Abstract

Typically, refractive lenses are used to focus rays of light, but an alternative way can be found by exploiting diffraction of light. It is well known that cascades of hard-edge apertures are able to focus light but with the great drawback of absorption losses. In this paper, we demonstrate that replacing hard-edge apertures with π-phase plates within a cascade greatly improves the focusing of collimated Gaussian beams. In addition, we propose a simple model to design this cascade, in particular to find the locations and the radii of the different optics once the focal length has been chosen. This model deduced from numerical simulation is useful for sizing cascades consisting of a high number of components and characterized by a strong focusing ability, without requiring a time-consuming optimization process.

© 2010 Optical Society of America

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  1. A. V. Baez, “Fresnel zone plate for optical image formation using extreme ultraviolet and soft x radiation,” J. Opt. Soc. Am. 51, 405–412 (1961).
    [CrossRef]
  2. G. Boivin, “Use of a fresnel zone plate for optical image information with short wavelength radiations,” Appl. Opt. 16, 1070–1073 (1977).
    [CrossRef] [PubMed]
  3. Y. A. Basov, D. V. Roshchupkin, and A. E. Yakshin, “Grazing incidence phase fresnel zone plate for x-ray focusing,” Opt. Commun. 109, 324–327 (1994).
    [CrossRef]
  4. V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
    [CrossRef]
  5. R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime—new insights,” Opt. Express 16, 9554–9564 (2008).
    [CrossRef] [PubMed]
  6. J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
    [CrossRef]
  7. V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
    [CrossRef]
  8. M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
    [CrossRef]
  9. N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
    [CrossRef]
  10. G. Otis, J.-L. Lachambre, and P. Lavigne, “Focusing of laser beams by a sequence of irises,” Appl. Opt. 18, 875–883 (1979).
    [CrossRef] [PubMed]
  11. R. R. Letfullin and T. F. George, “Optical effect of diffractive multifocal focusing of radiation on a bicomponent diffraction system,” Appl. Opt. 39, 2545–2550 (2000).
    [CrossRef]
  12. R. R. Letfullin, O. A. Zayakin, and T. F. George, “Theoretical and experimental investigations of the effect of diffractive multifocal focusing of radiation,” Appl. Opt. 40, 2138–2147 (2001).
    [CrossRef]
  13. R. R. Letfullin and O. A. Zayakin, “Diffractive focusing of a Gaussian beam,” J. Russ. Laser Res. 23, 148–160 (2002).
    [CrossRef]
  14. J. T. Foley, R. R. Letfullin, and T. F. George, Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2005), Vol. 139, Chap. 14, “The Diffractive Multifocal Focusing Effect,” pp. 289–318.
  15. R. R. Letfullin, T. F. George, and A. Siahmakoun, “Diffractive multifocal focusing of De Broglie matter waves,” J. Nanophotonics 1, 013553 (2007).
    [CrossRef]
  16. R. R. Letfullin, T. F. George, A. Siahmakoun, and M. J. McInerney, “De Broglie wave lens,” Opt. Eng. 47, 028001 (2008).
    [CrossRef]
  17. R. Bourouis, K. Aït-Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
    [CrossRef]
  18. G. Zhao, X. Ji, and B. Lu, “Approximate analytical propagation equations of Gaussian beams through hard-aperture optics,” Optik (Stuttgart) 114, 241–245 (2003).
  19. J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
    [CrossRef]
  20. S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
    [CrossRef]
  21. L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 25, 33–54 (1996).
  22. W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
    [CrossRef]
  23. T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
    [CrossRef]

2009

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[CrossRef]

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

2008

R. R. Letfullin, T. F. George, A. Siahmakoun, and M. J. McInerney, “De Broglie wave lens,” Opt. Eng. 47, 028001 (2008).
[CrossRef]

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

R. G. Mote, S. F. Yu, B. K. Ng, W. Zhou, and S. P. Lau, “Near-field focusing properties of zone plates in visible regime—new insights,” Opt. Express 16, 9554–9564 (2008).
[CrossRef] [PubMed]

2007

R. R. Letfullin, T. F. George, and A. Siahmakoun, “Diffractive multifocal focusing of De Broglie matter waves,” J. Nanophotonics 1, 013553 (2007).
[CrossRef]

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

2005

J. T. Foley, R. R. Letfullin, and T. F. George, Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2005), Vol. 139, Chap. 14, “The Diffractive Multifocal Focusing Effect,” pp. 289–318.

2003

G. Zhao, X. Ji, and B. Lu, “Approximate analytical propagation equations of Gaussian beams through hard-aperture optics,” Optik (Stuttgart) 114, 241–245 (2003).

2002

R. R. Letfullin and O. A. Zayakin, “Diffractive focusing of a Gaussian beam,” J. Russ. Laser Res. 23, 148–160 (2002).
[CrossRef]

2001

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

R. R. Letfullin, O. A. Zayakin, and T. F. George, “Theoretical and experimental investigations of the effect of diffractive multifocal focusing of radiation,” Appl. Opt. 40, 2138–2147 (2001).
[CrossRef]

2000

1997

R. Bourouis, K. Aït-Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

1996

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 25, 33–54 (1996).

1995

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

1994

Y. A. Basov, D. V. Roshchupkin, and A. E. Yakshin, “Grazing incidence phase fresnel zone plate for x-ray focusing,” Opt. Commun. 109, 324–327 (1994).
[CrossRef]

1988

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

1987

V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
[CrossRef]

1979

1977

1970

S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[CrossRef]

1961

Ait-Ameur, K.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

Aït-Ameur, K.

R. Bourouis, K. Aït-Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

Alda, J.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[CrossRef]

Aristov, V.

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

Aristov, V. V.

V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
[CrossRef]

Baez, A. V.

Basov, Y. A.

Y. A. Basov, D. V. Roshchupkin, and A. E. Yakshin, “Grazing incidence phase fresnel zone plate for x-ray focusing,” Opt. Commun. 109, 324–327 (1994).
[CrossRef]

V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
[CrossRef]

Boivin, G.

Boudebs, G.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

Bourouis, R.

R. Bourouis, K. Aït-Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

Breazeale, M. A.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Cagniot, E.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

Collins, S. A.

S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[CrossRef]

Dhayalan, V.

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

Dhez, P.

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

Foley, J. T.

J. T. Foley, R. R. Letfullin, and T. F. George, Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2005), Vol. 139, Chap. 14, “The Diffractive Multifocal Focusing Effect,” pp. 289–318.

Fromager, M.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

George, T. F.

R. R. Letfullin, T. F. George, A. Siahmakoun, and M. J. McInerney, “De Broglie wave lens,” Opt. Eng. 47, 028001 (2008).
[CrossRef]

R. R. Letfullin, T. F. George, and A. Siahmakoun, “Diffractive multifocal focusing of De Broglie matter waves,” J. Nanophotonics 1, 013553 (2007).
[CrossRef]

J. T. Foley, R. R. Letfullin, and T. F. George, Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2005), Vol. 139, Chap. 14, “The Diffractive Multifocal Focusing Effect,” pp. 289–318.

R. R. Letfullin, O. A. Zayakin, and T. F. George, “Theoretical and experimental investigations of the effect of diffractive multifocal focusing of radiation,” Appl. Opt. 40, 2138–2147 (2001).
[CrossRef]

R. R. Letfullin and T. F. George, “Optical effect of diffractive multifocal focusing of radiation on a bicomponent diffraction system,” Appl. Opt. 39, 2545–2550 (2000).
[CrossRef]

Godin, T.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

Guerin, P.

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

Idir, M.

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

Ingber, L.

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 25, 33–54 (1996).

Isoyan, A.

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

Ji, P.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

Ji, X.

G. Zhao, X. Ji, and B. Lu, “Approximate analytical propagation equations of Gaussian beams through hard-aperture optics,” Optik (Stuttgart) 114, 241–245 (2003).

Kohn, V.

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

Kuyumchyan, A.

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

Lachambre, J.-L.

Ladan, F.

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

Ladjouze, H.

R. Bourouis, K. Aït-Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

Lau, S. P.

Lavigne, P.

Letfullin, R. R.

R. R. Letfullin, T. F. George, A. Siahmakoun, and M. J. McInerney, “De Broglie wave lens,” Opt. Eng. 47, 028001 (2008).
[CrossRef]

R. R. Letfullin, T. F. George, and A. Siahmakoun, “Diffractive multifocal focusing of De Broglie matter waves,” J. Nanophotonics 1, 013553 (2007).
[CrossRef]

J. T. Foley, R. R. Letfullin, and T. F. George, Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2005), Vol. 139, Chap. 14, “The Diffractive Multifocal Focusing Effect,” pp. 289–318.

R. R. Letfullin and O. A. Zayakin, “Diffractive focusing of a Gaussian beam,” J. Russ. Laser Res. 23, 148–160 (2002).
[CrossRef]

R. R. Letfullin, O. A. Zayakin, and T. F. George, “Theoretical and experimental investigations of the effect of diffractive multifocal focusing of radiation,” Appl. Opt. 40, 2138–2147 (2001).
[CrossRef]

R. R. Letfullin and T. F. George, “Optical effect of diffractive multifocal focusing of radiation on a bicomponent diffraction system,” Appl. Opt. 39, 2545–2550 (2000).
[CrossRef]

Liu, W.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

Lu, B.

G. Zhao, X. Ji, and B. Lu, “Approximate analytical propagation equations of Gaussian beams through hard-aperture optics,” Optik (Stuttgart) 114, 241–245 (2003).

McInerney, M. J.

R. R. Letfullin, T. F. George, A. Siahmakoun, and M. J. McInerney, “De Broglie wave lens,” Opt. Eng. 47, 028001 (2008).
[CrossRef]

Mirone, A.

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

Mote, R. G.

Ng, B. K.

Otis, G.

Päivänranta, B.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

Passilly, N.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

Redkin, S. V.

V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
[CrossRef]

Rico-García, J. M.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[CrossRef]

Roshchupkin, D. V.

Y. A. Basov, D. V. Roshchupkin, and A. E. Yakshin, “Grazing incidence phase fresnel zone plate for x-ray focusing,” Opt. Commun. 109, 324–327 (1994).
[CrossRef]

Salgado-Remacha, F. J.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[CrossRef]

Sanchez-Brea, L. M.

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[CrossRef]

Sergienko, N.

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

Shulakov, E.

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

Siahmakoun, A.

R. R. Letfullin, T. F. George, A. Siahmakoun, and M. J. McInerney, “De Broglie wave lens,” Opt. Eng. 47, 028001 (2008).
[CrossRef]

R. R. Letfullin, T. F. George, and A. Siahmakoun, “Diffractive multifocal focusing of De Broglie matter waves,” J. Nanophotonics 1, 013553 (2007).
[CrossRef]

Snigirev, A.

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

Snigirev, A. A.

V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
[CrossRef]

Snigireva, I.

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

Soullie, G.

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

Stamnes, J. J.

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

Wen, J. J.

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

Yakshin, A. E.

Y. A. Basov, D. V. Roshchupkin, and A. E. Yakshin, “Grazing incidence phase fresnel zone plate for x-ray focusing,” Opt. Commun. 109, 324–327 (1994).
[CrossRef]

Yang, J.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

Yu, S. F.

Yunkin, V. A.

V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
[CrossRef]

Zayakin, O. A.

Zhao, G.

G. Zhao, X. Ji, and B. Lu, “Approximate analytical propagation equations of Gaussian beams through hard-aperture optics,” Optik (Stuttgart) 114, 241–245 (2003).

Zhou, W.

Appl. Opt.

Appl. Phys. B: Lasers Opt.

T. Godin, M. Fromager, B. Päivänranta, N. Passilly, G. Boudebs, E. Cagniot, and K. Ait-Ameur, “Considerations about z-scan sensitivity improvement: theory versus experiments,” Appl. Phys. B: Lasers Opt. 95, 579–587 (2009).
[CrossRef]

Contr. Cybernet.

L. Ingber, “Adaptive simulated annealing (ASA): lessons learned,” Contr. Cybernet. 25, 33–54 (1996).

J. Acoust. Soc. Am.

W. Liu, P. Ji, and J. Yang, “Development of a simple and accurate approximation method for the Gaussian beam expansion technique,” J. Acoust. Soc. Am. 123, 3516 (2008).
[CrossRef]

J. J. Wen and M. A. Breazeale, “A diffraction beam field expressed as the superposition of Gaussian beams,” J. Acoust. Soc. Am. 83, 1752–1756 (1988).
[CrossRef]

J. Mod. Opt.

R. Bourouis, K. Aït-Ameur, and H. Ladjouze, “Optimization of the Gaussian beam flattening using a phase-plate,” J. Mod. Opt. 44, 1417–1427 (1997).
[CrossRef]

J. Nanophotonics

R. R. Letfullin, T. F. George, and A. Siahmakoun, “Diffractive multifocal focusing of De Broglie matter waves,” J. Nanophotonics 1, 013553 (2007).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

S. A. Collins, Jr., “Lens-system diffraction integral written in terms of matrix optics,” J. Opt. Soc. Am. A 60, 1168–1177 (1970).
[CrossRef]

J. Russ. Laser Res.

R. R. Letfullin and O. A. Zayakin, “Diffractive focusing of a Gaussian beam,” J. Russ. Laser Res. 23, 148–160 (2002).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. A

V. Aristov, A. Isoyan, V. Kohn, A. Kuyumchyan, E. Shulakov, A. Snigirev, and I. Snigireva, “Study of optical properties of x-ray system based on two zone plates,” Nucl. Instrum. Methods Phys. Res. A 575, 238–241 (2007).
[CrossRef]

V. V. Aristov, Y. A. Basov, S. V. Redkin, A. A. Snigirev, and V. A. Yunkin, “Bragg zone plates for hard x-ray focusing,” Nucl. Instrum. Methods Phys. Res. A 261, 72–74 (1987).
[CrossRef]

Opt. Commun.

M. Idir, A. Mirone, G. Soullie, P. Guerin, F. Ladan, and P. Dhez, “2d focusing with an off-axis elliptical Bragg-Fresnel multilayer lens and application to x-ray imaging,” Opt. Commun. 119, 633–642 (1995).
[CrossRef]

N. Sergienko, V. Dhayalan, and J. J. Stamnes, “Comparison of focusing properties of conventional and diffractive lenses,” Opt. Commun. 194, 225–234 (2001).
[CrossRef]

J. Alda, J. M. Rico-García, F. J. Salgado-Remacha, and L. M. Sanchez-Brea, “Diffractive performance of square Fresnel zone plates,” Opt. Commun. 282, 3402–3407 (2009).
[CrossRef]

Y. A. Basov, D. V. Roshchupkin, and A. E. Yakshin, “Grazing incidence phase fresnel zone plate for x-ray focusing,” Opt. Commun. 109, 324–327 (1994).
[CrossRef]

Opt. Eng.

R. R. Letfullin, T. F. George, A. Siahmakoun, and M. J. McInerney, “De Broglie wave lens,” Opt. Eng. 47, 028001 (2008).
[CrossRef]

Opt. Express

Optik (Stuttgart)

G. Zhao, X. Ji, and B. Lu, “Approximate analytical propagation equations of Gaussian beams through hard-aperture optics,” Optik (Stuttgart) 114, 241–245 (2003).

Other

J. T. Foley, R. R. Letfullin, and T. F. George, Tribute to Emil Wolf: Science and Engineering Legacy of Physical Optics (SPIE Press, 2005), Vol. 139, Chap. 14, “The Diffractive Multifocal Focusing Effect,” pp. 289–318.

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Figures (2)

Fig. 1
Fig. 1

Cascade of m phase plates.

Fig. 2
Fig. 2

Experimental–numerical comparison.

Tables (5)

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Table 1 B min = 10 mm

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Table 2 B min = 50 mm

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Table 3 B min = 10 mm and θ = π

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Table 4 Numerical Validation of Our Model with the First Test Case Mentioned in Section 3 ( B min = f = 10 mm )

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Table 5 Numerical Validation of Our Model for Various Wavelengths ( w 0 = 1 mm , f = 50 mm ) a

Equations (37)

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τ ( ρ a i ) = { exp ( j θ i ) , 0 ρ a i , 1 , ρ > a i , }
Ψ ( ρ , z ) = Γ ( z ) exp [ ρ 2 w ( z ) 2 ] exp [ j k ρ 2 2 R ( z ) ] ,
Γ ( z ) = 2 π 1 w ( z ) exp [ j γ ( z ) ] exp [ j k ( z z 0 ) ] .
w ( z ) = w 0 [ 1 + ( z z 0 z R ) 2 ] 1 2 ,
R ( z ) = ( z z 0 ) [ 1 + ( z R z z 0 ) 2 ] ,
γ ( z ) = arctan ( z z 0 z R ) ,
E 1 ( r , z ) = j k B 1 exp ( j k z ) exp ( j k D 1 2 B 1 r 2 ) × 0 + τ ( ρ a 1 ) Ψ ( ρ , 0 ) exp ( j k A 1 2 B 1 ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ = Γ ( 0 ) j k B 1 exp ( j k z ) exp ( j k D 1 2 B 1 r 2 ) × 0 + τ ( ρ a 1 ) exp ( Q ( 1 ) ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ ,
Q ( 1 ) = 1 w ( 0 ) 2 + j k 2 [ A 1 B 1 + 1 R ( 0 ) ] ,
E 1 ( r , z ) = Γ ( 0 ) j k B 1 exp ( j k z ) exp ( j k D 1 2 B 1 r 2 ) × { [ exp ( j θ 1 ) 1 ] 0 a 1 exp ( Q ( 1 ) ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ + 0 + exp ( Q ( 1 ) ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ } .
circ ( ρ a ) = { 1 , 0 ρ a , 0 , ρ > a , }
E 1 ( r , z ) = Γ ( 0 ) j k B 1 exp ( j k z ) exp ( j k D 1 2 B 1 r 2 ) { [ exp ( j θ 1 ) 1 ] 0 + circ ( ρ a 1 ) exp ( Q ( 1 ) ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ + 0 + exp ( Q ( 1 ) ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ } .
circ ( ρ a ) n = 0 N 1 F n exp ( G n a 2 ρ 2 ) ,
E 1 ( r , z ) Γ ( 0 ) j k B 1 exp ( j k z ) exp ( j k D 1 2 B 1 r 2 ) { [ exp ( j θ 1 ) 1 ] n 1 = 0 N 1 [ F n 1 0 + exp ( Q n 1 ( 1 ) ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ ] + 0 + exp ( Q ( 1 ) ρ 2 ) J 0 ( k B 1 r ρ ) ρ d ρ } ,
Q n 1 ( 1 ) = Q ( 1 ) + G n 1 a 1 2 .
{ n 1 < N , ( F n 1 ( 1 ) , G n 1 ( 1 ) ) = ( F n 1 , G n 1 ) , ( F N ( 1 ) , G N ( 1 ) ) = ( 1 exp ( j θ 1 ) 1 , 0 ) . }
n 1 N , Q n 1 ( 1 ) = Q ( 1 ) + G n 1 ( 1 ) a 1 2 .
0 + exp ( p t 2 ) J 0 ( b t ) t d t = 1 2 p exp ( b 2 4 p ) , Re ( p ) > 0 ,
E 1 ( r , z ) Γ ( 0 ) j k B 1 exp ( j k z ) exp ( j k D 1 2 B 1 r 2 ) [ exp ( j θ 1 ) 1 ] n 1 = 0 N F n 1 ( 1 ) 2 Q n 1 ( 1 ) exp ( k 2 4 Q n 1 ( 1 ) B 1 2 r 2 ) .
E 2 ( r , z ) Γ ( 0 ) j k B 1 j k B 2 exp ( j k z ) exp ( j k D 2 2 B 2 r 2 ) [ exp ( j θ 1 ) 1 ] [ exp ( j θ 2 ) 1 ] × n 1 = 0 N F n 1 ( 1 ) 2 Q n 1 ( 1 ) n 2 = 0 N F n 2 ( 2 ) 2 Q n 2 ( 2 ) exp ( k 2 4 Q n 2 ( 2 ) B 2 2 r 2 ) ,
Q n 2 ( 2 ) = k 2 4 Q n 1 ( 1 ) B 1 2 + j k 2 [ D 1 B 1 + A 2 B 2 ] + G n 2 ( 2 ) a 2 2 .
E m ( r , z ) Γ ( 0 ) j k B 1 j k B 2 j k B m exp ( j k z ) exp ( j k D m 2 B m r 2 ) [ exp ( j θ 1 ) 1 ] [ exp ( j θ 2 ) 1 ] [ exp ( j θ m ) 1 ] n 1 = 0 N F n 1 ( 1 ) 2 Q n 1 ( 1 ) n 2 = 0 N F n 2 ( 2 ) 2 Q n 2 ( 2 ) n m = 0 N F n m ( m ) 2 Q n m ( m ) exp ( k 2 4 Q n m ( m ) B m 2 r 2 ) ,
Q n m ( m ) = k 2 4 Q n m 1 ( m 1 ) B m 1 2 + j k 2 [ D m 1 B m 1 + A m B m ] + G n m ( m ) a m 2 .
a 2 2 = λ N f B 2 ,
a 1 2 = λ N f ( B 1 + B 2 ) ,
a 1 2 = λ N B 1 ,
1 = N f N + α 2 ,
a 1 a 2 a m ,
β 1 ,
a 1 > a 2 > > a m ,
a k = [ N f λ z k ] 1 2 .
a k = [ N f λ z k ] 1 2 = [ N λ ( z k z k 1 ) ] 1 2 , k > 0 ,
z k = ( N N N f ) k f , k 0 .
N N N f > 2 .
N f < N < 2 N f .
N f λ z k w 0 2 .
( N N N f ) k w 0 2 N f λ f .
m ¯ = ln ( w 0 2 N f λ f ) ln ( N N N f ) ,

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