Abstract

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

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  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]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2011

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. SPIE 8076, 80760F.1-80760F.10 (2011).

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]

2010

2009

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]

2008

2005

2003

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-Letters 42, 866–868 (2003).

2002

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]

2000

1997

1996

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]

1994

J. F. Clauser and S. F. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994).
[CrossRef] [PubMed]

1991

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]

1979

J. Jahns and A. W. Lohmann, “Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28(3), 263–267 (1979).
[CrossRef]

1975

1964

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]

1948

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

1928

J. Thibaud, “Soft x-ray emission and absorption spectra with tangential grating,” Nature 121(3044), 321–322 (1928).
[CrossRef]

1836

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

Ahn, M.

Alferness, R.

Anderson, E. H.

Assoufid, L.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

Auxier, J.

Baruchel, J.

Baumann, J.

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]

Bennett, E. E.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

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. Express 18(26), 27481–27492 (2010).
[CrossRef] [PubMed]

Bunk, O.

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]

Cambie, R.

Chabior, M.

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]

Chang, B. J.

Chang, C. H.

Chilla, J. L. A.

Clauser, J. F.

J. F. Clauser and S. F. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994).
[CrossRef] [PubMed]

Cloetens, P.

da Silva, P.

David, C.

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]

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]

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]

De Martino, C.

Diaz, A.

Donath, T.

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]

Druart, G.

Ekstrom, C. R.

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]

Filevich, J.

Finkenthal, M.

Franks, A.

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]

Fu, S. J.

Goncharenko, A. M.

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]

Guérineau, N.

Guigay, J. P.

Gullikson, E. M.

Hamaishi, Y.

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-Letters 42, 866–868 (2003).

Heilmann, R. K.

Hempel, E.

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]

Hoheisel, M.

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]

Hong, Y. L.

Idir, M.

Jahns, J.

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]

J. Jahns and A. W. Lohmann, “Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28(3), 263–267 (1979).
[CrossRef]

Kanizay, K.

Kawamoto, S.

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-Letters 42, 866–868 (2003).

Keith, D. W.

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]

Kemble, C. K.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

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. Express 18(26), 27481–27492 (2010).
[CrossRef] [PubMed]

Khilo, N. A.

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]

Koyama, I.

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-Letters 42, 866–868 (2003).

Lau, E.

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

Leith, E. N.

Li, S. F.

J. F. Clauser and S. F. Li, “Talbot-vonLau atom interferometry with cold slow potassium,” Phys. Rev. A 49(4), R2213–R2216 (1994).
[CrossRef] [PubMed]

Liu, C.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

Liu, Y.

Liu, Z. K.

Lohmann, A. W.

J. Jahns and A. W. Lohmann, “Lau effect (a diffraction experiment with incoherent illumination),” Opt. Commun. 28(3), 263–267 (1979).
[CrossRef]

Lynch, S. K.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

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. Express 18(26), 27481–27492 (2010).
[CrossRef] [PubMed]

Marconi, M. C.

Mazilu, D.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

Mercère, P.

Mohr, J.

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]

Moldovan, N.

Momose, A.

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-Letters 42, 866–868 (2003).

Morgan, N. Y.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

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. Express 18(26), 27481–27492 (2010).
[CrossRef] [PubMed]

Nohammer, B.

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]

Padmore, H. A.

Pfeiffer, F.

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]

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]

Popescu, S.

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]

Primot, J.

Pritchard, D. E.

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]

Reznikova, E.

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]

Rizzi, J.

Rocca, J. J.

Salmassi, F.

Sayce, L. A.

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]

Schattenburg, M. L.

Schlenker, M.

Schuster, M.

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]

Solak, H. H.

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]

Stampanoni, M.

Stutman, D.

Suzuki, Y.

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-Letters 42, 866–868 (2003).

Takai, K.

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-Letters 42, 866–868 (2003).

Talbot, H. F.

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

Tan, X.

Testorf, M.

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]

Thibaud, J.

J. Thibaud, “Soft x-ray emission and absorption spectra with tangential grating,” Nature 121(3044), 321–322 (1928).
[CrossRef]

Turchette, Q. A.

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]

Vincent, G.

Voronov, D. L.

Warwick, T.

Weitkamp, T.

Wen, H.

Wen, H. H.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

Xiao, X.

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. SPIE 8076, 80760F.1-80760F.10 (2011).

Xu, X. D.

Yashchuk, V. V.

Ziegler, E.

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]

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]

Zipp, L.

Ann. Phys. (Leipzig)

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

Appl. Opt.

Appl. Phys. Lett.

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Nature

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Figures (8)

Fig. 1
Fig. 1

An illustration of a Talbot-Lau grating interferometer. The two gratings and the detector plane are parallel. The light source can be polychromatic with low spatial coherence and still produces interference fringes on the detector plane if the appropriate grating periods and geometric layout are used.

Fig. 2
Fig. 2

Coordinate system and general interferometer configuration described by the theoretical derivations. The pitch angles of the 3 principal planes relative to the axis of the incident beam are θ, θ1 and θ2. The coordinates within the principal planes are (xn, yn) and the Y axes of the planes are all perpendicular to the incident beam axis and parallel to each other. The para-axial distances between the source and the 3 principal planes are Rj. In the case of a parallel incident beam the distance R0 is set to infinity.

Fig. 3
Fig. 3

Transformation of an extended source in the XZ plane as part of the process to map the oblique incidence configuration (C3) to an equivalent normal incidence configuration. With an extended source, the para-axial distance between any point on the source and the G0 grating must be maintained between the two configurations. This condition is met by the shear transformation of the source in the pitch (XZ) plane.

Fig. 4
Fig. 4

Compensating for a small difference δ1 between the pitch angles of G0 and G1 gratings by moving the detector plane along the beam axis over a distance of ΔR2. The quantitative relationship between the pitch angle difference and the compensating movement of the detector plane is given in Eq. (66).

Fig. 5
Fig. 5

The relationship between the pitch angle error of the G1 grating (relative to the G0 grating) and the translational movement of the detector plane that compensates for that error. The three curves are for three different x-ray beam incident angles of 30°, 45° and 60°. Other parameters of the interferometer are given in the text.

Fig. 6
Fig. 6

Illustration of a grazing angle reflective grating interferometer in black line drawings, and its equivalent transmission configuration (ETC) in red line drawings. In the ETC the detector is represented by its reflection with respect to the G1 grating surface, and both the detector and the G1 grating further reflected with respect to the G0 surface. In the real reflection configuration, the intersections of the beam axis with the G1 grating and the detector plane are (x1, z1) and (x2, z2), while the equivalent points in the ETC are (xT1, zT1) and (xT2, zT2). To find the new detector position (x’2, z’2) that compensates for a small pitch angle difference δ1 between the G1 and G0 gratings, we first find the new detector position (x’T2, z’T2) in the ETC according to Eq. (66), then reflect it with respect to the G0 surface and the tilted G1 surface.

Fig. 7
Fig. 7

The relationship between the pitch angle error of the G1 grating (relative to the G0 grating) and the movement of the detector plane that compensates for that error in a grazing incidence reflective interferometer. (a) The required axial movement of the detector as a function of the pitch angle error for three different grazing incident angles of the x-ray beam. (b) The required vertical movement of the detector as a function of the pitch angle error does not change with the incident angle. Other parameters of the reflective interferometer are given in the text.

Fig. 8
Fig. 8

X-ray reflective interferometer demonstration of compensating for a small pitch angle difference between the G0 and G1 gratings by moving the detector phosphor screen. (a) In the initial setup, the image on the detector screen consists of the primary interferences fringes on the left, and large-period secondary fringes (alignment aids) on the right margin. The intensity profile of a center cross-section of the primary fringes is shown. The fringe visibility as defined in the text is 12.3%. The scale bar indicates 20 µm in the vertical direction perpendicular to the x-ray beam. (b) After making a small change of the G1 pitch angle, the primary fringes are obliterated. (c) Moving the detector screen by 4.2 cm along the x-ray beam axis recovers the primary fringes, and the fringe visibility returns to 12.9%. The movement distance indicates that the the pitch angle change we introduced was 0.19 milliradians (39 arc seconds).

Tables (1)

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Table 1 Conversion Between Equivalent Interferometer Configurations

Equations (70)

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A 4 ( x 2 , y 2 | R j , θ n , G m )~ d x 0 d y 0 d x 1 d y 1 exp(ik| r 0 |) | r 0 | G 0 ( x 0 , y 0 )( n 0 e 1 ) exp(ik| r 1 |) | r 1 | G 1 ( x 1 , y 1 )( n 1 e 2 ) exp(ik| r 2 |) | r 2 | ,
A 4 ( x 2 , y 2 | R j , θ n , G m )~ d x 0 d y 0 d x 1 d y 1 1 R 0 R 1 R 2 exp[ik(| r 0 |+| r 1 |+| r 2 |)] G 0 ( x 0 , y 0 ) G 1 ( x 1 , y 1 )
δ 1 = θ 1 θ, δ 2 = θ 2 θ,
| r 0 |= [ ( R 0 cosθ+ x 0 ) 2 + ( R 0 sinθ) 2 + y 0 2 ] 1/2 R 0 + x 0 cosθ+ 1 2 x 0 2 sin 2 θ R 0 + 1 2 y 0 2 R 0 ;
| r 1 | [ ( R 1 cosθ+ x 1 x 0 ) 2 + ( R 1 sinθ x 1 δ 1 ) 2 + ( y 1 y 0 ) 2 ] 1/2 R 1 +( x 1 x 0 )cosθ x 1 δ 1 sinθ+ 1 2 ( x 1 x 0 ) 2 sin 2 θ R 1 + 1 2 ( y 1 y 0 ) 2 R 1 + ( x 1 x 0 ) x 1 δ 1 sinθcosθ R 1 ;
| r 2 | [ ( R 2 cosθ+ x 2 x 1 ) 2 + ( R 2 sinθ x 2 δ 2 + x 1 δ 1 ) 2 + ( y 2 y 1 ) 2 ] 1/2 R 2 +( x 2 x 1 )cosθ( x 2 δ 2 x 1 δ 1 )sinθ + 1 2 ( x 2 x 1 ) 2 sin 2 θ R 2 + 1 2 ( y 2 y 1 ) 2 R 2 + ( x 2 x 1 )( x 2 δ 2 x 1 δ 1 )sinθcosθ R 2 .
ϕ 4 ( x 2 , y 2 | R j , θ n , G m )=k 0 2 | r j | ~ 0 2 R j +(cosθ δ 2 sinθ) x 2 + 1 2 sin 2 θ[ x 0 2 ( 1 R 0 + 1 R 1 )+ x 1 2 ( 1 R 1 + 1 R 2 )+ x 2 2 R 2 2 x 0 x 1 R 1 2 x 1 x 2 R 2 ] +sinθcosθ[ x 1 2 δ 1 ( 1 R 1 + 1 R 2 )+ x 2 2 δ 2 R 2 x 0 x 1 δ 1 R 1 x 1 x 2 ( δ 1 + δ 2 ) R 2 ] + 1 2 [ y 0 2 ( 1 R 0 + 1 R 1 )+ y 1 2 ( 1 R 1 + 1 R 2 )+ y 2 2 R 2 2 y 0 y 1 R 1 2 y 1 y 2 R 2 ].
ϕ 4 ( x 2 , y 2 | R j , θ n , G m )~ ϕ C ( x 2 , y 2 ) + 1 2 sin 2 θ[ x 0 2 ( 1 R 0 + 1 R 1 )+ x 1 2 ( 1 R 1 + 1 R 2 ) 2 x 0 x 1 R 1 2 x 1 x 2 R 2 ] +sinθcosθ[ x 1 2 δ 1 ( 1 R 1 + 1 R 2 ) x 0 x 1 δ 1 R 1 x 1 x 2 ( δ 1 + δ 2 ) R 2 ] + 1 2 [ y 0 2 ( 1 R 0 + 1 R 1 )+ y 1 2 ( 1 R 1 + 1 R 2 ) 2 y 0 y 1 R 1 2 y 1 y 2 R 2 ],
ϕ C ( x 2 , y 2 )= 0 2 R j +(cosθ δ 2 sinθ) x 2 + 1 2 sin 2 θ x 2 2 R 2 +sinθcosθ x 2 2 δ 2 R 2 + 1 2 y 2 2 R 2 .
C1:( R 0 =, θ n =90°, G m ),
C2:(Finite R 0 , θ n =90°, G m ).
ϕ 1 ( x 2 , y 2 | R 0 =, θ n =90°, G m )~ ϕ C ( x 2 , y 2 ) + 1 2 [ x 0 2 1 R 1 + x 1 2 ( 1 R 1 + 1 R 2 ) 2 x 0 x 1 R 1 2 x 1 x 2 R 2 ] + 1 2 [ y 0 2 1 R 1 + y 1 2 ( 1 R 1 + 1 R 2 ) 2 y 0 y 1 R 1 2 y 1 y 2 R 2 ],
ϕ 2 ( x 2 , y 2 | R j , θ n =90°, G m )~ ϕ C ( x 2 , y 2 ) + 1 2 [ x 0 2 ( 1 R 0 + 1 R 1 )+ x 1 2 ( 1 R 1 + 1 R 2 ) 2 x 0 x 1 R 1 2 x 1 x 2 R 2 ] + 1 2 [ y 0 2 ( 1 R 0 + 1 R 1 )+ y 1 2 ( 1 R 1 + 1 R 2 ) 2 y 0 y 1 R 1 2 y 1 y 2 R 2 ].
x ' 0 2 R ' 1 = x 0 2 ( 1 R 0 + 1 R 1 ),
x ' 0 x ' 1 R 1 = x 0 x 1 R 1 ,
x ' 1 x ' 2 R 2 = x 1 x 2 R 2 ,
x ' 1 2 ( 1 R 1 + 1 R 2 )=x ' 1 2 ( 1 R 1 + 1 R 2 ),
x ' 1 = x 1 .
x ' 0 R ' 1 = x 0 R 1 .
R ' 1 = R 1 (1+ R 1 / R 0 ),
x ' 0 = x 0 (1+ R 1 / R 0 ).
R ' 2 = R 2 R 0 + R 1 R 0 + R 1 + R 2 .
x ' 2 = R 0 + R 1 R 0 + R 1 + R 2 x 2 .
G ' m (x ' m ,y ' m )= G m ( x m , y m ).
G ' 0 ( x 0 , y 0 )= G 0 ( x 0 /(1+ R 1 / R 0 ), y 0 /(1+ R 1 / R 0 )).
G ' 1 ( x 1 , y 1 )= G 1 ( x 1 , y 1 ).
I 2 ( x 2 , y 2 )~I ' 1 (x ' 2 ,y ' 2 ),
I 2 ( x 2 , y 2 )~I ' 1 ( R 0 + R 1 R 0 + R 1 + R 2 x 2 , R 0 + R 1 R 0 + R 1 + R 2 y 2 ).
C3:(finite R 0 , θ n =θ, G m ).
ϕ 3 ( x 2 , y 2 | R j , θ n =θ, G m )~ ϕ C ( x 2 , y 2 ) + 1 2 sin 2 θ[ x 0 2 ( 1 R 0 + 1 R 1 )+ x 1 2 ( 1 R 1 + 1 R 2 ) 2 x 0 x 1 R 1 2 x 1 x 2 R 2 ] + 1 2 [ y 0 2 ( 1 R 0 + 1 R 1 )+ y 1 2 ( 1 R 1 + 1 R 2 ) 2 y 0 y 1 R 1 2 y 1 y 2 R 2 ],
x ' j = x j sinθ
G ' m ( x m , y m )= G m ( x m /sinθ, y m ).
I 3 ( x 2 , y 2 )~I ' 2 (x ' 2 ,y ' 2 ),
I 3 ( x 2 , y 2 )~I ' 2 ( x 2 sinθ, y 2 ).
C4:(finite R 0 , θ 0 =θ, θ 1 =θ+ δ 1 , θ 2 =θ+ δ 2 , G m ).
x ' 0 = x 0 ,
x ' 1 = x 1 ,
x ' 0 2 ( 1 R 0 + 1 R 1 )= x 0 2 ( 1 R 0 + 1 R 1 ),
x ' 0 x ' 1 R 1 = x 0 x 1 R 1 (1+ δ 1 cotθ),
x ' 1 2 ( 1 R 1 + 1 R 2 )= x 1 2 ( 1 R 1 + 1 R 2 )(1+2 δ 1 cotθ),
x ' 1 x ' 2 R 2 = x 1 x 2 R 2 [1+( δ 1 + δ 2 )cotθ].
R ' 1 R 1 (1 δ 1 cotθ).
R ' 0 R 0 (1+ R 0 R 1 δ 1 cotθ).
R ' 2 R 2 [1(2+ R 2 R 1 ) δ 1 cotθ].
x ' 2 = x 2 [1(1+ R 2 R 1 ) δ 1 cotθ+ δ 2 cotθ].
y ' 0 2 ( 1 R ' 0 + 1 R ' 1 )= y 0 2 ( 1 R 0 + 1 R 1 ),
y ' 0 y ' 1 R ' 1 = y 0 y 1 R 1 ,
y ' 1 y ' 2 R ' 2 = y 1 y 2 R 2 ,
y ' 1 2 ( 1 R ' 1 + 1 R ' 2 )= y 1 2 ( 1 R 1 + 1 R 2 ),
y ' 1 y ' 2 R ' 2 = y 1 y 2 R 2 .
y ' 0 = y 0 .
y ' 1 = y 1 (1 δ 1 cotθ).
y ' 2 = y 2 [1(1+ R 2 R 1 ) δ 1 cotθ].
G ' 0 ( x 0 , y 0 )= G 0 ( x 0 , y 0 ),
G ' 1 ( x 1 , y 1 )= G 1 ( x 1 ,(1+ δ 1 cotθ) y 1 ).
I 4 ( x 2 , y 2 )~I ' 3 ([1(1+ R 2 R 1 ) δ 1 cotθ+ δ 2 cotθ] x 2 ,[1(1+ R 2 R 1 ) δ 1 cotθ] y 2 ),
R 1 R 2 /( R 1 + R 2 )=a P 1 2 /λ,
( R 1 + R 2 )/ R 2 =b P 0 / P 1 ,
P ' m = P m sinθ.
R 1 R 2 /( R 1 + R 2 )/ sin 2 θ=a P 1 2 /λ,
( R 1 + R 2 )/ R 2 =b P 0 / P 1 .
R 1 R 2 /( R 1 + R 2 )/ sin 2 θ=(12 δ 1 cotθ)a P 1 2 /λ,
( R 1 + R 2 )/ R 2 =(1+ δ 1 cotθ)b P 0 / P 1 .
R 2A =(1+Δ) R 2 .
( R 1 + R 2A )/ R 2A =[1Δ R 1 /( R 1 + R 2 )](1+ δ 1 cotθ)b P 0 / P 1 .
Δ R 2 = R 2 δ 1 cotθ( R 1 + R 2 )/ R 1 .
( x 1 , z 1 )=( R 1 cosθ, R 1 sinθ), ( x 2 , z 2 )=( R 1 cosθ+ R 2 cosθ, R 1 sinθ R 2 sinθ), ( x T1 , z T1 )=( R 1 cosθ, R 1 sinθ), ( x T2 , z T2 )=( R 1 cosθ+ R 2 cosθ, R 1 sinθ R 2 sinθ).
( x T2 , z T2 )=( R 1 cosθ+ R 2 cosθ+Δ R 2 cosθ, R 1 sinθ R 2 sinθΔ R 2 sinθ),
( x 2 , z 2 )=( x 2 +Δ x 2 , z 2 +Δ z 2 ), Δ x 2 = δ 1 R 2 (1+ R 2 / R 1 )cotθ, Δ z 2 = δ 1 R 2 (3+ R 2 / R 1 )cosθ.
λ 2 R P 3 cotθ<<1,

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