Abstract

The near-normal-incidence efficiencies of a 2400-groove/mm holographic master grating, a replica grating, and a multilayer grating are modeled in the soft-x-ray-extreme-ultraviolet (EUV) regions and are compared with efficiencies that are measured with synchrotron radiation. The efficiencies are calculated by the computer program PCGrate, which is based on a rigorous modified integral method. The theory of our integral method is described both for monolayer and multilayer gratings designated for the soft-x-ray-EUV-wavelength range. The calculations account for the groove profile as determined from atomic force microscopy with a depth scaling in the case of the multilayer grating and an average random microroughness (0.7 nm) for the short wavelengths. The refractive indices of the grating substrate and coatings have been taken from different sources because of the wide range of the wavelengths (4.5–50 nm). The measured peak absolute efficiency of 10.4% in the second diffraction order at a wavelength of 11.4 nm is achieved for the multilayer grating and is in good agreement with a computed value of ∼11.5%. Rigorous modeling of the efficiencies of three similar gratings is in good overall agreement with the measured efficiency over a wide wavelength region. Additional calculations have indicated that relatively high normal incidence efficiency (of at least several percent) and large angular dispersion in the higher orders can be achieved in the 4.5–10.5-nm range by application of various multilayer coatings.

© 2002 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  6. J. F. Seely, L. I. Goray, W. R. Hunter, J. C. Rife, “Thin-film interference effects of a normal-incidence grating in the 100–350-Å wavelength region,” Appl. Opt. 38, 1251–1258 (1999).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  9. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
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    [CrossRef]
  15. L. I. Goray, “Rigorous integral method in application to computing diffraction on relief gratings working in wavelength range from microwaves to x ray,” in Application and Theory of Periodic Structures, T. Jannson, N. C. Gallagher, eds., Proc. SPIE2532, 427–433 (1995).
    [CrossRef]
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    [CrossRef]
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  19. A. Sammar, J.-M. André, B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the XUV region,” Opt. Commun. 86, 245–254 (1991).
    [CrossRef]
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    [CrossRef]
  23. L. I. Goray, “Modified integral method for weak convergence problems of light scattering on relief grating,” in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. I. Sutherland, D. W. Prather, I. Cindrich, eds., Proc. SPIE4291, 1–12 (2001).
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    [CrossRef]
  28. D. A. Content, “Grating groove metrology and efficiency predictions from the soft-x-ray to the far infrared,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A. M. Larar, M. G. Mlynczak, eds., Proc. SPIE4485 (to be published).
  29. L. I. Goray, “Modified integral method and real electromagnetic properties of echelles,” in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. I. Sutherland, D. W. Prather, I. Cindrich, eds., Proc. SPIE4291, 13–24 (2001).
    [CrossRef]
  30. M. Nevière, F. Montiel, “Soft-x-ray multilayer coated echelle gratings: electromagnetic and phenomenological study,” J. Opt. Soc. Am. A 13, 811–818 (1996).
    [CrossRef]
  31. A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
    [CrossRef]
  32. S. Yu. Sadov, L. I. Goray are preparing a manuscript to be called “The modified integral method for gratings covered with thin or thick layers of arbitrary shape.”
  33. A. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 221–236 (1985).
    [CrossRef]
  34. W. Jark, “Enhancement of diffraction grating efficiencies in soft-x-ray region by multilayer coating,” Opt. Commun. 60, 201–205 (1986).
    [CrossRef]
  35. S. Bajt, “Molybdenum-ruthenium/beryllium multilayer coatings,” J. Vac. Sci. Technol. A 18, 557–559 (2000).
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2001 (2)

2000 (1)

S. Bajt, “Molybdenum-ruthenium/beryllium multilayer coatings,” J. Vac. Sci. Technol. A 18, 557–559 (2000).
[CrossRef]

1999 (1)

1997 (3)

1996 (1)

1995 (1)

1994 (3)

M. Nevière, “Bragg-Fresnel multilayer gratings electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
[CrossRef]

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

H. A. Podmore, V. Martynov, K. Holis, “The use of diffraction efficiency theory in the design of soft-x-ray monochromators,” Nucl. Instrum. Methods A 347, 206–215 (1994).
[CrossRef]

1993 (2)

A. Sammar, J.-M. André, “Diffraction of multilayer gratings and zone plates in the x-ray region using the Born approximation,” J. Opt. Soc. Am. A 10, 600–613 (1993).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

1991 (3)

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

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

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

1986 (1)

W. Jark, “Enhancement of diffraction grating efficiencies in soft-x-ray region by multilayer coating,” Opt. Commun. 60, 201–205 (1986).
[CrossRef]

1980 (1)

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

Agafonov, Yu.

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

V. V. Martynov, H. A. Padmore, Yu. Agafonov, A. Yuakshin, “X-ray multilayer gratings with very high diffraction efficiency,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 2–8 (1997).
[CrossRef]

Agafonov, Yu. A.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

André, J.-M.

A. Sammar, J.-M. André, “Diffraction of multilayer gratings and zone plates in the x-ray region using the Born approximation,” J. Opt. Soc. Am. A 10, 600–613 (1993).
[CrossRef]

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

Arakawa, A. T.

A. T. Arakawa, T. A. Callcott, Y. C. Chang, “Beryllium (Be),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, New York, 1991), pp. 421–433.

Bajt, S.

Barbee, T. W.

Brunel, M.

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

Callcott, T. A.

A. T. Arakawa, T. A. Callcott, Y. C. Chang, “Beryllium (Be),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, New York, 1991), pp. 421–433.

Chang, Y. C.

A. T. Arakawa, T. A. Callcott, Y. C. Chang, “Beryllium (Be),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, New York, 1991), pp. 421–433.

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. Walker, eds., Proc. SPIE2515, 240–245 (1995).
[CrossRef]

Content, D. A.

D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Conference on Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).
[CrossRef]

D. A. Content, “Grating groove metrology and efficiency predictions from the soft-x-ray to the far infrared,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A. M. Larar, M. G. Mlynczak, eds., Proc. SPIE4485 (to be published).

Cruddace, R. G.

Dhez, P.

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings: theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 142–149 (1985).
[CrossRef]

Ely, R.

Erko, A.

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

Erko, A. I.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

Erofeev, V. I.

V. I. Erofeev, N. V. Kovalenko, “Method of eigenvectors for numerical studies of multilayer gratings,” X-Ray Sci. Technol. 7, 75 (1997).

Flamand, J.

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

Goray, L. I.

J. F. Seely, L. I. Goray, W. R. Hunter, J. C. Rife, “Thin-film interference effects of a normal-incidence grating in the 100–350-Å wavelength region,” Appl. Opt. 38, 1251–1258 (1999).
[CrossRef]

M. P. Kowalski, J. F. Seely, L. I. Goray, W. R. Hunter, J. C. Rife, “Comparison of the calculated and the measured efficiencies of a normal-incidence grating in the 125–225-Å wavelength range,” Appl. Opt. 36, 8939–8943 (1997).
[CrossRef]

L. I. Goray, “Modified integral method and real electromagnetic properties of echelles,” in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. I. Sutherland, D. W. Prather, I. Cindrich, eds., Proc. SPIE4291, 13–24 (2001).
[CrossRef]

L. I. Goray, “Modified integral method for weak convergence problems of light scattering on relief grating,” in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. I. Sutherland, D. W. Prather, I. Cindrich, eds., Proc. SPIE4291, 1–12 (2001).
[CrossRef]

J. F. Seely, L. I. Goray, “Normal incidence multilayer gratings for the extreme ultraviolet region: experimental measurements and computational modeling,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 364–370 (1999).
[CrossRef]

S. Yu. Sadov, L. I. Goray are preparing a manuscript to be called “The modified integral method for gratings covered with thin or thick layers of arbitrary shape.”

L. I. Goray, “Nonscalar properties of high groove frequency gratings for soft-x-ray and XUV regions: the integral equation method,” in X-Ray and UV Detectors, R. B. Hoover, M. W. Tate, eds., Proc. SPIE2278, 173–177 (1994).
[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, “Rigorous integral method in application to computing diffraction on relief gratings working in wavelength range from microwaves to x ray,” in Application and Theory of Periodic Structures, T. Jannson, N. C. Gallagher, eds., Proc. SPIE2532, 427–433 (1995).
[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. Walker, eds., Proc. SPIE2515, 240–245 (1995).
[CrossRef]

Heidemann, K. F.

Heinzmann, U.

Holis, K.

H. A. Podmore, V. Martynov, K. Holis, “The use of diffraction efficiency theory in the design of soft-x-ray monochromators,” Nucl. Instrum. Methods A 347, 206–215 (1994).
[CrossRef]

Hunter, W. R.

Jark, W.

W. Jark, “Enhancement of diffraction grating efficiencies in soft-x-ray region by multilayer coating,” Opt. Commun. 60, 201–205 (1986).
[CrossRef]

Kleineberg, U.

Kovalenko, N. V.

V. I. Erofeev, N. V. Kovalenko, “Method of eigenvectors for numerical studies of multilayer gratings,” X-Ray Sci. Technol. 7, 75 (1997).

Kowalski, M. P.

Lynch, A. W.

A. W. Lynch, W. R. Hunter, “Molybdenum (Mo),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 303–313.

Martynov, V.

H. A. Podmore, V. Martynov, K. Holis, “The use of diffraction efficiency theory in the design of soft-x-ray monochromators,” Nucl. Instrum. Methods A 347, 206–215 (1994).
[CrossRef]

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

Martynov, V. V.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

V. V. Martynov, H. A. Padmore, Yu. Agafonov, A. Yuakshin, “X-ray multilayer gratings with very high diffraction efficiency,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 2–8 (1997).
[CrossRef]

Menke, D.

Montcalm, C.

Montiel, F.

Nevière, M.

M. Nevière, F. Montiel, “Soft-x-ray multilayer coated echelle gratings: electromagnetic and phenomenological study,” J. Opt. Soc. Am. A 13, 811–818 (1996).
[CrossRef]

M. Nevière, “Bragg-Fresnel multilayer gratings electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
[CrossRef]

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

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings: theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 142–149 (1985).
[CrossRef]

Osterried, K.

Padmore, H. A.

V. V. Martynov, H. A. Padmore, Yu. Agafonov, A. Yuakshin, “X-ray multilayer gratings with very high diffraction efficiency,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 2–8 (1997).
[CrossRef]

Pardo, B.

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

Podmore, H. A.

H. A. Podmore, V. Martynov, K. Holis, “The use of diffraction efficiency theory in the design of soft-x-ray monochromators,” Nucl. Instrum. Methods A 347, 206–215 (1994).
[CrossRef]

Pomp, A.

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

Rife, J. C.

Roschupkin, D. V.

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

Rosenbluth, A. E.

A. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 221–236 (1985).
[CrossRef]

Sammar, A.

A. Sammar, J.-M. André, “Diffraction of multilayer gratings and zone plates in the x-ray region using the Born approximation,” J. Opt. Soc. Am. A 10, 600–613 (1993).
[CrossRef]

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

Seely, J. F.

C. Montcalm, S. Bajt, J. F. Seely, “MoRu-Be multilayer-coated grating with 10.4% normal-incidence efficiency near the 11.4-nm wavelength,” Opt. Lett. 26, 125–127 (2001).
[CrossRef]

J. F. Seely, C. Montcalm, S. Bajt, “High-efficiency MoRu-Be multilayer-coated gratings operating near normal incidence in the 11.1–12.0-nm wavelength range,” Appl. Opt. 40, 5565–5574 (2001).
[CrossRef]

J. F. Seely, L. I. Goray, W. R. Hunter, J. C. Rife, “Thin-film interference effects of a normal-incidence grating in the 100–350-Å wavelength region,” Appl. Opt. 38, 1251–1258 (1999).
[CrossRef]

J. F. Seely, M. P. Kowalski, R. G. Cruddace, K. F. Heidemann, U. Heinzmann, U. Kleineberg, K. Osterried, D. Menke, J. C. Rife, W. R. Hunter, “Multilayer-coated laminar grating with 16% normal-incidence efficiency in the 150-Å region,” Appl. Opt. 36, 8206–8213 (1997).
[CrossRef]

M. P. Kowalski, J. F. Seely, L. I. Goray, W. R. Hunter, J. C. Rife, “Comparison of the calculated and the measured efficiencies of a normal-incidence grating in the 125–225-Å wavelength range,” Appl. Opt. 36, 8939–8943 (1997).
[CrossRef]

J. F. Seely, R. G. Cruddace, M. P. Kowalski, W. R. Hunter, T. W. Barbee, J. C. Rife, R. Ely, K. G. Stilt, “Polarization and efficiency of a concave multilayer grating in the 135–250-Å region and in normal-incidence and Seya-Namioka mounts,” Appl. Opt. 34, 7347–7354 (1995).
[CrossRef] [PubMed]

J. F. Seely, “Multilayer grating for the extreme ultraviolet spectrometer (EIS),” in X-Ray Optics, Instruments, and Missions IV, R. B. Hoover, A. B. C. Walker, eds., Proc. SPIE4138, 174–181 (2000).
[CrossRef]

J. F. Seely, L. I. Goray, “Normal incidence multilayer gratings for the extreme ultraviolet region: experimental measurements and computational modeling,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 364–370 (1999).
[CrossRef]

Spiller, A.

A. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 221–236 (1985).
[CrossRef]

Stilt, K. G.

Vidal, B.

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings: theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 142–149 (1985).
[CrossRef]

Vincent, P.

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings: theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 142–149 (1985).
[CrossRef]

Yu. Sadov, S.

S. Yu. Sadov, L. I. Goray are preparing a manuscript to be called “The modified integral method for gratings covered with thin or thick layers of arbitrary shape.”

Yuakshin, A.

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

V. V. Martynov, H. A. Padmore, Yu. Agafonov, A. Yuakshin, “X-ray multilayer gratings with very high diffraction efficiency,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 2–8 (1997).
[CrossRef]

Appl. Opt. (5)

J. Mod. Opt. (1)

A. Pomp, “The integral method for coated gratings: computational cost,” J. Mod. Opt. 38, 109–120 (1991).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. Vac. Sci. Technol. A (1)

S. Bajt, “Molybdenum-ruthenium/beryllium multilayer coatings,” J. Vac. Sci. Technol. A 18, 557–559 (2000).
[CrossRef]

Nucl. Instrum. Methods (1)

M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
[CrossRef]

Nucl. Instrum. Methods A (3)

H. A. Podmore, V. Martynov, K. Holis, “The use of diffraction efficiency theory in the design of soft-x-ray monochromators,” Nucl. Instrum. Methods A 347, 206–215 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, M. Brunel, “Multilayer grating efficiency: numerical and physical experiments,” Nucl. Instrum. Methods A 333, 599–606 (1993).
[CrossRef]

V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, Yu. Agafonov, A. Erko, A. Yuakshin, “Comparison of modal and diffirential methods for multilayer gratings,” Nucl. Instrum. Methods A 339, 617–625 (1994).
[CrossRef]

Opt. Commun. (2)

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

W. Jark, “Enhancement of diffraction grating efficiencies in soft-x-ray region by multilayer coating,” Opt. Commun. 60, 201–205 (1986).
[CrossRef]

Opt. Lett. (1)

X-Ray Sci. Technol. (1)

V. I. Erofeev, N. V. Kovalenko, “Method of eigenvectors for numerical studies of multilayer gratings,” X-Ray Sci. Technol. 7, 75 (1997).

Other (19)

L. I. Goray, “Modified integral method for weak convergence problems of light scattering on relief grating,” in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. I. Sutherland, D. W. Prather, I. Cindrich, eds., Proc. SPIE4291, 1–12 (2001).
[CrossRef]

Internet site, http://www.pcgrate.com .

J. F. Seely, L. I. Goray, “Normal incidence multilayer gratings for the extreme ultraviolet region: experimental measurements and computational modeling,” in X-Ray Optics, Instruments, and Missions II, R. B. Hoover, A. B. Walker, eds., Proc. SPIE3766, 364–370 (1999).
[CrossRef]

J. F. Seely, “Multilayer grating for the extreme ultraviolet spectrometer (EIS),” in X-Ray Optics, Instruments, and Missions IV, R. B. Hoover, A. B. C. Walker, eds., Proc. SPIE4138, 174–181 (2000).
[CrossRef]

D. A. Content, “Diffraction grating groove analysis used to predict efficiency and scatter performance,” in Conference on Gradient Index, Miniature, and Diffractive Optical Systems, A. D. Kathman, ed., Proc. SPIE3778, 19–30 (1999).
[CrossRef]

D. A. Content, “Grating groove metrology and efficiency predictions from the soft-x-ray to the far infrared,” in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A. M. Larar, M. G. Mlynczak, eds., Proc. SPIE4485 (to be published).

L. I. Goray, “Modified integral method and real electromagnetic properties of echelles,” in Diffractive and Holographic Technologies for Integrated Photonic Systems, R. I. Sutherland, D. W. Prather, I. Cindrich, eds., Proc. SPIE4291, 13–24 (2001).
[CrossRef]

R. Petit, ed., Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

L. I. Goray, “Nonscalar properties of high groove frequency gratings for soft-x-ray and XUV regions: the integral equation method,” in X-Ray and UV Detectors, R. B. Hoover, M. W. Tate, eds., Proc. SPIE2278, 173–177 (1994).
[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, “Rigorous integral method in application to computing diffraction on relief gratings working in wavelength range from microwaves to x ray,” in Application and Theory of Periodic Structures, T. Jannson, N. C. Gallagher, eds., Proc. SPIE2532, 427–433 (1995).
[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. Walker, eds., Proc. SPIE2515, 240–245 (1995).
[CrossRef]

B. Vidal, P. Vincent, P. Dhez, M. Nevière, “Thin films and gratings: theories used to optimize the high reflectivity of mirrors and gratings for x-ray optics,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 142–149 (1985).
[CrossRef]

V. V. Martynov, H. A. Padmore, Yu. Agafonov, A. Yuakshin, “X-ray multilayer gratings with very high diffraction efficiency,” in Gratings and Grating Monochromators for Synchrotron Radiation, W. R. McKinney, C. A. Palmer, eds., Proc. SPIE3150, 2–8 (1997).
[CrossRef]

A. 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 Tables54, 181-342 (1993). Updated optical constants were obtained from the Internet site, http://cindy.lbl.gov/optical_constants .
[CrossRef]

A. T. Arakawa, T. A. Callcott, Y. C. Chang, “Beryllium (Be),” in Handbook of Optical Constants of Solids II, E. D. Palik, ed. (Academic, New York, 1991), pp. 421–433.

A. W. Lynch, W. R. Hunter, “Molybdenum (Mo),” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 303–313.

S. Yu. Sadov, L. I. Goray are preparing a manuscript to be called “The modified integral method for gratings covered with thin or thick layers of arbitrary shape.”

A. Spiller, A. E. Rosenbluth, “Determination of thickness errors and boundary roughness from the measured performance of a multilayer coating,” in Applications of Thin-Film Multilayered Structures to Figured X-Ray Optics, G. F. Marshall, ed., Proc. SPIE563, 221–236 (1985).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Groove profiles derived from the AFM images of the 2400-groove/mm gratings: curve 1, 7.5-nm depth master AFM profile; curve 2, 9.0-nm depth replica AFM profile; curve 3, 6.6-nm depth scaled replica AFM profile. The depth is relative to the period.

Fig. 2
Fig. 2

Measured efficiencies of the multilayer grating at a wavelength of 11.37 nm. The angle of incidence is 13.9°.

Fig. 3
Fig. 3

Comparison of the measured (data points) and calculated (curves) master grating efficiencies for the indicated diffraction orders in the 12.5–22.5-nm wavelength range.

Fig. 4
Fig. 4

Comparison of the measured (data points) and calculated (curves) replica grating efficiencies for the indicated diffraction orders in the 12.5–22.5-nm wavelength range.

Fig. 5
Fig. 5

Peak efficiencies (a) measured and (b) calculated at an angle of incidence of 13.9° for the multilayer grating for the indicated diffraction orders in the 11.1–12.0-nm wavelength range.

Fig. 6
Fig. 6

Comparison of the measured (data points) and calculated (curves) multilayer grating efficiencies for the indicated diffraction orders in the wavelength range from 25 to 50 nm.

Fig. 7
Fig. 7

Calculated relative efficiencies of the multilayer grating for the indicated diffraction orders in the 4.5–10.5-nm wavelength range.

Equations (25)

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ΔU+k2U=0,
Ui=expik+sinθx-cosθyez,
U=Ui+Ud.
je=jeδx-xδy-fxez,
ΔE+k+2E=-iωμ0jeδx-xδy-fx.
Ed=S+=S Γ+x, y, sΦsds,
Γ+x, y, s=-expiαnx-x+iγn+|y -fx|/γn+2id,
Ψx, fx=Φx, fxexp-iα0x×1+fx21/2,G+x, y, s=expiα0x-expiKnx-x +iγn+|y-fx|/γn+2id.
Ed=d G+x, x, y, fxΨx, fxdx.
E+x, fx=Eix, fx+d G+x, x, fx, fxΨx, fxdx.
Ht+-Ht-=je,
dE/dn+-dE/dn-=-iωμ0je.
dE/dn++dE/dn-/2=dEix, fx/dn+ddG+x, x, fx, fx/dnΨx, fxdx.
dEx, fx/dn+=dEix, fx/dn+0.5Ψ×x, fxexpiα0x/1+fx21/2+ddG+×x, x, fx, fx/dn×Ψx, fxdx.
dG+x, x, y, fx/dn=n=-expiα0x/2d1+fx21/2×signfx-fx-fxαn/γn+expiKnx-x+iγn+|fx-fx|,
γn-=k-2-αn21/2, Imγn-0,
Eix/2+dG-x, xdEix/dn+EixdG-x, x/dndx=-0.5 d G-x, xΨxdx-0.5 d G+x, xΨxdx- dG+x, xdG-x, x/dn-G-x, xdG+x, x/dnΨxdxdx,
An+=-0.5i/dγn+dexp-iγn+fx-inKxΨx, fxdx.
En+=|An+|2γn+/γi,
WA=S-=Sw-nds,
EA=0.5 dReE-dE/dn-*/iωμ0ds,
E1s1/2=Ei/2+S1 G1+s1, s1dE1s1/dn-dEis1/dnds1 +S1dG1+s1, s1/dnE1s1 -Eis1ds1,Eksk/2=Sk Gk-sk, skdEksk/dndsk -SkdGk-sk, sk/dnEkskdsk +Sk+1 Gk+1+sk, sk+1dEk+1sk+1/dndsk+1 +Sk+1dGk+1+sk, sk+1/dn ×Ek+1sk+1dsk+1,Ek+1sk+1/2=Sk+1 Gk+1+sk+1, sk+1×dEk+1sk/dndsk +Sk+1dGk+1+sk+1, sk+1/dn ×Ek+1sk+1dsk+1 -Sk Gk-sk+1, skdEksk/dndsk -SkdGk-sk+1, sk/dnEkskdsk,EKsK/2=-SKGK-sK, sKdEKsK/dndsK -SKdGK-sK, sK/dnEK-sKdsK, k=1, K-1,
dEis1/dn=-iγi+α0df1xs1/ds1 ×exp-iγif1xs1,
E1x, y=n=- AnexpinKx+iγ1,n+y.
An=0.5/dS1-iγ1,n+dE1s1/dn -dEis1/dn+dxs1/ds1-df1s1/ds1αn/γ1,n+E1s1 -Eis1exp-iγ1,n+f1s1-inKxs1ds1.

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