Analytic theory of alternate multilayer gratings operating in single-order regime

Xiaowei Yang; Igor V. Kozhevnikov; Qiushi Huang; Hongchang Wang; Matthew Hand; Kawal Sawhney; Zhanshan Wang

doi:10.1364/OE.25.015987

Analytic theory of alternate multilayer gratings operating in single-order regime

Xiaowei Yang, Igor V. Kozhevnikov, Qiushi Huang, Hongchang Wang, Matthew Hand, Kawal Sawhney, and Zhanshan Wang

Optics Express
Vol. 25,
Issue 14,
pp. 15987-16001
(2017)
•https://doi.org/10.1364/OE.25.015987

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Xiaowei Yang, Igor V. Kozhevnikov, Qiushi Huang, Hongchang Wang, Matthew Hand, Kawal Sawhney, and Zhanshan Wang, "Analytic theory of alternate multilayer gratings operating in single-order regime," Opt. Express 25, 15987-16001 (2017)

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Related Topics
Table of Contents Category
- Diffraction and Gratings
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Diffraction gratings (050.1950)
- Diffraction theory (050.1960)
- Multilayers (230.4170)
- X-rays, soft x-rays, extreme ultraviolet (EUV) (340.7480)

About this Article
History
- Original Manuscript: April 4, 2017
- Revised Manuscript: June 13, 2017
- Manuscript Accepted: June 23, 2017
- Published: June 28, 2017

Accessible

Open Access

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.

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References

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Publication

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[Crossref] [PubMed]
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[Crossref]
E. Spiller, “Low-Loss Reflection Coatings Using Absorbing Materials,” Appl. Phys. Lett. 20(9), 365–367 (1972).
[Crossref]
T. W. Barbee., “Multilayers For X-Ray Optics,” Opt. Eng. 25(8), 898–915 (1986).
[Crossref]
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]
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]
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]
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[Crossref] [PubMed]
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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. 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).
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[Crossref] [PubMed]
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[Crossref] [PubMed]
S. Bajt, H. N. Chapman, A. Aquila, and E. Gullikson, “High-efficiency x-ray gratings with asymmetric-cut multilayers,” J. Opt. Soc. Am. A 29(3), 216–230 (2012).
[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]
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, 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]
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]
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).
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www.pcgrate.com .
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]
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]
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]
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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]
I. V. Kozhevnikov and A. V. Vinogradov, “Basic Formulae of XUV Multilayer Optics,” Phys. Scr. T 17, 137–145 (1987).
[Crossref]
D. L. Windt, “IMD—Software for modeling the optical properties of multilayer films,” Comput. Phys. 12(4), 360–370 (1998).
[Crossref]

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]

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)

S. Bajt, H. N. Chapman, A. Aquila, and E. Gullikson, “High-efficiency x-ray gratings with asymmetric-cut multilayers,” J. Opt. Soc. Am. A 29(3), 216–230 (2012).
[Crossref] [PubMed]

2011 (2)

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]

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]

2010 (2)

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]

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]

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)

M. Ishino, P. A. Heimann, H. Sasai, M. Hatayama, H. Takenaka, K. Sano, E. M. Gullikson, and M. Koike, “Development of multilayer laminar-type diffraction gratings to achieve high diffraction efficiencies in the 1-8 keV energy region,” Appl. Opt. 45(26), 6741–6745 (2006).
[Crossref] [PubMed]

1998 (2)

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

1997 (2)

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]

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]

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)

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]

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.

S. Bajt, H. N. Chapman, A. Aquila, and E. Gullikson, “High-efficiency x-ray gratings with asymmetric-cut multilayers,” J. Opt. Soc. Am. A 29(3), 216–230 (2012).
[Crossref] [PubMed]

Bajt, S.

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]

S. Bajt, H. N. Chapman, A. Aquila, and E. Gullikson, “High-efficiency x-ray gratings with asymmetric-cut multilayers,” J. Opt. Soc. Am. A 29(3), 216–230 (2012).
[Crossref] [PubMed]

Barbee, T. W.

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

Bastiaens, B.

Bastiaens, H. M. J.

Bijkerk, F.

Bizeuil, C.

Boller, K.

Boller, K.-J.

Bridou, F.

Brons, C.

Cabrini, S.

Cambie, R.

Capitanio, B.

Cauchon, G.

Chapman, H. N.

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]

S. Bajt, H. N. Chapman, A. Aquila, and E. Gullikson, “High-efficiency x-ray gratings with asymmetric-cut multilayers,” J. Opt. Soc. Am. A 29(3), 216–230 (2012).
[Crossref] [PubMed]

Choueikani, F.

Cloup, A. L.

Cruddace, R. G.

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.

Dhez, P.

Dhuey, S. D.

Eggenstein, F.

Erko, A.

Gautier, J.

Goray, L. I.

Gullikson, E.

S. Bajt, H. N. Chapman, A. Aquila, and E. Gullikson, “High-efficiency x-ray gratings with asymmetric-cut multilayers,” J. Opt. Soc. Am. A 29(3), 216–230 (2012).
[Crossref] [PubMed]

Gullikson, E. M.

Gwalt, G.

Haase, A.

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]

Hand, M.

Hatayama, M.

Hegeman, P.

Heidemann, K. F.

Heimann, P. A.

Heinzmann, U.

Huang, Q.

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]

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.

Huskens, J.

Idir, M.

Ishino, M.

Jourdain, E.

Kaulich, B.

Kleineberg, U.

Koike, M.

Kowalski, M. P.

Kozhevnikov, I.

Kozhevnikov, I. V.

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]

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

Krishnan, B.

Kruijs, R.

Krumrey, M.

Kutz, O.

Lagarde, B.

Lemke, S.

Louis, E.

Medvedev, V.

Meltchakov, E.

Menke, D.

Mercere, P.

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.

Osterried, K.

Otero, E.

Packe, I.

Padmore, H. A.

Polack, F.

Prasciolu, M.

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]

Ravet-Krill, M.-F.

Rife, J. C.

Roulliay, M.

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.

Schäfers, F.

Scholze, F.

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]

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.

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.

Waberski, Ch.

Wang, H.

Wang, Z.

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]

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.

Yang, X.

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]

Yashchuk, V. V.

Zeldovich, B. Y.

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]

Zeschke, T.

AIP Adv. (1)

AIP Conf. Proc. (1)

Appl. Opt. (3)

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]

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)

Comput. Phys. (1)

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Fig. 1 Schematic of X-ray diffraction from an alternate multilayer grating.

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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).

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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).

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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).

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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).

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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).

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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) = \sum_{n = - \infty}^{+ \infty} U_{n} \exp (\frac{2 i π n x}{D}), U_{0} = Γ, U_{n \neq 0} = \frac{1 - \exp (- 2 i π n Γ)}{2 i π n} .

E (x, z) = \sum_{n = - \infty}^{+ \infty} 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,

\begin{array}{l} \frac{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)] \cdot \sum_{m \neq n} U_{n - m} F_{m} (z) \end{array}

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) = \sum_{j = - \infty}^{+ \infty} u_{j} e^{\frac{2 i π j z}{d}}, u_{0} = \bar{χ} = γ χ_{A} + (1 - γ) χ_{S}, u_{j \neq 0} = (χ_{A} - χ_{S}) \frac{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} = \sqrt{k^{2} - q_{n}^{2}}

\frac{d A_{n}}{d z} \exp (i κ_{n} z) + \frac{d C_{n}}{d z} \exp (- i κ_{n} z) = 0.

{\begin{cases} \frac{d A_{n} (z)}{d z} = - \frac{i k^{2}}{2 κ_{n}} \sum_{j}^{} [w_{j} e^{2 i π j z / d} (A_{n} + C_{n} e^{- 2 i κ_{n} z}) + \sum_{m \neq 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})] \\ \frac{d C_{n} (z)}{d z} = \frac{i k^{2}}{2 κ_{n}} \sum_{j}^{} [w_{j} e^{2 i π j z / d} (A_{n} e^{2 i κ_{n} z} + C_{n}) + \sum_{m \neq 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})] \end{cases}

v_{j} = u_{j} [1 - \exp (- \frac{2 i π j h}{d})], w_{j} = u_{j} - (​ 1 - Γ) v_{j} = u_{j} [Γ + (​ 1 - Γ) \exp (- \frac{2 i π j h}{d})] .

A_{n} (0) = δ_{n, 0}; C_{n} (L) = 0,

\sin^{2} θ_{n} > > \bar{χ} = \sin^{2} θ_{С}; n = 0, \pm 1, \dots

κ_{0} + κ_{n} \approx 2 π j / d, n \neq 0, i .e . \sin θ_{0} + \sin θ_{n} \approx j λ / d,

{\begin{cases} \frac{d A_{0} (z)}{d z} = - \frac{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}] \\ \frac{d C_{n} (z)}{d z} = \frac{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}] \end{cases}

R_{n} = {| \frac{U_{+} \tan h (S L)}{b \tan h (S L) - i \sqrt{U_{+} U_{-} - b^{2}}} |}^{2}, n \neq 0,

\begin{array}{l} S = \frac{k}{2 \sqrt{\sin θ_{0} \sin θ_{n}}} \sqrt{U_{+} U_{-} - b^{2}}; \\ b = \bar{χ} \frac{\sin θ_{0} + \sin θ_{n}}{2 \sqrt{\sin θ_{0} \sin θ_{n}}} - \sqrt{\sin θ_{0} \sin θ_{n}} (\sin θ_{0} + \sin θ_{n} - \frac{j λ}{d}); \\ U_{\pm} \equiv u_{\pm j} U_{\mp n} [1 - \exp (\mp 2 i π j \frac{h}{d})] \\ = (χ_{A} - χ_{S}) \frac{1 - \exp (\mp 2 i π j γ)}{2 π j} \frac{1 - \exp (\pm 2 i π n Γ)}{2 π n} [1 - \exp (\mp 2 i π j \frac{h}{d})] \end{array}

\begin{array}{l} \frac{j λ}{2 d} = \frac{\sin θ_{0} + \sin θ_{n}}{2} - \frac{Re \bar{χ} (\sin θ_{0} + \sin θ_{n})}{4 \sin θ_{0} \sin θ_{n}} \\ + \frac{Re (χ_{A} - χ_{S})}{\sin θ_{0} + \sin θ_{n}} \cdot \frac{Im (χ_{A} - χ_{S})}{Im \bar{χ}} \cdot \frac{\sin^{2} (π j γ)}{{(π j)}^{2}} \cdot \frac{\sin^{2} (π n Γ)}{{(π n)}^{2}} \cdot 4 \sin (π j \frac{h}{d}) \end{array}

R_{n} = \frac{1 - V}{1 + V}, V= \sqrt{\frac{1 - y^{2}}{1 + f^{2} y^{2}}}, n \neq 0,

\begin{array}{l} y = P_{1} \cdot P_{2} \cdot P_{3}; \\ P_{1} = \frac{\sin (π j γ)}{π j (γ + g)}, P_{2} = \frac{2 \sqrt{\sin θ_{0} \sin θ_{n}}}{\sin θ_{0} + \sin θ_{n}}, P_{3} = \frac{2}{π} \frac{\sin (π n Γ)}{n} \sin (π j \frac{h}{d}) \end{array}

f = \frac{Re (χ_{A} - χ_{S})}{Im (χ_{A} - χ_{S})}, g= \frac{Im χ_{S}}{Im (χ_{A} - χ_{S})} .

κ_{0} \approx π j / d, i .e . 2 \sin θ_{0} \approx j λ / d .

{\begin{cases} \frac{d A_{0} (z)}{d z} = - \frac{i k^{2}}{2 κ_{0}} [A_{0} w_{0} + w_{j} C_{0} e^{2 i (π j z / d - κ_{0}) z}] \\ \frac{d C_{0} (z)}{d z} = \frac{i k^{2}}{2 κ_{0}} [C_{0} w_{0} + w_{- j} A_{0} e^{- 2 i (π j z / d - κ_{0}) z}] \end{cases}

R_{0} = {| \frac{w_{j} \tan h (S L)}{b \tan h (S L) - i \sqrt{w_{j} w_{- j} - b^{2}}} |}^{2}; T_{0} = {| \frac{\sqrt{w_{j} w_{- j} - b^{2}}}{b \sin h (S L) - i \sqrt{w_{j} w_{- j} - b^{2}} \cos h (S L)} |}^{2},

b = \bar{χ} - \sin θ_{0} \cdot (2 \sin θ_{0} - \frac{j λ}{d}); S = \frac{k}{2 \sin θ_{0}} \sqrt{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 = \sum_{j = - \infty}^{+ \infty} w_{j} e^{\frac{2 i π j z}{d}}; w_{j} = \frac{u_{j}}{2} [1 + \exp (- \frac{2 i π j h}{d})],

T_{0} = {| \exp (- S L) |}^{2} = \exp (- \frac{k L}{\sin θ_{0}} Im \bar{χ}),

Δ θ = {(Δ θ)}_{M M} \cdot 2 \frac{\sin (π n Γ)}{π n} \sin (\frac{π j h}{d}) \cdot \frac{\sin (2 θ_{0})}{\sin (θ_{0} + θ_{n})} \sqrt{\frac{\sin θ_{n}}{\sin θ_{0}}},

2 j Γ D \cdot {(Δ θ)}_{M M} < < d,

Abstract

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