Abstract

The efficiency of a diffraction grating was measured near normal incidence in the 125–225-Å wavelength range with synchrotron radiation. The grating pattern had 2400 grooves/mm and was recorded on a concave fused-silica blank by a holographic technique. The grooves were shaped by ion-beam etching to provide a facet with a blaze angle of 2.5° as determined by atomic force microscopy. Because of the characteristics of the etching process the groove profile was approximately triangular, with the other facet inclined at an angle of 5.5° to the surface. The measured efficiency was compared with the efficiency calculated by a computer program, small enough to run on a personal computer, that solved the periodic boundary-value problem corresponding to electromagnetic radiation incident on a diffraction grating with finite conductivity. The calculation was based on the nominal groove profile that was determined by atomic force microscopy. The measured and the calculated efficiencies were in good agreement. This investigation indicates that the diffraction efficiency of a normal-incidence grating can be calculated in the soft-x-ray region with a personal computer.

© 1997 Optical Society of America

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References

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  1. D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
    [CrossRef]
  2. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
    [CrossRef]
  3. R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980), and references therein.
    [CrossRef]
  4. E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
    [CrossRef] [PubMed]
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    [CrossRef]
  6. J. F. Meekins, M. P. Kowalski, R. G. Cruddace, “Efficiency measurements of reflection gratings in the 100–300 Å band,” Appl. Opt. 28, 1369–1377 (1989).
    [CrossRef] [PubMed]
  7. M. Nevière, “Multilayer coated gratings for x-ray diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–1473 (1991).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  11. J. C. Rife, H. R. Sadeghi, W. R. Hunter, “Upgrades and recent performance of the grating/crystal monochromator,” Rev. Sci. Instrum. 60, 2064–2067 (1989).
    [CrossRef]
  12. W. R. Hunter, J. C. Rife, “An ultrahigh vacuum reflectometer/goniometer for use with synchrotron radiation,” Nucl. Instrum. Methods A 246, 465–468 (1986).
    [CrossRef]
  13. B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
    [CrossRef]

1994 (1)

1993 (1)

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

1991 (1)

1989 (2)

J. F. Meekins, M. P. Kowalski, R. G. Cruddace, “Efficiency measurements of reflection gratings in the 100–300 Å band,” Appl. Opt. 28, 1369–1377 (1989).
[CrossRef] [PubMed]

J. C. Rife, H. R. Sadeghi, W. R. Hunter, “Upgrades and recent performance of the grating/crystal monochromator,” Rev. Sci. Instrum. 60, 2064–2067 (1989).
[CrossRef]

1986 (1)

W. R. Hunter, J. C. Rife, “An ultrahigh vacuum reflectometer/goniometer for use with synchrotron radiation,” Nucl. Instrum. Methods A 246, 465–468 (1986).
[CrossRef]

1978 (1)

1977 (1)

1974 (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

1972 (1)

D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
[CrossRef]

Chernov, B. C.

L. I. Goray, B. C. Chernov, “Comparison of rigorous methods for x-ray and XUV grating diffraction analysis,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. C. Walker, eds., Proc. SPIE2515, 240–245 (1995).
[CrossRef]

Cruddace, R. G.

Davis, J. C.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Goray, L. I.

L. I. Goray, “Numerical analysis for relief gratings working in the soft x-ray and XUV region by the integral equation method,” in X-Ray and UV Detectors, R. B. Hoover, M. W. Tate, eds., Proc. SPIE2278, 168–172 (1994).
[CrossRef]

L. I. Goray, B. C. Chernov, “Comparison of rigorous methods for x-ray and XUV grating diffraction analysis,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. C. Walker, eds., Proc. SPIE2515, 240–245 (1995).
[CrossRef]

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Henke, B. L.

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

Hunter, W. R.

J. C. Rife, H. R. Sadeghi, W. R. Hunter, “Upgrades and recent performance of the grating/crystal monochromator,” Rev. Sci. Instrum. 60, 2064–2067 (1989).
[CrossRef]

W. R. Hunter, J. C. Rife, “An ultrahigh vacuum reflectometer/goniometer for use with synchrotron radiation,” Nucl. Instrum. Methods A 246, 465–468 (1986).
[CrossRef]

Kowalski, M. P.

Loewen, E. G.

Maystre, D.

E. G. Loewen, M. Nevière, D. Maystre, “Grating efficiency theory as it applies to blazed and holographic gratings,” Appl. Opt. 16, 2711–2721 (1977).
[CrossRef] [PubMed]

D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
[CrossRef]

Meekins, J. F.

Nevière, M.

Petit, R.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Rife, J. C.

J. C. Rife, H. R. Sadeghi, W. R. Hunter, “Upgrades and recent performance of the grating/crystal monochromator,” Rev. Sci. Instrum. 60, 2064–2067 (1989).
[CrossRef]

W. R. Hunter, J. C. Rife, “An ultrahigh vacuum reflectometer/goniometer for use with synchrotron radiation,” Nucl. Instrum. Methods A 246, 465–468 (1986).
[CrossRef]

Sadeghi, H. R.

J. C. Rife, H. R. Sadeghi, W. R. Hunter, “Upgrades and recent performance of the grating/crystal monochromator,” Rev. Sci. Instrum. 60, 2064–2067 (1989).
[CrossRef]

Vincent, P.

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Appl. Opt. (3)

At. Data Nucl. Data Tables (1)

B. L. Henke, E. M. Gullikson, J. C. Davis, “X-ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, Z = 1-92,” At. Data Nucl. Data Tables 54, 181–342 (1993).
[CrossRef]

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

Nouv. Rev. Opt. (1)

M. Nevière, P. Vincent, R. Petit, “Sur la théorie du réseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
[CrossRef]

Nucl. Instrum. Methods A (1)

W. R. Hunter, J. C. Rife, “An ultrahigh vacuum reflectometer/goniometer for use with synchrotron radiation,” Nucl. Instrum. Methods A 246, 465–468 (1986).
[CrossRef]

Opt. Commun. (1)

D. Maystre, “Sur la diffraction d’une onde plane par un reseau metallique de conductivite finie,” Opt. Commun. 6, 50–54 (1972).
[CrossRef]

Rev. Sci. Instrum. (1)

J. C. Rife, H. R. Sadeghi, W. R. Hunter, “Upgrades and recent performance of the grating/crystal monochromator,” Rev. Sci. Instrum. 60, 2064–2067 (1989).
[CrossRef]

Other (3)

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, New York, 1980), and references therein.
[CrossRef]

L. I. Goray, “Numerical analysis for relief gratings working in the soft x-ray and XUV region by the integral equation method,” in X-Ray and UV Detectors, R. B. Hoover, M. W. Tate, eds., Proc. SPIE2278, 168–172 (1994).
[CrossRef]

L. I. Goray, B. C. Chernov, “Comparison of rigorous methods for x-ray and XUV grating diffraction analysis,” in X-Ray and Extreme Ultraviolet Optics, R. B. Hoover, A. B. C. Walker, eds., Proc. SPIE2515, 240–245 (1995).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

AFM image of the grating. The vertical scale has been exaggerated to reveal the texture of the grooves.

Fig. 2
Fig. 2

Histogram of the pixel heights that were derived from one grating period of the AFM image.

Fig. 3
Fig. 3

Histogram of the blaze angles that were derived from one grating period of the AFM image.

Fig. 4
Fig. 4

Representative scaled groove profile derived from the AFM image and used in the calculation of the grating efficiency.

Fig. 5
Fig. 5

Measured grating efficiency for a wavelength of 146 Å and an angle of incidence of 10°. The inside (m > 0) and the outside (m < 0) diffraction orders are indicated.

Fig. 6
Fig. 6

Comparison of the measured (data points) and the calculated (solid curves) grating efficiencies for the indicated diffraction orders.

Fig. 7
Fig. 7

Comparison of the (a) measured and (b) calculated grating efficiencies for the indicated diffraction orders.

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