Abstract

An extended set of coupled wave equations were derived to describe non-idealized lamellar multilayer grating structures with properties as obtained with state-of-the-art fabrication techniques. These generalized equations can include all relevant effects describing the influence of passivation and contamination layers, non-rectangular lamel profiles and sidewall scalloping. The calculations showed that passivation and contamination plays an important role in that it may significantly reduce peak reflectivity. However, we also derived a condition for layer thicknesses having negligible effects. Slightly positive tapered lamel profiles are shown to further reduce the bandwidth as compared to a rectangular lamel profile, whereas negative tapers significantly increased the bandwidth. The influence of intriguing effects, such as the sidewall scalloping caused by Bosch Deep Reactive Ion Etching, are also modeled. We identified the signature of such scalloping as additional side peaks in the reflectivity spectrum and present parameters with which these can be effectively suppressed.

© 2013 OSA

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  1. 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. Exp.18, 16234–16242 (2010).
    [CrossRef]
  2. 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. Exp.19, 9172–9184 (2011).
    [CrossRef]
  3. A. Sammar, J.-M. André, and B. Pardo, “Diffraction and scattering by lamellar amplitude multilayer gratings in the x-uv region,” Opt. Comm.86, 245–254 (1991).
    [CrossRef]
  4. A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
    [CrossRef]
  5. R. Benbalagh, “Monochromateurs multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV,” Ph.D. thesis, Universite Pierre et Marie Curie (2003).
  6. P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
    [CrossRef]
  7. R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
    [CrossRef]
  8. R. van der Meer, B. Krishnan, M. J. de Boer, and F. Bijkerk, are preparing a manuscript to be called “Improved etch anisotropy using thin-layer mixed-material.”
  9. H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
    [CrossRef]
  10. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray database of x-ray interactions with matter,” http://henke.lbl.gov/optical_constants/
  11. B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, z=1-92,” At. Data Nucl. Data Tables54, 181 – 342 (1993).
    [CrossRef]
  12. 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 Advances3, 012103 (2013).
    [CrossRef]
  13. I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of xuv multilayer optics,” Phys. Scripta Vol. T17, 137–145 (1987).
    [CrossRef]
  14. L. L. Balakireva and I. V. Kozhevnikov, “Two-period multilayer mirrors for the soft x-ray region,” Proc. of SPIE2453, 67–79 (1995).
    [CrossRef]
  15. I. V. Kozhevnikov, S. Yulin, T. Feigl, and N. Kaiser, “Effect of anomalous transmittance in EUV multilayer optics,” Opt. Commun.2813025–3031 (2008).
    [CrossRef]

2013

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 Advances3, 012103 (2013).
[CrossRef]

2012

P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
[CrossRef]

2011

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

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

2010

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. Exp.18, 16234–16242 (2010).
[CrossRef]

2009

H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
[CrossRef]

2008

I. V. Kozhevnikov, S. Yulin, T. Feigl, and N. Kaiser, “Effect of anomalous transmittance in EUV multilayer optics,” Opt. Commun.2813025–3031 (2008).
[CrossRef]

1995

L. L. Balakireva and I. V. Kozhevnikov, “Two-period multilayer mirrors for the soft x-ray region,” Proc. of SPIE2453, 67–79 (1995).
[CrossRef]

1993

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, z=1-92,” At. Data Nucl. Data Tables54, 181 – 342 (1993).
[CrossRef]

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

1991

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

1987

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of xuv multilayer optics,” Phys. Scripta Vol. T17, 137–145 (1987).
[CrossRef]

Agafonov, Y. A.

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

André, J.-M.

P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
[CrossRef]

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

Balakireva, L. L.

L. L. Balakireva and I. V. Kozhevnikov, “Two-period multilayer mirrors for the soft x-ray region,” Proc. of SPIE2453, 67–79 (1995).
[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 Advances3, 012103 (2013).
[CrossRef]

Bastiaens, H. M. J.

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. Exp.19, 9172–9184 (2011).
[CrossRef]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

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. Exp.18, 16234–16242 (2010).
[CrossRef]

Benbalagh, R.

R. Benbalagh, “Monochromateurs multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV,” Ph.D. thesis, Universite Pierre et Marie Curie (2003).

Bijkerk, F.

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 Advances3, 012103 (2013).
[CrossRef]

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

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

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. Exp.18, 16234–16242 (2010).
[CrossRef]

R. van der Meer, B. Krishnan, M. J. de Boer, and F. Bijkerk, are preparing a manuscript to be called “Improved etch anisotropy using thin-layer mixed-material.”

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 Advances3, 012103 (2013).
[CrossRef]

Boller, K.-J.

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. Exp.19, 9172–9184 (2011).
[CrossRef]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

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. Exp.18, 16234–16242 (2010).
[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 Advances3, 012103 (2013).
[CrossRef]

Brons, G. C. S.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

Brunel, M.

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

Coudevylle, J. R.

P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
[CrossRef]

Davis, J. C.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, z=1-92,” At. Data Nucl. Data Tables54, 181 – 342 (1993).
[CrossRef]

de Boer, M. J.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
[CrossRef]

R. van der Meer, B. Krishnan, M. J. de Boer, and F. Bijkerk, are preparing a manuscript to be called “Improved etch anisotropy using thin-layer mixed-material.”

Elwenspoek, M. C.

H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
[CrossRef]

Erko, A. I.

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

Feigl, T.

I. V. Kozhevnikov, S. Yulin, T. Feigl, and N. Kaiser, “Effect of anomalous transmittance in EUV multilayer optics,” Opt. Commun.2813025–3031 (2008).
[CrossRef]

Gullikson, E. M.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, z=1-92,” At. Data Nucl. Data Tables54, 181 – 342 (1993).
[CrossRef]

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 Advances3, 012103 (2013).
[CrossRef]

Hegeman, P. E.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

Henke, B. L.

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, z=1-92,” At. Data Nucl. Data Tables54, 181 – 342 (1993).
[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 Advances3, 012103 (2013).
[CrossRef]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

Isac, N.

P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
[CrossRef]

Jansen, H. V.

H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
[CrossRef]

Jonnard, P.

P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
[CrossRef]

Kaiser, N.

I. V. Kozhevnikov, S. Yulin, T. Feigl, and N. Kaiser, “Effect of anomalous transmittance in EUV multilayer optics,” Opt. Commun.2813025–3031 (2008).
[CrossRef]

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 Advances3, 012103 (2013).
[CrossRef]

Kozhevnikov, I. V.

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. Exp.19, 9172–9184 (2011).
[CrossRef]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

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. Exp.18, 16234–16242 (2010).
[CrossRef]

I. V. Kozhevnikov, S. Yulin, T. Feigl, and N. Kaiser, “Effect of anomalous transmittance in EUV multilayer optics,” Opt. Commun.2813025–3031 (2008).
[CrossRef]

L. L. Balakireva and I. V. Kozhevnikov, “Two-period multilayer mirrors for the soft x-ray region,” Proc. of SPIE2453, 67–79 (1995).
[CrossRef]

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of xuv multilayer optics,” Phys. Scripta Vol. T17, 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 Advances3, 012103 (2013).
[CrossRef]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

R. van der Meer, B. Krishnan, M. J. de Boer, and F. Bijkerk, are preparing a manuscript to be called “Improved etch anisotropy using thin-layer mixed-material.”

Le Guen, K.

P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
[CrossRef]

Louwerse, M. C.

H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
[CrossRef]

Martynov, V. V.

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

Pardo, B.

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

Roschupkin, D. V.

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

Sammar, A.

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

Unnikrishnan, S.

H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
[CrossRef]

van der Meer, R.

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 Advances3, 012103 (2013).
[CrossRef]

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

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

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. Exp.18, 16234–16242 (2010).
[CrossRef]

R. van der Meer, B. Krishnan, M. J. de Boer, and F. Bijkerk, are preparing a manuscript to be called “Improved etch anisotropy using thin-layer mixed-material.”

van der Wiel, W. G.

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

Vidal, B.

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

Vincent, P.

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

Vinogradov, A. V.

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of xuv multilayer optics,” Phys. Scripta Vol. T17, 137–145 (1987).
[CrossRef]

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 Advances3, 012103 (2013).
[CrossRef]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

Yulin, S.

I. V. Kozhevnikov, S. Yulin, T. Feigl, and N. Kaiser, “Effect of anomalous transmittance in EUV multilayer optics,” Opt. Commun.2813025–3031 (2008).
[CrossRef]

AIP Advances

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 Advances3, 012103 (2013).
[CrossRef]

At. Data Nucl. Data Tables

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-Ray interactions: photoabsorption, scattering, transmission, and reflection at E = 50-30,000 eV, z=1-92,” At. Data Nucl. Data Tables54, 181 – 342 (1993).
[CrossRef]

J. Micromech. Microeng

H. V. Jansen, M. J. de Boer, S. Unnikrishnan, M. C. Louwerse, and M. C. Elwenspoek, “Black silicon method x: a review on high speed and selective plasma etching of silicon with profile control: an in-depth comparison between bosch and cryostat drie processes as a roadmap to next generation equipment,” J. Micromech. Microeng. 19, 033001 (2009).
[CrossRef]

J. X-ray Spect

P. Jonnard, K. Le Guen, J.-M. André, J. R. Coudevylle, and N. Isac, “An etched multilayer as a dispersive element in a curved--crystal spectrometer: implementation and performance,” J. X-ray Spect. 44, 308–312 (2012).
[CrossRef]

Nucl. Instr. Meth. Phys. Res. A

A. I. Erko, B. Vidal, P. Vincent, Y. A. Agafonov, V. V. Martynov, D. V. Roschupkin, and M. Brunel, “Multilayer gratings efficiency: numerical and physical experiments,” Nucl. Instr. Meth. Phys. Res. A333, 599–606 (1993).
[CrossRef]

Opt. Comm.

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

Opt. Commun.

I. V. Kozhevnikov, S. Yulin, T. Feigl, and N. Kaiser, “Effect of anomalous transmittance in EUV multilayer optics,” Opt. Commun.2813025–3031 (2008).
[CrossRef]

Opt. Exp.

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. Exp.18, 16234–16242 (2010).
[CrossRef]

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

Phys. Scripta Vol. T

I. V. Kozhevnikov and A. V. Vinogradov, “Basic formulae of xuv multilayer optics,” Phys. Scripta Vol. T17, 137–145 (1987).
[CrossRef]

Proc. of SPIE

L. L. Balakireva and I. V. Kozhevnikov, “Two-period multilayer mirrors for the soft x-ray region,” Proc. of SPIE2453, 67–79 (1995).
[CrossRef]

R. van der Meer, B. Krishnan, I. V. Kozhevnikov, M. J. de Boer, B. Vratzov, H. M. J. Bastiaens, J. Huskens, W. G. van der Wiel, P. E. Hegeman, G. C. S. Brons, K.-J. Boller, and F. Bijkerk, “Improved resolution for soft-x-ray monochromatization using lamellar multilayer gratings,” Proc. of SPIE8139, 81390Q–8 (2011).
[CrossRef]

Other

R. van der Meer, B. Krishnan, M. J. de Boer, and F. Bijkerk, are preparing a manuscript to be called “Improved etch anisotropy using thin-layer mixed-material.”

B. L. Henke, E. M. Gullikson, and J. C. Davis, “X-ray database of x-ray interactions with matter,” http://henke.lbl.gov/optical_constants/

R. Benbalagh, “Monochromateurs multicouches à bande passante étroite et à faible fond continu pour le rayonnement X-UV,” Ph.D. thesis, Universite Pierre et Marie Curie (2003).

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Figures (7)

Fig. 1
Fig. 1

Schematic cross section of a highly idealized LMG. An incident beam from the left (In), under grazing angle Θ0, is reflected from the multilayer and diffracted into multiple orders (Out) under grazing angle Φj by the grating structure. The multilayer is built up from N bi-layers (thickness d) consisting of an absorber (A) and spacer material (S) with thickness-ratio γ. The grating structure is defined by the period D and lamel width ΓD (i.e., Γ < 1 is the ratio of the lamel width to the grating period).

Fig. 2
Fig. 2

Schematic cross section of a general shaped lamel. Curves 1 and 2 indicate the sidewalls of the ML structure and are described by the functions ±p(z). A cover layer with thickness h is indicated in red by curves 3 and 4. Γtop and Γbottom indicate the Γ-ratio at the top and bottom of the lamel, respectively.

Fig. 3
Fig. 3

Calculations of the diffraction efficiencies of the 0th (reflectivity) and ±1st orders for various thicknesses of a CnF2n cover layer on a Mo/B4C LMG at the incident SXR energy E of 183.4 eV. The cover layers result in a negligible increase in bandwidth, but can strongly reduce the peak reflectivity and result in a shift of peak position for thicker layers. The parameters of the LMG are D = 300 nm, Γ = 0.3, N = 150, d = 6 nm and γ = 0.33. The values of the complex polarizability used are χ(Mo) = 2.61 · 10−2i5.77 · 10−3 and χ(B4C) = 4.43 · 10−3i1.08 · 10−3.

Fig. 4
Fig. 4

Calculated 0th and ±1st diffraction order efficiencies for rectangular as well as positively and negatively tapered lamel profiles with the same average Γ-ratio of 0.3 at the incident energy of 183.4 eV. The LMG parameters are the same as for Fig. 3.

Fig. 5
Fig. 5

(a) Scanning Electron Microscope image of a grating (D = 200 nm and Γ = 0.5) fabricated in silicon that shows the typical sidewall scalloping inherent to Bosch DRIE. (b) Schematic representation of a the lamel profile used in the calculations. The scallops are described by the scallop amplitude a and scallop length l. Typical values for both scallop dimensions are between a few nm’s up to several tens of nm.

Fig. 6
Fig. 6

0th and ±1st order efficiencies for various scallop widths a at a scallop length l of 150 nm at the incident energy of 183.4 eV. The additional scallop harmonics of the specular reflection are indicated by red arrows. The LMG parameters are the same as for Fig. 3.

Fig. 7
Fig. 7

0th and ±1st order efficiencies for various scallop widths (a) at the worst scallop length (lW) of 205 nm for the incident energy of 183.4 eV. The scallop harmonics of the diffraction orders (indicated by red arrows) clearly overlap with the main Bragg peak of the specular reflection, resulting in a large decrease in peak reflectivity for the scallop width of 80 nm. The LMG parameters are the same as for Fig. 3.

Equations (19)

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ε ( x , z ) = 1 if z < 0 ; ε ( x , z ) = 1 χ ( z ) U ( x , z ) χ cov V ( x , z ) if 0 z H ε ( x , z ) = 1 χ sub if z > H
U ( x , z ) = n = + U n ( z ) exp ( 2 i π n x / D ) ; V ( x , z ) = n = + V n ( z ) exp ( 2 i π n x / D ) ,
U n ( z ) = 1 π n sin ( 2 π n p ( z ) D ) ; V n ( z ) = 2 π n sin ( π n h D ) cos [ π n ( 2 p ( z ) D + h D ) ]
E ( x , z ) = n = + F n ( z ) exp ( i q n x ) ; q n = q 0 + 2 π n D ; q 0 = k cos Θ 0 ; k = 2 π λ .
F n ( z ) + κ n 2 F n ( z ) = k 2 m [ χ ( z ) U n m ( z ) + χ cov V n m ( z ) ] F m ( z ) ,
F n ( 0 ) + i κ n F n ( 0 ) = 2 i κ n δ n , 0 ; F n ( L ) i κ n ( s ) F n ( L ) = 0 ,
F n ( z ) = A n ( z ) exp ( i κ n z ) + C n ( z ) exp ( i κ n z ) ,
d A n ( z ) d z exp ( i κ n z ) + d C n ( z ) d z exp ( i κ n z ) = 0 .
d A n ( z ) d z = i k 2 2 κ n m W n m ( z ) [ A m ( z ) exp ( i ( κ m + κ n ) z ) + C m ( z ) exp ( i ( κ m κ n ) z ) ]
d C n ( z ) d z = + i k 2 2 κ n m W n m ( z ) [ A m ( z ) exp ( i ( κ m + κ n ) z ) + C m ( z ) exp ( i ( κ m κ n ) z ) ] ,
W n m ( z ) = χ ( z ) U n m ( z ) + χ cov V n m ( z ) ,
A n ( 0 ) = δ n , 0 ; C n ( L ) = 0 ,
Γ D Δ Θ M M d ,
F 0 ( z ) + k 2 ( sin 2 ( Θ 0 ) χ eff ( z ) ) F 0 ( z ) = 0 ,
χ eff ( z ) = χ ( z ) 2 p ( z ) / D + χ cov 2 h / D .
h Γ D 2 Im ( χ S ) Im ( χ cov ) .
χ eff ( z ) Γ χ ¯ + a D χ ¯ sin ( 2 π z d 1 ) + ( χ A χ S ) sin ( π γ ) π . [ 2 Γ cos ( 2 π z d 0 π γ ) + a D sin ( 2 π z d 2 π γ ) a D sin ( 2 π z d 3 π γ ) ]
λ = 2 d 0 sin ( Θ 0 ( 0 ) ) λ = d 2 ( sin ( Θ 0 ( 0 ) ) + sin ( Φ + 1 ( 2 ) ) ) λ = D ( cos ( Θ 0 ( 0 ) ) cos ( Φ + 1 ( 2 ) ) ) .
l w D tan ( Θ 0 ( 0 ) ) ,

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