Abstract

The self-imaging phenomenon is investigated as the basis for designing diffractive optical elements to generate three-dimensional diffraction patterns. The phase-only diffractive element is related to the intensity distribution at a finite and discrete set of Fresnel diffraction planes by use of the matrix formalism of the fractional Talbot effect. This description provides a framework to determine the degrees of freedom which can be exploited for design. It also helps to identify inherent symmetries of periodic wavefronts, which limit the set of intensity patterns that can be implemented. A simulated annealing algorithm is used to exploit the design freedom. Our discussion includes an example to illustrate observations applicable to a more general class of design problems.

© 2006 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. N. Mait, "Understanding diffractive optic design in the scalar domain," J. Opt. Soc. Am. A 12, 2145-2158 (1995).
    [CrossRef]
  2. M. C. King, A. M. Noll, and D. H. Berry, "A new approach to computer generated holograms," Appl. Opt. 9, 471-475 (1970).
    [CrossRef] [PubMed]
  3. T. Yatagai, "Steroscopic approach to 3-D display using computer generated holograms," Appl. Opt. 15, 2722-2729 (1976).
    [CrossRef] [PubMed]
  4. C. Frere, D. Leseberg, and O. Bryngdahl, "Computer-generated holograms of three-dimensional objects composed of line segments," J. Opt. Soc. Am. A 3, 726-730 (1986).
    [CrossRef]
  5. S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
    [CrossRef]
  6. R. Piestun and J. Shamir, "Control of wave-front propagation with diffractive elements," Opt. Lett. 19, 771-773 (1994).
    [CrossRef] [PubMed]
  7. R. Piestun, B. Spektor, and J. Shamir, "Pattern generation with an extended focal depth," Appl. Opt. 37, 5394-5398 (1998).
    [CrossRef]
  8. R. Liu, B.-Z. Dong, G.-Z. Yang, and B.-Y. Gu, "Generation of pseudo-nondiffractive beams with use of diffractive phase elments designed by the conjugate gradient method," J. Opt. Soc. Am. A 15, 144-151 (1998).
    [CrossRef]
  9. U. Levy, D. Mendlovic, Z. Zalevsky, G. Shabtay, and E. Marom, "Iterative algorithm for determining optimal beam profiles in a three-dimensional space," Appl. Opt. 38, 6732-6736 (1999).
    [CrossRef]
  10. N. Guerineau, B. Harchaoui, J. Primot, and K. Heggarty, "Generation of achromatic and propagation invariant spot arrays by us of continuous self-imaging gratings," Opt. Lett. 26, 411-413 (2001).
    [CrossRef]
  11. J. Courtial, G. Whyte, Z. Bouchal, and J. Wagner, "Iterative algorithm for holographic shaping of non-diffracting and self-imaging light beams," Opt. Express 14, 2108-2116 (2006).
    [CrossRef] [PubMed]
  12. J. T. Winthrop and C. R. Worthington, "Theory of Fresnel Images. I. Plane Periodic Objects in Monochromatic Light," J. Opt. Soc. Am. 55, 373-381 (1965).
    [CrossRef]
  13. A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik 79, 41-45 (1988).
  14. V. Arrizon, J. G. Ibarra, and J. Ojeda-Castaneda, "Matrix formulation of the Fresnel transform of complex transmittance gratings," J. Opt. Soc. Am. A 13, 2414-2422 (1996).
    [CrossRef]
  15. M. Testorf, V. Arrizon, and J. Ojeda-Castaneda, "Numerical optimization of phase-only elements based on the fractional Talbot effect," J. Opt. Soc. Am. A 16, 97-105 (1999).
    [CrossRef]
  16. K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed., vol. XXVII, pp. 2-108 (Elsevier, Amsterdam, 1989).
  17. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, chap. 10.9, pp. 444-455, 2nd ed. (Cambridge University Press, New York, 1992).
  18. M. S. Kim, M. R. Feldman, and C. C. Guest, "Optimum coding of binary phase-only filters with a simulated annealing algorithm," Opt. Lett. 14, 545-547 (1989).
    [CrossRef] [PubMed]
  19. V. Arrizon, E. Lopez-Olazagasti, and A. Serrano-Heredia, "Talbot array illuminators with optimum compression ratio," Opt. Lett. 21, 233-235 (1996).
    [CrossRef] [PubMed]
  20. V. Arrizon and M. Testorf, "Efficieny limit of spatially quantized Fourier array illuminators," Opt. Lett. 22, 197-199 (1997).
    [CrossRef] [PubMed]

2006 (1)

2004 (1)

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

1999 (1)

1998 (2)

1997 (1)

1996 (1)

1995 (1)

1994 (1)

1989 (1)

1988 (1)

A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik 79, 41-45 (1988).

1986 (1)

1976 (1)

1970 (1)

1965 (1)

Arrizon, V.

Berry, D. H.

Bouchal, Z.

Bryngdahl, O.

Courtial, J.

Dong, B.-Z.

Feldman, M. R.

Frere, C.

Gu, B.-Y.

Guest, C. C.

Jeon, S.

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Kim, M. S.

King, M. C.

Leseberg, D.

Levy, U.

Liu, R.

Lohmann, A. W.

A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik 79, 41-45 (1988).

Lopez-Olazagasti, E.

Mait, J. N.

Maria, J.

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Marom, E.

Meitl, M.

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Menard, E.

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Mendlovic, D.

Noll, A. M.

Park, J.-U.

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Piestun, R.

Rogers, J. A.

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Serrano-Heredia, A.

Shabtay, G.

Shamir, J.

Spektor, B.

Testorf, M.

Wagner, J.

Whyte, G.

Winthrop, J. T.

Worthington, C. R.

Yang, G.-Z.

Yatagai, T.

Zalevsky, Z.

Zaumseil, J.

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Adv. Mater. (1)

S. Jeon, E. Menard, J.-U. Park, J. Maria, M. Meitl, J. Zaumseil, and J. A. Rogers, "Three-dimensional nanofabrication with rubber stamps and conformable photomasks," Adv. Mater. 16(15), 1369-1373 (2004).
[CrossRef]

Appl. Opt. (4)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Opt. Express (1)

Opt. Lett. (4)

Optik (1)

A. W. Lohmann, "An array illuminator based on the Talbot effect," Optik 79, 41-45 (1988).

Other (5)

V. Arrizon, J. G. Ibarra, and J. Ojeda-Castaneda, "Matrix formulation of the Fresnel transform of complex transmittance gratings," J. Opt. Soc. Am. A 13, 2414-2422 (1996).
[CrossRef]

M. Testorf, V. Arrizon, and J. Ojeda-Castaneda, "Numerical optimization of phase-only elements based on the fractional Talbot effect," J. Opt. Soc. Am. A 16, 97-105 (1999).
[CrossRef]

K. Patorski, "The self-imaging phenomenon and its applications," in Progress in Optics, E.Wolf, ed., vol. XXVII, pp. 2-108 (Elsevier, Amsterdam, 1989).

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes in C, chap. 10.9, pp. 444-455, 2nd ed. (Cambridge University Press, New York, 1992).

N. Guerineau, B. Harchaoui, J. Primot, and K. Heggarty, "Generation of achromatic and propagation invariant spot arrays by us of continuous self-imaging gratings," Opt. Lett. 26, 411-413 (2001).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1.
Fig. 1.

Basic geometry of the optical setup.

Fig. 2.
Fig. 2.

TAIs with extended focal depth. The intensity is specified in (a) one, (b) two, and (c) three diffraction planes. For the optimum DOE one period of the intensity in all specified planes (upper part) and the intensity distribution in x and z (lower part) is shown.

Fig. 3.
Fig. 3.

Average intensity confined to (a) 1/4, (b) 1/2 of the transverse period. Solid lines refer to DOE with L = 8 discrete phase levels, the dashed green line corresponds to L = 16.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

u ( x , zN , M ) = q = 0 Q 1 c q g ( x qd Q ) .
u n = q = 0 Q 1 D n , q g q ,
c q = 1 Q exp [ ( q 2 Q 1 4 ) ]
u n 2 = q , q′ = 0 Q 1 D n , q , D n , q′ * g q g q′ * ,

Metrics