Abstract

We present a compact analytical formulation for the fractional Talbot effect at the paraxial domain of a finite grating. Our results show that laterally shifted distorted images of the grating basic cell form the Fresnel field at a fractional Talbot plane of the grating. Our formulas give the positions of those images and show that they are given by the convolution of the nondistorted cells (modulated by a quadratic phase factor) with the Fourier transform of the finite-grating pupil.

© 2001 Optical Society of America

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References

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  1. 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]
  2. J. P. Guigay, “On the Fresnel diffraction by one-dimensional periodic objects, with application to structure determination of phase objects,” Opt. Acta 18, 677–682 (1971).
    [CrossRef]
  3. V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
    [CrossRef] [PubMed]
  4. C. Zhou, L. Liu, “Simple equations for the calculation of a multilevel phase grating for Talbot array illumination,” Opt. Commun. 115, 40–44 (1995).
    [CrossRef]
  5. M. Testorf, J. Ojeda-Castañeda, “Fractional Talbot effect: analysis in phase space,” J. Opt. Soc. Am. A 13, 119–125 (1996).
    [CrossRef]
  6. V. Arrizón, G. Rojo-Velázquez, Juan G. Ibarra, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
    [CrossRef]
  7. A. W. Lohmann, J. A. Thomas, “Making an array illuminator based on the Talbot effect,” Appl. Opt. 29, 4337–4340 (1990).
    [CrossRef] [PubMed]
  8. W. Klaus, Y. Arimoto, K. Kodate, “Talbot array illuminators providing spatial intensity and phase modulation,” J. Opt. Soc. Am. A 14, 1092–1102 (1997).
    [CrossRef]
  9. T. J. Suleski, “Generation of Lohmann images from binary-phase Talbot array illuminators,” Appl. Opt. 36, 4686–4691 (1997).
    [CrossRef] [PubMed]
  10. P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22, 1059–1061 (1997).
    [CrossRef] [PubMed]
  11. S. Nowak, Ch. Kurtsiefer, T. Pfau, C. David, “High-order Talbot fringes for atomic matter waves,” Opt. Lett. 22, 1430–1432 (1997).
    [CrossRef]
  12. F. Mitschke, V. Morgner, “The temporal Talbot effect,” Opt. Photon. News, June1998, pp. 45–47.
    [CrossRef]
  13. O. Carnal, Q. A. Turchette, H. J. Kimble, “Near-field imaging with two transmission gratings for submicrometer localization of atoms,” Phys. Rev. A 51, 3079–3087 (1995).
    [CrossRef] [PubMed]
  14. J. F. Clauser, M. W. Reinsch, “New theoretical and experimental results in Fresnel optics with applications to matter-wave and x-ray interferometry,” Appl. Phys. B 54, 380–395 (1992).
    [CrossRef]
  15. K. Patorski, “The self-imaging phenomenon and its applications,” in Progress in Optics XXVII, E. Wolf, ed. (Elsevier, Amsterdam, 1989), pp. 1–108.
  16. F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, San Diego, Calif., 1994), Vol. 2, pp. 139–148.

2000 (1)

V. Arrizón, G. Rojo-Velázquez, Juan G. Ibarra, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

1998 (1)

F. Mitschke, V. Morgner, “The temporal Talbot effect,” Opt. Photon. News, June1998, pp. 45–47.
[CrossRef]

1997 (4)

1996 (1)

1995 (2)

O. Carnal, Q. A. Turchette, H. J. Kimble, “Near-field imaging with two transmission gratings for submicrometer localization of atoms,” Phys. Rev. A 51, 3079–3087 (1995).
[CrossRef] [PubMed]

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

1994 (1)

1992 (1)

J. F. Clauser, M. W. Reinsch, “New theoretical and experimental results in Fresnel optics with applications to matter-wave and x-ray interferometry,” Appl. Phys. B 54, 380–395 (1992).
[CrossRef]

1990 (1)

1971 (1)

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

1965 (1)

Arimoto, Y.

Arrizón, V.

V. Arrizón, G. Rojo-Velázquez, Juan G. Ibarra, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

V. Arrizón, J. Ojeda-Castañeda, “Multilevel phase gratings for array illuminators,” Appl. Opt. 33, 5925–5931 (1994).
[CrossRef] [PubMed]

Baruchel, J.

Carnal, O.

O. Carnal, Q. A. Turchette, H. J. Kimble, “Near-field imaging with two transmission gratings for submicrometer localization of atoms,” Phys. Rev. A 51, 3079–3087 (1995).
[CrossRef] [PubMed]

Clauser, J. F.

J. F. Clauser, M. W. Reinsch, “New theoretical and experimental results in Fresnel optics with applications to matter-wave and x-ray interferometry,” Appl. Phys. B 54, 380–395 (1992).
[CrossRef]

Cloetens, P.

David, C.

De Martino, C.

Gori, F.

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, San Diego, Calif., 1994), Vol. 2, pp. 139–148.

Guigay, J. P.

P. Cloetens, J. P. Guigay, C. De Martino, J. Baruchel, M. Schlenker, “Fractional Talbot imaging of phase gratings with hard x rays,” Opt. Lett. 22, 1059–1061 (1997).
[CrossRef] [PubMed]

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

Ibarra, Juan G.

V. Arrizón, G. Rojo-Velázquez, Juan G. Ibarra, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

Kimble, H. J.

O. Carnal, Q. A. Turchette, H. J. Kimble, “Near-field imaging with two transmission gratings for submicrometer localization of atoms,” Phys. Rev. A 51, 3079–3087 (1995).
[CrossRef] [PubMed]

Klaus, W.

Kodate, K.

Kurtsiefer, Ch.

Liu, L.

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

Lohmann, A. W.

Mitschke, F.

F. Mitschke, V. Morgner, “The temporal Talbot effect,” Opt. Photon. News, June1998, pp. 45–47.
[CrossRef]

Morgner, V.

F. Mitschke, V. Morgner, “The temporal Talbot effect,” Opt. Photon. News, June1998, pp. 45–47.
[CrossRef]

Nowak, S.

Ojeda-Castañeda, J.

Patorski, K.

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

Pfau, T.

Reinsch, M. W.

J. F. Clauser, M. W. Reinsch, “New theoretical and experimental results in Fresnel optics with applications to matter-wave and x-ray interferometry,” Appl. Phys. B 54, 380–395 (1992).
[CrossRef]

Rojo-Velázquez, G.

V. Arrizón, G. Rojo-Velázquez, Juan G. Ibarra, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

Schlenker, M.

Suleski, T. J.

Testorf, M.

Thomas, J. A.

Turchette, Q. A.

O. Carnal, Q. A. Turchette, H. J. Kimble, “Near-field imaging with two transmission gratings for submicrometer localization of atoms,” Phys. Rev. A 51, 3079–3087 (1995).
[CrossRef] [PubMed]

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 illumination,” Opt. Commun. 115, 40–44 (1995).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. B (1)

J. F. Clauser, M. W. Reinsch, “New theoretical and experimental results in Fresnel optics with applications to matter-wave and x-ray interferometry,” Appl. Phys. B 54, 380–395 (1992).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Acta (1)

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

Opt. Commun. (1)

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

Opt. Lett. (2)

Opt. Photon. News (1)

F. Mitschke, V. Morgner, “The temporal Talbot effect,” Opt. Photon. News, June1998, pp. 45–47.
[CrossRef]

Opt. Rev. (1)

V. Arrizón, G. Rojo-Velázquez, Juan G. Ibarra, “Fractional Talbot effect: compact description,” Opt. Rev. 7, 129–131 (2000).
[CrossRef]

Phys. Rev. A (1)

O. Carnal, Q. A. Turchette, H. J. Kimble, “Near-field imaging with two transmission gratings for submicrometer localization of atoms,” Phys. Rev. A 51, 3079–3087 (1995).
[CrossRef] [PubMed]

Other (2)

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

F. Gori, “Why is the Fresnel transform so little known?” in Current Trends in Optics, J. C. Dainty, ed. (Academic, San Diego, Calif., 1994), Vol. 2, pp. 139–148.

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Figures (2)

Fig. 1
Fig. 1

(a) Binary grating with fill factor of 1/10 at the plane z=0 and portions of the intensity distribution of the Fresnel pattern at the distance z=ZT/4 for the grating with (b) 55 periods and (c) 5 periods.

Fig. 2
Fig. 2

Intensities at the centers of diffraction spots for extended domains of the diffraction patterns corresponding to the grating in Fig. 1 with (a) 55 periods and (b) 5 periods.

Equations (29)

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FRz[t(x)]1iλzt(x)Ez(x),
t(x)=n=-t0(x-nd),
FRz[t(x)]=L=-C(L, M, N)t0(x-Ld/N),
C(L, M, N)=1Nm=0N-1 expi2πNm(L-Mm).
C(L, M, N)=0,even N/2,odd L,
C(L, M, N)=0,odd N/2,even L,
C(L, M, N)0,otherwise.
t(x)=t(x)w(x),
FRz[t(x)]=1λzEz(x)[Ez*(x)FRz[t(x)]]Wxλz,
FRz[t(x)]=L=-C(L, M, N)ΨL(x),
ΨL(x)=1λzEz(x)[Ez*(x)t0(x-Ld/N)]Wxλz.
P4MNdΔt0.
ΨL(x)-Ez*(ξ)t0ξ-LdNWx-ξλzdξ.
ΨL(x=Ld/N){τ˜(u)ω(λzu)}u=Ld/Nλz,
ΨL(x=Ld/N)τ˜xλzw(x)x=Ld/N.
dλ(P+P)N4M.
FRz[f(x)g(x)]
=1iλzEz(x)-f(ξ)g(ξ)Ez(ξ)exp-i2πξxλzdξ.
FRz[f(x)g(x)]=1iλzEz(x)[I(u)G(u)],
I(u)=-f(ξ)Ez(ξ)exp(-i2πξu)dξ.
I(u)G(u)-I(u)G(u-u)du
=1λz-IxλzGx-xλzdx
=1λzIxλzGxλz,
I(x/λz)=iλzEz*(x)FRz[f(x)].
FRz[f(x)g(x)]=1λzEz(x){Ez*(x)FRz[f(x)]}Gxλz,
t(x)=L=-QQtL rectx-Lpa,
Ψ(x)=1iλzL=-QQtL expiπL2p2λz×exp-i2πLpxλzI(x, L),
I(x, L)=- rectξaexpiπξ2λz×exp-i2πξλz(x-Lp)dξ,
Ψ(x)=aiλzL=-QQtL expiπL2p2λz×exp-i2πLpxλz×sincaλz(x-Lp).

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