Abstract

Studying the limitations of sharpness in self-images of the Talbot effect leads us to abandon the use of the paraxial assumption. In this respect, we will clarify the self-image concept and show experimentally and theoretically the influence of “nonparaxial effects” on the self-image. The boundary between paraxial and nonparaxial scalar theory is also clarified in this context. The Rayleigh criterion in aberration diffraction theory is adapted for explicit estimations of this boundary and the maximum sharpness of lines projected by use of the Talbot effect.

© 2004 Optical Society of America

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References

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  1. H. F. Talbot, “Facts relating to optical science. No IV,” Philos. Mag. 9, 401–407 (1836).
  2. N. Guérineau, J. Primot, M. Tauvy, M. Caes, “Modulation transfer function measurement of an infrared focal plane array using the self-imaging property of a canted periodic target,” Appl. Opt. 38, 631–637 (1999).
    [CrossRef]
  3. W. D. Montgomery, “Self-imaging objects of infinite aperture,” J. Opt. Soc. Am. 57, 772–778 (1967).
    [CrossRef]
  4. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  5. P. Szwaykowski, V. Arrizon, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
    [CrossRef] [PubMed]
  6. 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]
  7. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989).
  8. P. M. Mejias, R. Martinez Herrero, “Diffraction by one-dimensional Ronchi grids: on the validity of the Talbot effect,” J. Opt. Soc. Am. A 8, 266–269 (1991).
    [CrossRef]
  9. Y. Cohen-Sabban, D. Joyeux, “Aberration-free nonparaxial self-imaging,” J. Opt. Soc. Am. 73, 707–719 (1983).
    [CrossRef]
  10. J. Turunen, E. Noponen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
    [CrossRef]
  11. J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
    [CrossRef]
  13. M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, UK, 1993), pp. 460–468.
  14. N. Guérineau, B. Harchaoui, K. Heggarty, J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–413 (2001).
    [CrossRef]
  15. N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
    [CrossRef]

2001 (1)

2000 (1)

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

1999 (1)

1993 (2)

P. Szwaykowski, V. Arrizon, “Talbot array illuminator with multilevel phase gratings,” Appl. Opt. 32, 1109–1114 (1993).
[CrossRef] [PubMed]

J. Turunen, E. Noponen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

1991 (1)

1990 (1)

1983 (1)

1969 (1)

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

1967 (1)

1965 (1)

1836 (1)

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

Arrizon, V.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, UK, 1993), pp. 460–468.

Caes, M.

Cohen-Sabban, Y.

Edgar, R. F.

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

Goodman, J.

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Guérineau, N.

Harchaoui, B.

N. Guérineau, B. Harchaoui, K. Heggarty, J. Primot, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–413 (2001).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

Heggarty, K.

Joyeux, D.

Lohmann, A. W.

Martinez Herrero, R.

Mejias, P. M.

Montgomery, W. D.

Noponen, E.

J. Turunen, E. Noponen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989).

Primot, J.

Szwaykowski, P.

Talbot, H. F.

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

Tauvy, M.

Thomas, J. A.

Turunen, J.

J. Turunen, E. Noponen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

Winthrop, J. T.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, UK, 1993), pp. 460–468.

Worthington, C. R.

Appl. Opt. (3)

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Acta (1)

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

Opt. Commun. (2)

J. Turunen, E. Noponen, “Electromagnetic theory of Talbot imaging,” Opt. Commun. 98, 132–140 (1993).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
[CrossRef]

Opt. Lett. (1)

Philos. Mag. (1)

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

Other (3)

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, Vol XXVII, E. Wolf, ed. (North-Holland, Amsterdam, 1989).

J. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

M. Born, E. Wolf, Principles of Optics, 6th (corrected) ed. (Pergamon, Oxford, UK, 1993), pp. 460–468.

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Figures (9)

Fig. 1
Fig. 1

Geometry of the problem.

Fig. 2
Fig. 2

(a) Binary transmittance t(x) of the grating with period d and slit width a, (b) corresponding amplitude spectrum |T(v)|.

Fig. 3
Fig. 3

Intensity profiles (normalized to the incident wave intensity I0=u0u0*) at several paraxial, Fourier self-image distances md2/λ for m=(a) 0, (b) 1, (c) 2, (d) 3 behind the grating of period d=300 μm, slit width a=3 μm, and at wavelength λ=632.8 nm. Profiles in (a) and (c) are shifted by d/2 for comparison.

Fig. 4
Fig. 4

Simulation of the intensity i(x, z) close to the first paraxial Fourier self-image according to (a), (c) nonparaxial and (b), (d) paraxial theory with the corresponding C(z) curves (image sharpness criterion). We consider the intensity in the plane z=zC as the nonparaxial Fourier self-image.

Fig. 5
Fig. 5

Criterion C versus slit width a at the first nonparaxial Fourier self-image Cmax (solid curve). With d=300 μm and λ=632.8 nm, we find an optimal slit width aopt of 7.95 μm (in dotted curve: curve C=d/a obtained with the paraxial assumption).

Fig. 6
Fig. 6

(a) Cmax versus a and d for λ=632.8 nm for the first nonparaxial Fourier self-image; (b) aopt (dotted curve) versus period d: pluses, aopt estimated by criterion C; dashed curve, estimation with the Rayleigh criterion.

Fig. 7
Fig. 7

Experimental setup.

Fig. 8
Fig. 8

Normalized intensity profiles of the first nonparaxial Fourier self-image on a period d for the grating used in our experiment: (a) simulation, (b) experiment.

Fig. 9
Fig. 9

C(z) close to the first Fourier self-image (zzT/2): The smooth curve is the nonparaxial scalar simulation; the curve with the error bars is the experimental data.

Equations (21)

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t(x)=pcpexpjp 2πd x,
cp=adsin cpad.
u(x, z)=u0pcp×expj 2πλ z1-pλd21/2expjp 2πd x.
1-pλd21/2=1-12pλd2.
u=u0expj 2πλ zpcp×exp-j2π zzT p2expjp 2πd x.
ur=u exp-j 2πλ z,
ur=u0pcpexp-j2π zzT p2expjp 2πd x,
ur=pcpup,
cpup=u0cpexp[jϕr(p)(z)]expjp 2πd x,
ϕr(p)(z)=2πλ z1-pλd21/2-1.
(Δυ)z=|I(υ, z)|2dυ|I(0, z)|2,
(Δυ)z=1dp|Dp|2|D0|2.
C(z)=d(Δυ)z=p|Dp(z)|2|D0(z)|2.
Cm zT2=da,
ϕp=2πλ z1-pλd21/2-1+p 2πd x.
ϕ0p=-2π md2λzT p2+p 2πd x=-πmp2+p 2πd x
ϕab(p)(z)=2πλ z1-pλd21/2-1+πmp2,
ϕab(pmax)m d2λ=2πλ m d2λ1-λa21/2-1+πmda2.
ϕab(pmax)m zT2=π4 mdλ2λa4π2.
aopt=(m/2)1/4λd0.84λd.
aopt0.55λd,

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