Abstract

Self-supporting transmission gratings with periods of 1 μm or below are used in combination with grazing-incidence telescopes in celestial x-ray astronomy. They can be produced with sizes up to only a few cm2; therefore, several hundreds or even thousands of individual elements are needed in order to cover the aperture of a telescope. This large number leads to the problem of characterization of the gratings regarding their x-ray performance. We demonstrate that spectrometry in the resonance domain using H polarization is a suitable method for the determination of the grating wire profile and deviations of the grating surface from a plane. Although developed originally for microwave applications it can be shown that the methods of strict solution of the Helmholtz equation are able to explain even small effects related to imperfections of periodic submicrometer structures.

© 1992 Optical Society of America

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  1. F. D. Seward, T. Chlebowski, J. P. Delvaille, J. P. Henry, S. M. Kahn, L. Van Speybroeck, J. Dijkstra, A. C. Brinkman, J. Heise, R. Mewe, J. Schrijver, “Calibration and efficiency of the Einstein objective grating spectrometer,” Appl. Opt. 21, 2012–2021 (1982).
    [CrossRef] [PubMed]
  2. C. R. Canizares, J. Kruper, “A sharp x-ray absorption feature in the BL Lacertae Object PKS 1255-304,” Astrophys. J. Lett. 278, L99–L100 (1984).
    [CrossRef]
  3. M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).
  4. A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).
  5. H. Bräuninger, H. Kraus, H. Dangschat, K. P. Beuermann, P. Predehl, J. Trümper, “Fabrication of transmission gratings for use in cosmic x-ray and XUV astronomy,” Appl. Opt. 18, 3502–3505 (1979).
    [CrossRef] [PubMed]
  6. M. C. Hutley, in Diffraction Gratings, N. H. March, H. N. Daglish, eds. (Academic, London, 1982), Chap. 5, pp. 140–146.
  7. P. Predehl, H. Bräuninger, W. Burkert, B. Aschenbach, J. Trümper, M. Kühne, P. Müller, “Transmission of grating spectrometer on SPEKTROSAT,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 265–272 (1988).
  8. M. C. Hutley, in Diffraction Gratings, N. H. March, H. N. Daglish, eds. (Academic, London, 1982), Chap. 5, pp. 157–163.
  9. J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
    [CrossRef]
  10. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  11. R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 26–30.
    [CrossRef]
  12. A. Hessel, A. A. Oliner, “A new theory of Wood’s anomalies on optical gratings,” Appl. Opt. 4, 1275–1297 (1965).
    [CrossRef]
  13. M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 123–157.
    [CrossRef]
  14. A. A. Volkov, B. P. Gorshunov, A. A. Irisov, G. V. Kozlov, S. P. Lebedev, “Electrodynamic properties of plane wire grids,” Int. J. Infrared Millimeter Waves 3, 19–43 (1982).
    [CrossRef]
  15. H. Bräuninger, P. Predehl, K. P. Beuermann, “Transmission grating efficiencies for wavelengths between 5.4 Å and 44.8 Å,” Appl. Opt. 18, 368–373 (1979).
    [CrossRef] [PubMed]
  16. H. W. Schnopper, L. P. Van Speybroeck, J. P. Delvaille, A. Epstein, E. Källne, R. Z. Bachrach, J. Dijkstra, L. Lantward, “Diffraction grating transmission efficiencies for XUV and soft x rays,” Appl. Opt. 16, 1088–1091 (1977).
    [PubMed]
  17. R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26, 2348–2354 (1987).
    [CrossRef] [PubMed]
  18. Y. Y. Teng, E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
    [CrossRef]
  19. J. Y. Suratteau, R. Petit, “Numerical study of perfectly conducting wire gratings in the resonance domain,” Int. J. Infrared Millimeter Waves 6, 831–865 (1985).
    [CrossRef]
  20. R. A. Depine, Universidad de Buenos Aires, Pabellon 1, 1428 Buenos Aires, Argentina (personal communication, 1990).

1987 (1)

1985 (2)

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

J. Y. Suratteau, R. Petit, “Numerical study of perfectly conducting wire gratings in the resonance domain,” Int. J. Infrared Millimeter Waves 6, 831–865 (1985).
[CrossRef]

1984 (1)

C. R. Canizares, J. Kruper, “A sharp x-ray absorption feature in the BL Lacertae Object PKS 1255-304,” Astrophys. J. Lett. 278, L99–L100 (1984).
[CrossRef]

1982 (2)

1981 (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

1979 (2)

1977 (1)

1967 (1)

Y. Y. Teng, E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

1965 (1)

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Aschenbach, B.

P. Predehl, H. Bräuninger, W. Burkert, B. Aschenbach, J. Trümper, M. Kühne, P. Müller, “Transmission of grating spectrometer on SPEKTROSAT,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 265–272 (1988).

Bachrach, R. Z.

Beuermann, K. P.

Bleeker, J. A. M.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Bräuninger, H.

Brinkman, A. C.

F. D. Seward, T. Chlebowski, J. P. Delvaille, J. P. Henry, S. M. Kahn, L. Van Speybroeck, J. Dijkstra, A. C. Brinkman, J. Heise, R. Mewe, J. Schrijver, “Calibration and efficiency of the Einstein objective grating spectrometer,” Appl. Opt. 21, 2012–2021 (1982).
[CrossRef] [PubMed]

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Burkert, W.

P. Predehl, H. Bräuninger, W. Burkert, B. Aschenbach, J. Trümper, M. Kühne, P. Müller, “Transmission of grating spectrometer on SPEKTROSAT,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 265–272 (1988).

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

Canizares, C. R.

C. R. Canizares, J. Kruper, “A sharp x-ray absorption feature in the BL Lacertae Object PKS 1255-304,” Astrophys. J. Lett. 278, L99–L100 (1984).
[CrossRef]

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Chlebowski, T.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Dangschat, H.

de Korte, P. A. J.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Delvaille, J. P.

Depine, R. A.

Dewey, D.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Dijkstra, J.

Dijkstra, J. H.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Epstein, A.

Gorshunov, B. P.

A. A. Volkov, B. P. Gorshunov, A. A. Irisov, G. V. Kozlov, S. P. Lebedev, “Electrodynamic properties of plane wire grids,” Int. J. Infrared Millimeter Waves 3, 19–43 (1982).
[CrossRef]

Heise, J.

F. D. Seward, T. Chlebowski, J. P. Delvaille, J. P. Henry, S. M. Kahn, L. Van Speybroeck, J. Dijkstra, A. C. Brinkman, J. Heise, R. Mewe, J. Schrijver, “Calibration and efficiency of the Einstein objective grating spectrometer,” Appl. Opt. 21, 2012–2021 (1982).
[CrossRef] [PubMed]

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Henry, J. P.

Hessel, A.

Hutley, M. C.

M. C. Hutley, in Diffraction Gratings, N. H. March, H. N. Daglish, eds. (Academic, London, 1982), Chap. 5, pp. 140–146.

M. C. Hutley, in Diffraction Gratings, N. H. March, H. N. Daglish, eds. (Academic, London, 1982), Chap. 5, pp. 157–163.

Irisov, A. A.

A. A. Volkov, B. P. Gorshunov, A. A. Irisov, G. V. Kozlov, S. P. Lebedev, “Electrodynamic properties of plane wire grids,” Int. J. Infrared Millimeter Waves 3, 19–43 (1982).
[CrossRef]

Kahn, S. M.

Källne, E.

Kozlov, G. V.

A. A. Volkov, B. P. Gorshunov, A. A. Irisov, G. V. Kozlov, S. P. Lebedev, “Electrodynamic properties of plane wire grids,” Int. J. Infrared Millimeter Waves 3, 19–43 (1982).
[CrossRef]

Kraus, H.

Kruper, J.

C. R. Canizares, J. Kruper, “A sharp x-ray absorption feature in the BL Lacertae Object PKS 1255-304,” Astrophys. J. Lett. 278, L99–L100 (1984).
[CrossRef]

Kühne, M.

P. Predehl, H. Bräuninger, W. Burkert, B. Aschenbach, J. Trümper, M. Kühne, P. Müller, “Transmission of grating spectrometer on SPEKTROSAT,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 265–272 (1988).

Lantward, L.

Lebedev, S. P.

A. A. Volkov, B. P. Gorshunov, A. A. Irisov, G. V. Kozlov, S. P. Lebedev, “Electrodynamic properties of plane wire grids,” Int. J. Infrared Millimeter Waves 3, 19–43 (1982).
[CrossRef]

Levine, A. M.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Markert, T. H.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

McPhedran, R. C.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Mewe, R.

F. D. Seward, T. Chlebowski, J. P. Delvaille, J. P. Henry, S. M. Kahn, L. Van Speybroeck, J. Dijkstra, A. C. Brinkman, J. Heise, R. Mewe, J. Schrijver, “Calibration and efficiency of the Einstein objective grating spectrometer,” Appl. Opt. 21, 2012–2021 (1982).
[CrossRef] [PubMed]

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Müller, P.

P. Predehl, H. Bräuninger, W. Burkert, B. Aschenbach, J. Trümper, M. Kühne, P. Müller, “Transmission of grating spectrometer on SPEKTROSAT,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 265–272 (1988).

Neviere, M.

M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 123–157.
[CrossRef]

Oliner, A. A.

Paerels, T.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Petit, R.

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

J. Y. Suratteau, R. Petit, “Numerical study of perfectly conducting wire gratings in the resonance domain,” Int. J. Infrared Millimeter Waves 6, 831–865 (1985).
[CrossRef]

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 26–30.
[CrossRef]

Predehl, P.

Schattenburg, M. L.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Schnopper, H. W.

Schrijver, J.

Seward, F. D.

Smith, H. I.

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

Stern, E. A.

Y. Y. Teng, E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

Suratteau, J. Y.

J. Y. Suratteau, R. Petit, “Numerical study of perfectly conducting wire gratings in the resonance domain,” Int. J. Infrared Millimeter Waves 6, 831–865 (1985).
[CrossRef]

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

Teng, Y. Y.

Y. Y. Teng, E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

Trümper, J.

H. Bräuninger, H. Kraus, H. Dangschat, K. P. Beuermann, P. Predehl, J. Trümper, “Fabrication of transmission gratings for use in cosmic x-ray and XUV astronomy,” Appl. Opt. 18, 3502–3505 (1979).
[CrossRef] [PubMed]

P. Predehl, H. Bräuninger, W. Burkert, B. Aschenbach, J. Trümper, M. Kühne, P. Müller, “Transmission of grating spectrometer on SPEKTROSAT,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 265–272 (1988).

van Rooijen, J. J.

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

Van Speybroeck, L.

Van Speybroeck, L. P.

Volkov, A. A.

A. A. Volkov, B. P. Gorshunov, A. A. Irisov, G. V. Kozlov, S. P. Lebedev, “Electrodynamic properties of plane wire grids,” Int. J. Infrared Millimeter Waves 3, 19–43 (1982).
[CrossRef]

Appl. Opt. (6)

Astrophys. J. Lett. (1)

C. R. Canizares, J. Kruper, “A sharp x-ray absorption feature in the BL Lacertae Object PKS 1255-304,” Astrophys. J. Lett. 278, L99–L100 (1984).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

J. Y. Suratteau, M. Cadilhac, R. Petit, “The perfectly conducting wire grating: computation of the diffracted field from Maxwell’s equations and Hamilton’s canonical system,” IEEE Trans. Antennas Propag. AP-33, 404–408 (1985).
[CrossRef]

Int. J. Infrared Millimeter Waves (2)

A. A. Volkov, B. P. Gorshunov, A. A. Irisov, G. V. Kozlov, S. P. Lebedev, “Electrodynamic properties of plane wire grids,” Int. J. Infrared Millimeter Waves 3, 19–43 (1982).
[CrossRef]

J. Y. Suratteau, R. Petit, “Numerical study of perfectly conducting wire gratings in the resonance domain,” Int. J. Infrared Millimeter Waves 6, 831–865 (1985).
[CrossRef]

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Phys. Rev. Lett. (1)

Y. Y. Teng, E. A. Stern, “Plasma radiation from metal grating surfaces,” Phys. Rev. Lett. 19, 511–514 (1967).
[CrossRef]

Other (8)

R. A. Depine, Universidad de Buenos Aires, Pabellon 1, 1428 Buenos Aires, Argentina (personal communication, 1990).

R. Petit, “A tutorial introduction,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 26–30.
[CrossRef]

M. Neviere, “The homogeneous problem,” in Electromagnetic Theory of Gratings, R. Petit, ed., Vol. 22 of Topics in Current Physics (Springer-Verlag, Berlin, 1980), pp. 123–157.
[CrossRef]

M. L. Schattenburg, C. R. Canizares, D. Dewey, A. M. Levine, T. H. Markert, H. I. Smith, “Transmission grating spectroscopy and the Advanced X-Ray Astrophysics Facility (AXAF),” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 210–218 (1988).

A. C. Brinkman, J. J. van Rooijen, J. A. M. Bleeker, J. H. Dijkstra, J. Heise, P. A. J. de Korte, R. Mewe, T. Paerels, “Low energy x-ray transmission grating spectrometer for AXAF,” in X-Ray Instrumentation in Astronomy, J. L. Culhane, ed., Proc. Soc. Photo-Opt. Instrum. Eng.597, 232–237 (1985).

M. C. Hutley, in Diffraction Gratings, N. H. March, H. N. Daglish, eds. (Academic, London, 1982), Chap. 5, pp. 140–146.

P. Predehl, H. Bräuninger, W. Burkert, B. Aschenbach, J. Trümper, M. Kühne, P. Müller, “Transmission of grating spectrometer on SPEKTROSAT,” in X-Ray Instrumentation in Astronomy II, L. Golub, ed., Proc. Soc. Photo-Opt. Instrum. Eng.982, 265–272 (1988).

M. C. Hutley, in Diffraction Gratings, N. H. March, H. N. Daglish, eds. (Academic, London, 1982), Chap. 5, pp. 157–163.

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Figures (8)

Fig. 1
Fig. 1

Explanation of the notations. A freestanding transmission grating is a structure of periodically positioned wires (top). The period is d. The thickness of the wires (perpendicular to grating plane) is h, and their width (in the plane) is b. Additional parameters are needed for nonrectangular wire cross sections. For a rhombus (bottom), s denotes the slope angle. The angle of incidence is always Θ.

Fig. 2
Fig. 2

Influence of the wire thickness on the transmittance. (a) Calculated intensity of zeroth-order diffraction versus wavelength for H-polarized light at normal incidence. The intensity is normalized to the incident beam and the wavelength is in units of the grating period d. A rectangular grating wire profile has been assumed. The width of the wire rectangles is always d/2, and the thicknesses are 0.3d (solid curve), 0.5d (dashed curve), and 0.7d (dotted curve). (b) The wavelength of the main resonance peak versus the thickness of the wire rectangles in units of d for three different wire widths: 0.3d (solid curve), 0.5d (dashed curve), and 0.7d (dotted curve).

Fig. 3
Fig. 3

The influence of the wire width on the transmittance. (a) Similar to Fig. 2(a) but for different wire widths. The thickness of the wire rectangles is always d/2 and the widths are 0.3d (solid curve), 0.5d (dashed curve), and 0.7d (dotted curve). (b) The calculated intensity at the wide shallow minimum versus the width of the wire rectangles in units of d for three different wire thicknesses: 0.3d (solid curve), 0.5d (dashed curve), and 0.7d (dotted curve).

Fig. 4
Fig. 4

Similar to Fig. 2(a) but for different angles of incidence Θ = 0° (solid curve), Θ = 1° (dashed curve), and Θ = 2° (dotted curve). For increasing angle, the two minima are shifted symmetrically away from the resonance minimum. The shift is proportional to Θ.

Fig. 5
Fig. 5

Difference between a rectangular wire cross section and a rhombus. (a) The transmittance of a grating (b = 0.5d, h = 0.45d, Θ = 0°) with a rhombus as the wire profile for different slope angles s = 90° (solid curve), s = 85° (dashed curve), and s = 80° (dotted curve). (b) The best fit of a transmittance curve for a rectangular cross section (h = 0.42d, b = 0.48d, dashed curve) to that of a rhombus profile [parameters as in (a); s = 85°, solid curve].

Fig. 6
Fig. 6

Optical setup used for determination of the grating period with a side view (upper) and top view (lower figure). A He–Cd laser (λ = 441.8 nm) has been utilized: 1, friction clutch; 2, measuring arm; 3, driver arm; 4, optical rotary encoder; 5, beam widener; 6, aperture stop; 7, grating; 8, lens; 9, reference detector; 10, detector; 11, stepper motor.

Fig. 7
Fig. 7

Comparison between theoretical and measured transmittance curves. (a) The measured transmittance of the grating U1 (solid curve) and the best-fit curve for a rhombus profile (h = 0.48d, b = 0.61d, s = 85°, dashed curve). (b) Same as Fig. 7(a) but the region around the Rayleigh anomaly is expanded. For the theoretical curve, an angle of incidence Θ = 0.45° has been assumed. (c) Measured transmittance of the grating U1 for different angles of incidence: Θ = 0° (solid curve), Θ = 1° (dashed curve), and Θ = 2° (dotted curve).

Fig. 8
Fig. 8

Transmittance of the grating G2P4. The dip at λ = 1.05d between the Rayleigh anomaly and the resonance maximum is caused by ripples and buckles on the grating plane.

Tables (1)

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Table I Comparison of X-Ray and Spectrophotometry Measurement & Results for Grating U1

Equations (1)

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k = [ ω c ] 2 1 ( ω ) 1 + 1 ( ω ) ,

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