Abstract

A grand challenge in soft x-ray spectroscopy is to drive the resolving power of monochromators and spectrometers from the 104 achieved routinely today to well above 105. This need is driven mainly by the requirements of a new technique that is set to have enormous impact in condensed matter physics, Resonant Inelastic X-ray Scattering (RIXS). Unlike x-ray absorption spectroscopy, RIXS is not limited by an energy resolution dictated by the core-hole lifetime in the excitation process. Using much higher resolving power than used for normal x-ray absorption spectroscopy enables access to the energy scale of soft excitations in matter. These excitations such as magnons and phonons drive the collective phenomena seen in correlated electronic materials such as high temperature superconductors. RIXS opens a new path to study these excitations at a level of detail not formerly possible. However, as the process involves resonant excitation at an energy of around 1 keV, and the energy scale of the excitations one would like to see are at the meV level, to fully utilize the technique requires the development of monochromators and spectrometers with one to two orders of magnitude higher energy resolution than has been conventionally possible. Here we investigate the detailed diffraction characteristics of multilayer blazed gratings. These elements offer potentially revolutionary performance as the dispersive element in ultra-high resolution x-ray spectroscopy. In doing so, we have established a roadmap for the complete optimization of the grating design. Traditionally 1st order gratings are used in the soft x-ray region, but we show that as in the optical domain, one can work in very high spectral orders and thus dramatically improve resolution without significant loss in efficiency.

© 2015 Optical Society of America

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  28. L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering,” J. Appl. Phys. 108(3), 033516 (2010).
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  35. See, for example, Yu. Shvyd’ko, X-Ray Optics. High-Resolution Applications (Springer, 2004).
  36. L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
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  37. D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge University, 1999).
  38. T. Warwick, Y.-D. Chuang, D. L. Voronov, and H. A. Padmore, “A multiplexed high-resolution imaging spectrometer for resonant inelastic soft X-ray scattering spectroscopy,” J. Synchrotron Radiat. 21(4), 736–743 (2014).
    [Crossref] [PubMed]
  39. V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
    [Crossref]
  40. R. M. Fechtchenko, A. V. Vinogradov, and D. L. Voronov, “Optical properties of sliced multilayer gratings,” Opt. Commun. 210(3-6), 179–186 (2002).
    [Crossref]

2014 (2)

T. Warwick, Y.-D. Chuang, D. L. Voronov, and H. A. Padmore, “A multiplexed high-resolution imaging spectrometer for resonant inelastic soft X-ray scattering spectroscopy,” J. Synchrotron Radiat. 21(4), 736–743 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. M. Gullikson, F. Salmassi, T. Warwick, and H. A. Padmore, “Enhancement of diffraction efficiency via higher-order operation of a multilayer blazed grating,” Opt. Lett. 39(11), 3157–3160 (2014).
[Crossref] [PubMed]

2013 (1)

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

2012 (1)

2011 (4)

2010 (5)

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

L. I. Goray, “Application of the boundary integral equation method to very small wavelength-to-period diffraction problems,” Waves Random Media 20(4), 569–586 (2010).
[Crossref]

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

L. I. Goray and G. Schmidt, “Solving conical diffraction grating problems with integral equations,” J. Opt. Soc. Am. A 27(3), 585–597 (2010).
[Crossref] [PubMed]

D. L. Voronov, M. Ahn, E. H. Anderson, R. Cambie, C. H. Chang, E. M. Gullikson, R. K. Heilmann, F. Salmassi, M. L. Schattenburg, T. Warwick, V. V. Yashchuk, L. Zipp, and H. A. Padmore, “High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths,” Opt. Lett. 35(15), 2615–2617 (2010).
[Crossref] [PubMed]

2008 (2)

H. Lin, L. Zhang, L. Li, Ch. Jin, H. Zhou, and T. Huo, “High-efficiency multilayer-coated ion-beam-etched blazed grating in the extreme-ultraviolet wavelength region,” Opt. Lett. 33(5), 485–487 (2008).
[Crossref] [PubMed]

M. Ahn, R. K. Heilmann, and M. L. Schattenburg, “Fabrication of 200 nm-period blazed transmission gratings on silicon-on-insulator wafers,” J. Vac. Sci. Technol. B 26(6), 2179–2182 (2008).
[Crossref]

2007 (1)

D. L. Voronov, R. Cambie, R. M. Feshchenko, E. Gullikson, H. A. Padmore, A. V. Vinogradov, and V. V. Yashchuk, “Development of an ultra-high resolution diffraction grating for soft X-rays,” Proc. SPIE 6705, 67050E (2007).
[Crossref]

2006 (2)

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

2005 (2)

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

L. I. Goray, “Scalar and electromagnetic properties of X-ray diffraction gratings,” Bull. Russ. Acad. Sci., Physics 69, 231–236 (2005).

2004 (1)

2002 (1)

R. M. Fechtchenko, A. V. Vinogradov, and D. L. Voronov, “Optical properties of sliced multilayer gratings,” Opt. Commun. 210(3-6), 179–186 (2002).
[Crossref]

1997 (1)

V. V. Martynov, H. A. Padmore, A. Yuakshin, and Y. A. Agafonov, “Lamellar multilayer gratings with very high diffraction efficiency,” Proc. SPIE 3150, 2–8 (1997).
[Crossref]

1996 (1)

1995 (1)

J. H. Underwood, C. Kh. Malek, E. M. Gullikson, and M. Krumrey, “Multilayer-coated echelle gratings for soft x-rays and extreme ultraviolet,” Rev. Sci. Instrum. 66(2), 2147–2150 (1995).
[Crossref]

1994 (1)

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

1991 (1)

1989 (1)

1985 (1)

1980 (1)

Y. Fujii, K. I. Aoyama, and J. I. Minowa, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. 16(2), 165–169 (1980).
[Crossref]

1977 (1)

1976 (1)

D. Maystre and R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7(3), 165–180 (1976).
[Crossref]

1963 (1)

A. P. Lukirskii and E. P. Savinov, “Use of diffraction gratings and echelettes in the ultrasoft x-ray region,” Opt. Spectrosc. 14, 147–151 (1963).A. P. Lukirskii, E. P. Savinov, and Yu. P. Shepelev, “Behaviour of gold and titanium coated echelettes in the 23.6-113 A° region,” Opt. Spectrosc. 15, 290–293 (1963).

Agafonov, Y. A.

V. V. Martynov, H. A. Padmore, A. Yuakshin, and Y. A. Agafonov, “Lamellar multilayer gratings with very high diffraction efficiency,” Proc. SPIE 3150, 2–8 (1997).
[Crossref]

Ahn, M.

Alatas, A.

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

Anderson, E. H.

Aoyama, K. I.

Y. Fujii, K. I. Aoyama, and J. I. Minowa, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. 16(2), 165–169 (1980).
[Crossref]

Barbee, T. W.

Bastiaens, H. M. J.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Analytic theory of soft x-ray diffraction by Lamellar Multilayer Gratings,” Opt. Express 19(10), 9172–9184 (2011).
[Crossref] [PubMed]

Bijkerk, F.

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Analytic theory of soft x-ray diffraction by Lamellar Multilayer Gratings,” Opt. Express 19(10), 9172–9184 (2011).
[Crossref] [PubMed]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Boller, K.-J.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Analytic theory of soft x-ray diffraction by Lamellar Multilayer Gratings,” Opt. Express 19(10), 9172–9184 (2011).
[Crossref] [PubMed]

Brons, G. C. S.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Bruccoleri, A.

Cabrini, S.

Cambie, R.

Chang, C. H.

Chang, C.-H.

Chuang, Y.-D.

T. Warwick, Y.-D. Chuang, D. L. Voronov, and H. A. Padmore, “A multiplexed high-resolution imaging spectrometer for resonant inelastic soft X-ray scattering spectroscopy,” J. Synchrotron Radiat. 21(4), 736–743 (2014).
[Crossref] [PubMed]

Cruddace, R. G.

De Boer, M. J.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Dhuey, S. D.

Fechtchenko, R. M.

R. M. Fechtchenko, A. V. Vinogradov, and D. L. Voronov, “Optical properties of sliced multilayer gratings,” Opt. Commun. 210(3-6), 179–186 (2002).
[Crossref]

Fedorenko, A. I.

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

Feshchenko, R. M.

D. L. Voronov, R. Cambie, R. M. Feshchenko, E. Gullikson, H. A. Padmore, A. V. Vinogradov, and V. V. Yashchuk, “Development of an ultra-high resolution diffraction grating for soft X-rays,” Proc. SPIE 6705, 67050E (2007).
[Crossref]

Fujii, Y.

Y. Fujii, K. I. Aoyama, and J. I. Minowa, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. 16(2), 165–169 (1980).
[Crossref]

Garrett, R.

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

Gentle, I.

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

Goray, L.

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

Goray, L. I.

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

L. I. Goray and G. Schmidt, “Solving conical diffraction grating problems with integral equations,” J. Opt. Soc. Am. A 27(3), 585–597 (2010).
[Crossref] [PubMed]

L. I. Goray, “Application of the boundary integral equation method to very small wavelength-to-period diffraction problems,” Waves Random Media 20(4), 569–586 (2010).
[Crossref]

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

L. I. Goray, “Scalar and electromagnetic properties of X-ray diffraction gratings,” Bull. Russ. Acad. Sci., Physics 69, 231–236 (2005).

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

Gullikson, E.

D. L. Voronov, R. Cambie, R. M. Feshchenko, E. Gullikson, H. A. Padmore, A. V. Vinogradov, and V. V. Yashchuk, “Development of an ultra-high resolution diffraction grating for soft X-rays,” Proc. SPIE 6705, 67050E (2007).
[Crossref]

Gullikson, E. M.

D. L. Voronov, E. M. Gullikson, F. Salmassi, T. Warwick, and H. A. Padmore, “Enhancement of diffraction efficiency via higher-order operation of a multilayer blazed grating,” Opt. Lett. 39(11), 3157–3160 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Ultra-high efficiency multilayer blazed gratings through deposition kinetic control,” Opt. Lett. 37(10), 1628–1630 (2012).
[Crossref] [PubMed]

R. K. Heilmann, M. Ahn, A. Bruccoleri, C.-H. Chang, E. M. Gullikson, P. Mukherjee, and M. L. Schattenburg, “Diffraction efficiency of 200-nm-period critical-angle transmission gratings in the soft x-ray and extreme ultraviolet wavelength bands,” Appl. Opt. 50(10), 1364–1373 (2011).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

D. L. Voronov, M. Ahn, E. H. Anderson, R. Cambie, C. H. Chang, E. M. Gullikson, R. K. Heilmann, F. Salmassi, M. L. Schattenburg, T. Warwick, V. V. Yashchuk, L. Zipp, and H. A. Padmore, “High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths,” Opt. Lett. 35(15), 2615–2617 (2010).
[Crossref] [PubMed]

J. H. Underwood, C. Kh. Malek, E. M. Gullikson, and M. Krumrey, “Multilayer-coated echelle gratings for soft x-rays and extreme ultraviolet,” Rev. Sci. Instrum. 66(2), 2147–2150 (1995).
[Crossref]

Hegeman, P. E.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Heidemann, K. F.

Heilmann, R. K.

Hunter, W. R.

Huo, T.

Huskens, J.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Jin, Ch.

Kierey, H.

Kleemann, B. H.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

Kondratenko, V. V.

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

Kowalski, M. P.

Kozhevnikov, I. V.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Analytic theory of soft x-ray diffraction by Lamellar Multilayer Gratings,” Opt. Express 19(10), 9172–9184 (2011).
[Crossref] [PubMed]

Krishnan, B.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Krumrey, M.

J. H. Underwood, C. Kh. Malek, E. M. Gullikson, and M. Krumrey, “Multilayer-coated echelle gratings for soft x-rays and extreme ultraviolet,” Rev. Sci. Instrum. 66(2), 2147–2150 (1995).
[Crossref]

Kuetgens, U.

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

Lenke, R.

Lerche, M.

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

Levashov, V. E.

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

Li, L.

Lin, H.

Lubov, M.

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

Lukirskii, A. P.

A. P. Lukirskii and E. P. Savinov, “Use of diffraction gratings and echelettes in the ultrasoft x-ray region,” Opt. Spectrosc. 14, 147–151 (1963).A. P. Lukirskii, E. P. Savinov, and Yu. P. Shepelev, “Behaviour of gold and titanium coated echelettes in the 23.6-113 A° region,” Opt. Spectrosc. 15, 290–293 (1963).

Malek, C. Kh.

J. H. Underwood, C. Kh. Malek, E. M. Gullikson, and M. Krumrey, “Multilayer-coated echelle gratings for soft x-rays and extreme ultraviolet,” Rev. Sci. Instrum. 66(2), 2147–2150 (1995).
[Crossref]

Martynov, V. V.

V. V. Martynov, H. A. Padmore, A. Yuakshin, and Y. A. Agafonov, “Lamellar multilayer gratings with very high diffraction efficiency,” Proc. SPIE 3150, 2–8 (1997).
[Crossref]

Maystre, D.

P. Philippe, S. Valette, O. M. Mendez, and D. Maystre, “Wavelength demultiplexer: using echelette gratings on silicon substrate,” Appl. Opt. 24(7), 1006–1011 (1985).
[Crossref] [PubMed]

D. Maystre and R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7(3), 165–180 (1976).
[Crossref]

Mendez, O. M.

Minowa, J. I.

Y. Fujii, K. I. Aoyama, and J. I. Minowa, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. 16(2), 165–169 (1980).
[Crossref]

Montiel, F.

Mukherjee, P.

Nevière, M.

Nugent, K.

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

Padmore, H. A.

T. Warwick, Y.-D. Chuang, D. L. Voronov, and H. A. Padmore, “A multiplexed high-resolution imaging spectrometer for resonant inelastic soft X-ray scattering spectroscopy,” J. Synchrotron Radiat. 21(4), 736–743 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. M. Gullikson, F. Salmassi, T. Warwick, and H. A. Padmore, “Enhancement of diffraction efficiency via higher-order operation of a multilayer blazed grating,” Opt. Lett. 39(11), 3157–3160 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Ultra-high efficiency multilayer blazed gratings through deposition kinetic control,” Opt. Lett. 37(10), 1628–1630 (2012).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

D. L. Voronov, M. Ahn, E. H. Anderson, R. Cambie, C. H. Chang, E. M. Gullikson, R. K. Heilmann, F. Salmassi, M. L. Schattenburg, T. Warwick, V. V. Yashchuk, L. Zipp, and H. A. Padmore, “High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths,” Opt. Lett. 35(15), 2615–2617 (2010).
[Crossref] [PubMed]

D. L. Voronov, R. Cambie, R. M. Feshchenko, E. Gullikson, H. A. Padmore, A. V. Vinogradov, and V. V. Yashchuk, “Development of an ultra-high resolution diffraction grating for soft X-rays,” Proc. SPIE 6705, 67050E (2007).
[Crossref]

V. V. Martynov, H. A. Padmore, A. Yuakshin, and Y. A. Agafonov, “Lamellar multilayer gratings with very high diffraction efficiency,” Proc. SPIE 3150, 2–8 (1997).
[Crossref]

Petit, R.

D. Maystre and R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7(3), 165–180 (1976).
[Crossref]

Philippe, P.

Poltseva, O. V.

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

Rathsfeld, A.

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

Rife, J. C.

Rüter, H. D.

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

Salmassi, F.

Savinov, E. P.

A. P. Lukirskii and E. P. Savinov, “Use of diffraction gratings and echelettes in the ultrasoft x-ray region,” Opt. Spectrosc. 14, 147–151 (1963).A. P. Lukirskii, E. P. Savinov, and Yu. P. Shepelev, “Behaviour of gold and titanium coated echelettes in the 23.6-113 A° region,” Opt. Spectrosc. 15, 290–293 (1963).

Schattenburg, M. L.

Schmidt, G.

L. I. Goray and G. Schmidt, “Solving conical diffraction grating problems with integral equations,” J. Opt. Soc. Am. A 27(3), 585–597 (2010).
[Crossref] [PubMed]

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

Shvyd’ko, Y. V.

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

Struk, I. I.

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

Underwood, J. H.

J. H. Underwood, C. Kh. Malek, E. M. Gullikson, and M. Krumrey, “Multilayer-coated echelle gratings for soft x-rays and extreme ultraviolet,” Rev. Sci. Instrum. 66(2), 2147–2150 (1995).
[Crossref]

Valette, S.

van der Meer, R.

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Analytic theory of soft x-ray diffraction by Lamellar Multilayer Gratings,” Opt. Express 19(10), 9172–9184 (2011).
[Crossref] [PubMed]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

van der Wiel, W. G.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Vinogradov, A. V.

D. L. Voronov, R. Cambie, R. M. Feshchenko, E. Gullikson, H. A. Padmore, A. V. Vinogradov, and V. V. Yashchuk, “Development of an ultra-high resolution diffraction grating for soft X-rays,” Proc. SPIE 6705, 67050E (2007).
[Crossref]

R. M. Fechtchenko, A. V. Vinogradov, and D. L. Voronov, “Optical properties of sliced multilayer gratings,” Opt. Commun. 210(3-6), 179–186 (2002).
[Crossref]

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

A. V. Vinogradov and B. Ya. Zeldovich, “X-ray and far UV multilayer mirrors: Principles and possibilities,” Appl. Opt. 16(1), 89–93 (1977).
[Crossref] [PubMed]

Voronov, D. L.

T. Warwick, Y.-D. Chuang, D. L. Voronov, and H. A. Padmore, “A multiplexed high-resolution imaging spectrometer for resonant inelastic soft X-ray scattering spectroscopy,” J. Synchrotron Radiat. 21(4), 736–743 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. M. Gullikson, F. Salmassi, T. Warwick, and H. A. Padmore, “Enhancement of diffraction efficiency via higher-order operation of a multilayer blazed grating,” Opt. Lett. 39(11), 3157–3160 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Ultra-high efficiency multilayer blazed gratings through deposition kinetic control,” Opt. Lett. 37(10), 1628–1630 (2012).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

D. L. Voronov, M. Ahn, E. H. Anderson, R. Cambie, C. H. Chang, E. M. Gullikson, R. K. Heilmann, F. Salmassi, M. L. Schattenburg, T. Warwick, V. V. Yashchuk, L. Zipp, and H. A. Padmore, “High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths,” Opt. Lett. 35(15), 2615–2617 (2010).
[Crossref] [PubMed]

D. L. Voronov, R. Cambie, R. M. Feshchenko, E. Gullikson, H. A. Padmore, A. V. Vinogradov, and V. V. Yashchuk, “Development of an ultra-high resolution diffraction grating for soft X-rays,” Proc. SPIE 6705, 67050E (2007).
[Crossref]

R. M. Fechtchenko, A. V. Vinogradov, and D. L. Voronov, “Optical properties of sliced multilayer gratings,” Opt. Commun. 210(3-6), 179–186 (2002).
[Crossref]

Vratzov, B.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Warwick, T.

T. Warwick, Y.-D. Chuang, D. L. Voronov, and H. A. Padmore, “A multiplexed high-resolution imaging spectrometer for resonant inelastic soft X-ray scattering spectroscopy,” J. Synchrotron Radiat. 21(4), 736–743 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. M. Gullikson, F. Salmassi, T. Warwick, and H. A. Padmore, “Enhancement of diffraction efficiency via higher-order operation of a multilayer blazed grating,” Opt. Lett. 39(11), 3157–3160 (2014).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Ultra-high efficiency multilayer blazed gratings through deposition kinetic control,” Opt. Lett. 37(10), 1628–1630 (2012).
[Crossref] [PubMed]

D. L. Voronov, E. H. Anderson, R. Cambie, S. Cabrini, S. D. Dhuey, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “A 10,000 groove/mm multilayer coated grating for EUV spectroscopy,” Opt. Express 19(7), 6320–6325 (2011).
[Crossref] [PubMed]

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

D. L. Voronov, M. Ahn, E. H. Anderson, R. Cambie, C. H. Chang, E. M. Gullikson, R. K. Heilmann, F. Salmassi, M. L. Schattenburg, T. Warwick, V. V. Yashchuk, L. Zipp, and H. A. Padmore, “High-efficiency 5000 lines/mm multilayer-coated blazed grating for extreme ultraviolet wavelengths,” Opt. Lett. 35(15), 2615–2617 (2010).
[Crossref] [PubMed]

Wilkins, S.

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

Yashchuk, V. V.

Yuakshin, A.

V. V. Martynov, H. A. Padmore, A. Yuakshin, and Y. A. Agafonov, “Lamellar multilayer gratings with very high diffraction efficiency,” Proc. SPIE 3150, 2–8 (1997).
[Crossref]

Yulin, S. A.

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

Zeldovich, B. Ya.

Zhang, L.

Zhao, J.

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

Zhou, H.

Zipp, L.

Zubarev, E. N.

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

AIP Conf. Proc. (1)

T. Warwick, H. A. Padmore, D. L. Voronov, V. V. Yashchuk, R. Garrett, I. Gentle, K. Nugent, and S. Wilkins, “A soft x-ray spectrometer using a highly dispersive multilayer grating,” AIP Conf. Proc. 1234, 776–780 (2010).
[Crossref]

Appl. Opt. (4)

Bull. Russ. Acad. Sci., Physics (1)

L. I. Goray, “Scalar and electromagnetic properties of X-ray diffraction gratings,” Bull. Russ. Acad. Sci., Physics 69, 231–236 (2005).

Commun. Comput. Phys. (1)

A. Rathsfeld, G. Schmidt, and B. H. Kleemann, “On a fast integral equation method for diffraction gratings,” Commun. Comput. Phys. 1, 984–1009 (2006).

IEEE J. Quantum Electron. (1)

Y. Fujii, K. I. Aoyama, and J. I. Minowa, “Optical demultiplexer using a silicon echelette grating,” IEEE J. Quantum Electron. 16(2), 165–169 (1980).
[Crossref]

J. Appl. Cryst. (1)

L. Goray and M. Lubov, “Nonlinear continuum growth model of multiscale reliefs as applied to rigorous analysis of multilayer short-wave scattering intensity. I. Gratings,” J. Appl. Cryst. 46(4), 926–932 (2013).
[Crossref] [PubMed]

J. Appl. Phys. (1)

L. I. Goray, “Application of the rigorous method to x-ray and neutron beam scattering,” J. Appl. Phys. 108(3), 033516 (2010).
[Crossref]

J. Opt. Soc. Am. A (3)

J. Synchrotron Radiat. (1)

T. Warwick, Y.-D. Chuang, D. L. Voronov, and H. A. Padmore, “A multiplexed high-resolution imaging spectrometer for resonant inelastic soft X-ray scattering spectroscopy,” J. Synchrotron Radiat. 21(4), 736–743 (2014).
[Crossref] [PubMed]

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

M. Ahn, R. K. Heilmann, and M. L. Schattenburg, “Fabrication of 200 nm-period blazed transmission gratings on silicon-on-insulator wafers,” J. Vac. Sci. Technol. B 26(6), 2179–2182 (2008).
[Crossref]

Nouv. Rev. Opt. (1)

D. Maystre and R. Petit, “Some recent theoretical results for gratings: application for their use in the very far ultraviolet region,” Nouv. Rev. Opt. 7(3), 165–180 (1976).
[Crossref]

Nucl. Instrum. Meth. A (1)

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

Opt. Commun. (2)

V. E. Levashov, E. N. Zubarev, A. I. Fedorenko, V. V. Kondratenko, O. V. Poltseva, S. A. Yulin, I. I. Struk, and A. V. Vinogradov, “High throughput and resolution compact spectrograph for the 124-250 A range based on MoSi2-Si sliced multilayer grating,” Opt. Commun. 109(1-2), 1–4 (1994).
[Crossref]

R. M. Fechtchenko, A. V. Vinogradov, and D. L. Voronov, “Optical properties of sliced multilayer gratings,” Opt. Commun. 210(3-6), 179–186 (2002).
[Crossref]

Opt. Express (2)

Opt. Lett. (5)

Opt. Spectrosc. (1)

A. P. Lukirskii and E. P. Savinov, “Use of diffraction gratings and echelettes in the ultrasoft x-ray region,” Opt. Spectrosc. 14, 147–151 (1963).A. P. Lukirskii, E. P. Savinov, and Yu. P. Shepelev, “Behaviour of gold and titanium coated echelettes in the 23.6-113 A° region,” Opt. Spectrosc. 15, 290–293 (1963).

Phys. Rev. Lett. (1)

Y. V. Shvyd’ko, M. Lerche, U. Kuetgens, H. D. Rüter, A. Alatas, and J. Zhao, “X-ray Bragg diffraction in asymmetric backscattering geometry,” Phys. Rev. Lett. 97(23), 235502 (2006).
[Crossref] [PubMed]

Proc. SPIE (3)

V. V. Martynov, H. A. Padmore, A. Yuakshin, and Y. A. Agafonov, “Lamellar multilayer gratings with very high diffraction efficiency,” Proc. SPIE 3150, 2–8 (1997).
[Crossref]

D. L. Voronov, R. Cambie, R. M. Feshchenko, E. Gullikson, H. A. Padmore, A. V. Vinogradov, and V. V. Yashchuk, “Development of an ultra-high resolution diffraction grating for soft X-rays,” Proc. SPIE 6705, 67050E (2007).
[Crossref]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. De Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. SPIE 8139, 81390Q (2011).
[Crossref]

Rev. Sci. Instrum. (1)

J. H. Underwood, C. Kh. Malek, E. M. Gullikson, and M. Krumrey, “Multilayer-coated echelle gratings for soft x-rays and extreme ultraviolet,” Rev. Sci. Instrum. 66(2), 2147–2150 (1995).
[Crossref]

Waves Random Media (1)

L. I. Goray, “Application of the boundary integral equation method to very small wavelength-to-period diffraction problems,” Waves Random Media 20(4), 569–586 (2010).
[Crossref]

Other (7)

L. I. Goray and G. Schmidt, “Boundary integral equation methods for conical diffraction and short waves,” in Gratings: Theory and Numerical Applications, E. Popov, ed. (Institute Fresnel, AMU, 2014), pp. 12.1–12.86, http://www.fresnel.fr/numerical-grating-book-2 .

D. Attwood, Soft X-Rays and Extreme Ultraviolet Radiation (Cambridge University, 1999).

http://henke.lbl.gov/

http://www.pcgrate.com/ .

J. Elschner, R. Hinder, A. Rathsfeld, and G. Schmidt, DIPOG Homepage, http://www.wias-berlin.de/software/DIPOG .

A. V. Vinogradov, I. A. Brytov, A. Ya. Grudsky, M. T. Kogan, I. V. Kozhevnikov, and V. A. Slemzin, Mirror X-Ray Optics (in Russian) (Mashinostroenie, 1989).

See, for example, Yu. Shvyd’ko, X-Ray Optics. High-Resolution Applications (Springer, 2004).

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Figures (13)

Fig. 1
Fig. 1 Diffraction geometry of a multilayer blazed grating: α - angle of incidence, β - diffraction angle for the mth blazed diffraction order, ϕ - blaze angle, θΒ -Bragg angle, d - grating period, Δ - d-spacing of a multilayer, N - normal to the grating plane.

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Fig. 2
Fig. 2 Diffraction efficiency calculated with MIM and GFEM for a MBG with a period of 200 nm, blaze angle of 6°, 40 W/B4C bi-layers having a multilayer period Δ of 3.06 nm and Γ-ratio of 0.5 illuminated at an angle of incidence of 83.25°(a). Diffraction efficiency calculated with MIM and the differential method for a MBG with a period of 360 nm, blaze angle of 5°, 30 Rh/C bi-layers having a multilayer period Δ of 3.3 nm and Γ-ratio of 0.33 illuminated with an angle of incidence of 82.8° (b).

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Fig. 3
Fig. 3 Schematic of a ML-coated blazed gratings for soft x-rays: (a) a long-period grating with the period larger than an extinction length of the radiation in the multilayer; (b) a short-period grating with the period much smaller than the extinction length. Incident light does not penetrate through the ML stack for the long-period gratings, and the bottom part of the grooves appears to be shadowed. The radiation penetrates through the many semi-transparent grooves of a short-period grating, and therefore a reduction of shadowing is expected.

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Fig. 4
Fig. 4 Calculated TE efficiency of MBGs with periods of 200 nm (solid curve) and 800 nm (dashed curve). The gratings having a blaze angle of 6° are coated with identical W/B4C multilayers composed of 40 bi-layers with a Δ-spacing of 3.06 nm and Γ-ratio of 0.5. An angle of incidence of 83.25° provides blazed condition for the 7th and 28th diffraction orders of the gratings respectively. Reflectance of the W/B4C multilayer is shown by a dotted curve. The grating efficiency predicted with the phenomenological approach [Eq. (1)] is depicted by a starred symbol.

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Fig. 5
Fig. 5 Dependence of the efficiency of W/B4C blazed gratings coated by 40 bi-layers on the grating period (circles). All the gratings have the same blaze angle of 6°, identical ML coating, and the same incidence angle of 83.25°. The efficiency calculated with the phenomenological approach is depicted with a grey line. The red curve is a fit of the efficiency data with the modified phenomenological function (4).

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Fig. 6
Fig. 6 Diffraction efficiency of the MBGs optimized for the 1st, 7th, and 9th blazed orders (see Table 2). All the gratings have a period of 200 nm, and are coated with the same W/B4C multilayer with Δ-spacing of 2.99 nm and Γ-ratio of 0.5, so that a Bragg angle of 12.75° was kept in all the cases. The arrow depicts a resonant wavelength calculated with no refraction taken into account.

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Fig. 7
Fig. 7 Efficiency of the positive 9th blazed order of a ML coated blazed grating (black solid curve) and the negative 9th blazed order (gray solid curve). The efficiency curves are shifted in opposite directions with respect to the ML reflectance curve (dashed curve) which corresponds to the case of symmetrical Bragg diffraction. The grating has a period of 200 nm, blaze angle of 7.12°, and is coated with a W/B4C multilayer with 51 bi-layers, a Δ-spacing of 2.75 nm and a Γ-ratio of 0.2.

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Fig. 8
Fig. 8 Dependence of relative diffraction efficiency on diffraction asymmetry (solid symbols and red line), calculated for ultra-dense MBGs listed in Table 3. The multilayer parameters (d-spacing, Γ-ratio, Bragg angle etc.) are the same for all the gratings. The diffraction efficiency exceeds the one calculated with the Maystre-Petit formula (green line). The dependence of parameter A from Eq. (4), which is a difference between the two calculations, is shown with a blue curve.

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Fig. 9
Fig. 9 (a) Transmitted energy (circles), absorbed energy (open circles), and absolute efficiency (triangles) for the MBG gratings with different blaze angles listed in the Table 4. (b) Dependences of diffraction efficiency on the number of bi-layers for the gratings with a blaze angle of 6° and 7.72° (see Table 4) are shown by curves with solid and open circles respectively; reflectance of the respective plane multilayer versus the number of bi-layers is shown with a curve with triangle symbols.

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Fig. 10
Fig. 10 Schematic of symmetric (a) and asymmetric (b) Bragg diffraction. Arrows show the path of soft x-rays reflected from an interface buried at the depth h in a multilayer. Since the absorption length is always longer for asymmetrical diffraction, absorption is always stronger for a MBG as compared to a correspondent flat ML.

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Fig. 11
Fig. 11 Relative diffraction efficiency of the MBGs listed in Table 3 versus offset of the effective blaze angle from the “geometrical” blaze angle.

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Fig. 12
Fig. 12 Reflectance of the MLs used in simulation of different MBGs listed in Table 5. The ML parameters listed in Table 5 were varied so that the average refraction index of the multilayers reduces progressively. As a result, the reflectance curves shift gradually towards longer wavelengths. The number of bi-layers was adjusted for each ML in order to obtain the same reflectance for all of the MLs.

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Fig. 13
Fig. 13 Efficiency of the 9th diffraction order of 200 nm period blazed gratings #1-#4 with a blaze angle of 7.12°, as listed in Table 5, coated with the multilayers shown in Fig. 9. The parameters of the gratings and the MLs were varied to investigate the dependence of MBG diffraction efficiency on ML d-spacing (the gratings #1 and #2), ML Γ-ratio (the gratings #2 and #3), and ML materials (the gratings #3 and #4).

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Tables (5)

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Table 1 The period and the corresponding blazed orders for the ML blazed gratings used for efficiency simulations shown in Fig. 5. All other parameters are the same for all the gratings: the blaze angle of 6°, the incident angle of 83.25°, a W/B4C multilayer with a Δ-spacing of 3.06 nm, a Γ-ratio of 0.5, and the number of bi-layers is 40.

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Table 2 Parameters of the MBGs optimized for the 1st, 7th, and 9th blazed orders. All the gratings have a period of 200 nm, and are coated with the same W/B4C multilayer with 40 bi-layers, a Δ-spacing of 2.99 nm and Γ-ratio of 0.5.

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Table 3 Parameters of 1st-order MBDs for relative efficiency calculations summarized in Fig. 8.

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Table 4 Absolute efficiency of the 1st blaze order MBGs with different blazed angles. All the gratings are coated with W/B4C multilayers composed of 25 bi-layers with d-spacing of 2.99 nm and Γ-ratio of 0.5.

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Table 5 Parameters of the 200 nm blazed gratings with a blaze angle of 7.12° coated with multilayers having different refractive properties. The average refraction index of the multilayers reduces progressively for the gratings #1 to #4.

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Equations (9)

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

E M a y s t r e = R × min [ cos α / cos β , cos β / cos α ]
α = β + 2 φ
Δ = d sin φ / m ,
E = R 0 { cos α / cos β + A exp [ − ( d / L ) 2 ] }
α + θ = 90 ° + φ ,
2 Δ sin θ = n λ ( 1 + ω s ) ,
ω a = ω s ( b − 1 ) / ( 2 b ) ,
b = sin ( − θ ± φ ) / sin ( θ ± φ )
( λ / Δ λ ) a = | b | ( λ / Δ λ ) s

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