Abstract

A rigorous electromagnetic analysis is developed for metallic gratings covered by a stack of pairs of dielectrics and absorbers. The thicknesses of the layers can be arbitrary. In particular, they can be much thinner than the groove depth, which permits the optimization of grating efficiencies for the soft-x-ray range. The non-scalar behavior of such gratings is established for high angles of incidence. Examples of performances under near-normal and grazing incidences are given.

© 1991 Optical Society of America

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References

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  1. T. W. Barbee, “Sputtered layered synthetic microstructure (LSM) dispersion elements,” in Low-Energy X-Ray Diagnostics 1981, D. T. Atwood, B. L. Henke, eds., Vol. 75 of the AIP Conference Proceedings (American Institute of Physics, Washington, D.C., 1981), p. 131;E. Spiller, “Reflective multilayer coatings for the far uv region,” Appl. Opt. 15, 2333–2338 (1976).
    [CrossRef] [PubMed]
  2. B. Vidal, P. Vincent, “Metallic multilayers for x rays using classical thin-film theory,” Appl. Opt. 23, 1794–1801 (1984).
    [CrossRef] [PubMed]
  3. 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]
  4. 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]
  5. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  6. 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. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
    [CrossRef]
  7. F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).
  8. H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
    [CrossRef]
  9. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
    [CrossRef]
  10. E. G. Loewen, M. Nevière, “Simple selection rules for VUV and XUV diffraction gratings,” Appl. Opt. 17, 1087–1092 (1978).
    [CrossRef] [PubMed]

1989 (2)

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

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]

1984 (1)

1978 (2)

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]

1950 (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

Abelès, F.

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

André, J. M.

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

Barbee, T. W.

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]

T. W. Barbee, “Sputtered layered synthetic microstructure (LSM) dispersion elements,” in Low-Energy X-Ray Diagnostics 1981, D. T. Atwood, B. L. Henke, eds., Vol. 75 of the AIP Conference Proceedings (American Institute of Physics, Washington, D.C., 1981), p. 131;E. Spiller, “Reflective multilayer coatings for the far uv region,” Appl. Opt. 15, 2333–2338 (1976).
[CrossRef] [PubMed]

Barchewitz, R.

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

Berrouane, H.

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

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. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

Fouchet, S.

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

Fugikawa, B. K.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
[CrossRef]

Henke, B. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
[CrossRef]

Hunter, W. R.

Khan Malek, C.

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

Ladan, F. R.

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

Lee, P.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
[CrossRef]

Loewen, E. G.

Maystre, D.

Nevière, M.

E. G. Loewen, M. Nevière, “Simple selection rules for VUV and XUV diffraction gratings,” Appl. Opt. 17, 1087–1092 (1978).
[CrossRef] [PubMed]

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]

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. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

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, C.

Rivoira, R.

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

Shimabukuro, R. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
[CrossRef]

Tanaka, T. J.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
[CrossRef]

Vidal, B.

B. Vidal, P. Vincent, “Metallic multilayers for x rays using classical thin-film theory,” Appl. Opt. 23, 1794–1801 (1984).
[CrossRef] [PubMed]

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. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

Vincent, P.

B. Vidal, P. Vincent, “Metallic multilayers for x rays using classical thin-film theory,” Appl. Opt. 23, 1794–1801 (1984).
[CrossRef] [PubMed]

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]

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. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

Ann. Phys. (Paris) (1)

F. Abelès, “Recherches sur la propagation des ondes électromagnétiques sinusoidales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).

Appl. Opt. (3)

J. Opt. Soc. Am. (1)

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]

X-Ray/EUV Optics for Astronomy and Microscopy (1)

H. Berrouane, J. M. André, C. Khan Malek, S. Fouchet, F. R. Ladan, R. Rivoira, R. Barchewitz, “Fabrication and test of soft x-ray multilayer diffraction gratings,” in X-Ray/EUV Optics for Astronomy and Microscopy, Proc. Soc. Photo-Opt. Instrum. Eng. 1160, 280–285 (1989).
[CrossRef]

Other (3)

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photoabsorption, scattering, and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
[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. Soc. Photo-Opt. Instrum. Eng.563, 142–149 (1985).
[CrossRef]

T. W. Barbee, “Sputtered layered synthetic microstructure (LSM) dispersion elements,” in Low-Energy X-Ray Diagnostics 1981, D. T. Atwood, B. L. Henke, eds., Vol. 75 of the AIP Conference Proceedings (American Institute of Physics, Washington, D.C., 1981), p. 131;E. Spiller, “Reflective multilayer coatings for the far uv region,” Appl. Opt. 15, 2333–2338 (1976).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Bare grating and notation.

Fig. 2
Fig. 2

Periodic step function α(x, y) for a given value y0 of y, over one period d.

Fig. 3
Fig. 3

Multilayer grating over one period d.

Fig. 4
Fig. 4

Shape of the periodic step functions α(x, y) for a multilayer grating at a given value y0 of y.

Fig. 5
Fig. 5

Arbitrary periodic step function.

Fig. 6
Fig. 6

Blazing of a multilayer echelette grating. The grazing angle α is equal to θφ. The incidence angle is π/2 − α, and |k|= 2π/λ.

Fig. 7
Fig. 7

Reflectivity and −1-order efficiency curves as functions of the grazing angle α for a multilayer plane stack (MLPS) and a multilayer grating (MLG).

Tables (1)

Tables Icon

Table 1 Absolute Efficiencies of the Multilayer Grating

Equations (29)

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Δ + α ( x , y ) = 0 ,
α ( x , y ) = { k 2 ν 1 2 y > g ( x ) k 2 ν 2 2 y < g ( x ) .
f ( x ) = 0 + k = 1 N σ k d ( x x k ) ,
σ k = y k + 1 y k , k ( 1 , n 1 ) , σ N = y 1 y N ,
f ( x ) = n f n exp ( inKx ) ,
f ( x ) = n inK f n exp ( inKx ) .
d ( x ) = 1 d n exp ( inKx ) .
f n = 1 2 i π n k = 1 N σ k exp ( inK x k ) , n 0
f n = 1 2 π n k = 1 N σ k [ sin ( n K x k ) + i cos ( n K x k ) ] .
f 0 = 1 d 0 d f ( x ) d x = y 1 ( x 1 + d x N ) + y 2 ( x 2 x 1 ) + y 3 ( x 3 x 2 ) + y N 1 ( x N 1 x N 2 ) + y N ( x N x N 1 ) , f 0 = 1 d [ y 1 d + k = 1 N 1 x k ( y k y k + 1 ) + x N ( y N y 1 ) ] f 0 = 1 d ( y 1 d k = 1 N σ k x k ) .
i = 1 N C e i > a .
( i ) y 0 < a : .
tes ( n ) = j = 1 n e j
X 1 , 1 = g 1 ( y ) , X 2 , 1 = g 2 ( y ) , X 1 , i = g 1 [ y tes ( i 1 ) ] , i ( 2 , N ) , X 2 , i = g 2 [ y tes ( i 1 ) ] , i ( 2 , N ) .
x k = { X 1 , k k ( 1 , N ) X 2 , 2 N k + 1 k ( N + 1 , 2 N ) .
σ 1 = ν h 2 ν 2 2 , σ 2 N = σ 1 , σ k = ( 1 ) k ( ν l 2 ν h 2 ) , y 1 = ν 2 2 .
( i i ) a < y 0 < i = 1 N C e i .
tes ( n ) = j = 1 n e j
tesa ( n ) = a + j = 1 n e j
X 1 , i = g 1 [ y tes ( N + i 1 ) ] , i ( 1 , N S ) , X 2 , i = g 2 [ y tes ( N + i 1 ) ] , i ( 1 , N S ) , x k = { X 1 , k k ( 1 , N S ) X 2 , 2 N S k + 1 k ( N S + 1 , 2 N S ) , σ k = ( 1 ) ( k + N ) ( ν 2 ν h 2 ) , k ( 1 , 2 N S ) , y 1 = { ν h 2 N = 2 p + 1 ν 2 N = 2 p ( p integer ) , ( iii ) i = 1 N C e i < y 0 < a + i = 1 N C e i .
tesa ( n ) = a + i = 1 n e i
N S 2 = 2 N S = 2 ( N N ) = 2 ( N N C 1 ) .
tes ( n ) = i = 1 n e i ,
X 1 , i = g 1 [ y tes ( N + i 1 ) ] , i ( 1 , N S ) , X 2 , i = g 2 [ y tes ( N + i 1 ) ] , i ( 1 , N S ) , x k = { X 1 , k k ( 1 , N S ) X 2 , 2 N S k + 1 k ( N S + 1 , 2 N S ) .
σ k = ( 1 ) k + N ( ν l 2 ν h 2 ) , k ( 1 , N S 1 ) , k ( N S + 2 , 2 N S ) , σ N S = ν 1 2 ν h 2 , σ N S + 1 = σ N S ,
y 1 = { ν h 2 N = 2 p + 1 ν l 2 N = 2 p .
λ = ( 2 D / n ) sin θ ,
λ = ( 2 d sin φ sin θ ) / m .
D / n = ( d sin φ ) / m .

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