Abstract

New devices combining the Bragg reflection from periodic multilayer structures with Fraunhofer or Fresnel diffraction arising from lateral patterning of the multilayer are now available for x-ray optics. Using the Green’s-function method, we establish an integral equation for the scattered amplitude that is valid in the framework of both Fraunhofer and Fresnel diffraction. The scattered amplitude is given in the first and the second Born approximations for multilayer mirrors, laminar and sawtooth-profile multilayer gratings, and linear multilayer zone plates. The main diffractive properties of these devices are deduced. The efficiencies are computed in the first and/or in the second Born approximation and are compared with efficiencies obtained from a rigorous electromagnetic theory when they are available.

© 1993 Optical Society of America

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References

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  1. P. Dhez, “X and X-UV multilayered optics: principles, fabrication methods, tests and applications,” Ann Phys. Fr. 15, 493–527 (1990).
    [CrossRef]
  2. C. Khan Malek, “A review of microfabrication technologies: applications to x-ray optics,” J. X-Ray Sci. Technol. 3, 45–67 (1991).
    [CrossRef]
  3. T. W. Barbee, “Combined microstructure x-ray optics,” Rev. Sci. Instrum. 60, 1588–1595 (1989).
    [CrossRef]
  4. T. W. Barbee, J. V. Bixler, D. D. Dietrich, “Performance of multilayer coated gratings at near normal incidence in the extreme ultraviolet,” Phys. Scr. 41, 740–744 (1990).
    [CrossRef]
  5. E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
    [CrossRef]
  6. J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
    [CrossRef]
  7. J. C. Rife, W R. Hunter, T. W. Barbee, R. G. Cruddace, “Multilayer-coated blazed grating performance in the soft x-ray region,” Appl. Opt. 28, 2984–2986 (1989).
    [CrossRef] [PubMed]
  8. J. C. Rife, T. W. Barbee, W. R. Hunter, R. G. Cruddace, “Performance of a tungsten/carbon multilayer-coated, blazed grating from 150 to 1700 eV,” Phys. Scr. 41, 418–421 (1990).
    [CrossRef]
  9. V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
    [CrossRef]
  10. B. Pardo, J-M. André, A. Sammar, “Dynamical theory of diffraction at in-depth multilayered gratings,” J. Opt. (Paris) 22, 141–148 (1991).
    [CrossRef]
  11. M. Nevière, “Multilayer coated gratings for x-ray diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–1473 (1991).
    [CrossRef]
  12. A. Sammar, J-M. André, B. Pardo, “Diffraction and scattering by lamellar multilayer grating in the X-UV region,” Opt. Commun. 86, 245–254 (1991).
    [CrossRef]
  13. G. Barton, Elements of Green’s Functions and Propagation: Potentials, Diffusion, and Waves (Clarendon, Oxford, 1989).
  14. W. K. Warburton, “On the diffraction properties of multilayer coated plane gratings,” Nucl. Instrum. Methods A 291, 278–285 (1990).
    [CrossRef]
  15. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
    [CrossRef]
  16. A. I. Erko, “Synthetized Bragg–Fresnel multilayer optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
    [CrossRef]
  17. J. Kirz, “Phase zone plates for x rays and the extreme UV,” J. Opt. Soc. Am. 64, 301–309 (1974).
    [CrossRef]
  18. B. Pardo, T. Megademini, J-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
    [CrossRef]

1992 (1)

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

1991 (5)

M. Nevière, “Multilayer coated gratings for x-ray diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–1473 (1991).
[CrossRef]

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

C. Khan Malek, “A review of microfabrication technologies: applications to x-ray optics,” J. X-Ray Sci. Technol. 3, 45–67 (1991).
[CrossRef]

B. Pardo, J-M. André, A. Sammar, “Dynamical theory of diffraction at in-depth multilayered gratings,” J. Opt. (Paris) 22, 141–148 (1991).
[CrossRef]

A. Sammar, J-M. André, B. Pardo, “Diffraction and scattering by lamellar multilayer grating in the X-UV region,” Opt. Commun. 86, 245–254 (1991).
[CrossRef]

1990 (5)

W. K. Warburton, “On the diffraction properties of multilayer coated plane gratings,” Nucl. Instrum. Methods A 291, 278–285 (1990).
[CrossRef]

P. Dhez, “X and X-UV multilayered optics: principles, fabrication methods, tests and applications,” Ann Phys. Fr. 15, 493–527 (1990).
[CrossRef]

T. W. Barbee, J. V. Bixler, D. D. Dietrich, “Performance of multilayer coated gratings at near normal incidence in the extreme ultraviolet,” Phys. Scr. 41, 740–744 (1990).
[CrossRef]

J. C. Rife, T. W. Barbee, W. R. Hunter, R. G. Cruddace, “Performance of a tungsten/carbon multilayer-coated, blazed grating from 150 to 1700 eV,” Phys. Scr. 41, 418–421 (1990).
[CrossRef]

A. I. Erko, “Synthetized Bragg–Fresnel multilayer optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
[CrossRef]

1989 (2)

1988 (2)

B. Pardo, T. Megademini, J-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

1982 (1)

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

1974 (1)

André, J-M.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

B. Pardo, J-M. André, A. Sammar, “Dynamical theory of diffraction at in-depth multilayered gratings,” J. Opt. (Paris) 22, 141–148 (1991).
[CrossRef]

A. Sammar, J-M. André, B. Pardo, “Diffraction and scattering by lamellar multilayer grating in the X-UV region,” Opt. Commun. 86, 245–254 (1991).
[CrossRef]

B. Pardo, T. Megademini, J-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

Aristov, V. V.

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

Bac, S.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Barbee, T. W.

T. W. Barbee, J. V. Bixler, D. D. Dietrich, “Performance of multilayer coated gratings at near normal incidence in the extreme ultraviolet,” Phys. Scr. 41, 740–744 (1990).
[CrossRef]

J. C. Rife, T. W. Barbee, W. R. Hunter, R. G. Cruddace, “Performance of a tungsten/carbon multilayer-coated, blazed grating from 150 to 1700 eV,” Phys. Scr. 41, 418–421 (1990).
[CrossRef]

T. W. Barbee, “Combined microstructure x-ray optics,” Rev. Sci. Instrum. 60, 1588–1595 (1989).
[CrossRef]

J. C. Rife, W R. Hunter, T. W. Barbee, R. G. Cruddace, “Multilayer-coated blazed grating performance in the soft x-ray region,” Appl. Opt. 28, 2984–2986 (1989).
[CrossRef] [PubMed]

Barchewitz, R.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Barton, G.

G. Barton, Elements of Green’s Functions and Propagation: Potentials, Diffusion, and Waves (Clarendon, Oxford, 1989).

Berrouane, H.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Bixler, J. V.

T. W. Barbee, J. V. Bixler, D. D. Dietrich, “Performance of multilayer coated gratings at near normal incidence in the extreme ultraviolet,” Phys. Scr. 41, 740–744 (1990).
[CrossRef]

Christensen, F. E.

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

Cruddace, R. G.

J. C. Rife, T. W. Barbee, W. R. Hunter, R. G. Cruddace, “Performance of a tungsten/carbon multilayer-coated, blazed grating from 150 to 1700 eV,” Phys. Scr. 41, 418–421 (1990).
[CrossRef]

J. C. Rife, W R. Hunter, T. W. Barbee, R. G. Cruddace, “Multilayer-coated blazed grating performance in the soft x-ray region,” Appl. Opt. 28, 2984–2986 (1989).
[CrossRef] [PubMed]

Dhez, P.

P. Dhez, “X and X-UV multilayered optics: principles, fabrication methods, tests and applications,” Ann Phys. Fr. 15, 493–527 (1990).
[CrossRef]

Dietrich, D. D.

T. W. Barbee, J. V. Bixler, D. D. Dietrich, “Performance of multilayer coated gratings at near normal incidence in the extreme ultraviolet,” Phys. Scr. 41, 740–744 (1990).
[CrossRef]

Erko, A. I.

A. I. Erko, “Synthetized Bragg–Fresnel multilayer optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
[CrossRef]

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

Fujikawa, B. K.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Henke, B. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Hunter, W R.

Hunter, W. R.

J. C. Rife, T. W. Barbee, W. R. Hunter, R. G. Cruddace, “Performance of a tungsten/carbon multilayer-coated, blazed grating from 150 to 1700 eV,” Phys. Scr. 41, 418–421 (1990).
[CrossRef]

Khan Malek, C.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

C. Khan Malek, “A review of microfabrication technologies: applications to x-ray optics,” J. X-Ray Sci. Technol. 3, 45–67 (1991).
[CrossRef]

Kirz, J.

Ladan, F. R.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Lamboy, P.

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

Lee, P.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Martinov, V. V.

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

Megademini, T.

B. Pardo, T. Megademini, J-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

Moreno, T.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Nevière, M.

Padmore, H. A.

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

Pardo, B.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

B. Pardo, J-M. André, A. Sammar, “Dynamical theory of diffraction at in-depth multilayered gratings,” J. Opt. (Paris) 22, 141–148 (1991).
[CrossRef]

A. Sammar, J-M. André, B. Pardo, “Diffraction and scattering by lamellar multilayer grating in the X-UV region,” Opt. Commun. 86, 245–254 (1991).
[CrossRef]

B. Pardo, T. Megademini, J-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

Puik, E. J.

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

Rife, J. C.

J. C. Rife, T. W. Barbee, W. R. Hunter, R. G. Cruddace, “Performance of a tungsten/carbon multilayer-coated, blazed grating from 150 to 1700 eV,” Phys. Scr. 41, 418–421 (1990).
[CrossRef]

J. C. Rife, W R. Hunter, T. W. Barbee, R. G. Cruddace, “Multilayer-coated blazed grating performance in the soft x-ray region,” Appl. Opt. 28, 2984–2986 (1989).
[CrossRef] [PubMed]

Rivoira, R.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Sammar, A.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

B. Pardo, J-M. André, A. Sammar, “Dynamical theory of diffraction at in-depth multilayered gratings,” J. Opt. (Paris) 22, 141–148 (1991).
[CrossRef]

A. Sammar, J-M. André, B. Pardo, “Diffraction and scattering by lamellar multilayer grating in the X-UV region,” Opt. Commun. 86, 245–254 (1991).
[CrossRef]

Schirmann, D.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Shimabukuro, R. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Tanaka, T. J.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

Troussel, P.

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

Van der Wiel, M. J.

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

Verhoeven, J.

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

Warburton, W. K.

W. K. Warburton, “On the diffraction properties of multilayer coated plane gratings,” Nucl. Instrum. Methods A 291, 278–285 (1990).
[CrossRef]

Ann Phys. Fr. (1)

P. Dhez, “X and X-UV multilayered optics: principles, fabrication methods, tests and applications,” Ann Phys. Fr. 15, 493–527 (1990).
[CrossRef]

Appl. Opt. (1)

At. Data Nucl. Data Tables (1)

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fujikawa, “Low-energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” At. Data Nucl. Data Tables 27, 1–144 (1982).
[CrossRef]

J. Opt. (Paris) (1)

B. Pardo, J-M. André, A. Sammar, “Dynamical theory of diffraction at in-depth multilayered gratings,” J. Opt. (Paris) 22, 141–148 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. X-Ray Sci. Technol. (3)

A. I. Erko, “Synthetized Bragg–Fresnel multilayer optics,” J. X-Ray Sci. Technol. 2, 297–316 (1990).
[CrossRef]

C. Khan Malek, “A review of microfabrication technologies: applications to x-ray optics,” J. X-Ray Sci. Technol. 3, 45–67 (1991).
[CrossRef]

E. J. Puik, M. J. Van der Wiel, P. Lamboy, J. Verhoeven, F. E. Christensen, H. A. Padmore, “Characterization of a multilayer coated laminar reflection grating at λ = 0.154 nm,” J. X-Ray Sci. Technol. 3, 19–34 (1991).
[CrossRef]

Nucl. Instrum. Methods A (1)

W. K. Warburton, “On the diffraction properties of multilayer coated plane gratings,” Nucl. Instrum. Methods A 291, 278–285 (1990).
[CrossRef]

Opt. Commun. (1)

A. Sammar, J-M. André, B. Pardo, “Diffraction and scattering by lamellar multilayer grating in the X-UV region,” Opt. Commun. 86, 245–254 (1991).
[CrossRef]

Phys. Scr. (2)

J. C. Rife, T. W. Barbee, W. R. Hunter, R. G. Cruddace, “Performance of a tungsten/carbon multilayer-coated, blazed grating from 150 to 1700 eV,” Phys. Scr. 41, 418–421 (1990).
[CrossRef]

T. W. Barbee, J. V. Bixler, D. D. Dietrich, “Performance of multilayer coated gratings at near normal incidence in the extreme ultraviolet,” Phys. Scr. 41, 740–744 (1990).
[CrossRef]

Rev. Phys. Appl. (2)

B. Pardo, T. Megademini, J-M. André, “X-UV synthetic interference mirrors: theoretical approach,” Rev. Phys. Appl. 23, 1579–1597 (1988).
[CrossRef]

V. V. Aristov, A. I. Erko, V. V. Martinov, “Principles of Bragg–Fresnel multilayer optics,” Rev. Phys. Appl. 23, 1623–1630 (1988).
[CrossRef]

Rev. Sci. Instrum. (2)

J-M. André, A. Sammar, C. Khan Malek, P. Troussel, S. Bac, R. Barchewitz, B. Pardo, H. Berrouane, T. Moreno, F. R. Ladan, R. Rivoira, D. Schirmann, “Multilayer gratings for the soft x-ray region,” Rev. Sci. Instrum. 63, 1399–1403 (1992).
[CrossRef]

T. W. Barbee, “Combined microstructure x-ray optics,” Rev. Sci. Instrum. 60, 1588–1595 (1989).
[CrossRef]

Other (1)

G. Barton, Elements of Green’s Functions and Propagation: Potentials, Diffusion, and Waves (Clarendon, Oxford, 1989).

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Figures (10)

Fig. 1
Fig. 1

Geometry of multilayer gratings.

Fig. 2
Fig. 2

Efficiency versus glancing angle θ0 of a laminar-amplitude multilayer grating. 10 Molybdenum–carbon bilayers, dc = 2 nm, dMo = 1 nm, grating pitch D = 360 nm, D′/D = Γ = 0.5, photon energy = 930 eV (a) Calculated from a rigorous theory, (b) calculated in the first Born approximation, (c) calculated in the second Born approximation.

Fig. 3
Fig. 3

Efficiency versus glancing angle θ0 of the fifth diffraction order of a multilayer SPG calculated in the first Born approximation in the blaze condition. Blaze angle δ = 2.38 deg, grating pitch D = 360 nm, D′/D = Γ = 1, same multilayer structure and photon energy as in Fig. 2.

Fig. 4
Fig. 4

Efficiency versus glancing angle θ0 of the fourth, fifth, and sixth diffraction orders of a multilayer SPG calculated in the first Born approximation in the blaze condition for the fifth order. Blaze angle δ = 2.38 deg, grating pitch D = 360 nm, D′/D = Γ = 0.5, same multilayer structure and photon energy as in Fig. 2.

Fig. 5
Fig. 5

Efficiency versus glancing angle θ0 of the fourth, fifth, and sixth diffraction orders of a multilayer SPG calculated in the second Born approximation in the blaze condition for the fifth order. Blaze angle 5 = 2.38 deg, grating pitch D = 360 nm, D′/D = Γ = 0.5, same multilayer structure and photon energy as in Fig. 2.

Fig. 6
Fig. 6

Geometry of linear multilayer zone plates. For positive zone plates, black strips are multilayer bars; for negative zone plates, white strips are multilayer bars.

Fig. 7
Fig. 7

Diffracted intensity versus distance of observation r in the direction of focalization of a positive zone plate with parameter a = 3150 nm and number of lines L = 60; multilayer structure and photon energy are the same as in Fig. 2.

Fig. 8
Fig. 8

Three-dimensional depiction of diffracted intensity versus the distance r around the first focus and the number of lines for the same parameters as in Fig. 7.

Fig. 9
Fig. 9

Diffracted intensity in a light-density plot, where the lighter regions are for higher intensity, recorded in the real (xz′) plane perpendicular to the y axis from a scanning of the ZP; parameters and structures are the same as in Fig. 7.

Fig. 10
Fig. 10

Diffracted intensity in a light-density plot with a polychromatic x-ray radiation at fixed glancing angle θ0 = 13.1 deg; parameters are the same as in Fig. 7.

Tables (1)

Tables Icon

Table 1 (−2, −1,0, +1, +2) Diffraction Peak Efficiency of a Laminar Amplitude Grating Consisting of 10 Molybdenum–Carbon Bilayersa

Equations (115)

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[ Δ + k 2 ( R ) ] φ = 0 ,
( E H 0 ) φ = V φ ,
φ ± = φ 0 + G 0 ± ( E ) V φ ± ,
( E H 0 ) φ = 0
( E H 0 ) G 0 ± ( R , R , E ) = δ ( R R ) .
G 0 ± ( E ) = lim s 0 + G 0 ( E ± i s )
G 0 ± ( R , R , E ) = exp ( ± i k | R R | ) 4 π | R R | .
φ ± = φ 0 + G 0 ± ( E ) V φ 0 + G 0 ± ( E ) V G 0 ± ( E ) V φ 0 + .
T ± ( E ) = V G 0 ± ( E ) ( E H 0 ) ,
T ± ( E ) = V + V G 0 ± ( E ) T ± ( E ) .
T ± ( E ) = V + V G 0 ± ( E ) V + V G 0 ± ( E ) V G 0 ± ( E ) V + .
φ ± = φ 0 + G 0 ± ( E ) T ± ( E ) φ 0 .
φ 0 = exp ( i k r ) ,
φ ± ( r ) = exp ( i k r ) 1 4 π d R exp ( ± i k | r R | ) | r R | T ± ( E ) × exp ( i k R ) .
φ ( r ) = exp ( i k r ) 1 4 π d R exp ( i k | r R | ) | r R | V ( R ) × exp ( i k r ) + 1 ( 4 π ) 2 d R exp ( i k | r R | ) | r R | V ( R ) × d R exp ( i k | R R | ) | R R | V ( R ) exp ( i k R ) .
φ ( r ) = exp ( i k r ) 1 4 π exp ( i k r ) r d R exp ( i k d R ) × exp ( i k R 2 2 r ) T ( E ) exp ( i k R ) ,
k d = k r r , k = | k d × R | 2 k R 2
φ ( r ) = exp ( i k r ) 1 4 π exp ( i k r ) r d R × exp ( i k R 2 2 r ) k d | R R | T ( E ) | k .
d R | R R | = 1 ,
φ ( r ) = exp ( i k r ) 1 4 π exp ( i k r ) r k d | T ( E ) | k ,
T ( E ) = + V G 0 ( E ) + V G 0 ( E ) V G 0 ( E ) + ,
( R , r ) = exp ( i k R 2 2 r ) V ( R ) .
φ ( r ) = exp ( i k r ) + exp ( i k r ) r f ( k d , k , r ) .
f ( k d , k , r ) = 1 4 π k d | T ( E ) | k .
f ( k d , k , r ) = 1 4 π ̂ ( k d k , r ) + lim s 0 + 1 ( 2 π ) d × d K V ̂ ( k d k ) K 2 k 2 + i s f ( K , k , r ) ,
G 0 ( E ) = lim s 0 + 1 K 2 k 2 + i s
V ̂ ( K ) = d R V ( R ) exp ( i K R ) ,
̂ ( K , r ) = d R ( R , r ) exp ( i K R ) .
f ( 1 ) ( Q , r ) = 1 4 π ̂ ( Q , r ) ,
Q = k d k .
f ( 2 ) ( k d , k , r ) = f ( 1 ) ( Q , r ) lim s 0 + 1 4 π 1 ( 2 π ) d × d K V ̂ ( k d K ) ̂ ( K k , r ) K 2 k 2 + i s
f ( n ) ( k d , k , r ) = f ( n 1 ) ( k d , k , r ) + lim s 0 + 1 ( 2 π ) d × d K V ̂ ( k d K ) f ( n 1 ) ( K k , r ) K 2 k 2 + i s
V ( R ) = [ 1 ( R ) ] k 2 .
V G ( R ) = V G ( x , z ) = V D ( x . z ) Π Π D ( x ) ,
V D ( x , z ) = n = 0 N ( V a + V b T a ) Θ ( z f n ( x ) ) × Θ ( g n ( x ) z ) Θ [ z f ( x ) ] Θ [ g ( x ) z ] ,
f n ( x ) = tan ( δ ) x + n d g n ( x ) = tan ( δ ) x + ( d a + n d ) f ( x ) = tan ( η ) x g ( x ) = tan ( η ) ( x D ) .
[ d a cos ( δ ) , d a sin ( δ ) ]
α = 1 tan ( γ ) + tan ( δ )
V ̂ G ( K ) = + d x + d z V D ( x , z ) × exp [ i ( K x x + K z z ] Π Π 2 π / D ( K x ) ,
V ̂ G ( K ) = g ( K x ) × ( 1 exp { i [ K x + K z tan ( δ ) ] tan ( γ ) α D } α [ K x + K z tan ( δ ) ] ) × n = 0 N exp { i n [ K x K z tan ( γ ) ] α d * } × ( V a 1 exp { i [ K x K z tan ( γ ) ] α d a * } [ K x K z tan ( γ ) ] + exp { i [ K x sin ( δ ) K z cos ( δ ) ] d a * } × V b 1 exp { i [ K x K z tan ( γ ) ] α d b * } [ K x K z tan ( γ ) ] ) ,
d * = d cos ( δ ) , d a * = d a cos ( δ ) , d b * = d b cos ( δ ) , g = 2 π D
n = 0 N exp { i n [ K x K z tan ( γ ) ] α d * } = U N + 1 { [ K x K z tan ( γ ) ] α d * 2 } × exp { i N [ K x K z tan ( γ ) ] α d * 2 } ,
U N ( x ) = sin ( N x ) sin x
V ̂ G ( K x , K z ) = n = δ ( K x n g ) A ( K x , K z ) ,
A ( K x , K z ) = ( 1 exp { i [ K x + K z tan ( δ ) ] tan ( γ ) α D } α [ K x + K z tan ( δ ) ] ) × U N + 1 { [ K x K z tan ( γ ) ] α d * 2 } × exp { i N [ K x K z tan ( γ ) ] α d * 2 } × ( V a 1 exp { i [ K x K z tan ( γ ) ] α d a * } [ K x K z tan ( γ ) ] + exp { i [ K x sin ( δ ) K z cos ( δ ) ] d a * } × V b 1 exp { i [ K x K z tan ( γ ) ] α d b * } [ K x K z tan ( γ ) ] ) .
f G ( 1 ) = V ̂ G ( Q ) 4 π = 1 4 π p A ( Q x , Q z ) δ ( Q x p g ) .
Q x = p g ,
Q z = ± [ k 2 ( k x + p g ) 2 ] 1 / 2 k z ,
G ( 1 ) ( p ) = [ 1 / ( 4 π ) 2 ] | A ( p ) | 2 .
D [ cos ( θ ) cos ( θ 0 ) ] = p λ .
[ Q x Q z tan ( γ ) ] α ( d * / 2 ) = m π ,
[ sin ( φ ) + sin ( φ 0 ) ] d = m λ ,
[ sin ( θ ) + sin ( θ 0 ) ] d = m λ .
2 sin ( θ 0 ) d = m λ .
Q x + Q z tan ( δ ) = 0 .
[ cos ( γ ) + cos ( δ ) ] [ cos ( φ ) cos ( φ 0 ) ] = 0 ;
2 d sin ( φ 0 Bl ) = m λ .
2 D sin ( δ ) sin ( φ 0 Bl ) = p λ .
D sin ( δ ) / d = p / m .
f G ( 2 ) = f G ( 1 ) 1 4 π 1 ( 2 π ) 2 n m + d K z + d K x × A ( k x d K x , k z d K z ) A ( K x k x , K z k z ) k x 2 + k z 2 k x 2 k z 2 × δ ( k x d K x n g ) δ ( K x k x m g ) .
p I ( p ) δ ( Q x p g ) ,
I ( p ) = n + d K z A ( n g , k z d K z ) A [ ( p n ) g , K z k z ] k z 2 [ k 2 ( k x d + n g ) 2 ]
I ( p ) = 2 π i n h [ k z d ( n ) ] A [ k z d ( p ) k z d ( n ) ] A [ k z d ( n ) k z ] g [ k z d ( n ) ] A [ k z d ( p ) + k z d ( n ) ] A [ k z d ( n ) k z ] 2 k z d ( n ) ,
k z d ( n ) = [ k 2 ( k x + n g ) 2 ] 1 / 2 ,
h ( x ) = sign ( x 2 ) + 3 4 , g ( x ) = sign ( x 2 ) + 1 4 ,
sign ( x ) = x / | x | .
f G ( 2 ) = f G ( 1 ) 1 4 π ( 2 π ) 2 p I ( p ) δ ( Q x p g ) .
G ( 2 ) ( p ) = [ 1 / ( 4 π ) 2 ] | A ( p ) + [ 1 / ( 2 π ) 2 ] I ( p ) | 2 .
σ tot = 4 π k Im [ f ( Q = 0 ) ] .
A ( K x , K z ) = A * ( K x , K z ) ,
Mo = ( 1 0.00178 + i 0.00065 ) 2 , c = ( 1 0.00057 + i 0.00006 ) 2 .
I p = rect [ x a ( 2 j + 1 ) 1 / 2 ] rect [ x a ( 2 j ) 1 / 2 ] ,
I n = rect { x a [ ( 2 ( j + 1 ) ] 1 / 2 } rect [ x a ( 2 j + 1 ) 1 / 2 ] ,
rect ( x b ) = { 1 if | x | < b 0 if | x | b .
V ̂ ( K ) = + d x + d z V ( x , z ) exp [ i ( K x x + K z z ) ] ,
̂ ( K , r ) = + d x + d z exp [ i k ( x 2 + z 2 ) 2 r ] V ( x , z ) × exp [ i ( K x x + K z z ) ] ,
V ( x , z ) = X ( x ) Z ( z ) ,
X ( x ) X + ( x ) = i = 0 L rect [ x / a ( 2 j + 1 ) 1 / 2 ] rect [ x / a ( 2 j ) 1 / 2 ]
X ( x ) X ( x ) = j = 0 L rect { x / a [ 2 ( j + 1 ) ] 1 / 2 } rect [ x / a ( 2 j + 1 ) 1 / 2 ]
Z ( z ) = n = 0 N ( V a + V b T a ) Θ ( z f n ) Θ ( g n z ) ,
( 0 , d a ) ,
V a = ( 1 a ) k 2 , V b = ( 1 b ) k 2 .
f n = n d , g n = n d + d a .
V ̂ ± ( K ) = V ̂ x ± ( K x ) V ̂ z ( K z ) ,
̂ ± ( K , r ) = ̂ x ± ( K x , r ) ̂ z ( K z , r ) ,
̂ x ± ( K x , r ) = exp ( i k 2 r ξ 2 ) × + exp ( i k 2 r ( x ξ ) 2 ) X ± ( x ) d x ,
ξ = ( K x r / k ) .
̂ z ( K z , r ) V ̂ z ( K z ) .
V ̂ x + ( K x ) = 2 a j = 0 j = L ( 1 ) j [ 2 j + 1 sinc ( 2 j + 1 a K x ) 2 j sinc ( 2 j a K x ) ] ,
V ̂ x ( K x ) = 2 a j = 0 j = L ( 1 ) j [ 2 j + 2 sinc ( 2 j + 2 a K x ) 2 j + 1 sinc ( 2 j + 1 a K x ) ] .
̂ x ± ( K x , r ) = ± exp ( i k 2 r ξ 2 ) j ( 1 ) j + 1 × a j a j exp [ i k 2 r ( x ξ ) 2 ] d x .
Fr ( w ) = C ( w ) + i S ( w ) = 0 w cos ( π 2 t 2 ) d t + i 0 w sin ( π 2 t 2 ) d t .
( k / 2 r ) ( x ξ ) 2 = ( π / 2 ) t 2 ,
̂ x ± ( K x , r ) = ± i exp ( i k 2 r ξ 2 ) π r k j ( 1 ) j + 1 × { Fr [ ( a j ξ ) k π r ] + Fr [ ( a j + ξ ) k π r ] } .
Fr ( x ) = ( 1 + i / 2 ) Erf [ ( π / 2 ) ( 1 i ) x ] ,
Erf ( x ) = 2 π 0 x exp ( t 2 ) d t ,
V ̂ z ( K z ) = [ V a 1 exp ( i K z d a ) K z + V b exp ( i K z d a ) 1 exp ( i K z d b ) K z ] U N + 1 ( K z d 2 ) .
f ZP ( 1 ) ( r ) = [ ̂ ( Q , r ) / 4 π ] ,
I ZP ( 1 ) ( r ) = | ̂ ( Q , r ) | 2 / ( 4 π ) 2 .
r 2 n + 1 a 2 sin 2 ( θ 0 ) ( 2 n + 1 ) λ ,
f ZP ( 2 ) ( r ) = f ZP ( 1 ) ( r ) 1 4 π 1 ( 2 π ) 2 × + d K z V ̂ z ( k z d K z ) V ̂ z ( K z k z ) × + d K x ̂ x ( k x d K x , r ) V ̂ x ( K x k x ) K x 2 + K z 2 k 2 ,
f ZP ( 2 ) ( r ) = f ZP ( 1 ) ( r ) + i ( 4 π ) 2 + d K z V ̂ z ( k z d K z ) V ̂ z ( K z k z ) × { h ( K z ) ̂ x [ k x d ( k 2 K x d ) 1 / 2 , r ] V ̂ x [ ( k 2 K z 2 ) 1 / 2 k x ] ( k 2 K z 2 ) 1 / 2 g ( K z ) ̂ x [ k x d + ( k 2 K z 2 ) 1 / 2 , r ] V ̂ x [ ( k 2 K z 2 ) 1 / 2 k x ] ( k 2 K z 2 ) 1 / 2 } .
K z = k tanh ( t ) .
z = m a 2 / 2 ( 2 p + 1 ) d ,
E 0 [ 0 , φ s ( x , z ) , 0 ] .
i k E 0 ( φ s z , 0 , φ s x ) .
P s = ( c / 8 π ) real [ E s H s * ] ,
P s = ( i c E 0 2 / 8 π k ) φ s φ s * .
P s = i c E 0 2 8 π k ( φ s φ s * r , φ s r φ s * θ , φ s r sin ψ φ s * φ ) .
φ s * r = 1 4 π d R exp ( i k d R ) r [ exp ( i k r ) r × exp ( i k R 2 2 R ) ] T * ( E ) exp ( i k R ) .
r [ exp ( i k r ) r exp ( i k R 2 2 r ) ] = i k exp ( i k r ) r exp ( i k R 2 2 r ) ( 1 λ i 2 π r R 2 r 2 ) .
P s ( c E 0 2 / 8 π r 2 ) | f | 2 ( r / r ) .
d I d Ω = I 0 | f | 2 ,
I 0 = c E 0 2 / 8 π .
| f | 2 = 1 I 0 d I d Ω .

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