Abstract

Based on the matrix description of the fractional Talbot effect, a new and effective method to numerically optimize diffractive optical elements that work in the Fresnel diffraction regime is described. When the investigation is restricted to spatially quantized phase-only gratings, diffraction can be described in terms of the fractional Talbot effect and the diffraction amplitude is efficiently evaluated from a finite set of sampling points. As an illustrating example we numerically optimize Talbot array illuminators. Our results show that a limited number of discrete phase levels does not imply a limited compression ratio but does lead to a reduced diffraction efficiency. Experimental results obtained from lithographically fabricated surface-relief gratings are compared with our theoretical designs.

© 1999 Optical Society of America

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References

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  1. A. W. Lohmann, D. P. Paris, “Binary Fraunhofer holograms generated by computer,” Appl. Opt. 6, 1739–1743 (1967).
    [CrossRef] [PubMed]
  2. H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
    [CrossRef]
  3. J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
    [CrossRef]
  4. O. Bryngdahl, F. Wyrowski, “Digital holography—computer generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. XXVIII.
  5. J. N. Mait, “Understanding diffractive optic design in the scalar domain,” J. Opt. Soc. Am. A 12, 2145–2158 (1995).
    [CrossRef]
  6. R. Piestun, B. Spektor, J. Shamir, “Wave fields in three dimensions: analysis and synthesis,” J. Opt. Soc. Am. A 13, 1837–1848 (1996).
    [CrossRef]
  7. R. G. Dorsch, A. W. Lohmann, S. Sinzinger, “Fresnel ping-pong algorithm for two-plane computer-generated hologram display,” Appl. Opt. 33, 869–875 (1994).
    [CrossRef] [PubMed]
  8. P. Pellat-Finet, “Fresnel diffraction and fractional-order Fourier transform,” Opt. Lett. 19, 1388–1390 (1994).
    [CrossRef] [PubMed]
  9. G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. (Bellingham) 28, 605–608 (1989).
    [CrossRef]
  10. J. T. Winthrop, C. R. Worthington, “Theory of Fresnel images. I. Plane periodic objects in monochromatic light,” J. Opt. Soc. Am. 55, 373–381 (1965).
    [CrossRef]
  11. V. Arrizón, J. Ojeda-Castañeda, “Fresnel diffraction of substructured gratings: matrix description,” Opt. Lett. 20, 118–120 (1995).
    [CrossRef] [PubMed]
  12. H. Hamam, J. L. de Bougrenet de la Tocnaye, “Efficient Fresnel-transform algorithm based on fractional Fresnel diffraction,” J. Opt. Soc. Am. A 12, 1920–1931 (1995).
    [CrossRef]
  13. M. Testorf, J. Jahns, “Planar-integrated Talbot array illuminators,” Appl. Opt. 37, 5399–5407 (1998).
    [CrossRef]
  14. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4, pp. 57–76.
  15. V. Arrizón, J. G. Ibarra, J. Ojeda-Castañeda, “Matrix formulation of the Fresnel transform of complex transmittance gratings,” J. Opt. Soc. Am. A 13, 2414–2422 (1996).
    [CrossRef]
  16. V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  17. M. Testorf, J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. A 13, 119–125 (1996).
    [CrossRef]
  18. H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
    [CrossRef]
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  20. A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).
  21. W. Klaus, Y. Arimoto, K. Kodate, “High performance Talbot array illuminators,” Appl. Opt. 37, 4357–4365 (1998).
    [CrossRef]
  22. V. Arrizón, J. G. Ibarra, “Trading visibility and opening ratio in Talbot arrays,” Opt. Commun. 112, 271–277 (1994).
    [CrossRef]
  23. J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. 15, 288–290 (1990).
    [CrossRef] [PubMed]
  24. V. Arrizón, E. López-Olazagasti, A. Serrano-Heredia, “Talbot array illuminators with optimum compression ratio,” Opt. Lett. 21, 233–235 (1996).
    [CrossRef] [PubMed]
  25. V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
    [CrossRef]
  26. T. J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36, 4686–4691 (1997).
    [CrossRef] [PubMed]
  27. F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
    [CrossRef]
  28. M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
    [CrossRef]

1998

1997

1996

1995

1994

1990

1989

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. (Bellingham) 28, 605–608 (1989).
[CrossRef]

1988

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

1980

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
[CrossRef]

1971

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

1967

1965

Arimoto, Y.

Arrizón, V.

Bryngdahl, O.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. XXVIII.

Dammann, H.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

de Bougrenet de la Tocnaye, J. L.

Dorsch, R. G.

Fienup, J. R.

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Goncharenko, A. M.

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4, pp. 57–76.

Görtler, K.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

Hamam, H.

Ibarra, J. G.

V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
[CrossRef]

V. Arrizón, J. G. Ibarra, J. Ojeda-Castañeda, “Matrix formulation of the Fresnel transform of complex transmittance gratings,” J. Opt. Soc. Am. A 13, 2414–2422 (1996).
[CrossRef]

V. Arrizón, J. G. Ibarra, “Trading visibility and opening ratio in Talbot arrays,” Opt. Commun. 112, 271–277 (1994).
[CrossRef]

Jahns, J.

M. Testorf, J. Jahns, “Planar-integrated Talbot array illuminators,” Appl. Opt. 37, 5399–5407 (1998).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

Khilo, N. A.

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

Klaus, W.

Kodate, K.

Leger, J. R.

Lohmann, A. W.

López-Olazagasti, E.

Mait, J. N.

Ojeda-Castañeda, J.

Paris, D. P.

Pellat-Finet, P.

Piestun, R.

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Serrano-Heredia, A.

V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
[CrossRef]

V. Arrizón, E. López-Olazagasti, A. Serrano-Heredia, “Talbot array illuminators with optimum compression ratio,” Opt. Lett. 21, 233–235 (1996).
[CrossRef] [PubMed]

Shamir, J.

Sinzinger, S.

Spektor, B.

Suleski, T. J.

Swanson, G. J.

J. R. Leger, G. J. Swanson, “Efficient array illuminator using binary-optics phase plates at fractional Talbot planes,” Opt. Lett. 15, 288–290 (1990).
[CrossRef] [PubMed]

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. (Bellingham) 28, 605–608 (1989).
[CrossRef]

Testorf, M.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Veldkamp, W. B.

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. (Bellingham) 28, 605–608 (1989).
[CrossRef]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Winthrop, J. T.

Worthington, C. R.

Wyrowski, F.

F. Wyrowski, “Diffractive optical elements: iterative calculation of quantized, blazed phase structures,” J. Opt. Soc. Am. A 7, 961–969 (1990).
[CrossRef]

O. Bryngdahl, F. Wyrowski, “Digital holography—computer generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. XXVIII.

Appl. Opt.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Commun.

H. Dammann, K. Görtler, “High-efficiency in-line multiple imaging by means of multiple phase holograms,” Opt. Commun. 3, 312–315 (1971).
[CrossRef]

H. Hamam, “Design of Talbot array illuminators,” Opt. Commun. 131, 359–370 (1996).
[CrossRef]

V. Arrizón, J. G. Ibarra, “Trading visibility and opening ratio in Talbot arrays,” Opt. Commun. 112, 271–277 (1994).
[CrossRef]

V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
[CrossRef]

M. Testorf, J. Jahns, N. A. Khilo, A. M. Goncharenko, “Talbot effect for oblique angle of light propagation,” Opt. Commun. 129, 167–172 (1996).
[CrossRef]

Opt. Eng. (Bellingham)

J. R. Fienup, “Iterative method applied to image reconstruction and to computer generated holograms,” Opt. Eng. (Bellingham) 19, 297–305 (1980).
[CrossRef]

G. J. Swanson, W. B. Veldkamp, “Diffractive optical elements for use in infrared systems,” Opt. Eng. (Bellingham) 28, 605–608 (1989).
[CrossRef]

Opt. Lett.

Optik (Stuttgart)

A. W. Lohmann, “An array illuminator based on the Talbot effect,” Optik (Stuttgart) 79, 41–45 (1988).

Other

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 4, pp. 57–76.

O. Bryngdahl, F. Wyrowski, “Digital holography—computer generated holograms,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1990), Vol. XXVIII.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

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Figures (9)

Fig. 1
Fig. 1

Optical system under investigation. Fresnel diffraction behind a phase-only grating yields the diffraction intensity at a desired fractional Talbot plane z.

Fig. 2
Fig. 2

(a) Phase profile and (b) intensity in a fractional Talbot plane of a spatially quantized phase-only grating.

Fig. 3
Fig. 3

One period of the phase distribution and the diffraction intensity of an optimized Talbot array illuminator (TAI) (z=zT /64). The phase is quantized into (a) L=8 and (b) L=4 equidistant levels.

Fig. 4
Fig. 4

Optimum diffraction efficiency of TAI’s versus compression ratio for L=2, 4, and 8.

Fig. 5
Fig. 5

Optimization of a split TAI (z=zT/27). The marked pixels are optimized in terms of diffraction efficiency and homogeneity. Optimization parameters and results are summarized in Table 1.

Fig. 6
Fig. 6

Same desired diffraction pattern as that in Fig. 5 obtained with the FPP algorithm. The constraint of a homogeneous background results in a small diffraction efficiency.

Fig. 7
Fig. 7

Enlarged view of a TAI fabricated as a surface-relief structure on a transparent substrate (z=zT/3,Q=3).

Fig. 8
Fig. 8

Sections of the diffraction patterns of two different TAI’s recorded with a CCD camera: (a) propagation distance z=zT/3 and Q=3, (b) propagation distance z=2zT/7 and Q=7.

Fig. 9
Fig. 9

Line scan of the diffraction intensity of a TAI with z=zT/16 and Q=8.

Tables (1)

Tables Icon

Table 1 Optimization Parameters and Results for the Examples in Figs. 5 and 6

Equations (9)

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

t(x)=rectxx0*q=0Q-1tqδ(x-qx0)*h=-δ(x-hd),
tq=exp(iϕq).
Φl=2πLlwithl=0, 1, , L-1.
N=QforNoddQ/2forNeven,
u(x, z=(M/N)zT)=rectxx0*p=0Q-1upδ(x-px0)*h=-δ(x-hd).
up=q=0Q-1Fp,qtqwithFp,q=C|p-q|.
Ca=1Qexp(iθa).
K0=Q-Ipixel,
K0=1Qq=1QQQ-Iq-Itune2.

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