “Multi-layer coated micro-grating array for x-ray phase-contrast imaging,” Proc. SPIE8076, 80760F.1-80760F.10 (2011).

Other

• J. D. Jackson, “Scalar diffraction theory” in Classical Electrodynamics (Wiley, New York 1998), pp. 478–481.

2011, Lynch, Proc. SPIE
2011, Rizzi, Opt. Lett.
2010, Voronov, Opt. Lett.
2010, Stutman, Appl. Opt.
2010, Kemble, Opt. Express
2009, Donath, J. Appl. Phys.
2008, Liu, Opt. Express
2005, Weitkamp, Opt. Express
2003, Momose, Jap. J. Appl. Phys. Part 2-Letters
2002, David, Appl. Phys. Lett.
2000, Filevich, Opt. Lett.
1997, Cloetens, Opt. Lett.
1996, Testorf, Opt. Commun.
1994, Clauser, Phys. Rev. A
1991, Keith, Phys. Rev. Lett.
1979, Jahns, Opt. Commun.
1975, Chang, Appl. Opt.
1964, Sayce, Proc. R. Soc. London Ser. A Math. Phys. Sci.
1948, Lau, Ann. Phys. (Leipzig)
1928, Thibaud, Nature
1836, Talbot, Philos. Mag.

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• Grazing angle Mach-Zehnder interferometer using reflective phase gratings and a polychromatic, un-collimated light source (OE)

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