Abstract

Fractional Talbot images of optical gratings acting as periodic phase objects have been obtained by use of x rays of 0.069-nm wavelength from a third-generation synchrotron radiation source. Quantitative evaluation of the data obtained as a function of defocusing distance provides information on the lateral coherence of the beam as well as on the phase modulation in the object.

© 1997 Optical Society of America

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References

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  1. H. F. Talbot, Phil. Mag., third series  9, 401 (1836).
  2. L. Liu, Appl. Opt. 28, 4668 (1989).
    [CrossRef] [PubMed]
  3. V. Arrizón and E. López-Olazagasti, J. Opt. Soc. Am. A 12, 801 (1995).
    [CrossRef]
  4. H. Hamam and J. L. De Bougrenet de la Tocnaye, J. Opt. Soc. Am. A 12, 1920 (1995).
    [CrossRef]
  5. A. I. Erko, V. V. Aristov, and B. Vidal, Diffraction X-ray Optics (IOP Publishing, Bristol, UK, 1996) ;V. V. Aristov, A. I. Erko, and V. V. Martynov, Opt. Commun. 53, 159 (1985); Opt. Spectrosc. (USSR) 64, 376 (1988).
    [CrossRef]
  6. R. Coïsson, Appl. Opt. 34, 904 (1995).
    [CrossRef]
  7. A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
    [CrossRef]
  8. P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
    [CrossRef]
  9. Ya. M. Hartman and A. Snigirev, in X-ray Microscopy IV (Bogorodskii Pechatnik, Moscow, 1994), pp. 429–432.
  10. M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).
  11. J. P. Guigay, Opt. Acta 18, 677 (1971).
    [CrossRef]
  12. V. Arrizón and J. Ojeda-Castañeda, J. Opt. Soc. Am. A 9, 1801 (1992).
    [CrossRef]
  13. H. Fujiwara, Opt. Acta 21, 861 (1974).
    [CrossRef]
  14. J. P. Guigay, Optik 49, 121 (1977).
  15. K. Fezzaa, F. Comin, S. Marchesini, R. Coïsson, and M. Belakhovsky, “X-ray interferometry at ESRF using two coherent beams from Fresnel mirrors,” J. X-Ray Sci. Technol. (to be published).

1996 (1)

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
[CrossRef]

1995 (4)

1992 (1)

1989 (1)

1977 (1)

J. P. Guigay, Optik 49, 121 (1977).

1974 (1)

H. Fujiwara, Opt. Acta 21, 861 (1974).
[CrossRef]

1971 (1)

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

1836 (1)

H. F. Talbot, Phil. Mag., third series  9, 401 (1836).

Aristov, V. V.

A. I. Erko, V. V. Aristov, and B. Vidal, Diffraction X-ray Optics (IOP Publishing, Bristol, UK, 1996) ;V. V. Aristov, A. I. Erko, and V. V. Martynov, Opt. Commun. 53, 159 (1985); Opt. Spectrosc. (USSR) 64, 376 (1988).
[CrossRef]

Arrizón, V.

Barrett, R.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
[CrossRef]

Baruchel, J.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
[CrossRef]

Belakhovsky, M.

K. Fezzaa, F. Comin, S. Marchesini, R. Coïsson, and M. Belakhovsky, “X-ray interferometry at ESRF using two coherent beams from Fresnel mirrors,” J. X-Ray Sci. Technol. (to be published).

Born, M.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Cloetens, P.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
[CrossRef]

Coïsson, R.

R. Coïsson, Appl. Opt. 34, 904 (1995).
[CrossRef]

K. Fezzaa, F. Comin, S. Marchesini, R. Coïsson, and M. Belakhovsky, “X-ray interferometry at ESRF using two coherent beams from Fresnel mirrors,” J. X-Ray Sci. Technol. (to be published).

Comin, F.

K. Fezzaa, F. Comin, S. Marchesini, R. Coïsson, and M. Belakhovsky, “X-ray interferometry at ESRF using two coherent beams from Fresnel mirrors,” J. X-Ray Sci. Technol. (to be published).

De Bougrenet de la Tocnaye, J. L.

Erko, A. I.

A. I. Erko, V. V. Aristov, and B. Vidal, Diffraction X-ray Optics (IOP Publishing, Bristol, UK, 1996) ;V. V. Aristov, A. I. Erko, and V. V. Martynov, Opt. Commun. 53, 159 (1985); Opt. Spectrosc. (USSR) 64, 376 (1988).
[CrossRef]

Fezzaa, K.

K. Fezzaa, F. Comin, S. Marchesini, R. Coïsson, and M. Belakhovsky, “X-ray interferometry at ESRF using two coherent beams from Fresnel mirrors,” J. X-Ray Sci. Technol. (to be published).

Fujiwara, H.

H. Fujiwara, Opt. Acta 21, 861 (1974).
[CrossRef]

Guigay, J. P.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
[CrossRef]

J. P. Guigay, Optik 49, 121 (1977).

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

Hamam, H.

Hartman, Ya. M.

Ya. M. Hartman and A. Snigirev, in X-ray Microscopy IV (Bogorodskii Pechatnik, Moscow, 1994), pp. 429–432.

Kohn, V.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Kuznetsov, S.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Liu, L.

López-Olazagasti, E.

Marchesini, S.

K. Fezzaa, F. Comin, S. Marchesini, R. Coïsson, and M. Belakhovsky, “X-ray interferometry at ESRF using two coherent beams from Fresnel mirrors,” J. X-Ray Sci. Technol. (to be published).

Ojeda-Castañeda, J.

Schelokov, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Schlenker, M.

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
[CrossRef]

Snigirev, A.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Ya. M. Hartman and A. Snigirev, in X-ray Microscopy IV (Bogorodskii Pechatnik, Moscow, 1994), pp. 429–432.

Snigireva, I.

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Talbot, H. F.

H. F. Talbot, Phil. Mag., third series  9, 401 (1836).

Vidal, B.

A. I. Erko, V. V. Aristov, and B. Vidal, Diffraction X-ray Optics (IOP Publishing, Bristol, UK, 1996) ;V. V. Aristov, A. I. Erko, and V. V. Martynov, Opt. Commun. 53, 159 (1985); Opt. Spectrosc. (USSR) 64, 376 (1988).
[CrossRef]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

Appl. Opt. (2)

J. Opt. Soc. Am. A (3)

J. Phys. D (1)

P. Cloetens, R. Barrett, J. Baruchel, J. P. Guigay, and M. Schlenker, J. Phys. D 29, 133 (1996); P. Cloetens, M. Pateyron-Salomé, J. Y. Buffière, G. Peix, J. Baruchel, F. Peyrin, and M. Schlenker, J. Appl. Phys. 81, 5878 (1997).
[CrossRef]

Opt. Acta (2)

J. P. Guigay, Opt. Acta 18, 677 (1971).
[CrossRef]

H. Fujiwara, Opt. Acta 21, 861 (1974).
[CrossRef]

Optik (1)

J. P. Guigay, Optik 49, 121 (1977).

Phil. Mag. (1)

H. F. Talbot, Phil. Mag., third series  9, 401 (1836).

Rev. Sci. Instrum. (1)

A. Snigirev, I. Snigireva, V. Kohn, S. Kuznetsov, and I. Schelokov, Rev. Sci. Instrum. 66, 5486 (1995).
[CrossRef]

Other (4)

Ya. M. Hartman and A. Snigirev, in X-ray Microscopy IV (Bogorodskii Pechatnik, Moscow, 1994), pp. 429–432.

M. Born and E. Wolf, Principles of Optics, 6th ed. (Pergamon, Oxford, 1980).

K. Fezzaa, F. Comin, S. Marchesini, R. Coïsson, and M. Belakhovsky, “X-ray interferometry at ESRF using two coherent beams from Fresnel mirrors,” J. X-Ray Sci. Technol. (to be published).

A. I. Erko, V. V. Aristov, and B. Vidal, Diffraction X-ray Optics (IOP Publishing, Bristol, UK, 1996) ;V. V. Aristov, A. I. Erko, and V. V. Martynov, Opt. Commun. 53, 159 (1985); Opt. Spectrosc. (USSR) 64, 376 (1988).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Fractional Talbot images obtained on a 2000-LPI grating at reduced defocusing distances z/ZT of (a) 0, (b) 1/8, (c) 1/4, (d) 1/2, and (e) 5/8. The curves represent the normalized intensity versus the coordinate x perpendicular to the lines with a vertical shift of 0.4 between them. a=12.7 µm, λ=0.069 nm, ZT=2a2/λ=4.68 m.

Fig. 2
Fig. 2

Amplitude of the fundamental of the intensity distribution versus the reduced distance for a 2000-LPI grating (filled circles) and a 4000-LPI grating (filled squares) obtained at an x-ray wavelength λ of 0.069  nm and with ZT of 4.68 and 1.17  m, respectively. The Fourier coefficient was scaled for the maximum possible contrast (respectively, 0.092 and 0.038). The fitted curves correspond to Eq.  (13) (a=12.7 µm, α0=0.7 µrad, and a=6.35 µm, α0=0.6 µrad).

Equations (13)

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ψx, z=iλz-1/2dξTx+ξexpiπξ2/λz,
ψx, z=2inn-1/2p=0,1,n-1Tx+pa/nCn, n, p,
Cn, n, p=k=0,1,2,2n-1expiπp+nk2/2nn
Cn, 1, p=1+-1pinexpiπp2/2n.
z=a2/λ,  ψ=Tx+a/2,
z=a2/2λ,  ψ=2i-1/2Tx+iTx+a/2,
z=3a2/2λ,  ψ=2i-1/2iTx+Tx+a/2.
Ix=1+sinφx-φx+a/2.
Im=S˜zfImcoh,
Sα=12πα0exp-α22α02
S˜zma=exp-2πα0zma2.
α0=aπ2zk2-z021/2lnI1z0I1zk1/2.
I1=2φ1 sin2πzZTexp-8πα0aλzZT2,

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