Abstract

We demonstrate the feasibility of measuring x-ray refractive indices by transparent edge diffraction without recourse to the Kramers–Kronig relations. The method requires a coherent x-ray source, a transparent sample with a straight edge, and a high resolution x-ray detector. Here, we use the aluminum Kα radiation originating from a laser-produced plasma to coherently illuminate the edge of thin aluminum and beryllium foils. The resulting diffraction patterns are recorded with an x-ray CCD camera. From least-squares fits of Fresnel diffraction modeling to the measured data we determine the refractive index of Al and Be at the wavelength of the Al Kα radiation (0.834 nm, 1.49 keV).

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References

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  1. D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge, 2007).
  2. D. C. Creagh, in International Tables for Crystallography,3rd ed. (Kluwer, 2004), Vol. C, pp. 241–258.
  3. D. C. Creagh, Aust. J. Phys. 28, 543 (1975).
    [CrossRef]
  4. F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
    [CrossRef]
  5. M. Deutsch and M. Hart, Phys. Rev. B 30, 643 (1984).
    [CrossRef]
  6. W.-K. Lee, P. Cloetens, and M. Schlenker, Acta Crystallogr. Sect. A 60, 58 (2004).
    [CrossRef]
  7. D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
    [CrossRef]
  8. J. Komrska, Opt. Acta 14, 127 (1967).
    [CrossRef]
  9. M. T. Tavassoly, S. R. Hosseini, A. M. Fard, and R. R. Naraghi, Appl. Opt. 51, 7170 (2012).
    [CrossRef]
  10. E. M. Gullikson, Index of Refraction http://henke.lbl.gov/optical_constants/ .
  11. B. Henke, E. Gullikson, and J. Davis, At. Data Nucl. Data Tables 54, 181 (1993).
    [CrossRef]

2012

2007

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

2004

W.-K. Lee, P. Cloetens, and M. Schlenker, Acta Crystallogr. Sect. A 60, 58 (2004).
[CrossRef]

1993

B. Henke, E. Gullikson, and J. Davis, At. Data Nucl. Data Tables 54, 181 (1993).
[CrossRef]

1992

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
[CrossRef]

1984

M. Deutsch and M. Hart, Phys. Rev. B 30, 643 (1984).
[CrossRef]

1975

D. C. Creagh, Aust. J. Phys. 28, 543 (1975).
[CrossRef]

1967

J. Komrska, Opt. Acta 14, 127 (1967).
[CrossRef]

Attwood, D.

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge, 2007).

Boschetto, D.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Clarke, R.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Cloetens, P.

W.-K. Lee, P. Cloetens, and M. Schlenker, Acta Crystallogr. Sect. A 60, 58 (2004).
[CrossRef]

Creagh, D. C.

D. C. Creagh, Aust. J. Phys. 28, 543 (1975).
[CrossRef]

D. C. Creagh, in International Tables for Crystallography,3rd ed. (Kluwer, 2004), Vol. C, pp. 241–258.

Davis, J.

B. Henke, E. Gullikson, and J. Davis, At. Data Nucl. Data Tables 54, 181 (1993).
[CrossRef]

Deutsch, M.

M. Deutsch and M. Hart, Phys. Rev. B 30, 643 (1984).
[CrossRef]

Fard, A. M.

Göbel, H.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
[CrossRef]

Gullikson, E.

B. Henke, E. Gullikson, and J. Davis, At. Data Nucl. Data Tables 54, 181 (1993).
[CrossRef]

Gullikson, E. M.

E. M. Gullikson, Index of Refraction http://henke.lbl.gov/optical_constants/ .

Hart, M.

M. Deutsch and M. Hart, Phys. Rev. B 30, 643 (1984).
[CrossRef]

Henke, B.

B. Henke, E. Gullikson, and J. Davis, At. Data Nucl. Data Tables 54, 181 (1993).
[CrossRef]

Hosseini, S. R.

Hou, B.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Komrska, J.

J. Komrska, Opt. Acta 14, 127 (1967).
[CrossRef]

Kumah, D.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Lee, W.-K.

W.-K. Lee, P. Cloetens, and M. Schlenker, Acta Crystallogr. Sect. A 60, 58 (2004).
[CrossRef]

Lengeler, B.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
[CrossRef]

Mordovanakis, A.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Mourou, G.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Naraghi, R. R.

Nees, J.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Rousse, A.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

Schlenker, M.

W.-K. Lee, P. Cloetens, and M. Schlenker, Acta Crystallogr. Sect. A 60, 58 (2004).
[CrossRef]

Schuster, M.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
[CrossRef]

Stanglmeier, F.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
[CrossRef]

Tavassoly, M. T.

Weber, W.

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
[CrossRef]

Acta Crystallogr. Sect. A

F. Stanglmeier, B. Lengeler, W. Weber, H. Göbel, and M. Schuster, Acta Crystallogr. Sect. A 48, 626 (1992).
[CrossRef]

W.-K. Lee, P. Cloetens, and M. Schlenker, Acta Crystallogr. Sect. A 60, 58 (2004).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

D. Boschetto, G. Mourou, A. Rousse, A. Mordovanakis, B. Hou, J. Nees, D. Kumah, and R. Clarke, Appl. Phys. Lett. 90, 011106 (2007).
[CrossRef]

At. Data Nucl. Data Tables

B. Henke, E. Gullikson, and J. Davis, At. Data Nucl. Data Tables 54, 181 (1993).
[CrossRef]

Aust. J. Phys.

D. C. Creagh, Aust. J. Phys. 28, 543 (1975).
[CrossRef]

Opt. Acta

J. Komrska, Opt. Acta 14, 127 (1967).
[CrossRef]

Phys. Rev. B

M. Deutsch and M. Hart, Phys. Rev. B 30, 643 (1984).
[CrossRef]

Other

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge, 2007).

D. C. Creagh, in International Tables for Crystallography,3rd ed. (Kluwer, 2004), Vol. C, pp. 241–258.

E. M. Gullikson, Index of Refraction http://henke.lbl.gov/optical_constants/ .

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Figures (2)

Fig. 1.
Fig. 1.

Sketch of Fresnel diffraction at a transparent half-plane of thickness d and complex refractive index n˜. Coherent illumination of the half-plane gives rise to a diffraction pattern that is recorded on the CCD. The shown pattern corresponds to T=0.8 and Δϕ=π. The coordinate on the detector is denoted by x, the source-edge distance by u, and the edge-detector distance by v.

Fig. 2.
Fig. 2.

Averaged profiles and corresponding least-squares fits of the diffraction patterns obtained for Al foils of thicknesses (a) 1.2 μm, (b) 1.8 μm, and (c) 2.5 μm as well as for a (d) 7.5 μm thick Be foil.

Tables (1)

Tables Icon

Table 1. Results for the Refractive Index Decrement δexp for Al and Be Obtained at λ=0.834nma

Equations (2)

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I=I02(X2+Y2),
X=T{cos(Δϕ)[1/2C(w)]+sin(Δϕ)[1/2S(w)]}+[1/2+C(w)],Y=T{cos(Δϕ)[1/2S(w)]sin(Δϕ)[1/2C(w)]}+[1/2+S(w)].

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