Abstract

When calculating the focusing properties of cylindrically symmetric focusing optics, numerical wave propagation calculations can be carried out using the quasi-discrete Hankel transform (QDHT). We describe here an implementation of the QDHT where a partial transform matrix can be stored to speed up repeated wave propagations over specified distances, with reduced computational memory requirements. The accuracy of the approach is then verified by comparison with analytical results, over propagation distances with both small and large Fresnel numbers. We then demonstrate the utility of this approach for calculating the focusing properties of Fresnel zone plate optics that are commonly used for x-ray imaging applications and point to future applications of this approach.

© 2015 Optical Society of America

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References

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2014 (4)

H. Yan, R. Conley, N. Bouet, and Y. S. Chu, “Hard x-ray nanofocusing by multilayer Laue lenses,” J. Phys. D 47, 263001 (2014).

S. Werner, S. Rehbein, P. Guttmann, and G. Schneider, “Three-dimensional structured on-chip stacked zone plates for nanoscale x-ray imaging with high efficiency,” Nano Res. 7, 1–8 (2014).

S.-C. Gleber, M. Wojcik, J. Liu, C. Roehrig, M. Cummings, J. Vila-Comamala, K. Li, B. Lai, D. Shu, and S. Vogt, “Fresnel zone plate stacking in the intermediate field for high efficiency focusing in the hard x-ray regime,” Opt. Express 22, 28142–28153 (2014).
[Crossref]

C. Pratsch, S. Rehbein, S. Werner, and G. Schneider, “Influence of random zone positioning errors on the resolving power of Fresnel zone plates,” Opt. Express 22, 30482–30491 (2014).
[Crossref]

2013 (1)

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

2011 (1)

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
[Crossref]

2010 (3)

M. J. Simpson and A. G. Michette, “The effects of manufacturing inaccuracies on the imaging properties of Fresnel zone plates,” Opt. Acta 30, 1455–1462 (2010).
[Crossref]

A. W. Norfolk and E. J. Grace, “Reconstruction of optical fields with the quasi-discrete Hankel transform,” Opt. Express 18, 10551–10556 (2010).
[Crossref]

A. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Photonics 4, 840–848 (2010).
[Crossref]

2009 (2)

P. Srisungsitthisunti, O. K. Ersoy, and X. Xu, “Beam propagation modeling of modified volume Fresnel zone plates fabricated by femtosecond laser direct writing,” J. Opt. Soc. Am. A 26, 188–194 (2009).
[Crossref]

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

2007 (1)

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

2004 (1)

2003 (1)

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

2002 (1)

Q. Luo and C. T. Law, “Discrete Bessel-based Arnoldi method for nonparaxial wave propagation,” IEEE Photon. Technol. Lett. 14, 50–52 (2002).
[Crossref]

2001 (2)

Q. Luo and C. T. Law, “Propagation of nonparaxial beams with a modified Arnoldi method,” Opt. Lett. 26, 1708–1710 (2001).
[Crossref]

S. D. Shastri, J. M. Maser, B. Lai, and J. Tys, “Microfocusing of 50 keV undulator radiation with two stacked zoneplates,” Opt. Commun. 197, 9–14 (2001).
[Crossref]

2000 (1)

1998 (1)

1997 (1)

1996 (1)

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

1993 (1)

B. X. Yang, “Fresnel and refractive lenses for x-rays,” Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993).

1992 (2)

R. P. Ratowsky, J. A. Fleck, and M. D. Feit, “Helmholtz beam propagation in rib waveguides and couplers by iterative Lanczos reduction,” J. Opt. Soc. Am. A 9, 265–273 (1992).
[Crossref]

J. Maser and G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[Crossref]

1989 (1)

1988 (1)

A. G. Michette, “X-ray microscopy,” Rep. Prog. Phys. 51, 1525–1606 (1988).
[Crossref]

1974 (1)

1957 (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).

1948 (1)

Attwood, D. T.

A. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Photonics 4, 840–848 (2010).
[Crossref]

Baez, A. V.

Bouet, N.

H. Yan, R. Conley, N. Bouet, and Y. S. Chu, “Hard x-ray nanofocusing by multilayer Laue lenses,” J. Phys. D 47, 263001 (2014).

Budai, J. D.

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
[Crossref]

Chen, M. Z.

Chen, W. Z.

Chu, Y. S.

H. Yan, R. Conley, N. Bouet, and Y. S. Chu, “Hard x-ray nanofocusing by multilayer Laue lenses,” J. Phys. D 47, 263001 (2014).

Conley, R.

H. Yan, R. Conley, N. Bouet, and Y. S. Chu, “Hard x-ray nanofocusing by multilayer Laue lenses,” J. Phys. D 47, 263001 (2014).

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Cowley, J. M.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).

Cummings, M.

David, C.

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Diaz, A.

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

Drakopoulos, M.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Ersoy, O. K.

Feit, M. D.

Feste, S.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Fleck, J. A.

Frehse, F.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Gleber, S.-C.

Goodman, J.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

Grace, E. J.

Guizar-Sicairos, M.

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

M. Guizar-Sicairos and J. C. Gutiérrez-Vega, “Computation of quasi-discrete Hankel transforms of integer order for propagating optical wave fields,” J. Opt. Soc. Am. A 21, 53–58 (2004).
[Crossref]

Gunzler, T.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Gutiérrez-Vega, J. C.

Guttmann, P.

S. Werner, S. Rehbein, P. Guttmann, and G. Schneider, “Three-dimensional structured on-chip stacked zone plates for nanoscale x-ray imaging with high efficiency,” Nano Res. 7, 1–8 (2014).

Handa, S.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Howells, M.

M. Howells, C. Jacobsen, and T. Warwick, “Principles and applications of zone plate x-ray microscopes,” in Science of Microscopy, P. W. Hawkes and J. C. H. Spence, eds., 1st ed. (Springer, 2006), Chap. 13.

Huang, M. C.

Huang, W. D.

Hunger, U.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Ice, G. E.

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
[Crossref]

Inagaki, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Ishikawa, T.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Jacobsen, C.

M. Howells, C. Jacobsen, and T. Warwick, “Principles and applications of zone plate x-ray microscopes,” in Science of Microscopy, P. W. Hawkes and J. C. H. Spence, eds., 1st ed. (Springer, 2006), Chap. 13.

Jefimovs, K.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Kang, H.

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Kewish, C. M.

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

Kimura, T.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Kirkpatrick, P.

Kirz, J.

Kohn, V.

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

Kuhlmann, M.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Kurapova, O.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Lai, B.

Law, C. T.

Lemoine, D.

Lengeler, B.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

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

Li, K.

Liu, C.

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Liu, J.

Luo, Q.

Macrander, A.

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Maser, J.

J. Maser and G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[Crossref]

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Maser, J. M.

S. D. Shastri, J. M. Maser, B. Lai, and J. Tys, “Microfocusing of 50 keV undulator radiation with two stacked zoneplates,” Opt. Commun. 197, 9–14 (2001).
[Crossref]

Matsuyama, S.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Michette, A. G.

M. J. Simpson and A. G. Michette, “The effects of manufacturing inaccuracies on the imaging properties of Fresnel zone plates,” Opt. Acta 30, 1455–1462 (2010).
[Crossref]

A. G. Michette, “X-ray microscopy,” Rep. Prog. Phys. 51, 1525–1606 (1988).
[Crossref]

A. G. Michette, Optical Systems for Soft X Rays (Plenum, 1986).

Mimura, H.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Moodie, A. F.

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).

Nishino, Y.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Norfolk, A. W.

Pang, J. W.

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
[Crossref]

Pilvi, T.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Pratsch, C.

Rabbe, J.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Ratowsky, R. P.

Rehbein, S.

S. Werner, S. Rehbein, P. Guttmann, and G. Schneider, “Three-dimensional structured on-chip stacked zone plates for nanoscale x-ray imaging with high efficiency,” Nano Res. 7, 1–8 (2014).

C. Pratsch, S. Rehbein, S. Werner, and G. Schneider, “Influence of random zone positioning errors on the resolving power of Fresnel zone plates,” Opt. Express 22, 30482–30491 (2014).
[Crossref]

Ritala, M.

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Roehrig, C.

Sakdinawat, A.

A. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Photonics 4, 840–848 (2010).
[Crossref]

Sano, Y.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Schmahl, G.

J. Maser and G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[Crossref]

Schneider, G.

S. Werner, S. Rehbein, P. Guttmann, and G. Schneider, “Three-dimensional structured on-chip stacked zone plates for nanoscale x-ray imaging with high efficiency,” Nano Res. 7, 1–8 (2014).

C. Pratsch, S. Rehbein, S. Werner, and G. Schneider, “Influence of random zone positioning errors on the resolving power of Fresnel zone plates,” Opt. Express 22, 30482–30491 (2014).
[Crossref]

Schroder, W.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Schroer, C.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Schug, C.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Shastri, S. D.

S. D. Shastri, J. M. Maser, B. Lai, and J. Tys, “Microfocusing of 50 keV undulator radiation with two stacked zoneplates,” Opt. Commun. 197, 9–14 (2001).
[Crossref]

Shu, D.

Simionovici, A.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Simpson, M. J.

M. J. Simpson and A. G. Michette, “The effects of manufacturing inaccuracies on the imaging properties of Fresnel zone plates,” Opt. Acta 30, 1455–1462 (2010).
[Crossref]

Snigirev, A.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

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

Snigireva, I.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

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

Somogyi, A.

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

Srisungsitthisunti, P.

Stephenson, G.

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Tamasaku, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Tys, J.

S. D. Shastri, J. M. Maser, B. Lai, and J. Tys, “Microfocusing of 50 keV undulator radiation with two stacked zoneplates,” Opt. Commun. 197, 9–14 (2001).
[Crossref]

Vila-Comamala, J.

S.-C. Gleber, M. Wojcik, J. Liu, C. Roehrig, M. Cummings, J. Vila-Comamala, K. Li, B. Lai, D. Shu, and S. Vogt, “Fresnel zone plate stacking in the intermediate field for high efficiency focusing in the hard x-ray regime,” Opt. Express 22, 28142–28153 (2014).
[Crossref]

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Vogt, S.

S.-C. Gleber, M. Wojcik, J. Liu, C. Roehrig, M. Cummings, J. Vila-Comamala, K. Li, B. Lai, D. Shu, and S. Vogt, “Fresnel zone plate stacking in the intermediate field for high efficiency focusing in the hard x-ray regime,” Opt. Express 22, 28142–28153 (2014).
[Crossref]

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Wang, S.

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

Warwick, T.

M. Howells, C. Jacobsen, and T. Warwick, “Principles and applications of zone plate x-ray microscopes,” in Science of Microscopy, P. W. Hawkes and J. C. H. Spence, eds., 1st ed. (Springer, 2006), Chap. 13.

Werner, S.

S. Werner, S. Rehbein, P. Guttmann, and G. Schneider, “Three-dimensional structured on-chip stacked zone plates for nanoscale x-ray imaging with high efficiency,” Nano Res. 7, 1–8 (2014).

C. Pratsch, S. Rehbein, S. Werner, and G. Schneider, “Influence of random zone positioning errors on the resolving power of Fresnel zone plates,” Opt. Express 22, 30482–30491 (2014).
[Crossref]

Wojcik, M.

S.-C. Gleber, M. Wojcik, J. Liu, C. Roehrig, M. Cummings, J. Vila-Comamala, K. Li, B. Lai, D. Shu, and S. Vogt, “Fresnel zone plate stacking in the intermediate field for high efficiency focusing in the hard x-ray regime,” Opt. Express 22, 28142–28153 (2014).
[Crossref]

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

Xu, X.

Yabashi, M.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Yamakawa, D.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Yamamura, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Yamauchi, K.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Yan, H.

H. Yan, R. Conley, N. Bouet, and Y. S. Chu, “Hard x-ray nanofocusing by multilayer Laue lenses,” J. Phys. D 47, 263001 (2014).

Yang, B. X.

B. X. Yang, “Fresnel and refractive lenses for x-rays,” Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993).

Yokoyama, H.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Yu, L.

Yumoto, H.

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Yun, W.

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

Zhu, Z. Z.

Acta Crystallogr. (1)

J. M. Cowley and A. F. Moodie, “The scattering of electrons by atoms and crystals. I. A new theoretical approach,” Acta Crystallogr. 10, 609–619 (1957).

Appl. Phys. Lett. (1)

C. Schroer, M. Kuhlmann, U. Hunger, T. Gunzler, O. Kurapova, S. Feste, F. Frehse, B. Lengeler, M. Drakopoulos, A. Somogyi, A. Simionovici, A. Snigirev, I. Snigireva, C. Schug, and W. Schroder, “Nanofocusing parabolic refractive x-ray lenses,” Appl. Phys. Lett. 82, 1485–1487 (2003).
[Crossref]

IEEE Photon. Technol. Lett. (1)

Q. Luo and C. T. Law, “Discrete Bessel-based Arnoldi method for nonparaxial wave propagation,” IEEE Photon. Technol. Lett. 14, 50–52 (2002).
[Crossref]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (4)

J. Phys. D (1)

H. Yan, R. Conley, N. Bouet, and Y. S. Chu, “Hard x-ray nanofocusing by multilayer Laue lenses,” J. Phys. D 47, 263001 (2014).

J. Synchrotron Radiat. (1)

J. Vila-Comamala, M. Wojcik, A. Diaz, M. Guizar-Sicairos, C. M. Kewish, S. Wang, and C. David, “Angular spectrum simulation of x-ray focusing by Fresnel zone plates,” J. Synchrotron Radiat. 20, 397–404 (2013).
[Crossref]

Nano Res. (1)

S. Werner, S. Rehbein, P. Guttmann, and G. Schneider, “Three-dimensional structured on-chip stacked zone plates for nanoscale x-ray imaging with high efficiency,” Nano Res. 7, 1–8 (2014).

Nat. Photonics (1)

A. Sakdinawat and D. T. Attwood, “Nanoscale x-ray imaging,” Nat. Photonics 4, 840–848 (2010).
[Crossref]

Nat. Phys. (1)

H. Mimura, S. Handa, T. Kimura, H. Yumoto, D. Yamakawa, H. Yokoyama, S. Matsuyama, K. Inagaki, K. Yamamura, Y. Sano, K. Tamasaku, Y. Nishino, M. Yabashi, T. Ishikawa, and K. Yamauchi, “Breaking the 10 nm barrier in hard-x-ray focusing,” Nat. Phys. 6, 122–125 (2009).
[Crossref]

Nature (1)

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

Nucl. Instrum. Methods Phys. Res. A (1)

B. X. Yang, “Fresnel and refractive lenses for x-rays,” Nucl. Instrum. Methods Phys. Res. A 328, 578–587 (1993).

Opt. Acta (1)

M. J. Simpson and A. G. Michette, “The effects of manufacturing inaccuracies on the imaging properties of Fresnel zone plates,” Opt. Acta 30, 1455–1462 (2010).
[Crossref]

Opt. Commun. (2)

J. Maser and G. Schmahl, “Coupled wave description of the diffraction by zone plates with high aspect ratios,” Opt. Commun. 89, 355–362 (1992).
[Crossref]

S. D. Shastri, J. M. Maser, B. Lai, and J. Tys, “Microfocusing of 50 keV undulator radiation with two stacked zoneplates,” Opt. Commun. 197, 9–14 (2001).
[Crossref]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. Lett. (1)

K. Jefimovs, J. Vila-Comamala, T. Pilvi, J. Rabbe, M. Ritala, and C. David, “Zone-doubling technique to produce ultrahigh-resolution x-ray optics,” Phys. Rev. Lett. 99, 264801 (2007).
[Crossref]

Rep. Prog. Phys. (1)

A. G. Michette, “X-ray microscopy,” Rep. Prog. Phys. 51, 1525–1606 (1988).
[Crossref]

Science (1)

G. E. Ice, J. D. Budai, and J. W. Pang, “The race to x-ray microbeam and nanobeam science,” Science 334, 1234–1239 (2011).
[Crossref]

Other (4)

M. Howells, C. Jacobsen, and T. Warwick, “Principles and applications of zone plate x-ray microscopes,” in Science of Microscopy, P. W. Hawkes and J. C. H. Spence, eds., 1st ed. (Springer, 2006), Chap. 13.

J. Goodman, Introduction to Fourier Optics, 3rd ed. (Roberts & Company, 2005).

J. Maser, G. Stephenson, S. Vogt, W. Yũn, A. Macrander, H. Kang, C. Liu, and R. Conley, “Multilayer Laue lenses as high-resolution x-ray optics,” in Design and Microfabrication of Novel X-ray Optics II, A. Snigirev and D. Mancini, eds. (SPIE, 2004), Vol. 5539, pp. 185–194.

A. G. Michette, Optical Systems for Soft X Rays (Plenum, 1986).

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Figures (8)

Fig. 1.
Fig. 1. Real space [Eq. (12)] and reciprocal space [Eq. (13)] propagators shown for two different propagation distances z with λ = 1 nm . In each case, the real part is shown in blue and the imaginary part in green. The dots are the positions of N = 1000 sampling points over a radius of R = 50 μm , for which Eq. (17) gives z 0 = 5 mm . The reciprocal space propagator is more slowly varying at short propagation distances, while the real space propagator is more slowly varying at longer distances.
Fig. 2.
Fig. 2. Sampling grid of the QDHT (red cross marks) and FFT (blue square marks). The QDHT sampling points are nonequally spaced and do not include α = 0 .
Fig. 3.
Fig. 3. Schematic of the partial transform matrix C N , M . If one wishes to calculate the wavefield over only a subset of M points on the output plane, a nonsymmetric matrix C N , M [Eq. (32)] can be used to reduce computational time and memory requirements in Eq. (33).
Fig. 4.
Fig. 4. Rapid QDHT calculation of the focusing of λ = 0.124 nm (10 keV) x rays by a fully absorptive Fresnel zone plate with 320 μm diameter and d r zp = 20 nm outermost zone width. The image shows the intensity as a function of radial distance r and axial and defocus distance Δ z , with a grid spacing of Δ r = 2 nm in radius and Δ z = 1 μm in defocus positions (intensities at negative radii are simply copied from the positive radii calculation points). The wavefield exiting the zone plate is used as the input wavefield to generate output wavefields at various distances.
Fig. 5.
Fig. 5. Comparison of the numerical QDHT calculation of far-field diffraction intensity of a pinhole using Eq. (21), versus the analytical result of the Airy pattern. The image on the left shows the radial intensity distribution along with the integral of intensity with radius, while the image on the right shows the intensity distribution along with the percentage difference from the analytical result. As is shown, the QDHT calculation with parameters as described in the text is accurate to within 0.12% of the expected analytical result.
Fig. 6.
Fig. 6. Comparison of the numerical QDHT calculation of the radial intensity distribution near the focus of a Fresnel zone plate ( D = 45 μm diameter, d r zp = 25 nm outermost zone width using λ = 0.124 nm or 10 keV x rays, yielding a focal length of f = D d r zp / λ = 9 mm ) using Eq. (21), versus the analytical result of the Airy pattern. The image on the left shows the radial intensity distribution along with the integral of intensity with radius, while the image on the right shows the intensity distribution along with the percentage difference from the analytical result.
Fig. 7.
Fig. 7. Verification that the QDHT preserves energy over sufficiently large radii of integration. The image on the left shows the intensity calculated from propagating a λ = 0.124 nm wavefield through a 5 μm diameter pinhole to a distance 50 cm downstream (Fresnel number F = 0.1 ). The image on the right shows the calculation for a binary absorption Fresnel zone plate with a diameter of 45 μm and outermost zone width of d r zp = 25 nm , at a distance of 9 mm which is one focal length away. As can be seen, about 1 / π 2 = 10 % of the total energy is located on the optical axis in the form of the first focal order, and about half of the transmitted energy is located near the optical axis in the form of the positive (converging) focal orders. The integrated intensity increases up to the point of the radial extent of the zone plate at 22.5 μm, with the remaining energy arriving in the -1 focal order (extending to twice the radius or 45 μm) until nearly all of the 50% nonabsorbed energy is contained within a 60 μm radius, as expected.
Fig. 8.
Fig. 8. Comparison of the focused intensity profile of a Fresnel zone plate (which is a far-distance calculation) as calculated using the near-distance convolution approach of Eq. (20), and the far-distance single Hankel transform approach of Eq. (21). The radial intensity distribution and energy integral is shown on the left, while the longitudinal intensity distribution about the focal point is shown on the right. These calculations assumed a zone plate with diameter D = 45 μm and outermost zone width d r zp = 25 nm , and an x-ray wavelength of λ = 0.124 nm , yielding a focal length of f = D d r zp / λ = 9 mm . The input plane sampling was done with N = 27,000 points over a radius of R = 54 μm , so that the distance z 0 for preferring near-distance or far-distance approaches [Eq. (17)] was z 0 = 1.7 mm . As can be seen, the far-distance method leads to a smooth intensity profile, whereas the near-distance method leads to irregularities in the intensity profile on the optical axis due to the fast oscillations in the reciprocal space propagator function, as shown in Fig. 1. The far-distance calculation approach works better for propagating by a distance of 9 mm when z 0 = 1.7 mm , but the near-distance approach still gives the correct overall intensity distribution at points away from the optical axis.

Equations (39)

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F { g ( x , y ) } = g ( x , y ) e i π ( x f x + y f y ) d x d y ,
F 1 { G ( f x , f y ) } = G ( f x , f y ) e i π ( x f x + y f y ) d f x d f y
ψ z ( x , y ) = i λ z h ( x , y ) F { ψ 0 ( x 0 , y 0 ) h ( x 0 , y 0 ) }
ψ z ( x , y ) = i λ z [ ψ ( x 0 , y 0 ) * h ( x , y ) ] = F 1 { F { ψ 0 ( x 0 , y 0 ) } H ( f x , f y ) } .
h ( x , y ) = e i 2 π λ z 2 + x 2 + y 2 e i π ( x 2 + y 2 ) / ( λ z ) ,
H ( f x , f y ) = i λ z F { h ( x , y ) } = e i 2 π λ z 2 λ 2 z 2 ( f x 2 + f y 2 ) e i π λ z ( f x 2 + f y 2 ) ,
f x = x λ z and f y = y λ z
H { g ( r ) } = 2 π 0 G ( ρ ) J 0 ( 2 π ρ r ) ρ d ρ ,
H 1 { G ( ρ ) } = 2 π 0 g ( r ) J 0 ( 2 π ρ r ) r d r ,
ψ z ( r ) = i λ z h ( r ) H { ψ 0 ( r 0 ) h ( r 0 ) }
ψ z ( r ) = H 1 { H { ψ 0 ( r 0 ) } H ( ρ ) } .
h ( r ) = e i 2 π λ z 2 + r 2 e i π r 2 / ( λ z ) ,
H ( ρ ) = i λ z H { h ( r ) } = e i 2 π λ z 2 λ 2 z 2 ρ 2 e i π λ z ρ 2 ,
ρ = r λ z
N real = R 2 λ z = N 2 Δ r 2 λ z ,
N Hankel = λ z P 2 = λ z 4 Δ r 2 .
z 0 = 2 R Δ r λ = 2 R 2 λ N .
F = a 2 λ L .
N = 2 R 2 λ z 0 = 2 F 0 ,
ψ 0 ( r 0 ) H { } × e i π λ z ρ 2 H 1 { } ψ z ( r ) ( z z 0 ) ,
ψ 0 ( r 0 ) × e i π r 0 2 / ( λ z ) H { } × i λ z e i π r 2 / ( λ z ) ψ z ( r ) ( z z 0 ) ,
f = 2 R d r zp λ > 2 R 2 λ N = z 0 ,
r α n / ( 2 π P ) in real space ,
ρ α m / ( 2 π R ) in reciprocal space
G ( α m P / S ) = 1 π P 2 n = 1 N g ( α n R / S ) J 1 2 ( α n ) J 0 ( α n α m / S ) ,
g ( α n R / S ) = 1 π R 2 m = 1 N G ( α m P / S ) J 1 2 ( α m ) J 0 ( α m α n / S ) ,
S = 2 π R P ,
g ( r ) = n = 1 g ( r n ) K n ( r ) ,
K n ( r ) = r n J 0 ( 2 π P r ) π P ( r n 2 r 2 ) J 1 ( 2 π P r n ) .
g ( r = 0 ) = n = 1 N g ( r n ) π P r n J 1 ( 2 π P r n ) ,
{ G ( ρ m ) = n = 1 N g ( r n ) π P 2 J 1 2 ( α n ) J 0 ( α n α m S ) g ( r n ) = m = 1 N G ( ρ m ) π R 2 J 1 2 ( α m ) J 0 ( α m α n S ) .
C m , n = J 0 ( α n α m α N + 1 ) J 1 2 ( α m ) ,
{ G ( ρ ) 1 , N = 1 π P 2 [ g ( r ) 1 , N · C N , N ] g ( r ) 1 , N = 1 π R 2 [ G ( ρ ) 1 , N · C N , N ] .
P = π N 2 π R = N 2 R
Δ ρ = 1 2 R .
R = ( λ z ) P = λ z 2 R N ,
Δ r = R N = λ z 2 R = λ z 2 N Δ r .
Δ r = D d r zp 2 R = D d r zp 2 N Δ r .
N = D d r zp 2 ( Δ r ) 2 .

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