Abstract

The original Talbot experiment in white light has been reconstituted, using an amplitude grating made of thin slits and a colour CCD camera and a model has been developed to describe the field diffracted by the grating illuminated in polychromatic light with a known spectral density. Above the historical interest of this study, the possibility of applying this effect to make spectral measurements is explored and a new concept of Talbot spectrometer is proposed.

© 2003 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. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol XXVII, pp 1–108.
  3. N. Guérineau, J. Primot, M. Tauvy, and 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]
  4. N. Guérineau, B. Harchaoui, and J. Primot, “Talbot experiment re-examined: demonstration of an achromatic and continuous self-imaging regime,” Opt. Commun. 180, 199–203 (2000).
    [CrossRef]
  5. W. Lohmann, “A new Fourier spectrometer consisting of a two-gratings-interferometer,” in Proceedings of the conference on optical instruments and techniques of London 1961, Habell ed., (Chapman & Hall, 1962), pp 58–61.
  6. H. Klages, “Fourier spectrometry based on grating resonances,” J. Phys. Colloque 2, C2–40 (1967).
  7. H. L. Kung, A. Bhatnagar, and D. A. B. Miller, “Transform spectrometer based on measuring the periodicity of Talbot self-images,” Opt. Lett. 26, 1645–1647 (2001).
    [CrossRef]
  8. G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47
  9. G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
    [CrossRef]
  10. R. F. Edgar, “The Fresnel diffraction images of periodic structures,” Opt. Acta 16, 281–287 (1969).
    [CrossRef]
  11. V. V. Aristov, A. I. Erko, and V. V. Martynov, “Optics and spectrometry based on the Talbot effect,” Opt. Spectrosc. (USSR)  64, 376–380 (1988).
  12. J.J. 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]

2001 (1)

2000 (2)

G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
[CrossRef]

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

1999 (1)

1988 (1)

V. V. Aristov, A. I. Erko, and V. V. Martynov, “Optics and spectrometry based on the Talbot effect,” Opt. Spectrosc. (USSR)  64, 376–380 (1988).

1969 (1)

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

1967 (1)

H. Klages, “Fourier spectrometry based on grating resonances,” J. Phys. Colloque 2, C2–40 (1967).

1965 (1)

1836 (1)

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

Aristov, V. V.

V. V. Aristov, A. I. Erko, and V. V. Martynov, “Optics and spectrometry based on the Talbot effect,” Opt. Spectrosc. (USSR)  64, 376–380 (1988).

Belonuchkin, V. E.

G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
[CrossRef]

G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47

Bhatnagar, A.

Caes, M.

Edgar, R. F.

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

Entin, M. A

G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
[CrossRef]

Erko, A. I.

V. V. Aristov, A. I. Erko, and V. V. Martynov, “Optics and spectrometry based on the Talbot effect,” Opt. Spectrosc. (USSR)  64, 376–380 (1988).

Eskin, N. I.

G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
[CrossRef]

Guérineau, N.

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

N. Guérineau, J. Primot, M. Tauvy, and 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]

Harchaoui, B.

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

Klages, H.

H. Klages, “Fourier spectrometry based on grating resonances,” J. Phys. Colloque 2, C2–40 (1967).

Kozel, S. M.

G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47

Kung, H. L.

Lohmann, W.

W. Lohmann, “A new Fourier spectrometer consisting of a two-gratings-interferometer,” in Proceedings of the conference on optical instruments and techniques of London 1961, Habell ed., (Chapman & Hall, 1962), pp 58–61.

Lokshin, G. R.

G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
[CrossRef]

G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47

Martynov, V. V.

V. V. Aristov, A. I. Erko, and V. V. Martynov, “Optics and spectrometry based on the Talbot effect,” Opt. Spectrosc. (USSR)  64, 376–380 (1988).

Miller, D. A. B.

Patorski, K.

K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol XXVII, pp 1–108.

Primot, J.

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

N. Guérineau, J. Primot, M. Tauvy, and 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]

Talbot, H. F.

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

Tauvy, M.

Uchenov, A. V.

G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
[CrossRef]

Winthrop, J.J.

Worthington, C. R.

Appl. Opt. (1)

J. Opt. Soc. Am. (1)

J. Phys. Colloque (1)

H. Klages, “Fourier spectrometry based on grating resonances,” J. Phys. Colloque 2, C2–40 (1967).

Opt. Acta (1)

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

Opt. Commun. (1)

N. Guérineau, B. Harchaoui, and 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)

Opt. Spectrosc. (2)

V. V. Aristov, A. I. Erko, and V. V. Martynov, “Optics and spectrometry based on the Talbot effect,” Opt. Spectrosc. (USSR)  64, 376–380 (1988).

G. R. Lokshin, A. V. Uchenov, M. A Entin, V. E. Belonuchkin, and N. I. Eskin, “On the spectra selectivity of Talbot and Lau effects,” Opt. Spectrosc. 89, 312–317 (2000).
[CrossRef]

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, E. Wolf, ed. (North-Holland, Amsterdam, 1989), Vol XXVII, pp 1–108.

W. Lohmann, “A new Fourier spectrometer consisting of a two-gratings-interferometer,” in Proceedings of the conference on optical instruments and techniques of London 1961, Habell ed., (Chapman & Hall, 1962), pp 58–61.

G. R. Lokshin, V. E. Belonuchkin, and S. M. Kozel, in Sixteenth Symposium on Holography, (Leningrad, 1985), p. 47

Supplementary Material (1)

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Figures (7)

Fig. 1.
Fig. 1.

Talbot-Lokshin spectrometer. Basic set-up.

Fig. 2:
Fig. 2:

Simulation of the field diffracted by a binary-amplitude grating in polychromatic light (c) of known spectral density (a), using the Fourier coefficients of the transverse intensity profile (b) (dotted line, Fourier coefficients in monochromatic light).

Fig. 3:
Fig. 3:

Simulation of the signal delivered by the Talbot-Lokshin spectrometer in monochromatic (a) or polychromatic (b) light.

Fig. 4.
Fig. 4.

Recommended spectrometer. Basic set-up.

Fig. 5:
Fig. 5:

Simulation of the signal delivered by the recommended spectrometer in monochromatic (a) or polychromatic (b) light and extraction of the spectral density (c).

Fig. 6.
Fig. 6.

(2.2 MB) Movie of the reconstituted Talbot experiment.

Fig. 7:
Fig. 7:

Experimental study in white light: (a) response of a line of CCD detectors versus propagation distance and extraction of the R-,G-, B- spectral responses (c) from the D1(z) curves.

Equations (24)

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Z = 2 d 2 Δ λ ,
z T = 2 d 2 λ .
t ( x ) = p = + C p exp ( 2 i π p x d ) .
C p = a d sinc ( pa d ) ,
u λ ( x , z ) = u i p max p max C p exp ( 2 i π p x d ) exp ( 2 i π z λ 1 ( λ p d ) 2 ) ,
I λ ( x , z ) = u i 2 2 p max 2 p max D p , λ ( z ) exp ( 2 i π p x d ) ,
D p , λ ( z ) = q C p + q C q exp [ 2 i π z λ ( 1 λ 2 ( p + q ) 2 d 2 1 λ 2 q 2 d 2 ) ]
Φ p + q Φ q = 2 i π z ( p 2 + 2 p q ) z T ,
D p odd ( even ) , λ ( z ) = m odd ( even ) C m p 2 C m + p 2 exp ( 2 i π m z z p ) .
I ( x , z ) = λ B ( λ ) I λ ( x , z ) d λ .
I ( x , z ) = p D p ( z ) exp ( 2 i π p x d ) ,
D p ( z ) = B ( λ ) D λ , p ( z ) d λ .
D p odd ( even ) ( z ) = m odd ( even ) C m p 2 C m + p 2 B ˜ ( m p z 2 d 2 ) ,
B ˜ ( f ) = λ B ( λ ) exp ( 2 i π λ f ) d λ .
S ( z ) = I ( 0 , z ) = p D p ( z )
δ z = a 2 λ .
λ δ λ = 2 d 2 a 2 .
Δ λ = λ a 2 d .
D 1 , λ ( z ) = m d m exp ( 2 i π m z z T ) ,
d m = C m 1 2 C m + 1 2 = ( a d ) 2 sinc ( m 1 ) a 2 d sinc ( m + 1 ) a 2 d C m 2
λ δ λ = d 2 a .
λ min λ max 2 .
B ext ( d 2 z ) = D 1 ( z ) M 1 ,
B corr ( d 2 z ) = D 1 ( z ) M 1 B ext ( 2 d 2 z ) × M 2 M 1 ,

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