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 theconference 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]

Appl. Opt.

J. Opt. Soc. Am.

J. Phys. Colloque

H. Klages, �??Fourier spectrometry based on grating resonances,�?? J. Phys. Colloque 2, C2-40 (1967).

Opt. Acta

R. F. Edgar, �??The Fresnel diffraction images of periodic structures,�?? Opt. Acta 16, 281-287 (1969).
[CrossRef]

Opt. Commun.

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.

Opt. Spectrosc.

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]

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).

Optical instr. and techn., London 1961

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

Philos. Mag.

H. F. Talbot, �??Facts relating to optical science. No IV,�?? Philos. Mag. 9, 401-407 (1836).

Progress in Optics

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.

Other

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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