Abstract

We have developed iterative algorithms for the calculation of holograms for non-diffracting or self-imaging light beams. Our methods make use of the special Fourier-space properties of the target beams. We demonstrate experimentally the holographic generation of perhaps the most challenging type of beam: a self-imaging beam shaped in more than one plane. Potential applications include the generation of light “crystals” for optical trapping and atomic diffraction studies.

© 2006 Optical Society of America

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References

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  1. J. Durnin, J. J. J. Miceli, and J. H. Eberly, "Diffraction-free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  2. J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
    [CrossRef]
  3. V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
    [CrossRef] [PubMed]
  4. D. McGloin, V. Garcés-Chávez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
    [CrossRef] [PubMed]
  5. E. Goldfain, "Optical biopsy with long-range nondiffracting beams," in Optical Biopsy III, R. R. Alfano, ed., Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), pp. 119-127 (2000).
  6. J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  7. S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
    [CrossRef]
  8. Z. Bouchal and J. Kyvalský, "Controllable 3D spatial localization of light fields synthesized by non-diffracting modes," J. Mod. Opt. 51, 157-176 (2004).
    [CrossRef]
  9. K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).
  10. E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, "3D interferometric optical tweezers using a single spatial light modulator," Opt. Express 13, 3777-3786 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3777.
    [CrossRef] [PubMed]
  11. R. Piestun and J. Shamir, "Control of wave-front propagation with diffractive elements," Opt. Lett. 19, 771-773 (1994).
    [CrossRef] [PubMed]
  12. V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Algorithm for the Generation of Non-diffracting Bessel Modes," J. Mod. Opt. 42, 1231-1239 (1995).
    [CrossRef]
  13. V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: rotating images," J. Mod. Opt. 44, 1409-1416 (1997).
    [CrossRef]
  14. M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5, 1341-1348 (2003).
    [CrossRef]
  15. Z. Bouchal, "Controlled spatial shaping of nondiffracting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002).
    [CrossRef]
  16. G. Indebetouw, "Quasi-self-imaging using aperiodic sequences," J. Opt. Soc. Am. A 9, 549-558 (1992).
    [CrossRef]
  17. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).
  18. M. A. Seldowitz, J. P. Allebach, and D. W. Sweeney, "Synthesis of digital holograms by direct binary search," Appl. Opt. 26, 2788-2798 (1987).
    [CrossRef] [PubMed]
  19. V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd, London, 1997).
  20. T. Haist, M. Schönleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
    [CrossRef]
  21. G. Shabtay, "Three-dimensional beam forming and Ewald’s surfaces," Opt. Commun. 226, 33-37 (2003).
    [CrossRef]
  22. G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 117 (2005).
    [CrossRef]
  23. CRL Opto Ltd., 1024 × 768 pixels, 13.9mm × 8.5mm active area.
  24. L. Santos, "Introduction to Focus Issue: Cold Atomic Gases in Optical Lattices," Optics Express 12, 2-3 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-2.
    [CrossRef] [PubMed]

2005 (2)

2004 (2)

L. Santos, "Introduction to Focus Issue: Cold Atomic Gases in Optical Lattices," Optics Express 12, 2-3 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-2.
[CrossRef] [PubMed]

Z. Bouchal and J. Kyvalský, "Controllable 3D spatial localization of light fields synthesized by non-diffracting modes," J. Mod. Opt. 51, 157-176 (2004).
[CrossRef]

2003 (3)

D. McGloin, V. Garcés-Chávez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
[CrossRef] [PubMed]

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5, 1341-1348 (2003).
[CrossRef]

G. Shabtay, "Three-dimensional beam forming and Ewald’s surfaces," Opt. Commun. 226, 33-37 (2003).
[CrossRef]

2002 (3)

Z. Bouchal, "Controlled spatial shaping of nondiffracting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002).
[CrossRef]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

2001 (1)

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

1997 (2)

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: rotating images," J. Mod. Opt. 44, 1409-1416 (1997).
[CrossRef]

T. Haist, M. Schönleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

1995 (1)

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Algorithm for the Generation of Non-diffracting Bessel Modes," J. Mod. Opt. 42, 1231-1239 (1995).
[CrossRef]

1994 (1)

1992 (1)

1989 (1)

K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).

1987 (3)

1972 (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Allebach, J. P.

Allison, I.

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

Arlt, J.

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Bouchal, Z.

Z. Bouchal and J. Kyvalský, "Controllable 3D spatial localization of light fields synthesized by non-diffracting modes," J. Mod. Opt. 51, 157-176 (2004).
[CrossRef]

Z. Bouchal, "Controlled spatial shaping of nondiffracting patterns and arrays," Opt. Lett. 27, 1376-1378 (2002).
[CrossRef]

Chávez-Cerda, S.

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

Cooper, J.

Courtial, J.

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, "3D interferometric optical tweezers using a single spatial light modulator," Opt. Express 13, 3777-3786 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3777.
[CrossRef] [PubMed]

G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 117 (2005).
[CrossRef]

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

Dennis, M. R.

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5, 1341-1348 (2003).
[CrossRef]

Dholakia, K.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
[CrossRef] [PubMed]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Durnin, J.

J. Durnin, J. J. J. Miceli, and J. H. Eberly, "Diffraction-free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact solutions for nondiffracting beams. I. The scalar theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. J. Miceli, and J. H. Eberly, "Diffraction-free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Garcés-Chávez, V.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
[CrossRef] [PubMed]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Gutiérrez-Vega, J.

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

Haist, T.

T. Haist, M. Schönleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

Indebetouw, G.

Jordan, P.

Khonina, S. N.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: rotating images," J. Mod. Opt. 44, 1409-1416 (1997).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Algorithm for the Generation of Non-diffracting Bessel Modes," J. Mod. Opt. 42, 1231-1239 (1995).
[CrossRef]

Kotlyar, V. V.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: rotating images," J. Mod. Opt. 44, 1409-1416 (1997).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Algorithm for the Generation of Non-diffracting Bessel Modes," J. Mod. Opt. 42, 1231-1239 (1995).
[CrossRef]

Kyvalský, J.

Z. Bouchal and J. Kyvalský, "Controllable 3D spatial localization of light fields synthesized by non-diffracting modes," J. Mod. Opt. 51, 157-176 (2004).
[CrossRef]

MacVicar, I.

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

McGloin, D.

D. McGloin, V. Garcés-Chávez, and K. Dholakia, "Interfering Bessel beams for optical micromanipulation," Opt. Lett. 28, 657-659 (2003).
[CrossRef] [PubMed]

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Melville, H.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

Miceli, J. J. J.

J. Durnin, J. J. J. Miceli, and J. H. Eberly, "Diffraction-free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

New, G.

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

O’Neil, A.

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

Padgett, M.

E. Schonbrun, R. Piestun, P. Jordan, J. Cooper, K. D. Wulff, J. Courtial, and M. Padgett, "3D interferometric optical tweezers using a single spatial light modulator," Opt. Express 13, 3777-3786 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-10-3777.
[CrossRef] [PubMed]

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

Patorski, K.

K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).

Piestun, R.

Santos, L.

L. Santos, "Introduction to Focus Issue: Cold Atomic Gases in Optical Lattices," Optics Express 12, 2-3 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-2.
[CrossRef] [PubMed]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Schonbrun, E.

Schönleber, M.

T. Haist, M. Schönleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

Seldowitz, M. A.

Shabtay, G.

G. Shabtay, "Three-dimensional beam forming and Ewald’s surfaces," Opt. Commun. 226, 33-37 (2003).
[CrossRef]

Shamir, J.

Sibbett, W.

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Soifer, V. A.

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: rotating images," J. Mod. Opt. 44, 1409-1416 (1997).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Algorithm for the Generation of Non-diffracting Bessel Modes," J. Mod. Opt. 42, 1231-1239 (1995).
[CrossRef]

Sweeney, D. W.

Tiziani, H. J.

T. Haist, M. Schönleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

Whyte, G.

G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 117 (2005).
[CrossRef]

Wulff, K. D.

Appl. Opt. (1)

J. Mod. Opt. (3)

Z. Bouchal and J. Kyvalský, "Controllable 3D spatial localization of light fields synthesized by non-diffracting modes," J. Mod. Opt. 51, 157-176 (2004).
[CrossRef]

V. V. Kotlyar, S. N. Khonina, and V. A. Soifer, "Algorithm for the Generation of Non-diffracting Bessel Modes," J. Mod. Opt. 42, 1231-1239 (1995).
[CrossRef]

V. V. Kotlyar, V. A. Soifer, and S. N. Khonina, "An algorithm for the generation of laser beams with longitudinal periodicity: rotating images," J. Mod. Opt. 44, 1409-1416 (1997).
[CrossRef]

J. Opt. B: Quantum Semiclass. Opt. (1)

S. Chávez-Cerda, M. Padgett, I. Allison, G. New, J. Gutiérrez-Vega, A. O’Neil, I. MacVicar, and J. Courtial, "Holographic generation and orbital angular momentum of high-order Mathieu beams," J. Opt. B: Quantum Semiclass. Opt. 4, S52-S57 (2002).
[CrossRef]

J. Opt. Soc. Am. A (2)

Nature (1)

V. Garcés-Chávez, D. McGloin, H. Melville, W. Sibbett, and K. Dholakia, "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam," Nature 419, 145-147 (2002).
[CrossRef] [PubMed]

New J. Phys. (2)

M. R. Dennis, "Braided nodal lines in wave superpositions," New J. Phys. 5, 1341-1348 (2003).
[CrossRef]

G. Whyte and J. Courtial, "Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm," New J. Phys. 7, 117 (2005).
[CrossRef]

Opt. Commun. (3)

T. Haist, M. Schönleber, and H. J. Tiziani, "Computer-generated holograms from 3D-objects written on twistednematic liquid crystal displays," Opt. Commun. 140, 299-308 (1997).
[CrossRef]

G. Shabtay, "Three-dimensional beam forming and Ewald’s surfaces," Opt. Commun. 226, 33-37 (2003).
[CrossRef]

J. Arlt, V. Garcés-Chávez, W. Sibbett, and K. Dholakia, "Optical micromanipulation using a Bessel light beam," Opt. Commun. 197, 239-245 (2001).
[CrossRef]

Opt. Express (1)

Opt. Lett. (3)

Optics Express (1)

L. Santos, "Introduction to Focus Issue: Cold Atomic Gases in Optical Lattices," Optics Express 12, 2-3 (2004). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-1-2.
[CrossRef] [PubMed]

Optik (1)

R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of the phase from image and diffraction plane pictures," Optik 35, 237-246 (1972).

Phys. Rev. Lett. (1)

J. Durnin, J. J. J. Miceli, and J. H. Eberly, "Diffraction-free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Progr. Opt. (1)

K. Patorski, "The self-imaging phenomenon and its applications," Progr. Opt. XXVII, 3-108 (1989).

Other (3)

E. Goldfain, "Optical biopsy with long-range nondiffracting beams," in Optical Biopsy III, R. R. Alfano, ed., Proceedings of the Society of Photo-Optical Instrumentation Engineers (SPIE), pp. 119-127 (2000).

CRL Opto Ltd., 1024 × 768 pixels, 13.9mm × 8.5mm active area.

V. Soifer, V. Kotlyar, and L. Doskolovich, Iterative Methods for Diffractive Optical Elements Computation (Taylor & Francis Ltd, London, 1997).

Supplementary Material (2)

» Media 1: MOV (297 KB)     
» Media 2: MOV (337 KB)     

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Figures (4)

Fig. 1.
Fig. 1.

Experimental generation of ND beams. A laser beam illuminates a thin ring aperture in the front focal plane (plane A) of the Fourier lens. Behind the lens, the beam is non-diffracting. The beam amplitude in the back focal plane of the Fourier lens (plane B) can be easily calculated as the Fourier transform (FT) of the amplitude in plane A. The intensity distributions on the far left and right illustrate the case of a single bright ring (uniform intensity, planar phase) at A, which leads to the simplest Bessel beam behind the lens.

Fig. 2.
Fig. 2.

Simulation of ND (a) and SI (b) light beams, calculated with the GS method, whose intensity cross-section has been shaped to resemble the target intensity shown in (a). In both cases the focal length of the Fourier lens is f = 1m; the radii of the Fourier-space rings are r =10mm (a) and r =6.06mm, 10mm, 12.77mm and 15.05mm (b), resulting in a self-imaging period of 20mm. Shown are intensity cross-sections over a 1mm×1mm area in different transverse planes. The finite width of each ring (the profile is Gaussian with a width of 300μm) limits the distance over which the beams are non-diffracting or diffraction-free to a finite distance; this can be seen in (c).

Fig. 3.
Fig. 3.

Simulated shaped ND and SI beams calculated with our DS method. (a), (b): intensity cross-sections of ND beams shaped into the constellation “Orion” (a diagram of the star position is inset). (c), (d): intensity cross-sections over two self-imaging periods (∆z = 40mm) of SI beams shaped into stretched unit cells of face-centred cubic (fcc, (c)) and body-centred cubic (bcc, (d)) “crystals” of bright spots; the merit points were located in the planes z = 0 and z = 10mm. In (a) both phase and intensity were modulated, in (b) and (d) only the phase (uniform intensity), in (c) only the intensity (uniform phase). In (a) and (b) r = 25mm (1 circle in the Fourier plane), in (c) and (d) r = 6.06mm, 10mm, 12.77mm and 15.05mm (4 circles); each circle contained 512 source points. In all cases f = 1m and λ = 633nm. All intensity cross-sections represent an area of approximately 0.5mm × 0.5mm. The additional multimedia material contains movies showing a computer rendition of the three-dimensional intensity distributions over slightly more than one self-imaging period of the fcc beam (308K) and the bcc beam (348K) as seen from different viewing angles. [Media 1] [Media 2]

Fig. 4.
Fig. 4.

Experimental setup for holographic generation of shaped self-imaging beams (a) and observed intensity distributions over an area of 3.2mm×3.2mm ((b); the corresponding modelled intensity distributions are those shown in Fig. 3(d)). Insets in (a) show the intensity hologram (corresponding to an area of 8.5mm×8.5mm) and the resulting diffraction orders in the Fourier plane, A; z 0 = 100mm is the z coordinate of plane B, into which the hologram plane, H, is imaged (after Fourier filtering). We use f 1 = 119mm and f 2 = 550mm. L 2 is the Fourier lens from Fig. 1.

Equations (5)

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k x 2 + k y 2 + k z 2 = k 2 .
k x 2 + k y 2 = k 2 k z 2 = const .
k z Δ z mod 2 π = Φ 0
i ( n i I i P + ε ) 1 ,
I ( x , y ) = [ u ( x , y ) exp ( ik sin ( α ) x ) + u * ( x , y ) exp ( ik sin ( α ) x ) + u 0 ] 2 .

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