Abstract

The problem of X-ray diffraction from multilayer-coated blazed diffraction gratings is analyzed. Invalidity of the conventional condition of maximal diffraction efficiency observed in previous experiments is explained theoretically. This is attributed to two factors: contribution of anti-blaze facets to diffraction efficiency and effect of strongly asymmetric diffraction. We demonstrate that a proper choice of the multilayer d-spacing allows to design grating with the diffraction efficiency close to the maximal possible one throughout the tender X-ray range (E∼1-5 keV). An optimization procedure is suggested for the first time to choose the optimal grating parameters and the operation diffraction order to obtain a high fix-focus constant and high diffraction efficiency simultaneously in a wide spectral range.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (2)

A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express 27(12), 16833–16846 (2019).
[Crossref]

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

2018 (2)

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

L. Goray, W. Jark, and D. Eichert, “Rigorous calculations and synchrotron radiation measurements of diffraction efficiencies for tender X-ray lamellar gratings: conical versus classical diffraction,” J. Synchrotron Radiat. 25(6), 1683–1693 (2018).
[Crossref]

2017 (2)

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

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express 25(14), 15987–16001 (2017).
[Crossref]

2016 (2)

2015 (4)

2014 (2)

2013 (2)

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
[Crossref]

2012 (1)

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

2011 (1)

2010 (2)

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

I. V. Kozhevnikov, R. van der Meer, H. J. M. 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]

2005 (2)

L. I. Gorai, “Scalar and electromagnetic properties of X-ray diffraction gratings,” Bull. Russ. Acad. Sci.: Phys. 69(2), 231–236 (2005).

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

1997 (1)

R. Follath and F. Senf, “New plane-grating monochromators for third generation synchrotron radiation light sources,” Nucl. Instrum. Methods Phys. Res., Sect. A 390(3), 388–394 (1997).
[Crossref]

1995 (2)

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft x-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

J. H. Underwood, C. K. 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]

1989 (1)

1987 (1)

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987).
[Crossref]

1985 (1)

S. V. Gaponov, V. M. Genkin, N. N. Salashchenko, and A. A. Fraerman, “Scattering of neutrons and x-radiation in the range 10-300 Å by periodic structures with rough boundaries,” JETP Lett. 41(2), 63–65 (1985).

Alimov, S.

Anderson, E. H.

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
[Crossref]

Arnold, T.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Bajt, S.

Barbee, T. W.

Bastiaens, H. J. M.

Betemps, R.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Bijkerk, F.

Boller, K.-J.

Braun, S.

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

Bridou, F.

Buchheim, J.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Cambie, R.

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

Chapman, H. N.

Chen, Q.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Choueikani, F.

Cruddace, R. G.

Delmotte, F.

Dhuey, S.

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

Eggenstein, F.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[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, C. 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]

Eichert, D.

L. Goray, W. Jark, and D. Eichert, “Rigorous calculations and synchrotron radiation measurements of diffraction efficiencies for tender X-ray lamellar gratings: conical versus classical diffraction,” J. Synchrotron Radiat. 25(6), 1683–1693 (2018).
[Crossref]

Erko, A.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[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, C. 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]

Facsko, S.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Feng, J.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express 27(12), 16833–16846 (2019).
[Crossref]

Firsov, .

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Flechsig, U.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Follath, R.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
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R. Follath and F. Senf, “New plane-grating monochromators for third generation synchrotron radiation light sources,” Nucl. Instrum. Methods Phys. Res., Sect. A 390(3), 388–394 (1997).
[Crossref]

Fraerman, A. A.

S. V. Gaponov, V. M. Genkin, N. N. Salashchenko, and A. A. Fraerman, “Scattering of neutrons and x-radiation in the range 10-300 Å by periodic structures with rough boundaries,” JETP Lett. 41(2), 63–65 (1985).

Frost, F.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Gaponov, S. V.

S. V. Gaponov, V. M. Genkin, N. N. Salashchenko, and A. A. Fraerman, “Scattering of neutrons and x-radiation in the range 10-300 Å by periodic structures with rough boundaries,” JETP Lett. 41(2), 63–65 (1985).

Gawlitza, P.

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

Genkin, V. M.

S. V. Gaponov, V. M. Genkin, N. N. Salashchenko, and A. A. Fraerman, “Scattering of neutrons and x-radiation in the range 10-300 Å by periodic structures with rough boundaries,” JETP Lett. 41(2), 63–65 (1985).

Giglia, A.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Gorai, L. I.

L. I. Gorai, “Scalar and electromagnetic properties of X-ray diffraction gratings,” Bull. Russ. Acad. Sci.: Phys. 69(2), 231–236 (2005).

Goray, L.

L. Goray, W. Jark, and D. Eichert, “Rigorous calculations and synchrotron radiation measurements of diffraction efficiencies for tender X-ray lamellar gratings: conical versus classical diffraction,” J. Synchrotron Radiat. 25(6), 1683–1693 (2018).
[Crossref]

Goray, L. I.

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]

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

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

Grenzer, J.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Grioni, M.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Gudat, W.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft x-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Gullikson, E. M.

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]

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]

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
[Crossref]

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

J. H. Underwood, C. K. 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]

Gwalt, G.

Haase, A.

Hand, M.

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express 25(14), 15987–16001 (2017).
[Crossref]

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

Hellwig, C.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft x-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Huang, H.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Huang, K.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Huang, Q.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express 27(12), 16833–16846 (2019).
[Crossref]

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

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express 25(14), 15987–16001 (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, C. 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]

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

Hunter, W. R.

Imhof, A.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Jark, W.

L. Goray, W. Jark, and D. Eichert, “Rigorous calculations and synchrotron radiation measurements of diffraction efficiencies for tender X-ray lamellar gratings: conical versus classical diffraction,” J. Synchrotron Radiat. 25(6), 1683–1693 (2018).
[Crossref]

Jia, Q.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Jonnard, P.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Jung, C.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft x-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Kaulich, B.

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

Knedel, J.

Kozhevnikov, I. V.

A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express 27(12), 16833–16846 (2019).
[Crossref]

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

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express 25(14), 15987–16001 (2017).
[Crossref]

M. Wen, I. V. Kozhevnikov, and Z. Wang, “Reflection of X-rays from a rough surface at extremely small grazing angles,” Opt. Express 23(19), 24220–24235 (2015).
[Crossref]

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

I. V. Kozhevnikov, R. van der Meer, H. J. M. Bastiaens, K.-J. Boller, and F. Bijkerk, “Analytic theory of soft X-rays diffraction by lamellar multilayer gratings,” Opt. Express 19(10), 9172–9184 (2011).
[Crossref]

I. V. Kozhevnikov, R. van der Meer, H. J. M. 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]

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of XUV multilayer optics,” Phys. Scr. T17, 137–145 (1987).
[Crossref]

Krempasky, J.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Kruijs, R.

Krumrey, M.

Kutz, O.

Lagarde, B.

Lemke, S.

Lemke, St.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Li, W.

Lin, J.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Liu, Z.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Löchel, B.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Louis, E.

Malek, C. K.

J. H. Underwood, C. K. 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]

Maystre, D.

D. Maystre, M. Neviere, and R. Petit, Experimental Verifications and Applications of the Theory, in Electromagnetic Theory of Gratings, edited by R. Petit, Springer: Berlin, 1980.

Meltchakov, E.

Mertin, M.

Meyer-Ilse, J.

Nelles, B.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Neviere, M.

D. Maystre, M. Neviere, and R. Petit, Experimental Verifications and Applications of the Theory, in Electromagnetic Theory of Gratings, edited by R. Petit, Springer: Berlin, 1980.

Ou, X.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Packe, I.

Padmore, H. A.

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]

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]

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]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

Patthey, L.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Peatman, W. B.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft x-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Petersen, H.

H. Petersen, C. Jung, C. Hellwig, W. B. Peatman, and W. Gudat, “Review of plane grating focusing for soft x-ray monochromators,” Rev. Sci. Instrum. 66(1), 1–14 (1995).
[Crossref]

Petit, R.

D. Maystre, M. Neviere, and R. Petit, Experimental Verifications and Applications of the Theory, in Electromagnetic Theory of Gratings, edited by R. Petit, Springer: Berlin, 1980.

Piazzalunga, A.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Pietag, F.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Polack, F.

Prasciolu, M.

Qi, R.

Raabe, J.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
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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).
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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).
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D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
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D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
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X. Yang, H. Wang, M. Hand, K. Sawhney, B. Kaulich, I. V. Kozhevnikov, Q. Huang, and Z. Wang, “Design of the multilayer based collimated plane grating monochromator for tender X-ray range,” J. Synchrotron Radiat. 24(1), 168–174 (2017).
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Schäfers, F.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
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V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
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V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
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A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express 27(12), 16833–16846 (2019).
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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, C. 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).
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Thomasset, M.

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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]

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]

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]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
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Waberski, C.

Waberski, Ch.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Wang, H.

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

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express 25(14), 15987–16001 (2017).
[Crossref]

Wang, X.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Wang, Z.

A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express 27(12), 16833–16846 (2019).
[Crossref]

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

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

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express 25(14), 15987–16001 (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, C. 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]

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

M. Wen, I. V. Kozhevnikov, and Z. Wang, “Reflection of X-rays from a rough surface at extremely small grazing angles,” Opt. Express 23(19), 24220–24235 (2015).
[Crossref]

Warwick, T.

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]

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]

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]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
[Crossref]

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

Wen, M.

Wolf, J.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[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, C. 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]

Wu, M.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Yang, X.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

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

X. Yang, I. V. Kozhevnikov, Q. Huang, H. Wang, M. Hand, K. Sawney, and Z. Wang, “Analytic theory of alternate multilayer gratings operating in single-order regime,” Opt. Express 25(14), 15987–16001 (2017).
[Crossref]

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

Yashchuk, V. V.

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]

D. L. Voronov, E. H. Anderson, R. Cambie, P. Gawlitza, L. I. Goray, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Development of near atomically perfect diffraction gratings for EUV and soft x-rays with very high efficiency and resolving power,” J. Phys.: Conf. Ser. 425(15), 152006 (2013).
[Crossref]

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
[Crossref]

D. L. Voronov, P. Gawlitza, R. Cambie, S. Dhuey, E. M. Gullikson, T. Warwick, S. Braun, V. V. Yashchuk, and H. A. Padmore, “Conformal growth of Mo/Si multilayers on grating substrates using collimated ion beam sputtering,” J. Appl. Phys. 111(9), 093521 (2012).
[Crossref]

You, T.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Yu, W.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Zeschke, T.

Zhang, S.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Zhang, Z.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

A. Sokolov, Q. Huang, F. Senf, J. Feng, S. Lemke, S. Alimov, J. Knedel, T. Zeschke, O. Kutz, T. Seliger, G. Gwalt, F. Schafers, F. Siewert, I. V. Kozhevnikov, R. Qi, Z. Zhang, W. Li, and Z. Wang, “Optimized highly-efficient multilayer-coated blazed gratings for the tender X-ray region,” Opt. Express 27(12), 16833–16846 (2019).
[Crossref]

Zhou, H.

Q. Huang, Q. Jia, J. Feng, H. Huang, X. Yang, J. Grenzer, K. Huang, S. Zhang, J. Lin, H. Zhou, T. You, W. Yu, S. Facsko, P. Jonnard, M. Wu, A. Giglia, Z. Zhang, Z. Liu, Z. Wang, X. Wang, and X. Ou, “Realization of wafer-scale nanogratings with sub-50 nm period through vacancy epitaxy,” Nat. Commun. 10(1), 2437 (2019).
[Crossref]

Zimoch, D.

V. N. Strocov, T. Schmitt, U. Flechsig, T. Schmidt, A. Imhof, Q. Chen, J. Raabe, R. Betemps, D. Zimoch, J. Krempasky, X. Wang, M. Grioni, A. Piazzalunga, and L. Patthey, “High-resolution soft X-ray beamline ADRESS at the Swiss Light Source for resonant inelastic X-ray scattering and angle-resolved photoelectron spectroscopies,” J. Synchrotron Radiat. 17(5), 631–643 (2010).
[Crossref]

Zizak, I.

F. Siewert, B. Löchel, J. Buchheim, and F. Eggenstein, A. Firsov, G. Gwalt, O. Kutz, St. Lemke, B. Nelles, I. Rudolph, F. Schäfers, T. Seliger, F. Senf, A. Sokolov, Ch. Waberski, J. Wolf, T. Zeschke, I. Zizak, R. Follath, T. Arnold, F. Frost, F. Pietag, and A. Erko, “Gratings for synchrotron and FEL beamlines: a project for the manufacture of ultra-precise gratings at Helmholtz Zentrum Berlin,” J. Synchrotron Radiat. 25(1), 91–99 (2018).
[Crossref]

Appl. Opt. (1)

Appl. Surf. Sci. (1)

D. L. Voronov, E. H. Anderson, E. M. Gullikson, F. Salmassi, T. Warwick, V. V. Yashchuk, and H. A. Padmore, “Control of surface mobility for conformal deposition of Mo–Si multilayers on saw-tooth substrates,” Appl. Surf. Sci. 284, 575–580 (2013).
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Bull. Russ. Acad. Sci.: Phys. (1)

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Figures (12)

Fig. 1.
Fig. 1. Two BMG models, where multilayer structure above anti-blaze facets is neglected (a) and taken into account (b).
Fig. 2.
Fig. 2. Diffraction efficiency of different orders n for gratings G1 (colored solid curves) versus the grazing incidence angle. Calculations were performed for two photon energies E = 1 keV (a) and 5 keV (b). The bilayer thickness was set to h = 6.16 nm (a) or 5.33 nm (b), and the number of bilayers N = 100. Results of numerical calculations with CWA are also shown in Fig. 2(a) assuming the Cr or C layer to be placed on the MS top (colored symbols or black dotted curves, respectively). In addition, the reflectivity of conventional Cr/C MM with the same bilayer thickness is presented.
Fig. 3.
Fig. 3. Variations in the peak diffraction efficiency of Сr/C BMG with dependence on the variation in geometrical grating parameters for two values of the photon energy.
Fig. 4.
Fig. 4. Parameters P2 and P3 and the product P2P3 versus the bilayer thickness h. Calculations were performed for Cr/C BMGs G0 (a, c) and G1 (b, d) at E = 1 keV (a, b) and 5 keV (c, d). Introduced parameters h0 = 6.2 nm, hmax = 7.2 nm, hc = 6.4 nm and the optimal bilayer thickness hopt = 5.3 nm are also shown in figure. (d) for clarity. The value of h = hopt indicated in Fig. 4(d) provides the maximal diffraction efficiency, h0 indicates the maximum of parameter P3, hmax and hc show the bi-layer thickness, when the incidence angle equals to zero or the critical angle of TER, respectively. Dashed curves in Fig. 4(b) show the contribution of blaze (1) and anti-blaze (2) facets to the parameter P3.
Fig. 5.
Fig. 5. -1st order diffraction efficiency of G0 and G1 gratings versus the grazing incidence angle at the photon energy E = 1 keV and for different bilayer thicknesses indicated in the graphs (in nm). Calculations were performed with analytic formulas (Eqs. (10)−(11), colored solid curves) and numerical calculations based on CWA (black dashed curves). Number of bilayers was set to N = 100. Cr layer was placed on the MS top.
Fig. 6.
Fig. 6. -1st order diffraction efficiency of G0 and G1 gratings versus the grazing incidence angle at the photon energy E = 5 keV and for different bilayer thicknesses indicated in the graphs (in nm). Calculations were performed with analytic formulas (Eqs. (10)−(11), colored solid curves) and numerical calculations based on CWA (black dashed curves). The -1st and 0th diffraction efficiencies at h = 7.1 nm were only calculated numerically, because the analytic approach is invalid inside the TER region. Number of bilayers was set to N = 100. C layer was placed on the MS top.
Fig. 7.
Fig. 7. Product P2P3 versus the bilayer thickness h for grating G1 (α = 0.98°) for three diffraction orders (n = -1, -2, -3) at photon energy E = 1 keV (a) and 5 keV (b). The parameter P2 was calculated via exact Eq. (16) and cut at the value of hc corresponding to the incoming wave incident at the critical angle of TER.
Fig. 8.
Fig. 8. Diffraction efficiency of the -1st-, -2nd-, and -3rd-order versus the grazing incidence angle at the photon energies E = 1 keV (a) and 5 keV (b). Calculations were performed for grating G1 using analytic formulas (Eqs. (10)−(11)) assuming perfectly smooth MS interfaces (solid curves) and rough interfaces with σ = 0.3 nm (dashed curves). Number and thickness of bilayers are given in Table 1. The dashed curves were shifted to the right by 0.5° (a) or 0.1° (b) for clarity.
Fig. 9.
Fig. 9. Diffraction efficiency Rn (a) and fix-focus constant parameter Cff (b) of grating G1 covered by Cr/C MS versus the photon energy. The calculations were performed for different diffraction orders, numbers of bilayers, and bilayer thicknesses indicated in Fig. 9(a). Curve number in Fig. 9(b) corresponds to that in Fig. 9(a). Curve 1 demonstrates the maximal possible -1st order diffraction efficiency with the optimal bilayer thickness chosen individually for the particular photon energy.
Fig. 10.
Fig. 10. Universal dependencies of (a) the optimal bilayer thickness h found under the condition θ0 = θc at E = Emax and (b) the product ηntan(αopt) found with Eq. (31) versus the ratio D/|n|. The calculations were performed for different grating operating intervals (Emin, Emax), where Emin = 1 keV was fixed, while Emax = 5, 2.5, and 1.5 keV (curves 1, 2, and 3, respectively).
Fig. 11.
Fig. 11. Diffraction efficiency (a), Cff-value (b), and the saturated thickness of the MS (c) versus the photon energy. The calculations were performed for different ratios D/|n| and different operation spectral intervals. The grating parameters are presented in Table 2. The calculations were performed with analytic Eqs. (15)−(16) (colored curves) and numerically with CWA (circles). The curve number in Fig. 11 corresponds to the line number in Table 2. For comparison, the peak reflectivity of conventional MM is also shown in Fig. 11(a) (curve MM).
Fig. 12.
Fig. 12. Measured reflectivity (symbols 1) of conventional Cr/C MM (d = 6.3 nm, dCr/d = 0.43, N = 35) and the experimental -1st-order diffraction efficiency of grating G1 versus the photon energy covered by the same MS (symbols 2). Curves 3 and 4 show the calculated reflectivity and diffraction efficiency of the perfect MM and perfect BMG. Curves 5 and 6 were calculated assuming the presence of 5.5 at.% argon in the carbon layers and short-scale roughness with rms values of σ = 0.48 nm (for MM) and σ = 0.60 nm (for BMG). Experimental data were taken from [2].

Tables (2)

Tables Icon

Table 1. Geometrical parameters and optical characteristics of G1 grating covered by Cr/C MS with bilayer thickness hopt and the number of bilayers N. The saturated number of bilayers Nsat is also indicated. The MS was optimized to obtain the maximal diffraction efficiency Rn for different orders n. The values of the grazing incidence angle θ0, the diffraction angle θn, and the fix-focus constant Cff, are also presented.

Tables Icon

Table 2. Geometrical parameters of BMGs, whose diffraction efficiencies and Cff-values versus the photon energy are shown in Fig. 11. The gratings were optimized for operation in different spectral intervals (Emin, Emax), where Emin = 1 keV was fixed. The grating ratio Γ was set to 0.7. The thickness ratio γ of Cr/C MS was equal to 0.4. The saturated thickness Lsat = Nsathopt. The first column shows the curve numbers in Fig. 11.

Equations (35)

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  D = D α + D β , H = D α tan α = D β tan β Γ = D α D = tan β tan α + tan β , 1 Γ = D β D = tan α tan α + tan β   h =   d α / cos α = d β / cos β
χ 0 ( x , z ) = { j u j exp ( 2 i π j z d α ) = j u j exp ( 2 i π j d α ( z cos α + x sin α ) ) , x [ D α , 0 ] j u j exp ( 2 i π j z d β ) = j u j exp ( 2 i π j d β ( z cos β x sin β ) ) , x [ 0 , D β ]
u j 0 = ( χ A χ S ) 1 exp ( 2 i π j γ ) 2 i π j ; u 0 = χ ¯ = γ χ A + ( 1 γ ) χ S
χ ( x , z ) = n U n ( z ) exp ( 2 i π n x D ) ,  where  U n ( z ) = 1 D D α D β χ 0 ( x , z ) exp ( 2 i π n x D ) d x
U n ( z ) = j u j v j , n exp ( 2 i π j z / h )
v j , n = exp [ i π Γ ( j D tan α / h n ) ] sin [ π Γ ( j D tan α / h n ) ] π ( j D tan α / h n ) + exp [ i π ( 1 Γ ) ( j D tan β / h + n ) ] sin [ π ( 1 Γ ) ( j D tan β / h + n ) ] π ( j D tan β / h + n )
v j , n (at  β  = 9 0 ) = exp [ i π ( j D sin α / d n ) ] sin [ π ( j D sin α / d n ) ] π ( j D sin α / d n )
v j , n = exp ( i π η + i π n / 2 ) [ Γ sin ( π η π n / 2 ) π η π n / 2 + ( 1 ) n ( 1 Γ ) sin ( π η + π n / 2 ) π η + π n / 2 ]
η = j D h tan α tan β tan α + tan β + n 2 tan α tan β tan α + tan β
Δ θ B M G << δ θ h / ( j D )
R n ( θ 0 , θ n , λ ) = | U + tanh ( S L ) b tanh ( S L ) i U + U b 2 | 2
  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 h )   U ± u ± j ν ± j , n
cos θ n = cos θ 0 + n λ / D
R n = | U + b i U + U b 2 | 2
  j λ 2 h = sin θ 0 + sin θ n 2 sin θ 0 + sin θ n 4 sin θ 0 sin θ n Re χ ¯ + 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 | ν j , n | 2
R n = 1 V 1 + V ; V = 1 y 2 1 + f 2 y 2
  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 = | ν j , n |   f = Re ( χ A χ S ) Im ( χ A χ S ) ; g = Im χ S Im ( χ A χ S )
P 2 = 2 C f f / 2 C f f ( 1 + C f f ) ( 1 + C f f )
( Δ θ ) B M G 2 | χ A χ S | sin ( θ 0 + θ n ) sin ( π j γ ) π j sin θ n sin θ 0 | ν j , n |
L s a t = N s a t h = 2.65 S = 2.65 λ π Im χ ¯ ( 1 y 2 ) ( 1 + f 2 y 2 ) 2 sin θ 0 sin θ n sin θ 0 + sin θ n
j D sin α / d = j D tan α / h = j H / h = | n |
h 0 = j D tan α | n | η n , where  η n 1 1 1 + 2 Γ ( 1 + π 2 n 2 6 Γ 1 Γ )  and  Γ 1 2
θ 0 2 θ n 2 2 n λ / D  and  θ 0 + θ n j λ / h
θ 0 j λ / ( 2 h ) + n h / ( j D )  and  θ n j λ / ( 2 h ) n h / ( j D )
P 2 1 ( 2 n h 2 j 2 λ D ) 2
h max = j 2 λ D / ( 2 | n | )
h min j D | n | < η n tan α < λ | n | 2 D
λ > 2 | n | h min 2 j 2 D
0.47 o < α < 1.13 o ,  if  | n | = 1 0.86 o < α < 1.46 o ,  if  | n | = 2 1.27 o < α < 1.76 o ,  if  | n | = 3
u ~ j = u j exp [ 2 ( π j σ / h ) 2 ]
C f f = sin θ n sin θ 0 θ n θ 0 1 + 2 | n | h 2 / ( j 2 D λ ) 1 2 | n | h 2 / ( j 2 D λ )
tan α opt = | n | h / ( j D η n )
h c ( λ ) λ θ c 2 ( λ ) + 2 | n | λ / D + θ c ( λ )
C f f ( λ min ) = sin θ n ( λ min ) sin θ c ( λ min ) 1 + 2 | n | λ min D θ c 2 ( λ min )
| n | h o p t 2 D λ min = 0.43 ± 0.03

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