Abstract

A uniform formulation for the self-imaging of gratings with any kind of partially coherent illumination is developed in terms of the cross mutual spectral density of the partial coherence theory. The formulation includes the time diffractive intensity distribution and the averaged diffractive intensity distribution at self-imaging distances and can be applied to both continuous and temporal illuminations with any kind of spectra. It is found that the averaged intensity distribution is related only to the intensity spectrum of illumination. The continuous polychromatic illumination and the ultrashort laser pulses with or without frequency chirp are then studied by a numerical stimulation. It is shown that the ultrashort laser pulse and the continuous polychromatic illuminations have similar averaged self-image distributions. Thus the Talbot effect may help in the study of the temporal and spectral characteristics of ultrashort laser pulses. An experiment with an LED is given, as well.

© 2003 Optical Society of America

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References

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  1. W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).
  2. A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
    [CrossRef]
  3. L. Liu, “Talbot and Lau effects on incident beams of arbitrary wavefront and their use,” Appl. Opt. 28, 4668–4678 (1989).
    [CrossRef] [PubMed]
  4. A. W. Lohmann, J. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  5. L. Liu, “Lau cavity and phase locking of laser arrays,” Opt. Lett. 14, 1312–1314 (1989).
    [CrossRef] [PubMed]
  6. H. Wang, C. H. Chou, J. L. Li, L. R. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
    [CrossRef]
  7. H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
    [CrossRef]
  8. L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).
  9. M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 2001).
  10. M. Bertolotti, A. Ferrari, L. Sereda, “Far-zone diffraction of polychromatic and nonstationary plane waves from a slit,” J. Opt. Soc. Am. B 12, 1519–1526 (1995).
    [CrossRef]
  11. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  12. P. Latimer, R. Crouse, “Talbot effect reinterpreted,” Appl. Opt. 31, 80–89 (1992).
    [CrossRef] [PubMed]
  13. J.-P. Geindre, P. Audebert, S. Rebibo, J.-C. Gauthier, “Single-shot spectral interferometry with chirped pulses,” Opt. Lett. 26, 1612–1614 (2001).
    [CrossRef]

2001

H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
[CrossRef]

J.-P. Geindre, P. Audebert, S. Rebibo, J.-C. Gauthier, “Single-shot spectral interferometry with chirped pulses,” Opt. Lett. 26, 1612–1614 (2001).
[CrossRef]

2000

H. Wang, C. H. Chou, J. L. Li, L. R. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[CrossRef]

1995

1992

1990

1989

1971

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

1836

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Audebert, P.

Bertolotti, M.

Born, M.

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 2001).

Chou, C. H.

H. Wang, C. H. Chou, J. L. Li, L. R. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[CrossRef]

Crouse, R.

Ferrari, A.

Gauthier, J.-C.

Geindre, J.-P.

Goodman, J. W.

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

Latimer, P.

Li, J. L.

H. Wang, C. H. Chou, J. L. Li, L. R. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[CrossRef]

Liu, L.

Liu, L. R.

H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
[CrossRef]

H. Wang, C. H. Chou, J. L. Li, L. R. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[CrossRef]

Lohmann, A. W.

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

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Mandel, L.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

Rebibo, S.

Sereda, L.

Silva, D. E.

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Talbot, W. H. F.

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Thomas, J.

Wang, H.

H. Wang, C. H. Chou, J. L. Li, L. R. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[CrossRef]

Wang, H. W.

H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
[CrossRef]

Wolf, E.

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 2001).

Xi, P.

H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
[CrossRef]

Zhao, S.

H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
[CrossRef]

Zhou, C. H.

H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
[CrossRef]

Appl. Opt.

J. Opt. A Pure Appl. Opt.

H. W. Wang, C. H. Zhou, S. Zhao, P. Xi, L. R. Liu, “The temporal Fresnel diffraction field of a grating illuminated by an ultrashort pulsed-laser beam,” J. Opt. A Pure Appl. Opt. 3, 159–163 (2001).
[CrossRef]

J. Opt. Soc. Am. B

Microwave Opt. Technol. Lett.

H. Wang, C. H. Chou, J. L. Li, L. R. Liu, “Talbot effect of a grating under ultrashort pulsed-laser illumination,” Microwave Opt. Technol. Lett. 25, 184–187 (2000).
[CrossRef]

Opt. Commun.

A. W. Lohmann, D. E. Silva, “An interferometer based on the Talbot effect,” Opt. Commun. 2, 413–415 (1971).
[CrossRef]

Opt. Lett.

Philos. Mag.

W. H. F. Talbot, “Facts relating to optical science,” Philos. Mag. 9, 401–407 (1836).

Other

L. Mandel, E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, Cambridge, UK, 1995).

M. Born, E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, Cambridge, UK, 2001).

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

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Figures (7)

Fig. 1
Fig. 1

Fresnel diffraction of a grating with any illumination.

Fig. 2
Fig. 2

Spatial profiles of self-images of a grating (d=200μm) with different spectral widths of continuous illumination: (a) at first Talbot distance and (b) at tenth Talbot distance.

Fig. 3
Fig. 3

Spatial profiles of self-images of a grating (d=200 μm) with different durations of laser pulse: (a) at first Talbot distance and (b) at twentieth Talbot distance.

Fig. 4
Fig. 4

The instantaneous intensity distribution of the grating (d=200 μm) illuminated by an ultrashort-pulse laser at the first Talbot distance varies with the spatial coordinate x.

Fig. 5
Fig. 5

Time profiles of diffractive intensity distribution at different Talbot distances.

Fig. 6
Fig. 6

Experimental setup.

Fig. 7
Fig. 7

Experimental self-image (points) and theoretical evaluation (solid curve).

Equations (45)

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s(r01, r02, ω1, ω2)=u(r01, ω1)u*(r02, ω2),
Γ(r01, r02, t1, t2)=--s(r01, r02, ω1, ω2)×exp[i(ω1t1-ω2t2)]dω1dω2.
I(r0, t)=Γ(r0, r0, t, t);
I(r0, t)=--s(r0, r0, ω1, ω2)×exp[i(ω1-ω2)t]dω1dω2.
s-(r01, r02, ω1, ω2)=s-(ω1, ω2).
s(r01, r02, ω1, ω2)=s-(ω1, ω2)T(r01)T*(r02).
s(r1, r2, ω1, ω2)
=u(r1, ω1)u*(r2, ω2)
=--s-(ω1, ω2)T(r01)T*(r02)K(r1; r01, ω1)
×K*(r2; r02, ω2)dr01dr02.
K(r; r0, ω)=ωn(ω)i2πczexpiωzn(ω)c×expi ωn(ω)2zc |r-r0|2,
s(r1, r2, ω1, ω2)
=--s-(ω1, ω2) ω1ω2n(ω1)n(ω2)4π2c2z2 T(r01)T*(r02)×expi [ωn(ω1)-ω2n(ω2)]zc×expi2zc [ω1n(ω1)|r1-r01|2-ω2n(ω2)|r2-r02|2]dr01dr02.
J(r1, r2, t)
=Γ(r1, r2, t, t)=--s(r1, r2, ω1, ω2)×exp[i(ω1-ω2)t]dω1dω2=----s-(ω1, ω2) ω1ω2n(ω1)n(ω2)4π2c2z2
×T(r01)T*(r02)expi2zc [ω1n(ω1)|r1-r01|2
-ω2n(ω2)|r2-r02|2]×expi[ω1n(ω1)-ω2n(ω2)]t-zc×dr01dr02dω1dω2.
I(r, t)=----s-(ω1, ω2)×ω1ω2n(ω1)n(ω2)4π2c2z2 T(r01)T*(r02)×expi2zc [ω1n(ω1)|r-r01|2-ω2n(ω2)|r-r02|2]×expi[ω1n(ω1)-ω2n(ω2)]t-zc×dr01dr02dω1dω2.
I(r, t)=----s-(ω1, ω2) ω1ω24π2ν2z2×T(r01)T*(r02)×expi2zν [ω1|r-r01|2-ω2|r-r02|2]×expi(ω1-ω2)t-zνdr01dr02dω1dω2,
I(r)=-I-(ω) ω24π2ν2z2×-T(r0)expiω2zν (|r-r0|2)dr02dω,
T(x0)=m=-rectx0-mdd/M,
T(x0)=m=-Cmexpi2πmx0d,
I(x, t)=--s-(ω1, ω2) ω1ω22πνz×mCmexpi2πxmdexp-i 2π2zνm2d2ω1×lClexpi2πxldexp-i 2π2zνl2d2ω2×expi(ω1-ω2)t-zνdω1dω2,
I(x)=-I-(ω) ω2πνzmCmexpi2πxmd×exp-i 2π2zνm2d2ω2dω.
zT=2k d2λ0,k=0, 1, 2, 3, ,
I(x, t)=--s-(ω1, ω2) ω1ω22πνzT×mCmexpi2πxmdexp-i 2π2zTνm2d2ω1×lClexpi2πxldexp-i 2π2zTνl2d2ω2×expi(ω1-ω2)t-zTνdω1dω2,
l(x)=-I-(ω) ω2kd2ω0×mCmexpi2πxmdexp-i 2πkm2ω0ω2dω.
I(x, t)
=I02πνzT1/2expiωt-zTν×mCmexpi2πxmdexp-i 2π2zTνm2d2ω2.
I(x)=I02kd2mCmexpi2πxmd2,
I(x)=I02kd2mrectx-mdd/M.
s-(ω1, ω2)=I(ω1)δ(ω1-ω2),
I(x, t)=- I(ω)ω2πνzTmCmexpi2πxmd×exp-i 2π2zTνm2d2ω2dω.
I(x)=- I(ω)ω2kd2ω0mCmexpi2πxmd×exp-i 2πkm2ω0ω2dω.
I(ω)=exp-ω-ω0ωT2,
u(t)=exp(-iω0t)exp-t22τ2,
u(ω)=τ2πexp-12 (ω-ω0)2τ2.
s-(ω1, ω2)
=τ22πexp-12 [(ω1-ω0)2+(ω2-ω0)2]τ2.
I(ω)=s-(ω, ω)=τ22πexp[-(ω-ω0)2τ2].
I(x, t)=- ωτ2π2πνzTexp-12 (ω-ω0)2τ2×expiωt-zTνmCmexpi2πxmd×exp-i 2π2zTνm2d2ωdω2.
I(x)=- τ22πexp[-(ω-ω0)2τ2] ω2kd2ω0×mCmexpi2πxmdexp-i 2πkm2ω0ω2dω.
u(t)=exp(-iω0t)exp-1+iC2tτ2,
u(ω)=A exp-τ22(1+iC) (ω-ω0)2,
Δω=(1+C2)1/2τ.

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