Abstract

A systematic analysis has been performed that predicts the existence of 36 cases in which 100% modulated, square-wave irradiance distributions can be generated in the Fresnel regime by simple binary-phase gratings. These types of distributions, referred to here as Lohmann images, have been previously predicted by researchers studying phase gratings known as Talbot array illuminators. Twenty of the cases are reported, to the best of my knowledge, for the first time. Sixteen of these new cases result in Lohmann images with twice the spatial frequency of the original grating. Experimental verifications of the theoretical predictions are presented.

© 1997 Optical Society of America

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References

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  1. A. W. Lohmann, “An array illuminator based on the Talbot-effect,” Optik 79, 41–45 (1988).
  2. 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]
  3. J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
    [CrossRef]
  4. W. H. F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).
  5. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  6. X. Y. Da, “Talbot effect and the array illuminators that are based on it,” Appl. Opt. 31, 2983–2986 (1992).
    [CrossRef] [PubMed]
  7. V. Arrizón, J. Ojeda-Castañeda, “Talbot array illuminators with binary phase gratings,” Opt. Lett. 18, 1–3 (1993).
    [CrossRef] [PubMed]
  8. P. Szwaykowski, V. Arrizón, “Talbot array illuminators with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [CrossRef] [PubMed]
  9. C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illuminators,” Opt. Commun. 115, 40–44 (1995).
    [CrossRef]
  10. J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects with applications to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
    [CrossRef]
  11. V. Arrizón, J. Ojeda-Castañeda, “Irradiance at Fresnel planes of a phase grating,” J. Opt. Soc. Am. A 9, 1801–1806 (1992).
    [CrossRef]
  12. V. Arrizón, J. G. Ibarra, “Trading visibility and opening ratio in Talbot arrays,” Opt. Commun. 112, 271–277 (1994).
    [CrossRef]
  13. V. Arrizón, J. G. Ibarra, A. Serrano-Heredia, “Split Talbot array illuminators,” Opt. Commun. 123, 63–70 (1996).
    [CrossRef]
  14. V. Arrizón, E. López-Olazagasti, “Binary phase grating for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
    [CrossRef]
  15. 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]
  16. R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
    [CrossRef]

1996 (1)

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

1995 (2)

V. Arrizón, E. López-Olazagasti, “Binary phase grating for array generation at 1/16 of Talbot length,” J. Opt. Soc. Am. A 12, 801–804 (1995).
[CrossRef]

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illuminators,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

1994 (1)

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

1993 (2)

1992 (2)

1990 (2)

1989 (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

1988 (1)

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

1971 (1)

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects with applications to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

1969 (1)

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

1965 (1)

1836 (1)

W. H. F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).

Arrizón, V.

Da, X. Y.

Downs, M. M.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Edgar, R. F.

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

Guigay, J. P.

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects with applications to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

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

Jahns, J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Leger, J. R.

Liu, L.

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illuminators,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

Lohmann, A. W.

A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
[CrossRef] [PubMed]

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

López-Olazagasti, E.

Ojeda-Castañeda, J.

Prise, M. E.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Serrano-Heredia, A.

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

Streibl, N.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Swanson, G. J.

Szwaykowski, P.

Talbot, W. H. F.

W. H. F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).

Thomas, J. A.

Walker, S. J.

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Winthrop, J. T.

Worthington, C. R.

Zhou, C.

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illuminators,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

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

Opt. Acta (2)

J. P. Guigay, “On Fresnel diffraction by one-dimensional periodic objects with applications to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
[CrossRef]

R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
[CrossRef]

Opt. Commun. (3)

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]

C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illuminators,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

Opt. Eng. (1)

J. Jahns, M. M. Downs, M. E. Prise, N. Streibl, S. J. Walker, “Dammann gratings for laser beam shaping,” Opt. Eng. 28, 1267–1275 (1989).
[CrossRef]

Opt. Lett. (2)

Optik (1)

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

Philos. Mag. (1)

W. H. F. Talbot, “Facts relating to optical science No. IV,” Philos. Mag. 9, 401–407 (1836).

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Figures (4)

Fig. 1
Fig. 1

(a) Simple binary-phase grating. (b) Schematic representation of a doubled Lohmann image from a binary-phase TAI.

Fig. 2
Fig. 2

Theoretical and experimental irradiance distributions from binary-phase TAI’s with 400-µm periods. These profiles correspond to (a) row 1, (b) row 5, (c) row 7, and (d) row 26 in Table 1.

Fig. 3
Fig. 3

CCD pictures of (a) single and (b) doubled Lohmann images from the same TAI. These irradiance distributions correspond to rows 5 and 7 in Table 1. The grating period d is 400 µm.

Fig. 4
Fig. 4

(a) Binary-phase TAI with w/d = 1/2 and ϕ = π. (b) Theoretical irradiance distribution arising from this TAI in the planes z = (2M - 1) Z T /16, corresponding to rows 29–36 in Table 1.

Tables (1)

Tables Icon

Table 1 Binary-Phase Grating Parameters for Obtaining Lohmann Images

Equations (18)

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ux;  z=1+expiϕ-1m=-m=am×exp-i2πm2z/ZTexpi2πmx/d,
am=wd sincmwd,
ux; z=h1m=-m=cm expi2πmx/d+h2×m=-m=dm expi2πmx/d,
ux; z=m=-m=cmh1+-1mh2expi2πmx/d,
cm=wd sincmwd,
1+expiϕ-1a0=h1+h2c0,  m=0,
expiϕ-1am exp-i2πm2z/ZT=cmh1+h2,  even m,
expiϕ-1am exp-i2πm2z/ZT=cmh1-h2,  odd m.
cosϕ=1-121c02am/cm2+a0-a02.
cosϕ=1-12a0=1-12w/d.
z/ZT=j/2n2-m2,
z/ZT=j/n2-m2,
z/ZT=j/k,
exp-i2πm2z/ZT=h1±h2expiϕ-1cmam,
exp-i2πm2z/ZT=A expiγeven mA expiηodd m,
tanγ=-sinϕcosϕ-11-2a0.
1+expiϕ-1a0=0.
ux; z=MZT/8=1+expiϕ-1exp-iMπ/4×odd mam expi2πmx/d

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