Abstract

The boundary integral equation code PCGrate-S(X) is used to analyze diffraction on Hubble Space Telescope Cosmic Origins Spectrograph gratings at different boundary shapes and layer thicknesses. An effect of resonance anomalies excited in nonconformal dielectric layers overcoated on the surface of metallic grating on the efficiency is studied for the first time to our knowledge. Refractive indices (RIs) for bulk MgF2 taken from well-known references are found to be not suitable for thin optical layers at wavelengths between 115 and 170nm. A method based on scale fitting of calculated and measured grating efficiencies is outlined for derivation of thin-film optical constants at hard to measure wavelengths. The calculated efficiency based on real boundary profiles and derived RIs of the G185M subwavelength grating is shown to fit within 9.6% or better to the measured data.

© 2006 Optical Society of America

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  1. J. C. Green, "Cosmic Origins Spectrograph," in UV, Optical, and IR Space Telescopes and Instruments, J.B.Breckinridge and P.Jakobsen, eds., Proc. SPIE 4013, 352-359 (2000).
  2. I. G. Kuznetsov, E. Wilkinson, D. A. Content, R. A. Boucarut, and T. J. Madison, "Grating efficiencies comparison study: calculations versus metrology for various types of high groove density gratings at VUV-UV wavelengths," in Optical Modeling and Performance Predictions, M.A.Kahan, ed., Proc. SPIE 5178, 267-277 (2004).
  3. R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).
  4. V. Dauer, "Optical constants of lithium fluoride thin films in the far ultraviolet," J. Opt. Soc. Am. B 17, 300-303 (2000).
    [CrossRef]
  5. E.Palik, ed., Handbook of Optical Constant of Solids (Academic, 1991), Vol. 2.
  6. J. Larruquert and R. Keski-Kuha, "Far ultraviolet optical properties of MgF2 films deposited by ion-beam sputtering and their application as protective coatings for Al," Opt. Commun. 215, 93-99 (2003).
    [CrossRef]
  7. See website at www.pcgrate.com.
  8. D. A. Content, P. Arsenovic, I. G. Kuznetsov, and T. Hadjimichael, "Grating groove metrology and efficiency predictions from the soft x-ray to the far infrared," in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 405-416 (2001).
  9. L. I. Goray and S. Yu. Sadov, "Numerical modeling of coated gratings in sensitive cases," in Diffractive Optics and Micro-Optics, R.Magnusson, ed., Vol. 75of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), pp. 365-379.
  10. D. Maystre, "A new general integral theory for dielectric coated gratings," J. Opt. Soc. Am. 68, 490-495 (1978).
    [CrossRef]
  11. A. Pomp, "The integral method for coated gratings: computational cost," J. Mod. Opt. 38, 109-120 (1991).
    [CrossRef]
  12. B. Kleemann, A. Mitreiter, and F. Wyrowski, "Integral equation method with parametrization of grating profile: theory and experiments," J. Mod. Opt. 43, 1323-1349 (1996).
    [CrossRef]
  13. J. T. Beale and M.-C. Lai, "A method for computing nearly singular integrals," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 1902-1925 (2001).
  14. K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind (Cambridge U. Press, 1997).
    [CrossRef]
  15. G. Elschner and I. Graham, "An optimal order collocation method for first kind boundary integral equations on polygons," Numer. Math. 70, 1-31 (1995).
    [CrossRef]
  16. R. Kress, "A Nyström method for boundary integral equations in domains with corners," Numer. Math. 58, 145-161 (1990).
    [CrossRef]
  17. P. Laubin, "High order convergence for collocation of second kind boundary integral equations on polygons," Numer. Math. 79, 107-140 (1998).
    [CrossRef]
  18. R. Kress, I. Sloan, and F. Stenger, "A sinc quadrature method for the double layer integral equation in planar domain with corners," J. Integral Equ. Appl. 10, 291-317 (1998).
    [CrossRef]
  19. C. M. Linton, "The Green's function for two-dimensional Helmholtz equation in periodic domains," J. Eng. Math. 33, 377-402 (1998).
    [CrossRef]
  20. R.Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).
    [CrossRef]
  21. A. Aho, J. Hopcroft, and J. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, 1976).
  22. D. Knuth, The Art of Computer Programming (Addison-Wesley, 1998), Vol. 2.
  23. I. G. Kuznetsov, D. A. Content, R. A. Boucarut, and T. J. Madison, "Design, performance and reliability of a high angular resolution, wide angular range, large aperture fully automated UV scatterometer," in Optical Spectroscopic Techniques, Remote Sensing, and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 417-428 (2001).
  24. J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).
  25. E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel-Dekker, 1997).
  26. American Institute of Physics Handbook (McGraw-Hill, 1972).
  27. R. Keski-Kuha, NASA GSFC, Code 551, Greenbelt, Maryland 20771 (personal communication 2005).
  28. E. Spiller, Soft X-ray Optics (SPIE Press, 1994).
    [CrossRef]
  29. L. I. Goray, "Modified integral method and real electromagnetic properties of echelles," in Diffractive and Holographic Technologies for Integrated Photonic Systems, R.I.Sutherland, D.W.Prather, and I.Cindrich, eds., Proc. SPIE 4291, 13-24 (2001).

2003 (1)

J. Larruquert and R. Keski-Kuha, "Far ultraviolet optical properties of MgF2 films deposited by ion-beam sputtering and their application as protective coatings for Al," Opt. Commun. 215, 93-99 (2003).
[CrossRef]

2001 (1)

J. T. Beale and M.-C. Lai, "A method for computing nearly singular integrals," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 1902-1925 (2001).

2000 (1)

1998 (3)

P. Laubin, "High order convergence for collocation of second kind boundary integral equations on polygons," Numer. Math. 79, 107-140 (1998).
[CrossRef]

R. Kress, I. Sloan, and F. Stenger, "A sinc quadrature method for the double layer integral equation in planar domain with corners," J. Integral Equ. Appl. 10, 291-317 (1998).
[CrossRef]

C. M. Linton, "The Green's function for two-dimensional Helmholtz equation in periodic domains," J. Eng. Math. 33, 377-402 (1998).
[CrossRef]

1996 (1)

B. Kleemann, A. Mitreiter, and F. Wyrowski, "Integral equation method with parametrization of grating profile: theory and experiments," J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

1995 (1)

G. Elschner and I. Graham, "An optimal order collocation method for first kind boundary integral equations on polygons," Numer. Math. 70, 1-31 (1995).
[CrossRef]

1991 (1)

A. Pomp, "The integral method for coated gratings: computational cost," J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

1990 (1)

R. Kress, "A Nyström method for boundary integral equations in domains with corners," Numer. Math. 58, 145-161 (1990).
[CrossRef]

1978 (1)

Aho, A.

A. Aho, J. Hopcroft, and J. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, 1976).

Arsenovic, P.

D. A. Content, P. Arsenovic, I. G. Kuznetsov, and T. Hadjimichael, "Grating groove metrology and efficiency predictions from the soft x-ray to the far infrared," in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 405-416 (2001).

Atkinson, K. E.

K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind (Cambridge U. Press, 1997).
[CrossRef]

Beale, J. T.

J. T. Beale and M.-C. Lai, "A method for computing nearly singular integrals," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 1902-1925 (2001).

Bonnemason, F.

R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).

Boucarut, R. A.

I. G. Kuznetsov, D. A. Content, R. A. Boucarut, and T. J. Madison, "Design, performance and reliability of a high angular resolution, wide angular range, large aperture fully automated UV scatterometer," in Optical Spectroscopic Techniques, Remote Sensing, and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 417-428 (2001).

I. G. Kuznetsov, E. Wilkinson, D. A. Content, R. A. Boucarut, and T. J. Madison, "Grating efficiencies comparison study: calculations versus metrology for various types of high groove density gratings at VUV-UV wavelengths," in Optical Modeling and Performance Predictions, M.A.Kahan, ed., Proc. SPIE 5178, 267-277 (2004).

Content, D. A.

I. G. Kuznetsov, E. Wilkinson, D. A. Content, R. A. Boucarut, and T. J. Madison, "Grating efficiencies comparison study: calculations versus metrology for various types of high groove density gratings at VUV-UV wavelengths," in Optical Modeling and Performance Predictions, M.A.Kahan, ed., Proc. SPIE 5178, 267-277 (2004).

I. G. Kuznetsov, D. A. Content, R. A. Boucarut, and T. J. Madison, "Design, performance and reliability of a high angular resolution, wide angular range, large aperture fully automated UV scatterometer," in Optical Spectroscopic Techniques, Remote Sensing, and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 417-428 (2001).

D. A. Content, P. Arsenovic, I. G. Kuznetsov, and T. Hadjimichael, "Grating groove metrology and efficiency predictions from the soft x-ray to the far infrared," in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 405-416 (2001).

Dauer, V.

V. Dauer, "Optical constants of lithium fluoride thin films in the far ultraviolet," J. Opt. Soc. Am. B 17, 300-303 (2000).
[CrossRef]

R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).

Elschner, G.

G. Elschner and I. Graham, "An optimal order collocation method for first kind boundary integral equations on polygons," Numer. Math. 70, 1-31 (1995).
[CrossRef]

Flamand, J.

R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).

Goray, L. I.

L. I. Goray, "Modified integral method and real electromagnetic properties of echelles," in Diffractive and Holographic Technologies for Integrated Photonic Systems, R.I.Sutherland, D.W.Prather, and I.Cindrich, eds., Proc. SPIE 4291, 13-24 (2001).

L. I. Goray and S. Yu. Sadov, "Numerical modeling of coated gratings in sensitive cases," in Diffractive Optics and Micro-Optics, R.Magnusson, ed., Vol. 75of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), pp. 365-379.

J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).

Graham, I.

G. Elschner and I. Graham, "An optimal order collocation method for first kind boundary integral equations on polygons," Numer. Math. 70, 1-31 (1995).
[CrossRef]

Grange, R.

R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).

Green, J. C.

J. C. Green, "Cosmic Origins Spectrograph," in UV, Optical, and IR Space Telescopes and Instruments, J.B.Breckinridge and P.Jakobsen, eds., Proc. SPIE 4013, 352-359 (2000).

Hadjimichael, T.

D. A. Content, P. Arsenovic, I. G. Kuznetsov, and T. Hadjimichael, "Grating groove metrology and efficiency predictions from the soft x-ray to the far infrared," in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 405-416 (2001).

Hopcroft, J.

A. Aho, J. Hopcroft, and J. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, 1976).

Keski-Kuha, R.

J. Larruquert and R. Keski-Kuha, "Far ultraviolet optical properties of MgF2 films deposited by ion-beam sputtering and their application as protective coatings for Al," Opt. Commun. 215, 93-99 (2003).
[CrossRef]

R. Keski-Kuha, NASA GSFC, Code 551, Greenbelt, Maryland 20771 (personal communication 2005).

Kjornrattanawanich, B.

J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).

Kleemann, B.

B. Kleemann, A. Mitreiter, and F. Wyrowski, "Integral equation method with parametrization of grating profile: theory and experiments," J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Knuth, D.

D. Knuth, The Art of Computer Programming (Addison-Wesley, 1998), Vol. 2.

Kress, R.

R. Kress, I. Sloan, and F. Stenger, "A sinc quadrature method for the double layer integral equation in planar domain with corners," J. Integral Equ. Appl. 10, 291-317 (1998).
[CrossRef]

R. Kress, "A Nyström method for boundary integral equations in domains with corners," Numer. Math. 58, 145-161 (1990).
[CrossRef]

Kuznetsov, I. G.

I. G. Kuznetsov, E. Wilkinson, D. A. Content, R. A. Boucarut, and T. J. Madison, "Grating efficiencies comparison study: calculations versus metrology for various types of high groove density gratings at VUV-UV wavelengths," in Optical Modeling and Performance Predictions, M.A.Kahan, ed., Proc. SPIE 5178, 267-277 (2004).

I. G. Kuznetsov, D. A. Content, R. A. Boucarut, and T. J. Madison, "Design, performance and reliability of a high angular resolution, wide angular range, large aperture fully automated UV scatterometer," in Optical Spectroscopic Techniques, Remote Sensing, and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 417-428 (2001).

D. A. Content, P. Arsenovic, I. G. Kuznetsov, and T. Hadjimichael, "Grating groove metrology and efficiency predictions from the soft x-ray to the far infrared," in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 405-416 (2001).

Lai, M.-C.

J. T. Beale and M.-C. Lai, "A method for computing nearly singular integrals," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 1902-1925 (2001).

Larruquert, J.

J. Larruquert and R. Keski-Kuha, "Far ultraviolet optical properties of MgF2 films deposited by ion-beam sputtering and their application as protective coatings for Al," Opt. Commun. 215, 93-99 (2003).
[CrossRef]

Laubin, P.

P. Laubin, "High order convergence for collocation of second kind boundary integral equations on polygons," Numer. Math. 79, 107-140 (1998).
[CrossRef]

Linton, C. M.

C. M. Linton, "The Green's function for two-dimensional Helmholtz equation in periodic domains," J. Eng. Math. 33, 377-402 (1998).
[CrossRef]

Loewen, E. G.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel-Dekker, 1997).

Madison, T. J.

I. G. Kuznetsov, D. A. Content, R. A. Boucarut, and T. J. Madison, "Design, performance and reliability of a high angular resolution, wide angular range, large aperture fully automated UV scatterometer," in Optical Spectroscopic Techniques, Remote Sensing, and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 417-428 (2001).

I. G. Kuznetsov, E. Wilkinson, D. A. Content, R. A. Boucarut, and T. J. Madison, "Grating efficiencies comparison study: calculations versus metrology for various types of high groove density gratings at VUV-UV wavelengths," in Optical Modeling and Performance Predictions, M.A.Kahan, ed., Proc. SPIE 5178, 267-277 (2004).

Maystre, D.

Mitreiter, A.

B. Kleemann, A. Mitreiter, and F. Wyrowski, "Integral equation method with parametrization of grating profile: theory and experiments," J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

Nevière, M.

R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).

Pomp, A.

A. Pomp, "The integral method for coated gratings: computational cost," J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

Popov, E.

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel-Dekker, 1997).

Sadov, S. Yu.

L. I. Goray and S. Yu. Sadov, "Numerical modeling of coated gratings in sensitive cases," in Diffractive Optics and Micro-Optics, R.Magnusson, ed., Vol. 75of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), pp. 365-379.

Saisse, M.

R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).

Seely, J. F.

J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).

Sloan, I.

R. Kress, I. Sloan, and F. Stenger, "A sinc quadrature method for the double layer integral equation in planar domain with corners," J. Integral Equ. Appl. 10, 291-317 (1998).
[CrossRef]

Spiller, E.

E. Spiller, Soft X-ray Optics (SPIE Press, 1994).
[CrossRef]

Stenger, F.

R. Kress, I. Sloan, and F. Stenger, "A sinc quadrature method for the double layer integral equation in planar domain with corners," J. Integral Equ. Appl. 10, 291-317 (1998).
[CrossRef]

Ullman, J.

A. Aho, J. Hopcroft, and J. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, 1976).

Uspenskii, Yu. A.

J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).

Vinogradov, A. V.

J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).

Wilkinson, E.

I. G. Kuznetsov, E. Wilkinson, D. A. Content, R. A. Boucarut, and T. J. Madison, "Grating efficiencies comparison study: calculations versus metrology for various types of high groove density gratings at VUV-UV wavelengths," in Optical Modeling and Performance Predictions, M.A.Kahan, ed., Proc. SPIE 5178, 267-277 (2004).

Windt, D. L.

J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).

Wyrowski, F.

B. Kleemann, A. Mitreiter, and F. Wyrowski, "Integral equation method with parametrization of grating profile: theory and experiments," J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

J. Eng. Math. (1)

C. M. Linton, "The Green's function for two-dimensional Helmholtz equation in periodic domains," J. Eng. Math. 33, 377-402 (1998).
[CrossRef]

J. Integral Equ. Appl. (1)

R. Kress, I. Sloan, and F. Stenger, "A sinc quadrature method for the double layer integral equation in planar domain with corners," J. Integral Equ. Appl. 10, 291-317 (1998).
[CrossRef]

J. Mod. Opt. (2)

A. Pomp, "The integral method for coated gratings: computational cost," J. Mod. Opt. 38, 109-120 (1991).
[CrossRef]

B. Kleemann, A. Mitreiter, and F. Wyrowski, "Integral equation method with parametrization of grating profile: theory and experiments," J. Mod. Opt. 43, 1323-1349 (1996).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Numer. Math. (3)

G. Elschner and I. Graham, "An optimal order collocation method for first kind boundary integral equations on polygons," Numer. Math. 70, 1-31 (1995).
[CrossRef]

R. Kress, "A Nyström method for boundary integral equations in domains with corners," Numer. Math. 58, 145-161 (1990).
[CrossRef]

P. Laubin, "High order convergence for collocation of second kind boundary integral equations on polygons," Numer. Math. 79, 107-140 (1998).
[CrossRef]

Opt. Commun. (1)

J. Larruquert and R. Keski-Kuha, "Far ultraviolet optical properties of MgF2 films deposited by ion-beam sputtering and their application as protective coatings for Al," Opt. Commun. 215, 93-99 (2003).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

J. T. Beale and M.-C. Lai, "A method for computing nearly singular integrals," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 38, 1902-1925 (2001).

Other (18)

K. E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind (Cambridge U. Press, 1997).
[CrossRef]

See website at www.pcgrate.com.

D. A. Content, P. Arsenovic, I. G. Kuznetsov, and T. Hadjimichael, "Grating groove metrology and efficiency predictions from the soft x-ray to the far infrared," in Optical Spectroscopic Techniques and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 405-416 (2001).

L. I. Goray and S. Yu. Sadov, "Numerical modeling of coated gratings in sensitive cases," in Diffractive Optics and Micro-Optics, R.Magnusson, ed., Vol. 75of OSA Trends in Optics and Photonics Series (Optical Society of America, 2002), pp. 365-379.

R.Petit, ed., Electromagnetic Theory of Gratings (Springer, 1980).
[CrossRef]

A. Aho, J. Hopcroft, and J. Ullman, The Design and Analysis of Computer Algorithms (Addison-Wesley, 1976).

D. Knuth, The Art of Computer Programming (Addison-Wesley, 1998), Vol. 2.

I. G. Kuznetsov, D. A. Content, R. A. Boucarut, and T. J. Madison, "Design, performance and reliability of a high angular resolution, wide angular range, large aperture fully automated UV scatterometer," in Optical Spectroscopic Techniques, Remote Sensing, and Instrumentation for Atmospheric and Space Research IV, A.M.Larar and M.G.Mlynczak, eds., Proc. SPIE 4485, 417-428 (2001).

J. F. Seely, L. I. Goray, D. L. Windt, B. Kjornrattanawanich, Yu. A. Uspenskii, and A. V. Vinogradov, "Extreme ultraviolet optical constants for the design and fabrication of multilayer gratings," in Optical Constants of Materials for UV to X-Ray Wavelengths, R.Soufli and J.F.Seely, eds., Proc. SPIE 5538, 43-52 (2004).

E. G. Loewen and E. Popov, Diffraction Gratings and Applications (Marcel-Dekker, 1997).

American Institute of Physics Handbook (McGraw-Hill, 1972).

R. Keski-Kuha, NASA GSFC, Code 551, Greenbelt, Maryland 20771 (personal communication 2005).

E. Spiller, Soft X-ray Optics (SPIE Press, 1994).
[CrossRef]

L. I. Goray, "Modified integral method and real electromagnetic properties of echelles," in Diffractive and Holographic Technologies for Integrated Photonic Systems, R.I.Sutherland, D.W.Prather, and I.Cindrich, eds., Proc. SPIE 4291, 13-24 (2001).

J. C. Green, "Cosmic Origins Spectrograph," in UV, Optical, and IR Space Telescopes and Instruments, J.B.Breckinridge and P.Jakobsen, eds., Proc. SPIE 4013, 352-359 (2000).

I. G. Kuznetsov, E. Wilkinson, D. A. Content, R. A. Boucarut, and T. J. Madison, "Grating efficiencies comparison study: calculations versus metrology for various types of high groove density gratings at VUV-UV wavelengths," in Optical Modeling and Performance Predictions, M.A.Kahan, ed., Proc. SPIE 5178, 267-277 (2004).

R. Grange, V. Dauer, M. Saisse, M. Nevière, J. Flamand, and F. Bonnemason, "6000-g/mm holographic flight gratings for the high resolution Far Ultraviolet Spectroscopic Explorer: efficiency, resolution and stray light measurements," in Theory and Practice of Surface-Relief Diffraction Gratings: Synchrotron and Other Applications, W.R.McKinney and C.A.Palmer, eds., Proc. SPIE 3450, 103-112 (1998).

E.Palik, ed., Handbook of Optical Constant of Solids (Academic, 1991), Vol. 2.

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Figures (9)

Fig. 1
Fig. 1

Slab grating.

Fig. 2
Fig. 2

AFM profilometry on a G185M grating performed by SPN Digital Nanoscope IIIA (a) before and (b) after Cr Al Mg F 2 coating. Vertical scale is the same in both graphs.

Fig. 3
Fig. 3

Five-boundary G185M grating model. Horizontal and vertical scales are different.

Fig. 4
Fig. 4

Measured (points) and calculated (curves) –1st-order efficiency of a G185M grating for nonpolarization plotted versus wavelength. Efficiency models calculated for Palik’s RIs and different geometry of a nonconformal Mg F 2 layer: border (Bor.), scaling factor (scale), vertical shift (shift). A heavier region of the horizontal axis indicates the G185M intended operational range.

Fig. 5
Fig. 5

Modeling of –1st-order efficiency of a G185M grating with a Mg F 2 conformal layer for different RIs and thicknesses versus wavelength.[5, 26] Thickness variations define the formation and positions of resonance anomalies: (a) TM polarization, (b) TE polarization. AIP, American Institute of Physics.

Fig. 6
Fig. 6

Modeling of –1st-order efficiency of a G185M grating with a nonconformal Mg F 2 layer for different RIs and geometry versus wavelength.[5, 26] AIP, American Institute of Physics.

Fig. 7
Fig. 7

Measured (points) and calculated (curves) –1st-order efficiency of a G185M grating for nonpolarization plotted versus wavelength.[5, 26, 27] Efficiency models calculated for the nonconformal Mg F 2 layer (border 2, scale of 1.04, shift of 68.5 nm ) and RIs of Al and Mg F 2 are taken from different sources. AIP, American Institute of Physics.

Fig. 8
Fig. 8

Imaginary part of Mg F 2 RIs in the FUV taken from different sources.

Fig. 9
Fig. 9

Measured (points) and calculated (curves) –1st-order efficiency of a G185M grating for nonpolarization plotted versus wavelength. Efficiency models calculated for accurately derived and linearly scaled (lin. func. of Im[RI]) RIs and different geometry of a nonconformal Mg F 2 layer.

Tables (2)

Tables Icon

Table 1 Correspondence between Operator Notations in Two Papers

Tables Icon

Table 2 Mg F 2 RIs for Evaporated Thin Films with Layer Thicknesses 40 nm Derived from Efficiency Modeling

Equations (17)

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u j = S j ̂ [ v j ] D j ̂ [ u j ] S j ̌ [ v j + ] + D j ̌ [ u j + ] .
r = S 0 + [ v 0 ] D 0 + [ u 0 ] ,
t = D N [ u N + ] S N [ v N + ] ,
exp { i α m ( x j x n ) + i β m y j y n } = { E m , j + E m , n + if y j y n ; E m , j E m , n if y j < y n . } .
E m , j ± = exp ( i α m x j ± i β m y j ) ,
X = B q M , 1 M < B .
X Y = { M M if q = q ( M M ) B 1 if q = q 1 0 if q < q 1 } .
α m = k 0 cos θ + 2 π m d , m = 0 , ± 1 , ± 2 , ,
β m ( j ) = ( k j 2 α m 2 ) 1 2 ,
u refl ( x , y ) = α m < k 0 c m + exp ( i α m x + i β m ( 0 ) y ) + evanescent waves
R = α m < k 0 c m + 2 β m ( 0 ) β 0 ( 0 ) .
u transm ( x , y ) = α m < k N c m exp ( i α m x i β m ( N ) y ) + evanescent waves ,
T = ( κ N κ 0 ) 2 α m < k N c m 2 β m ( N ) β 0 ( 0 ) .
κ j = { 1 for TE polarization ( ν j ν j + 1 ) 2 = ε j ε j + 1 for TM polarization } .
R + T = 1 .
A = 1 ( R + T ) 0
I A = 1 β 0 ( 0 ) d [ Im Γ 0 u 0 + ( v 0 + ) * d s ( κ N κ 0 ) 2 Im Γ N 1 u N 1 ( v N 1 ) * d s ] .

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