Abstract

High resolution while maintaining high peak reflectivities can be achieved for Lamellar Multilayer Amplitude Gratings (LMAG) in the soft-x-ray (SXR) region. Using the coupled waves approach (CWA), it is derived that for small lamellar widths only the zeroth diffraction order needs to be considered for LMAG performance calculations, referred to as the single-order regime. In this regime, LMAG performance can be calculated by assuming a conventional multilayer mirror with decreased density, which significantly simplifies the calculations. Novel analytic criteria for the design of LMAGs are derived from the CWA and it is shown, for the first time, that the resolution of an LMAG operating in the single-order regime is not limited by absorption as in conventional multilayer mirrors. It is also shown that the peak reflectivity of an LMAG can then still be as high as that of a conventional multilayer mirror (MM). The performance of LMAGs operating in the single-order regime are thus only limited by technological factors.

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  1. A. E. Yakshin, R. W. E. van de Kruijs, I. Nedelcu, E. Zoethout, E. Louis, F. Bijkerk, H. Enkisch, S. Müllender, and M. J. Lercel, “Enhanced reflectance of interface engineered Mo/Si multilayers produced by thermal particle deposition”, 651701:1–9, Proc SPIE, 6517 (2007)
  2. R. A. M. Keski-Kuha and A. M. Ritva, “Layered synthetic microstructure technology considerations for the extreme ultraviolet,” Appl. Opt. 23(20), 3534 (1984).
    [CrossRef] [PubMed]
  3. B. Vidal, P. Vincent, P. Dhez, and M. Neviere, “Thin films and gratings - Theories to optimize the high reflectivity of mirrors and gratings for X-ray optics ”, 142–149, Proc. SPIE, 563 (1985)
  4. R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
    [CrossRef]
  5. A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
    [CrossRef]
  6. T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. 6(12), 1869–1883 (1989).
    [CrossRef]
  7. A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
    [CrossRef]
  8. L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods 536(1-2), 211–221 (2005).
    [CrossRef]
  9. A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
    [CrossRef]
  10. K. Krastev, J.-M. André, and R. Barchewitz, “Further applications of a recursive modal method for calculating the efficiencies of X-UV multilayer gratings,” J. Opt. Soc. Am. A 13(10), 2027 (1996).
    [CrossRef]
  11. L. I. Goray, and J. F. Seely, “Wavelength separation of plus and minus orders of soft-x-ray-EUV multilayer-coated gratings at near-normal incidence”, 81–91, Proc. SPIE, 5900 (2005)
  12. A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
    [CrossRef]
  13. V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
    [CrossRef]
  14. R. Benbalagh, “Monochromateurs Multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV”, PhD Thesis, University of Paris VI, Paris, 2003.
  15. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T 137–14517, (1987).
    [CrossRef]
  16. R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980.

2005 (2)

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods 536(1-2), 211–221 (2005).
[CrossRef]

2004 (1)

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

1996 (1)

1994 (1)

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

1993 (2)

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

1991 (1)

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

1989 (1)

T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. 6(12), 1869–1883 (1989).
[CrossRef]

1987 (1)

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T 137–14517, (1987).
[CrossRef]

1984 (1)

Agafonov, A. Yu

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

Agafonov, Yu. A.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

André, J.-M.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

K. Krastev, J.-M. André, and R. Barchewitz, “Further applications of a recursive modal method for calculating the efficiencies of X-UV multilayer gratings,” J. Opt. Soc. Am. A 13(10), 2027 (1996).
[CrossRef]

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

Andres, M. V.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

Barchewitz, R.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

K. Krastev, J.-M. André, and R. Barchewitz, “Further applications of a recursive modal method for calculating the efficiencies of X-UV multilayer gratings,” J. Opt. Soc. Am. A 13(10), 2027 (1996).
[CrossRef]

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Benbalagh, R.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Boria, V. E.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

Brunel, M.

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

Coves, A.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

Erko, A. I.

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

Erko, A.I.

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

Filatova, E. O.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Gil, J.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

Gimeno, B.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

Goray, L. I.

L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods 536(1-2), 211–221 (2005).
[CrossRef]

Guérin, P.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Jonnard, P.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Julié, G.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Keski-Kuha, R. A. M.

Khan Malek, C.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Kozhevnikov, I. V.

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T 137–14517, (1987).
[CrossRef]

Krastev, K.

Ladan, F. R.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Marmoret, R.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Martynov, V. V.

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

Mollard, L.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Ouahabi, M.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Pardo, B.

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

Peng, T.

T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. 6(12), 1869–1883 (1989).
[CrossRef]

Rémond, C.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Ritva, A. M.

Rivoira, R.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

Rolland, G.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Roschupkin, D. V.

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

Sammar, A.

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

San Blas, A. A.

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

Troussel, P.

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Vidal, B.

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

Vincent, P.

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

Vinogradov, A. V.

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T 137–14517, (1987).
[CrossRef]

Yakshin, A.

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Antenn. Propag. (1)

A. Coves, B. Gimeno, J. Gil, M. V. Andres, A. A. San Blas, and V. E. Boria, “Full-wave analysis of dielectric frequency-selective surfaces using a vectorial modal method,” IEEE Trans. Antenn. Propag. 52(8), 2091–2099 (2004).
[CrossRef]

J. Opt. (1)

A. Sammar, M. Ouahabi, R. Barchewitz, J.-M. André, R. Rivoira, C. Khan Malek, F. R. Ladan, and P. Guérin, “Theoretical and experimental study of soft X-ray diffraction by a lamellar multilayer amplitude grating ,” J. Opt. 24(1), 37–41 (1993).
[CrossRef]

J. Opt. Soc. Am. (1)

T. Peng, “Rigorous formulation of scattering and guidance by dielectric grating waveguides: general case of oblique incidence,” J. Opt. Soc. Am. 6(12), 1869–1883 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nucl. Instrum. Methods (2)

L. I. Goray, “Numerical analysis of the efficiency of multilayer-coated gratings using integral method,” Nucl. Instrum. Methods 536(1-2), 211–221 (2005).
[CrossRef]

R. Benbalagh, J.-M. André, R. Barchewitz, P. Jonnard, G. Julié, L. Mollard, G. Rolland, C. Rémond, P. Troussel, R. Marmoret, and E. O. Filatova, “Lamellar multilayer amplitude grating as soft-X-ray Bragg monochromator,” Nucl. Instrum. Methods 541(3), 590–597 (2005).
[CrossRef]

Nucl. Instrum. Methods Phys. Res. (2)

A. I. Erko, B. Vidal, P. Vincent, Yu. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instrum. Methods Phys. Res. 333(2-3), 599–606 (1993).
[CrossRef]

V. V. Martynov, B. Vidal, P. Vincent, M. Brunel, D. V. Roschupkin, A. Yu Agafonov, A.I. Erko, and A. Yakshin, “Comparison of modal and differential methods for multilayer gratings,” Nucl. Instrum. Methods Phys. Res. 339(3), 617–625 (1994).
[CrossRef]

Opt. Commun. (1)

A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the X-UV region,” Opt. Commun. 86(2), 245–254 (1991).
[CrossRef]

Phys. Scr. T (1)

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T 137–14517, (1987).
[CrossRef]

Other (5)

R. Petit, Electromagnetic Theory of Gratings, Springer-Verlag, Berlin, 1980.

R. Benbalagh, “Monochromateurs Multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV”, PhD Thesis, University of Paris VI, Paris, 2003.

L. I. Goray, and J. F. Seely, “Wavelength separation of plus and minus orders of soft-x-ray-EUV multilayer-coated gratings at near-normal incidence”, 81–91, Proc. SPIE, 5900 (2005)

A. E. Yakshin, R. W. E. van de Kruijs, I. Nedelcu, E. Zoethout, E. Louis, F. Bijkerk, H. Enkisch, S. Müllender, and M. J. Lercel, “Enhanced reflectance of interface engineered Mo/Si multilayers produced by thermal particle deposition”, 651701:1–9, Proc SPIE, 6517 (2007)

B. Vidal, P. Vincent, P. Dhez, and M. Neviere, “Thin films and gratings - Theories to optimize the high reflectivity of mirrors and gratings for X-ray optics ”, 142–149, Proc. SPIE, 563 (1985)

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the cross section of an LMAG. (a): An incident beam from the left (In), under grazing angle θ0, is reflected from the multilayer and diffracted into multiple orders (Out) by the grating structure. The multilayer is built up from N bi-layers with thickness d. Each bilayer consists of an absorber material (A) with thickness γd and a spacer material (S) with thickness (1−γ)d. The grating structure of the LMAG is defined by the grating period D and lamel width ΓD. (b): The normalized function U(x) is used to describe the lamellar profile.

Fig. 2.
Fig. 2.

Zeroth order diffraction efficiency (reflectivity) of a Mo/B4C LMAG versus the grazing angle of an incident beam at SXR energy E = 183.4 eV for increasing number of diffraction orders taken into account when solving Eq. (4). Parameters of the LMAG: D = 2 µm, Γ = 0.3, N = 150, d = 6 nm, γ = 0.33. The values of the complex polarizability χ = 1 − ε used for calculations: χ(Mo) = 2.61·10−2i·5.77·10−3 and χ(B4C)=4.43·10−3 −i·1.08·10−3

Fig. 3.
Fig. 3.

Diffraction efficiencies of higher orders at E = 183.4 eV versus the grazing angle of the incident beam. Parameters of the LMAG are the same as for Fig. 2. At all times, 15 diffraction orders (up to ± 7th order) were taken into account in the calculations.

Fig. 4.
Fig. 4.

Diffraction efficiency (at E = 183.4 eV) of the zeroth (LMAG 0) and first (LMAG ± 1) diffraction orders of a Mo/B4C LMAG versus the grazing angle of the incident beam. The grating period D = 0.3 µm and the rest of the LMAG parameters are the same as for Fig. 2. In the calculations, 11 diffraction orders were taken into account. The reflectivity of a conventional Mo/B4C multilayer mirror consisting of materials with decreased density is also shown (MM), which has an excellent agreement with the zeroth order LMAG diffraction efficiency.

Fig. 5.
Fig. 5.

Reflectivity for the zeroth diffraction order versus grazing angle (at E = 183 eV) for a conventional Mo/B4C multilayer mirror (1) and three Mo/B4C LMAGs (2–4) with the same lamellar width ΓD = 70 nm, but different Γ’s, D’s and N. (2): Γ = 1/2, N = 200, D = 140 nm; (3): Γ = 1/3, N = 300, D = 210 nm; (4): Γ = 1/10, N = 1000, D = 700 nm. The other LMAG parameters are the same as for Fig. 2. 11 diffraction orders were taken into account in the calculations.

Equations (9)

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ε ( x , 0 z L ) = 1 χ ( z ) U ( x ; Γ , D ) ; ε ( x , z < 0 ) = 1 ; ε ( x , z > L ) = ε sub = const
U ( x ; Γ , D ) = Σ n = + U n exp ( 2 i π n x D ) , U 0 = Γ , U n 0 = [ 1 exp ( 2 i π n Γ ) ] ( 2 i π n )
E ( x , z ) = Σ n = + F n ( z ) exp ( i q n x ) ; q n = q 0 + 2 π n D ; q 0 = k cos θ 0 ; k = 2 π λ
F n ( z ) + κ n 2 F n ( z ) = k 2 χ ( z ) Σ m U n m F m ( z )
F n ( 0 ) + i κ n F n ( 0 ) = 2 i κ n δ n , 0 ; F n ( L ) i κ n ( s ) F n ( L ) = 0
F 0 ( z ) + κ 0 2 F 0 ( z ) = k 2 χ ( z ) F 0 ( z )
F 0 ( z ) + κ 0 2 F 0 ( z ) = k 2 Γ χ ( z ) F 0 ( z )
Γ D · Δ θ MM d
( Δ θ MM ) min = 2 Im χ s sin ( 2 θ 0 )

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