Abstract

High-blaze-angle multilayer coated echelette gratings used in very high (i.e., from 10th to 60th) orders and operating with not too small (≈ 5°) glancing angles of incidence turn out to be necessary for achieving ultrahigh resolution and collecting important flux in x-ray fluorescence spectroscopy. Predicting their diffracting behavior appears to be a challenge to grating theoreticians, because such gratings are associated with the three main difficulties encountered in grating-efficiency numerical computation, namely, low wavelength-to-groove-spacing ratio, large modulation, and multiple interpenetrating profiles. We propose a method to resolve the problem that combines the differential theory of gratings, the R-matrix propagation algorithm, and an asymmetric truncation of the Fourier series of the field. The numerical results are checked against many criteria. For a particular mounting they are compared with those given by a phenomenological formula that was developed for bare echelette gratings and turns out still to apply for multilayer-coated gratings. The theory is developed not only for periodic stacks but also for stacks made of layers with varying thicknesses, such as those used in supermirrors.

© 1996 Optical Society of America

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References

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  1. M. Nevière, J. Flamand, “Electromagnetic theory as it applies to x-ray and XUV gratings,” Nucl. Instrum. Methods 172, 273–279 (1980).
    [CrossRef]
  2. M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
    [CrossRef]
  3. M. Nevière, P. Vincent, R. Petit, “Sur la théorie du reseau conducteur et ses applications à l’optique,” Nouv. Rev. Opt. 5, 65–77 (1974).
    [CrossRef]
  4. M. Nevière, “Multilayer coated gratings for x-ray diffraction: differential theory,” J. Opt. Soc. Am. A 8, 1468–1473 (1991).
    [CrossRef]
  5. M. Nevière, “Bragg–Fresnel multilayer gratings: electromagnetic theory,” J. Opt. Soc. Am. A 11, 1835–1845 (1994).
    [CrossRef]
  6. M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
    [CrossRef]
  7. F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
    [CrossRef]
  8. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  9. D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
    [CrossRef]
  10. J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
    [CrossRef]
  11. L. F. Sandre, J. M. Elson, “Extinction theorem analysis of diffraction anomalies in overcoated gratings,” J. Opt. Soc. Am. A 8, 763–777 (1991).
    [CrossRef]
  12. L. Li, “A multilayer modal method for diffraction gratings of arbitrary profile, depth, and conductivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  13. L. Li, “Multilayer-coated diffraction gratings: differential method of Chandezon et al. revisited,” J. Opt. Soc. Am. A 11, 2816–2828 (1994).
    [CrossRef]
  14. H. Bremmer, “The W. K. B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
    [CrossRef]
  15. D. M. Pai, K. A. Awada, “Analysis of dielectric gratings of arbitrary profiles and thicknesses,” J. Opt. Soc. Am. A 8, 755–762 (1991).
    [CrossRef]
  16. M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
    [CrossRef]
  17. L. Li, “Bremmer series, R-matrix propagation algorithm, and numerical modeling of diffraction gratings,” J. Opt. Soc. Am. A 11, 2829–2836 (1994).
    [CrossRef]
  18. F. Abelès, “Recherches sur la propagation des ondes electromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces,” Ann. Phys. (Paris) 5, 596–640, 706–782 (1950).
  19. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  20. C. Rife, W. R. Hunter, T. W. Barbie, R. G. Gruddace, “Multilayer-coated blazed grating performance in the soft x-ray region,” Appl. Opt. 28, 2984–2986 (1989).
    [CrossRef] [PubMed]
  21. D. L. Windt, “The optical properties of 21 thin film materials in the 10 eV to 500 eV photon energy region,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1987).
  22. B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photo absorption, scattering and reflection,” in Vol. 27 of Atomic and Nuclear Data Tables (American Institute of Physics, New York, 1982), pp. 1–144.
    [CrossRef]
  23. D. Maystre, R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7, 165–180 (1976).
    [CrossRef]
  24. D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 179–225.
  25. M. Nevière, D. Maystre, W. R. Hunter, “On the use of classical and conical diffraction mountings for XUV gratings,” J. Opt. Soc. Am. 68, 1106–1113 (1978).
    [CrossRef]
  26. W. Jark, “Soft x-ray efficiencies: reciprocity theorem, blaze maximum and isoefficiency curves,” Nucl. Instrum. Methods A 266, 414–421 (1988).
    [CrossRef]
  27. A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
    [CrossRef]

1994 (5)

1993 (1)

1991 (3)

1989 (1)

1988 (2)

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

W. Jark, “Soft x-ray efficiencies: reciprocity theorem, blaze maximum and isoefficiency curves,” Nucl. Instrum. Methods A 266, 414–421 (1988).
[CrossRef]

1982 (2)

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[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]

1978 (2)

1976 (3)

D. Maystre, R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7, 165–180 (1976).
[CrossRef]

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

1974 (1)

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

1951 (1)

H. Bremmer, “The W. K. B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

1950 (1)

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

Awada, K. A.

Barbie, T. W.

Bremmer, H.

H. Bremmer, “The W. K. B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

Bruijn, M. P.

A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
[CrossRef]

Chandezon, J.

Cornet, G.

den Boggende, A. J. F.

A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
[CrossRef]

Dupuis, M. T.

Elson, J. M.

Flamand, J.

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

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

Fugikawa, B. K.

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

Gruddace, R. G.

Henke, B. L.

B. L. Henke, P. Lee, T. J. Tanaka, R. L. Shimabukuro, B. K. Fugikawa, “Low energy x-ray interaction coefficients: photo absorption, 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.

Jark, W.

W. Jark, “Soft x-ray efficiencies: reciprocity theorem, blaze maximum and isoefficiency curves,” Nucl. Instrum. Methods A 266, 414–421 (1988).
[CrossRef]

Lee, P.

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

Lerner, J. M.

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Li, L.

Light, J. C.

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Maystre, D.

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

M. Nevière, D. Maystre, W. R. Hunter, “On the use of classical and conical diffraction mountings for XUV gratings,” J. Opt. Soc. Am. 68, 1106–1113 (1978).
[CrossRef]

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

D. Maystre, R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7, 165–180 (1976).
[CrossRef]

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 179–225.

Montiel, F.

Nevière, M.

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

F. Montiel, M. Nevière, “Differential theory of gratings: extension to deep gratings of arbitrary profile and permittivity through the R-matrix propagation algorithm,” J. Opt. Soc. Am. A 11, 3241–3250 (1994).
[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, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

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

M. Nevière, D. Maystre, W. R. Hunter, “On the use of classical and conical diffraction mountings for XUV gratings,” J. Opt. Soc. Am. 68, 1106–1113 (1978).
[CrossRef]

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

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 179–225.

Padmore, H. A.

A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
[CrossRef]

Pai, D. M.

Petit, R.

D. Maystre, R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7, 165–180 (1976).
[CrossRef]

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

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 179–225.

Puik, E. J.

A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
[CrossRef]

Rife, C.

Sandre, L. F.

Shimabukuro, R. L.

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

Verhoeven, J.

A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
[CrossRef]

Vincent, P.

M. Nevière, P. Vincent, “Differential theory of gratings: answer to an objection on its validity for TM polarization,” J. Opt. Soc. Am. A 5, 1522–1524 (1988).
[CrossRef]

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

Walker, R. B.

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Windt, D. L.

D. L. Windt, “The optical properties of 21 thin film materials in the 10 eV to 500 eV photon energy region,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1987).

Zeijlemaker, H.

A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
[CrossRef]

Zvijac, D. J.

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Ann. Phys. (Paris) (1)

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

Appl. Opt. (1)

Chem. Phys. (1)

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976).
[CrossRef]

Commun. Pure Appl. Math. (1)

H. Bremmer, “The W. K. B. approximation as the first term of a geometric-optical series,” Commun. Pure Appl. Math. 4, 105–115 (1951).
[CrossRef]

J. Chem. Phys. (1)

J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (9)

Nouv. Rev. Opt. (2)

D. Maystre, R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7, 165–180 (1976).
[CrossRef]

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

Nucl. Instrum. Methods (2)

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

M. Nevière, J. Flamand, J. M. Lerner, “Optimization of gratings for soft x-ray monochromators,” Nucl. Instrum. Methods 195, 183–189 (1982).
[CrossRef]

Nucl. Instrum. Methods A (1)

W. Jark, “Soft x-ray efficiencies: reciprocity theorem, blaze maximum and isoefficiency curves,” Nucl. Instrum. Methods A 266, 414–421 (1988).
[CrossRef]

Opt. Commun. (1)

M. Nevière, F. Montiel, “Deep gratings: a combination of the differential theory and the multiple reflection series,” Opt. Commun. 108, 1–7 (1994).
[CrossRef]

Other (4)

D. L. Windt, “The optical properties of 21 thin film materials in the 10 eV to 500 eV photon energy region,” Ph.D. dissertation (University of Colorado, Boulder, Colo., 1987).

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

A. J. F. den Boggende, M. P. Bruijn, J. Verhoeven, H. Zeijlemaker, E. J. Puik, H. A. Padmore, “A broad-band multilayer coated blazed grating for x-ray wavelengths below 0.6 nm,” in Advanced X-Ray/EUV Radiation Sources and Applications, J. P. Knauer, G. K. Shenoy, Proc. SPIE1345, 189–197 (1990).
[CrossRef]

D. Maystre, M. Nevière, R. Petit, “Experimental verifications and applications of the theory,” in Electromagnetic Theory of Gratings, R. Petit, ed. (Springer-Verlag, Berlin, 1980), pp. 179–225.

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Figures (8)

Fig. 1
Fig. 1

Bare grating and notation. θ, angle of incidence; g(x), groove shape function.

Fig. 2
Fig. 2

Multilayer grating over one period d. The thickness of the multilayer is assumed to be lower than the groove depth a. gi(x), groove shape function.

Fig. 3
Fig. 3

Blazing of a multilayer-coated echelle grating. The order selected is symmetrical to the incident-wave direction with respect to the normal on the large facet. For an echelle, m typically lies between 10 and 60.

Fig. 4
Fig. 4

Reflectivity as a function of wavelength of a 20-C–W bilayer stack deposited over a Si substrate under 15.7° grazing incidence θ; the top layer is W; the ratio Γ of the thickness of W over the period D of the multilayer is 0.4.

Fig. 5
Fig. 5

Theoretical absolute efficiency ℰmc (dashed curve) in the mcth diffracted order and sum of absolute efficiencies of all propagating orders (solid curve) as a function of wavelength (in micrometers) for a multilayer coated echelle grating.

Fig. 6
Fig. 6

Comparison of the efficiency predicted with the phenomenological formula (solid curve) and the computed efficiencies (dotted curve) for the grating of Fig. 5 when wavelength (in micrometers) is varied.

Fig. 7
Fig. 7

Blazing performances in the usual (dashed curve) and echelle regimes (solid and dotted curves) (wavelength is in nanometers).

Fig. 8
Fig. 8

Minus-1st-order efficiency of a 525-gr/mm, 0.5°-blaze-angle gold grating coated with a 13-layer, nonperiodic stack of Rh and C films as a function of wavelength (in angstroms) under 88.9° incidence.

Tables (2)

Tables Icon

Table 1 Predicted (mp) and Calculated (mc) Blazed Order, Efficiency of the Blazed Order, and Sum of Propagating-Order Efficiencies As Functions of the Wavelength

Tables Icon

Table 2 Distribution of the Layers in the Multilayer Stack

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

Δ E + α ( x , y ) E = 0 ,
α ( x , y ) = ω 2 μ 0 ( x , y ) ,
f ( x ) = n f n exp ( i n K x ) ,             K = 2 π / d ,
f 0 = 1 d ( y 1 d - k = 1 N σ k x k ) ,
f n = 1 2 π n k = 1 N σ k [ sin ( n K x k ) + i cos ( n K x k ) ] ,
σ k = y k + 1 - y k ,             k [ 1 , N - 1 ] , σ N = y 1 - y N .
ν ˜ h = { ν h if N C = 2 q + 1 , q : integer ν l if N C = 2 q , ν ˜ l = { ν l if N C = 2 q + 1 , q : integer ν h if N C = 2 q .
y 0 < j = 1 N C e j .
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 ) ,             k ( 2 , 2 N - 1 ) , y 1 = ν 2 2 .
j = 1 N C e j < y 0 < a .
σ N = ν 1 2 - ν h 2 ,             σ N + 1 = - σ N .
a < y 0 < a + j = 1 N C e j .
X 1 , i = g 1 [ y - tes a ( N + i - 1 ) + a ] ,             i ( 1 , N S ) , X 2 , i = g 2 [ y - tes a ( N + i - 1 ) + a ] ,             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 if N = 2 p + 1 ν ˜ l 2 if N = 2 p ,             p :             integer .
λ = ( 2 D / n ) sin θ ,
λ = ( 2 d sin ϕ sin θ ) m ,
D n = d sin ϕ m .
E m = R ( θ ) × min [ sin α sin ( θ + ϕ ) , sin ( θ + ϕ ) sin α ] ,

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