Abstract

We report a multilens X-ray interferometer consisting of six parallel arrays of planar compound refractive lenses, each of which creates a diffraction limited beam under coherent illumination. Overlapping such coherent beams produces an interference pattern demonstrating substantially strong longitudinal functional dependence. The interference fringe pattern produced by multilens interferometer was described by Talbot imaging formalism. Theoretical analysis of the interference pattern formation was carried out and corresponding computer simulations were performed. The proposed multilens interferometer was experimentally tested at ID06 ESRF beamline in the X-ray energy range from 10 to 30 keV. The experimentally recorded fractional Talbot images are in a good agreement with computer simulations.

© 2014 Optical Society of America

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References

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    [Crossref]
  2. B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
    [Crossref]
  3. B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
    [Crossref]
  4. V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
    [Crossref]
  5. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
    [Crossref]
  6. A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).
  7. I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
    [Crossref]
  8. A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).
  9. G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
    [Crossref] [PubMed]
  10. V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. 216(4-6), 247–260 (2003).
    [Crossref]
  11. A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
    [Crossref]
  12. M. Drakopoulos, A. Snigirev, I. Snigireva, and J. Schilling, “X-ray high-resolution diffraction using refractive lenses,” Appl. Phys. Lett. 86(1), 014102 (2005).
    [Crossref]
  13. A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
    [Crossref]
  14. A. Bosak, I. Snigireva, K. S. Napolskii, and A. Snigirev, “High-resolution transmission X-ray microscopy: a new tool for mesoscopic materials,” Adv. Mater. 22(30), 3256–3259 (2010).
    [Crossref] [PubMed]
  15. P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
    [Crossref]
  16. A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
    [Crossref] [PubMed]
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  22. M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
    [Crossref]
  23. Ya. M. Hartman and A. Snigirev, “Some examples of high energy X-rays phase contrast,” in X-ray Microscopy IV (Bogorodskii Pechatnik, Chernogolovka, Moscow region, Russia), 429– 432 (1994).
  24. A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
    [Crossref]
  25. 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]
  26. 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]
  27. A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
    [Crossref] [PubMed]
  28. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
    [Crossref]
  29. V. Kohn, I. Snigireva, and A. Snigirev, “Direct Measurement of Transverse Coherence Length of Hard X Rays from Interference Fringes,” Phys. Rev. Lett. 85(13), 2745–2748 (2000).
    [Crossref] [PubMed]
  30. C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

2013 (1)

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

2012 (2)

A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
[Crossref] [PubMed]

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

2011 (1)

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

2010 (1)

A. Bosak, I. Snigireva, K. S. Napolskii, and A. Snigirev, “High-resolution transmission X-ray microscopy: a new tool for mesoscopic materials,” Adv. Mater. 22(30), 3256–3259 (2010).
[Crossref] [PubMed]

2009 (2)

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

2007 (1)

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

2006 (1)

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

2005 (3)

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[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]

M. Drakopoulos, A. Snigirev, I. Snigireva, and J. Schilling, “X-ray high-resolution diffraction using refractive lenses,” Appl. Phys. Lett. 86(1), 014102 (2005).
[Crossref]

2004 (1)

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

2003 (2)

V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. 216(4-6), 247–260 (2003).
[Crossref]

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

2001 (1)

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

2000 (2)

V. Kohn, I. Snigireva, and A. Snigirev, “Direct Measurement of Transverse Coherence Length of Hard X Rays from Interference Fringes,” Phys. Rev. Lett. 85(13), 2745–2748 (2000).
[Crossref] [PubMed]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

1999 (2)

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

1997 (1)

1996 (2)

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[Crossref]

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

1995 (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

1965 (1)

1960 (1)

J. M. Cowley and A. F. Moodie, “Fourier Images IV: The Phase Grating,” Pro. Ph. So. 76(3), 378–384 (1960).
[Crossref]

1881 (1)

L. Rayleigh, “XXV. On copying diffraction-gratings, and on some phenomena connected therewith,” Phil. Mag. Series 5 11(67), 196–205 (1881).
[Crossref]

1836 (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 3 9, 401–407 (1836).

Aristov, V.

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Baruchel, J.

Benner, B.

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

Berry, M. V.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

Bohic, S.

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

Bosak, A.

A. Bosak, I. Snigireva, K. S. Napolskii, and A. Snigirev, “High-resolution transmission X-ray microscopy: a new tool for mesoscopic materials,” Adv. Mater. 22(30), 3256–3259 (2010).
[Crossref] [PubMed]

Boye, P.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Burghammer, M.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Bytchkov, A.

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

Chukalina, M.

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

Cloetens, P.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “Fourier Images IV: The Phase Grating,” Pro. Ph. So. 76(3), 378–384 (1960).
[Crossref]

Curfs, C.

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

David, C.

De Martino, C.

Detlefs, C.

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

Di Michiel, M.

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

Diaz, A.

Dolbnya, I. P.

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

Drakopoulos, M.

M. Drakopoulos, A. Snigirev, I. Snigireva, and J. Schilling, “X-ray high-resolution diffraction using refractive lenses,” Appl. Phys. Lett. 86(1), 014102 (2005).
[Crossref]

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

Eliseev, A. A.

A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
[Crossref] [PubMed]

Ershov, P.

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

Falkenberg, G.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

Feldkamp, J.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Gleyzolle, H.

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

Goikhman, A.

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

Grigoriev, M.

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Grigoriev, M. B.

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

Guigay, J. P.

Günzler, T. F.

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

Hamaishi, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

Hart, D. C. T.

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

Hoffmann, M.

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Hoppe, R.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

Imhof, A.

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

Kawamoto, S.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

Klein, S.

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. 216(4-6), 247–260 (2003).
[Crossref]

V. Kohn, I. Snigireva, and A. Snigirev, “Direct Measurement of Transverse Coherence Length of Hard X Rays from Interference Fringes,” Phys. Rev. Lett. 85(13), 2745–2748 (2000).
[Crossref] [PubMed]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Koyama, I.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

Kuchler, M.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Kuhlmann, M.

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

Kurapova, O.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Kuznetsov, S.

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Lengeler, B.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[Crossref]

Momose, A.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “Fourier Images IV: The Phase Grating,” Pro. Ph. So. 76(3), 378–384 (1960).
[Crossref]

Moussaid, A.

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

Napolskii, K. S.

A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
[Crossref] [PubMed]

A. Bosak, I. Snigireva, K. S. Napolskii, and A. Snigirev, “High-resolution transmission X-ray microscopy: a new tool for mesoscopic materials,” Adv. Mater. 22(30), 3256–3259 (2010).
[Crossref] [PubMed]

Nuesse, D.

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

Patommel, J.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Petukhov, A. V.

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

Pfeiffer, F.

Rau, C.

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Rayleigh, L.

L. Rayleigh, “XXV. On copying diffraction-gratings, and on some phenomena connected therewith,” Phil. Mag. Series 5 11(67), 196–205 (1881).
[Crossref]

Richwin, M.

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

Riekel, C.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Rossat, M.

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

Roth, T.

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

Samberg, D.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

Sapoletova, N. A.

A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
[Crossref] [PubMed]

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Schilling, J.

M. Drakopoulos, A. Snigirev, I. Snigireva, and J. Schilling, “X-ray high-resolution diffraction using refractive lenses,” Appl. Phys. Lett. 86(1), 014102 (2005).
[Crossref]

Schlenker, M.

Schröder, W.

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

Schroer, C.

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

Schroer, C. G.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

Schropp, A.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

Seiboth, F.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

Shabelnikov, L.

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Simionovici, A. S.

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

Snigirev, A.

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
[Crossref] [PubMed]

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

A. Bosak, I. Snigireva, K. S. Napolskii, and A. Snigirev, “High-resolution transmission X-ray microscopy: a new tool for mesoscopic materials,” Adv. Mater. 22(30), 3256–3259 (2010).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

M. Drakopoulos, A. Snigirev, I. Snigireva, and J. Schilling, “X-ray high-resolution diffraction using refractive lenses,” Appl. Phys. Lett. 86(1), 014102 (2005).
[Crossref]

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. 216(4-6), 247–260 (2003).
[Crossref]

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

V. Kohn, I. Snigireva, and A. Snigirev, “Direct Measurement of Transverse Coherence Length of Hard X Rays from Interference Fringes,” Phys. Rev. Lett. 85(13), 2745–2748 (2000).
[Crossref] [PubMed]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Snigireva, I.

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
[Crossref] [PubMed]

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

A. Bosak, I. Snigireva, K. S. Napolskii, and A. Snigirev, “High-resolution transmission X-ray microscopy: a new tool for mesoscopic materials,” Adv. Mater. 22(30), 3256–3259 (2010).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

M. Drakopoulos, A. Snigirev, I. Snigireva, and J. Schilling, “X-ray high-resolution diffraction using refractive lenses,” Appl. Phys. Lett. 86(1), 014102 (2005).
[Crossref]

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. 216(4-6), 247–260 (2003).
[Crossref]

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

V. Kohn, I. Snigireva, and A. Snigirev, “Direct Measurement of Transverse Coherence Length of Hard X Rays from Interference Fringes,” Phys. Rev. Lett. 85(13), 2745–2748 (2000).
[Crossref] [PubMed]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

Stampanoni, M.

Stephan, S.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

Suzuki, Y.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

Takai, K.

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

Talbot, H. F.

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 3 9, 401–407 (1836).

Thijssen, J. H. J.

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

Tummler, J.

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

van Blaaderen, A.

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

van der Hart, A.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Vaughan, G.

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

Vaughan, G. B. M.

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

Vincze, L.

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

Voges, E.

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Weitkamp, T.

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]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Wellenreuther, G.

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

Winthrop, J. T.

Worthington, C. R.

Wright, J.

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

Wright, J. P.

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

Yunkin, V.

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

Ziegler, E.

Adv. Mater. (1)

A. Bosak, I. Snigireva, K. S. Napolskii, and A. Snigirev, “High-resolution transmission X-ray microscopy: a new tool for mesoscopic materials,” Adv. Mater. 22(30), 3256–3259 (2010).
[Crossref] [PubMed]

AIP Conf. Proc. (1)

I. Snigireva, A. Snigirev, V. Yunkin, M. Drakopoulos, M. Grigoriev, S. Kuznetsov, M. Chukalina, M. Hoffmann, D. Nuesse, and E. Voges, “X-ray Si-based Integrated Lens System for Wide Range of Hard Synchrotron Radiation,” AIP Conf. Proc. 705, 708–711 (2004).
[Crossref]

Angew. Chem. Int. Ed. Engl. (1)

A. A. Eliseev, N. A. Sapoletova, I. Snigireva, A. Snigirev, and K. S. Napolskii, “Electrochemical X-ray photolithography,” Angew. Chem. Int. Ed. Engl. 51(46), 11602–11605 (2012).
[Crossref] [PubMed]

Appl. Phys. Lett. (5)

B. Lengeler, C. G. Schroer, M. Richwin, J. Tummler, M. Drakopoulos, A. Snigirev, and I. Snigireva, “A microscope for hard X-rays based on parabolic compound refractive lenses,” Appl. Phys. Lett. 74(26), 3924 (1999).
[Crossref]

V. Aristov, M. Grigoriev, S. Kuznetsov, L. Shabelnikov, V. Yunkin, T. Weitkamp, C. Rau, I. Snigireva, A. Snigirev, M. Hoffmann, and E. Voges, “X-ray refractive planar lens with minimized absorption,” Appl. Phys. Lett. 77(24), 4058 (2000).
[Crossref]

C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Kuchler, “Hard X-ray nanoprobe based on refractive X-ray lenses,” Appl. Phys. Lett. 87(12), 124103 (2005).
[Crossref]

A. Schropp, R. Hoppe, J. Patommel, D. Samberg, F. Seiboth, S. Stephan, G. Wellenreuther, G. Falkenberg, and C. G. Schroer, “Hard x-ray scanning microscopy with coherent radiation: Beyond the resolution of conventional x-ray microscopes,” Appl. Phys. Lett. 100(25), 253112 (2012).
[Crossref]

M. Drakopoulos, A. Snigirev, I. Snigireva, and J. Schilling, “X-ray high-resolution diffraction using refractive lenses,” Appl. Phys. Lett. 86(1), 014102 (2005).
[Crossref]

J. Appl. Cryst. (2)

A. V. Petukhov, J. H. J. Thijssen, D. C. T. Hart, A. Imhof, A. van Blaaderen, I. P. Dolbnya, A. Snigirev, A. Moussaid, and I. Snigireva, “Microradian X-ray diffraction in colloidal photonic crystals,” J. Appl. Cryst. 39(2), 137–144 (2006).
[Crossref]

P. Ershov, S. Kuznetsov, I. Snigireva, V. Yunkin, A. Goikhman, and A. Snigirev, “Fourier crystal diffractometry based on refractive optics,” J. Appl. Cryst. 46(5), 1475–1480 (2013).
[Crossref]

J. Mod. Opt. (1)

M. V. Berry and S. Klein, “Integer, fractional and fractal Talbot effects,” J. Mod. Opt. 43(10), 2139–2164 (1996).
[Crossref]

J. Opt. Soc. Am. (1)

J. Synchrotron Radiat. (2)

B. Lengeler, C. Schroer, J. Tummler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, “Imaging by parabolic refractive lenses in the hard X-ray range,” J. Synchrotron Radiat. 6(6), 1153–1167 (1999).
[Crossref]

G. B. M. Vaughan, J. P. Wright, A. Bytchkov, M. Rossat, H. Gleyzolle, I. Snigireva, and A. Snigirev, “X-ray transfocators: focusing devices based on compound refractive lenses,” J. Synchrotron Radiat. 18(2), 125–133 (2011).
[Crossref] [PubMed]

JPCS (1)

A. Snigirev, I. Snigireva, G. B. M. Vaughan, J. Wright, M. Rossat, A. Bytchkov, and C. Curfs, “High energy X-ray transfocator based on Al parabolic refractive lenses for focusing and collimation,” JPCS 186, 012073 (2009).

Jpn. J. Appl. Phys. (1)

A. Momose, S. Kawamoto, I. Koyama, Y. Hamaishi, K. Takai, and Y. Suzuki, “Demonstration of X-ray Talbot interferometry,” Jpn. J. Appl. Phys. 42(7B7B), L866–L868 (2003).
[Crossref]

Nature (1)

A. Snigirev, V. Kohn, I. Snigireva, and B. Lengeler, “A compound refractive lens for focusing high-energy X-rays,” Nature 384(6604), 49–51 (1996).
[Crossref]

Opt. Commun. (1)

V. Kohn, I. Snigireva, and A. Snigirev, “Diffraction theory of imaging with X-ray compound refractive lens,” Opt. Commun. 216(4-6), 247–260 (2003).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phil. Mag. Series 3 (1)

H. F. Talbot, “Facts relating to optical science,” Phil. Mag. Series 3 9, 401–407 (1836).

Phil. Mag. Series 5 (1)

L. Rayleigh, “XXV. On copying diffraction-gratings, and on some phenomena connected therewith,” Phil. Mag. Series 5 11(67), 196–205 (1881).
[Crossref]

Phys. Rev. Lett. (2)

A. Snigirev, I. Snigireva, V. Kohn, V. Yunkin, S. Kuznetsov, M. B. Grigoriev, T. Roth, G. Vaughan, and C. Detlefs, “X-Ray Nanointerferometer Based on Si Refractive Bilenses,” Phys. Rev. Lett. 103(6), 064801 (2009).
[Crossref] [PubMed]

V. Kohn, I. Snigireva, and A. Snigirev, “Direct Measurement of Transverse Coherence Length of Hard X Rays from Interference Fringes,” Phys. Rev. Lett. 85(13), 2745–2748 (2000).
[Crossref] [PubMed]

Pro. Ph. So. (1)

J. M. Cowley and A. F. Moodie, “Fourier Images IV: The Phase Grating,” Pro. Ph. So. 76(3), 378–384 (1960).
[Crossref]

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, “On the possibilities of x-ray phase contrast microimaging by coherent high-energy synchrotron radiation,” Rev. Sci. Instrum. 66(12), 5486–5492 (1995).
[Crossref]

SPIE (2)

C. G. Schroer, B. Lengeler, B. Benner, T. F. Günzler, M. Kuhlmann, A. S. Simionovici, S. Bohic, M. Drakopoulos, A. Snigirev, I. Snigireva, and W. Schröder, “Microbeam Production Using Compound Refractive Lenses: Beam Characterization and Applications,” SPIE 4499, 52–63 (2001).

A. Snigirev, I. Snigireva, M. Grigoriev, V. Yunkin, M. Di Michiel, S. Kuznetsov, and G. Vaughan, “Silicon planar lenses for high-energy x-ray nanofocusing,” SPIE 6705, 670506 (2007).

Other (2)

K. Patorski, “I The Self-Imaging Phenomenon and its Applications,” in Progress in Optics, E. Wolf, ed. 1–108 (1989).

Ya. M. Hartman and A. Snigirev, “Some examples of high energy X-rays phase contrast,” in X-ray Microscopy IV (Bogorodskii Pechatnik, Chernogolovka, Moscow region, Russia), 429– 432 (1994).

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Figures (5)

Fig. 1
Fig. 1 (a) Schematic view of sixlens interferometer. (b) Scanning Electron Microscope image of the sixlens interferometers fabricated at the Si substrate.
Fig. 2
Fig. 2 (a) An interference pattern generated by a hexalens, recorded with 12 keV X-rays. (b) The intensity variation obtained for the line though the centre of the fringe pattern. The contrast visibility is approximately 71%. (c) The comparison of the intensity variations of the interference pattern experimentally measured and calculated one.
Fig. 3
Fig. 3 (a) Scanning Electron Microscopy image of bilens and sixlens interferometers and (b) comparison of the fringe patterns produced by bilens and sixlens interferometers.
Fig. 4
Fig. 4 (a) Experimental set-up fort the hexalens test with the point secondary source produced by a Si planar refractive lens. (b) An interference pattern generated by a hexalens, recorded at a 1 Å wavelength. (c) The intensity variation obtained for the line though the centre of the fringe pattern.
Fig. 5
Fig. 5 Calculated interference pattern generated by sixlens interferometer for 10 μm source size at 30 keV X-rays.

Tables (1)

Tables Icon

Table 1 Parameters of sixlens interferometers

Equations (12)

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P ( x , z ) = 1 ( i λ z ) 1 2 exp ( i π x 2 λ z ) ,
T ( x )   =   e x p ( i K   [ δ i β ]   t ( x ) ) ,
t k ( x ) = N ( x x k ) 2 / R , x k = d ( k M + 1 2 ) , k = 1 , ... M
I ( x ) = | a ( x 0 | 2 , a ( x 0 ) = d x 1 P ( x 0 x 1 , z r ) T ( x 1 ) , x 0 = x z 0 / z t
r k r j = r k j ( 0 ) = d 2 z [ ( k j ) ( k + j 2 M + 1 2 ) ]
r k r j = r k j ( 0 ) x d z n ( k j ) = r k j ( 0 ) λ n x d ( k j )
K f ( x , z n ) = M + 2 m = 1 M 1 ( M m ) cos ( 2 π m Λ x ) , Λ = d n
α = ( 2 π Λ ) 2 M 2 1 12 , w t = 1.665 α 1 / 2 = Λ 0.918 ( M 2 1 ) 1 / 2
K 1 f ( 0 , s ) = M + 2 m = 1 M 1 l = ( M m 1 ) / 2 ( M m 1 ) / 2 cos ( 2 π m l s )
K 1 f ( 0 , s ) = M 2 ( 2 π s ) 2 C , C = m = 1 M 1 m 2 l = ( M m 1 ) / 2 ( M m 1 ) / 2 l 2 = M 2 ( M 2 1 ) ( M 2 4 ) 720
w l = d 2 w s λ M 2 7.111 Λ 2 λ M 2 8.438 w t 2 λ ,
1 z f + z n + 1 z 0 = 1 z f ( ) + z n , z f = R 2 δ N , z n = d 2 λ n , z f = z f 1 z f / z 0

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