Abstract

Using the coupled wave approach (CWA), we introduce the analytical theory for alternate multilayer grating (AMG) operating in the single-order regime, in which only one diffraction order is excited. Differing from previous study analogizing AMG to crystals, we conclude that symmetrical structure, or equal thickness of the two multilayer materials, is not the optimal design for AMG and may result in significant reduction in diffraction efficiency. The peculiarities of AMG compared with other multilayer gratings are analyzed. An influence of multilayer structure materials on diffraction efficiency is considered. The validity conditions of analytical theory are also discussed.

© 2017 Optical Society of America

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References

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    [Crossref] [PubMed]
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2017 (2)

X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of a multilayer-based collimated plane-grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
[Crossref] [PubMed]

Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4(1), 011104 (2017).
[Crossref]

2016 (3)

2015 (3)

2014 (2)

2013 (4)

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

R. van der Meer, I. V. Kozhevnikov, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Extended theory of soft x-ray reflection for realistic lamellar multilayer gratings,” Opt. Express 21(11), 13105–13117 (2013).
[Crossref] [PubMed]

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

2012 (1)

2011 (2)

2010 (2)

2007 (1)

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

2006 (1)

1998 (2)

1997 (2)

1990 (1)

R. G. Cruddace, T. W. Barbee, J. C. Rife, and W. R. Hunter, “Measurements of the normal-incidence X-ray reflectance of a molybdenum-silicon multilayer deposited on a 2000 l/mm grating,” Phys. Scr. 41(4), 396–399 (1990).
[Crossref]

1987 (1)

I. V. Kozhevnikov and A. V. Vinogradov, “Basic Formulae of XUV Multilayer Optics,” Phys. Scr. T 17, 137–145 (1987).
[Crossref]

1986 (1)

T. W. Barbee., “Multilayers For X-Ray Optics,” Opt. Eng. 25(8), 898–915 (1986).
[Crossref]

1977 (1)

1972 (1)

E. Spiller, “Low-Loss Reflection Coatings Using Absorbing Materials,” Appl. Phys. Lett. 20(9), 365–367 (1972).
[Crossref]

Anderson, E. H.

Aquila, A.

Bajt, S.

Barbee, T. W.

R. G. Cruddace, T. W. Barbee, J. C. Rife, and W. R. Hunter, “Measurements of the normal-incidence X-ray reflectance of a molybdenum-silicon multilayer deposited on a 2000 l/mm grating,” Phys. Scr. 41(4), 396–399 (1990).
[Crossref]

T. W. Barbee., “Multilayers For X-Ray Optics,” Opt. Eng. 25(8), 898–915 (1986).
[Crossref]

Bastiaens, B.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Bastiaens, H. M. J.

Bijkerk, F.

Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4(1), 011104 (2017).
[Crossref]

F. Senf, F. Bijkerk, F. Eggenstein, G. Gwalt, Q. Huang, R. Kruijs, O. Kutz, S. Lemke, E. Louis, M. Mertin, I. Packe, I. Rudolph, F. Schäfers, F. Siewert, A. Sokolov, J. M. Sturm, Ch. Waberski, Z. Wang, J. Wolf, T. Zeschke, and A. Erko, “Highly efficient blazed grating with multilayer coating for tender X-ray energies,” Opt. Express 24(12), 13220–13230 (2016).
[Crossref] [PubMed]

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

R. van der Meer, I. V. Kozhevnikov, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Extended theory of soft x-ray reflection for realistic lamellar multilayer gratings,” Opt. Express 21(11), 13105–13117 (2013).
[Crossref] [PubMed]

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]

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express 18(15), 16234–16242 (2010).
[Crossref] [PubMed]

Bizeuil, C.

Boller, K.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Boller, K.-J.

Bridou, F.

F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B4C/Mo2C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. 39(7), 2141–2144 (2014).
[Crossref] [PubMed]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

Brons, C.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Cabrini, S.

Cambie, R.

Capitanio, B.

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

Cauchon, G.

Chapman, H. N.

Choueikani, F.

F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B4C/Mo2C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. 39(7), 2141–2144 (2014).
[Crossref] [PubMed]

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

Cloup, A. L.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Cruddace, R. G.

M. P. Kowalski, R. G. Cruddace, J. F. Seely, J. C. Rife, K. F. Heidemann, U. Heinzmann, U. Kleineberg, K. Osterried, D. Menke, and W. R. Hunter, “Efficiency of a multilayer-coated, ion-etched laminar holographic grating in the 14.5 16.0-nm wavelength region,” Opt. Lett. 22(11), 834–836 (1997).
[Crossref] [PubMed]

R. G. Cruddace, T. W. Barbee, J. C. Rife, and W. R. Hunter, “Measurements of the normal-incidence X-ray reflectance of a molybdenum-silicon multilayer deposited on a 2000 l/mm grating,” Phys. Scr. 41(4), 396–399 (1990).
[Crossref]

Cunsolo, A.

Y. V. Shvyd’ko, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6(3), 196–199 (2010).
[Crossref]

Delcamp, E.

Delmotte, F.

F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B4C/Mo2C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. 39(7), 2141–2144 (2014).
[Crossref] [PubMed]

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Dhez, P.

Dhuey, S. D.

Eggenstein, F.

Erko, A.

Gautier, J.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Goray, L. I.

Gullikson, E.

Gullikson, E. M.

Gwalt, G.

Haase, A.

Hand, M.

X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of a multilayer-based collimated plane-grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
[Crossref] [PubMed]

Hatayama, M.

Hegeman, P.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Heidemann, K. F.

Heimann, P. A.

Heinzmann, U.

Huang, Q.

Huang, X.

Y. V. Shvyd’ko, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6(3), 196–199 (2010).
[Crossref]

Hunter, W. R.

M. P. Kowalski, R. G. Cruddace, J. F. Seely, J. C. Rife, K. F. Heidemann, U. Heinzmann, U. Kleineberg, K. Osterried, D. Menke, and W. R. Hunter, “Efficiency of a multilayer-coated, ion-etched laminar holographic grating in the 14.5 16.0-nm wavelength region,” Opt. Lett. 22(11), 834–836 (1997).
[Crossref] [PubMed]

R. G. Cruddace, T. W. Barbee, J. C. Rife, and W. R. Hunter, “Measurements of the normal-incidence X-ray reflectance of a molybdenum-silicon multilayer deposited on a 2000 l/mm grating,” Phys. Scr. 41(4), 396–399 (1990).
[Crossref]

Huskens, J.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Idir, M.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

A. Mirone, E. Delcamp, M. Idir, G. Cauchon, F. Polack, P. Dhez, and C. Bizeuil, “Numerical and experimental study of grating efficiency for synchrotron monochromators,” Appl. Opt. 37(25), 5816–5822 (1998).
[Crossref] [PubMed]

Ishino, M.

Jourdain, E.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Kaulich, B.

X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of a multilayer-based collimated plane-grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
[Crossref] [PubMed]

Kleineberg, U.

Koike, M.

Kowalski, M. P.

Kozhevnikov, I.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Kozhevnikov, I. V.

X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of a multilayer-based collimated plane-grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
[Crossref] [PubMed]

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, K. Sawhney, and Z. Wang, “Wideband multilayer gratings for the 17-25 nm spectral region,” Opt. Express 24(13), 15079–15092 (2016).
[Crossref] [PubMed]

X. Yang, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Unified analytical theory of single-order soft x-ray multilayer gratings,” J. Opt. Soc. Am. B 32(4), 506–522 (2015).
[Crossref]

R. van der Meer, I. V. Kozhevnikov, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Extended theory of soft x-ray reflection for realistic lamellar multilayer gratings,” Opt. Express 21(11), 13105–13117 (2013).
[Crossref] [PubMed]

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]

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express 18(15), 16234–16242 (2010).
[Crossref] [PubMed]

I. V. Kozhevnikov and A. V. Vinogradov, “Basic Formulae of XUV Multilayer Optics,” Phys. Scr. T 17, 137–145 (1987).
[Crossref]

Krishnan, B.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Kruijs, R.

Krumrey, M.

F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B4C/Mo2C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. 39(7), 2141–2144 (2014).
[Crossref] [PubMed]

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

Kutz, O.

Lagarde, B.

F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B4C/Mo2C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. 39(7), 2141–2144 (2014).
[Crossref] [PubMed]

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Lemke, S.

Louis, E.

Medvedev, V.

Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4(1), 011104 (2017).
[Crossref]

Meltchakov, E.

F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B4C/Mo2C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. 39(7), 2141–2144 (2014).
[Crossref] [PubMed]

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

Menke, D.

Mercere, P.

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

Mertin, M.

Meyer-Ilse, J.

Mills, D. M.

D. M. Mills, “X-ray Optics Developments at the APS for the Third Generation of High-Energy Synchrotron Radiation Sources,” J. Synchrotron Radiat. 4(3), 117–124 (1997).
[Crossref] [PubMed]

Mirone, A.

Ohresser, P.

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

Osterried, K.

Otero, E.

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

Packe, I.

Padmore, H. A.

Polack, F.

F. Choueikani, B. Lagarde, F. Delmotte, M. Krumrey, F. Bridou, M. Thomasset, E. Meltchakov, and F. Polack, “High-efficiency B4C/Mo2C alternate multilayer grating for monochromators in the photon energy range from 0.7 to 3.4 keV,” Opt. Lett. 39(7), 2141–2144 (2014).
[Crossref] [PubMed]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

A. Mirone, E. Delcamp, M. Idir, G. Cauchon, F. Polack, P. Dhez, and C. Bizeuil, “Numerical and experimental study of grating efficiency for synchrotron monochromators,” Appl. Opt. 37(25), 5816–5822 (1998).
[Crossref] [PubMed]

Prasciolu, M.

Ravet-Krill, M.-F.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Rife, J. C.

M. P. Kowalski, R. G. Cruddace, J. F. Seely, J. C. Rife, K. F. Heidemann, U. Heinzmann, U. Kleineberg, K. Osterried, D. Menke, and W. R. Hunter, “Efficiency of a multilayer-coated, ion-etched laminar holographic grating in the 14.5 16.0-nm wavelength region,” Opt. Lett. 22(11), 834–836 (1997).
[Crossref] [PubMed]

R. G. Cruddace, T. W. Barbee, J. C. Rife, and W. R. Hunter, “Measurements of the normal-incidence X-ray reflectance of a molybdenum-silicon multilayer deposited on a 2000 l/mm grating,” Phys. Scr. 41(4), 396–399 (1990).
[Crossref]

Roulliay, M.

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Rudolph, I.

Said, A. H.

Y. V. Shvyd’ko, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6(3), 196–199 (2010).
[Crossref]

Salmassi, F.

Sano, K.

Sasai, H.

Sawhney, K.

X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of a multilayer-based collimated plane-grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
[Crossref] [PubMed]

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, K. Sawhney, and Z. Wang, “Wideband multilayer gratings for the 17-25 nm spectral region,” Opt. Express 24(13), 15079–15092 (2016).
[Crossref] [PubMed]

Schäfers, F.

Scholze, F.

Seely, J. F.

Senf, F.

Shvyd’ko, Y. V.

Y. V. Shvyd’ko, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6(3), 196–199 (2010).
[Crossref]

Siewert, F.

Sokolov, A.

Spiller, E.

E. Spiller, “Low-Loss Reflection Coatings Using Absorbing Materials,” Appl. Phys. Lett. 20(9), 365–367 (1972).
[Crossref]

Stoupin, S.

Y. V. Shvyd’ko, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6(3), 196–199 (2010).
[Crossref]

Sturm, J. M.

Takenaka, H.

Thomasset, M.

van de Kruijs, R.

Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4(1), 011104 (2017).
[Crossref]

van der Meer, R.

Vinogradov, A. V.

I. V. Kozhevnikov and A. V. Vinogradov, “Basic Formulae of XUV Multilayer Optics,” Phys. Scr. T 17, 137–145 (1987).
[Crossref]

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

Voronov, D. L.

Vratzov, B.

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

Waberski, Ch.

Wang, H.

X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of a multilayer-based collimated plane-grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
[Crossref] [PubMed]

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, K. Sawhney, and Z. Wang, “Wideband multilayer gratings for the 17-25 nm spectral region,” Opt. Express 24(13), 15079–15092 (2016).
[Crossref] [PubMed]

Wang, Z.

Warwick, T.

Windt, D. L.

D. L. Windt, “IMD—Software for modeling the optical properties of multilayer films,” Comput. Phys. 12(4), 360–370 (1998).
[Crossref]

Wolf, J.

Yakshin, A.

Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4(1), 011104 (2017).
[Crossref]

Yang, X.

Yashchuk, V. V.

Zeldovich, B. Y.

Zeschke, T.

AIP Adv. (1)

R. van der Meer, I. Kozhevnikov, B. Krishnan, J. Huskens, P. Hegeman, C. Brons, B. Vratzov, B. Bastiaens, K. Boller, and F. Bijkerk, “Single-order operation of lamellar multilayer gratings in the soft x-ray spectral range,” AIP Adv. 3(1), 012103 (2013).
[Crossref]

AIP Conf. Proc. (1)

F. Polack, B. Lagarde, M. Idir, A. L. Cloup, E. Jourdain, M. Roulliay, F. Delmotte, J. Gautier, and M.-F. Ravet-Krill, “Alternate Multilayer Gratings with Enhanced Diffraction Efficiency in the 500–5000 eV Energy Domain,” AIP Conf. Proc. 879, 489–492 (2007).
[Crossref]

Appl. Opt. (3)

Appl. Phys. Lett. (1)

E. Spiller, “Low-Loss Reflection Coatings Using Absorbing Materials,” Appl. Phys. Lett. 20(9), 365–367 (1972).
[Crossref]

Appl. Phys. Rev. (1)

Q. Huang, V. Medvedev, R. van de Kruijs, A. Yakshin, E. Louis, and F. Bijkerk, “Spectral tailoring of nanoscale EUV and soft x-ray multilayer optics,” Appl. Phys. Rev. 4(1), 011104 (2017).
[Crossref]

Comput. Phys. (1)

D. L. Windt, “IMD—Software for modeling the optical properties of multilayer films,” Comput. Phys. 12(4), 360–370 (1998).
[Crossref]

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

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

J. Phys. Conf. Ser. (2)

B. Lagarde, F. Choueikani, B. Capitanio, P. Ohresser, E. Meltchakov, F. Delmotte, M. Krumrey, and F. Polack, “High efficiency multilayer gratings for monochromators in the energy range from 500 eV to 2500 eV,” J. Phys. Conf. Ser. 425(15), 152012 (2013).
[Crossref]

F. Choueikani, F. Delmotte, F. Bridou, B. Lagarde, P. Mercere, E. Otero, P. Ohresser, and F. Polack, “Preparation for B4C/Mo2C multilayer deposition of alternate multilayer gratings with high efficiency in the 0.5-2.5 keV energy range,” J. Phys. Conf. Ser. 425(15), 152007 (2013).
[Crossref]

J. Synchrotron Radiat. (2)

D. M. Mills, “X-ray Optics Developments at the APS for the Third Generation of High-Energy Synchrotron Radiation Sources,” J. Synchrotron Radiat. 4(3), 117–124 (1997).
[Crossref] [PubMed]

X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of a multilayer-based collimated plane-grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
[Crossref] [PubMed]

Nat. Phys. (1)

Y. V. Shvyd’ko, S. Stoupin, A. Cunsolo, A. H. Said, and X. Huang, “High-reflectivity high-resolution X-ray crystal optics with diamonds,” Nat. Phys. 6(3), 196–199 (2010).
[Crossref]

Opt. Eng. (1)

T. W. Barbee., “Multilayers For X-Ray Optics,” Opt. Eng. 25(8), 898–915 (1986).
[Crossref]

Opt. Express (9)

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, L. I. Goray, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “High-order multilayer coated blazed gratings for high resolution soft x-ray spectroscopy,” Opt. Express 23(4), 4771–4790 (2015).
[Crossref] [PubMed]

M. Prasciolu, A. Haase, F. Scholze, H. N. Chapman, and S. Bajt, “Extended asymmetric-cut multilayer X-ray gratings,” Opt. Express 23(12), 15195–15204 (2015).
[Crossref] [PubMed]

D. L. Voronov, F. Salmassi, J. Meyer-Ilse, E. M. Gullikson, T. Warwick, and H. A. Padmore, “Refraction effects in soft x-ray multilayer blazed gratings,” Opt. Express 24(11), 11334–11344 (2016).
[Crossref] [PubMed]

F. Senf, F. Bijkerk, F. Eggenstein, G. Gwalt, Q. Huang, R. Kruijs, O. Kutz, S. Lemke, E. Louis, M. Mertin, I. Packe, I. Rudolph, F. Schäfers, F. Siewert, A. Sokolov, J. M. Sturm, Ch. Waberski, Z. Wang, J. Wolf, T. Zeschke, and A. Erko, “Highly efficient blazed grating with multilayer coating for tender X-ray energies,” Opt. Express 24(12), 13220–13230 (2016).
[Crossref] [PubMed]

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, K. Sawhney, and Z. Wang, “Wideband multilayer gratings for the 17-25 nm spectral region,” Opt. Express 24(13), 15079–15092 (2016).
[Crossref] [PubMed]

I. V. Kozhevnikov, R. van der Meer, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “High-resolution, high-reflectivity operation of lamellar multilayer amplitude gratings: identification of the single-order regime,” Opt. Express 18(15), 16234–16242 (2010).
[Crossref] [PubMed]

R. van der Meer, I. V. Kozhevnikov, H. M. J. Bastiaens, K.-J. Boller, and F. Bijkerk, “Extended theory of soft x-ray reflection for realistic lamellar multilayer gratings,” Opt. Express 21(11), 13105–13117 (2013).
[Crossref] [PubMed]

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]

Opt. Lett. (3)

Phys. Scr. (1)

R. G. Cruddace, T. W. Barbee, J. C. Rife, and W. R. Hunter, “Measurements of the normal-incidence X-ray reflectance of a molybdenum-silicon multilayer deposited on a 2000 l/mm grating,” Phys. Scr. 41(4), 396–399 (1990).
[Crossref]

Phys. Scr. T (1)

I. V. Kozhevnikov and A. V. Vinogradov, “Basic Formulae of XUV Multilayer Optics,” Phys. Scr. T 17, 137–145 (1987).
[Crossref]

Other (2)

www.pcgrate.com .

Electromagnetic Theory of Gratings, (Springer, 1980), R.Petit, Ed.

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Figures (6)

Fig. 1
Fig. 1 Schematic of X-ray diffraction from an alternate multilayer grating.
Fig. 2
Fig. 2 Diffraction efficiency of different orders (from −3rd to + 3rd) at 278 eV photon energy (s-polarized radiation case) versus grazing incidence angle for Cr/C AMG with different Г ratio: Г = 1/2 for (a) and Г = 0.4 for (b). Other geometrical parameters are: D = 300 nm, d = 5 nm, γ = 0.4, N = 150, h = d/2. Results shown in black dashed curves were obtained by numerical calculations basing on the CWA method. The colored solid lines represent analytical results obtained from Eq. (15).
Fig. 3
Fig. 3 −1st order diffraction efficiency of Cr/C AMG and BMG and reflectivity of corresponding MM versus grazing incidence angle at 3keV photon energy (s-polarized radiation case). The geometrical grating parameters are: D = 300 nm, d = 5 nm, γ = 0.4 N = 100. For AMG, Г = 1/2, h = d/2, and for BMG, the blaze angle is 0.95°. Results shown in black dashed curves were obtained by numerical calculations basing on the CWA method. The colored solid lines represent analytical results obtained with Eq. (15).
Fig. 4
Fig. 4 −1st order diffraction efficiency of W/C AMG and BMG and reflectivity of corresponding MM versus grazing incidence angle at 278 eV photon energy (s-polarized radiation case). The γ ratio is equal to 0.15 (a) or 0.5 (b). Other grating geometrical parameters are: D = 300 nm, N = 200, d = 5 nm. For AMG, Г = 1/2, h = d/2, and for BMG, the blaze angle is 0.95°. Results shown in black dashed curves were obtained by numerical calculations basing on the CWA method. The colored solid lines represent analytical results obtained with Eq. (15).
Fig. 5
Fig. 5 Diffraction efficiency of different orders (from −4th to + 4th) at 278 eV photon energy versus grazing incidence angle for Cr/C AMG (s-polarized radiation case). Geometrical parameters are: D = 1200 nm, Г = 0.4, d = 5 nm, γ = 0.4, N = 150, h = d/2. Results shown in solid curves were obtained by numerical calculations basing on the CWA method. The black dashed curves represent the analytical calculations of ± 1st diffraction order efficiency obtained with Eq. (15).
Fig. 6
Fig. 6 Diffraction efficiency (0th and −1st order) of Cr/C AMG (a) and reflectivity of corresponding MM (b) at 200 eV photon energy versus grazing incidence angle (s-polarized radiation case). Geometrical parameters are: D = 300 nm, Г = 1/2, h = d/2, d = 15 nm, γ = 0.4, N = 40. Curves 1 were calculated with analytical formulae. Curves 2, 3 were calculated by numerical integration of Eqs. (4)-(5) taking into account 21 diffraction orders (−10th to + 10th) (graph a) or with the IMD software [32] (graph b). The uppermost layer of multilayer structure was Cr (curves 2) or C (curves 3).

Equations (31)

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1 ε ( x , z ) = χ ( x , z ) = χ M L ( z ) U ( x ) + χ M L ( z h ) [ 1 U ( x ) ] ,
U ( x ) = 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 ) = n = + F n ( z ) exp ( i q n x ) , q 0 = k cos θ 0 , q n = q 0 + 2 π n / D = k cos θ n , k = 2 π / λ ,
d 2 F n ( z ) d z 2 + κ n 2 F n ( z ) = = k 2 [ Γ χ M L ( z ) + ( 1 Γ ) χ M L ( z h ) ] F n ( z ) + k 2 [ χ M L ( z ) χ M L ( z h ) ] m n 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 + h ) i κ n ( s ) F n ( L + h ) = 0 ,
R n = | F n ( 0 ) δ n , 0 | 2 Re ( κ n / κ 0 ) .
F 0 + k 2 [ sin 2 θ 0 χ M L ( z ) ] F 0 = 0.
χ M L ( z ) = j = + u j e 2 i π j z d , u 0 = χ ¯ = γ χ A + ( 1 γ ) χ S , u j 0 = ( χ A χ S ) 1 exp ( 2 i π j γ ) 2 i π j ,
F n ( z ) = A n ( z ) exp ( i κ n z ) + C n ( z ) exp ( i κ n z ) , κ n = k 2 q n 2
d A n d z exp ( i κ n z ) + d C n d z exp ( i κ n z ) = 0.
{ d A n ( z ) d z = i k 2 2 κ n j [ w j e 2 i π j z / d ( A n + C n e 2 i κ n z ) + m n U n m v j e 2 i π j z / d ( A m e i ( κ m κ n ) z + C m e i ( κ m + κ n ) z ) ] d C n ( z ) d z = i k 2 2 κ n j [ w j e 2 i π j z / d ( A n e 2 i κ n z + C n ) + m n U n m v j e 2 i π j z / d ( A m e i ( κ m + κ n ) z + C m e i ( κ n κ m ) z ) ]
v j = u j [ 1 exp ( 2 i π j h d ) ] , w j = u j ( 1 Γ ) v j = u j [ Γ + ( 1 Γ ) exp ( 2 i π j h d ) ] .
A n ( 0 ) = δ n , 0 ; C n ( L ) = 0 ,
sin 2 θ n > > χ ¯ = sin 2 θ С ; n = 0 , ± 1 ,
κ 0 + κ n 2 π j / d , n 0 , i .e . sin θ 0 + sin θ n j λ / d ,
{ d A 0 ( z ) d z = i k 2 2 κ 0 [ A 0 w 0 + v j U n C n e 2 i π j z / d i ( κ 0 + κ n ) z ] d C n ( z ) d z = i k 2 2 κ n [ C n w 0 + v j U n A 0 e 2 i π j z / d + i ( κ 0 + κ n ) z ]
R n = | U + tan h ( S L ) b tan h ( S L ) i U + U b 2 | 2 , n 0 ,
S = k 2 sin θ 0 sin θ n U + U b 2 ; b = χ ¯ sin θ 0 + sin θ n 2 sin θ 0 sin θ n sin θ 0 sin θ n ( sin θ 0 + sin θ n j λ d ) ; U ± u ± j U n [ 1 exp ( 2 i π j h d ) ] = ( χ A χ S ) 1 exp ( 2 i π j γ ) 2 π j 1 exp ( ± 2 i π n Γ ) 2 π n [ 1 exp ( 2 i π j h d ) ]
j λ 2 d = sin θ 0 + sin θ n 2 Re χ ¯ ( sin θ 0 + sin θ n ) 4 sin θ 0 sin θ n + Re ( χ A χ S ) sin θ 0 + sin θ n Im ( χ A χ S ) Im χ ¯ sin 2 ( π j γ ) ( π j ) 2 sin 2 ( π n Γ ) ( π n ) 2 4 sin ( π j h d )
R n = 1 V 1 + V , V= 1 y 2 1 + f 2 y 2 , n 0 ,
y = P 1 P 2 P 3 ; P 1 = sin ( π j γ ) π j ( γ + g ) , P 2 = 2 sin θ 0 sin θ n sin θ 0 + sin θ n , P 3 = 2 π sin ( π n Γ ) n sin ( π j h d )
f = Re ( χ A χ S ) Im ( χ A χ S ) , g= Im χ S Im ( χ A χ S ) .
κ 0 π j / d , i .e . 2 sin θ 0 j λ / d .
{ d A 0 ( z ) d z = i k 2 2 κ 0 [ A 0 w 0 + w j C 0 e 2 i ( π j z / d κ 0 ) z ] d C 0 ( z ) d z = i k 2 2 κ 0 [ C 0 w 0 + w j A 0 e 2 i ( π j z / d κ 0 ) z ]
R 0 = | w j tan h ( S L ) b tan h ( S L ) i w j w j b 2 | 2 ; T 0 = | w j w j b 2 b sin h ( S L ) i w j w j b 2 cos h ( S L ) | 2 ,
b = χ ¯ sin θ 0 ( 2 sin θ 0 j λ d ) ; S = k 2 sin θ 0 w j w j b 2 .
Γ = 1 / 2 and j h = d / 2
χ e f f ( z ) = [ χ M L ( z ) + χ M L ( z h ) ] / 2 = j = + w j e 2 i π j z d ; w j = u j 2 [ 1 + exp ( 2 i π j h d ) ] ,
T 0 = | exp ( S L ) | 2 = exp ( k L sin θ 0 Im χ ¯ ) ,
Δ θ = ( Δ θ ) M M 2 sin ( π n Γ ) π n sin ( π j h d ) sin ( 2 θ 0 ) sin ( θ 0 + θ n ) sin θ n sin θ 0 ,
2 j Γ D ( Δ θ ) M M < < d ,

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