Abstract

A numerical approach based on the boundary-element method (BEM) is described for the analysis of plane-wave diffraction from groove-type gratings. First, the diffraction problem is exactly analyzed as a two-medium boundary-value problem. Further, for metallic gratings, a simple method in which an approximate boundary condition using the surface impedance is combined with the BEM is proposed. Both cases of the TE- and TM-wave incidences are systematically formulated. Numerical examples are presented for dielectric holographic gratings and metallic Fourier gratings, and the validity of the BEM and the effectiveness of the surface-impedance approximation are confirmed.

© 1990 Optical Society of America

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  1. S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
    [CrossRef]
  2. D. E. Tremain, K. K. Mei, “Application of the unimoment method to scattering from periodic dielectric structures,” J. Opt. Soc. Am. 68, 775–783 (1978).
    [CrossRef]
  3. K. Yasuura, M. Tomita, “Numerical analysis of plane wave scattering from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J61-B, 662–669 (1978).
  4. K. C. Chang, V. Stah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
    [CrossRef]
  5. M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
    [CrossRef]
  6. J. Yamakita, K. Rokushima, “Scattering of plane waves from dielectric gratings with deep grooves,” Trans. Inst. Electron. Commun. Eng. Jpn. J66-B, 375–382 (1983).
  7. Y. Nakata, M. Koshiba, M. Suzuki, “Finite-element analysis of plane wave diffraction from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J69-C, 1503–1511 (1986).
  8. L. B. Mashev, E. K. Popov, E. G. Loewen, “Asymmetrical trapezoidal grating efficiency,” Appl. Opt. 26, 2864–2866 (1987).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. M. G. Moharam, T. K. Gaylord, G. T. Sincerbox, H. Werlich, B. Yung, “Diffraction characteristics of photoresist surface-relief gratings,” Appl. Opt. 23, 3214–3220 (1984).
    [CrossRef] [PubMed]
  11. K. J. Ilcisin, R. Fedosejevs, “Direct production of gratings on plastic substrates using 248-nm KrF laser radiation,” Appl. Opt. 26, 396–400 (1987).
    [CrossRef] [PubMed]
  12. G. M. Whitman, D. M. Leskiw, F. Schwering, “Rigorous theory of scattering by perfectly conducting periodic surfaces with trapezoidal height profile. TE and TM polarization,” J. Opt. Soc. Am. 70, 1495–1503 (1980).
    [CrossRef]
  13. P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
    [CrossRef]
  14. J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
    [CrossRef]
  15. M. Breidne, D. Maystre, “Variational theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. 72, 499–506 (1982).
    [CrossRef]
  16. S. L. Chuang, J. A. Kong, “Wave scattering and guidance by dielectric waveguides with periodic surfaces,” J. Opt. Soc. Am. 73, 669–679 (1983).
    [CrossRef]
  17. R. Petit, M. Cadilhac, “Form of the electromagnetic field in the groove region of a perfectly conducting echelette grating,” J. Opt. Soc. Am. 73, 963–965 (1983).
    [CrossRef]
  18. A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt. Soc. Am. 73, 1028–1041 (1983).
    [CrossRef]
  19. Y. Okuno, T. Matsuda, “Mode-matching method with a higher-order smoothing procedure for the numerical solution of diffraction by a grating,” J. Opt. Soc. Am. 73, 1305–1311 (1983).
    [CrossRef]
  20. P. Peterson, A. Gavrielides, “Power losses in lamellar gratings subject to mixed boundary conditions,” Appl. Opt. 23, 4045–4050 (1984).
    [CrossRef] [PubMed]
  21. M. Dahleh, R. Nevels, L. Tsang, “Plane-wave diffraction by a dielectric-coated corrugated surface,” J. Appl. Phys. 58, 646–650 (1985).
    [CrossRef]
  22. K. Yasuura, M. Murayama, “Numerical analysis of diffraction from a sinusoidal metal grating,” Trans. Inst. Commun. Eng. Jpn. J69-B, 198–205 (1986).
  23. D. Agassi, T. F. George, “Convergent scheme for light scattering from an arbitrary deep metallic grating,” Phys. Rev. B 33, 2393–2400 (1986).
    [CrossRef]
  24. M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
    [CrossRef]
  25. Y. Okuno, T. Matsuda, “Efficient technique for the numerical solution of diffraction by a Fourier grating,” J. Opt. Soc. Am. A 4, 465–472 (1987).
    [CrossRef]
  26. R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26, 2348–2354 (1987).
    [CrossRef] [PubMed]
  27. Y. Nakata, M. Koshiba, “Finite-element analysis of plane wave diffraction from metallic gratings with arbitrary complex permittivity,” Trans. Inst. Electron. Inform. Commun. Eng. J70-C, 1513–1522 (1987).
  28. M. K. Moaveni, “Diffraction characteristics of metallic reflection gratings,” Proc. Inst. Electr. Eng. Part J 135, 318–324 (1988).
  29. L.-J. Stanković, S. Jovićević, “Modified least squares method with application to diffraction and eigenvalue problems,” Proc. Inst. Electr. Eng. Part H 135, 339–343 (1988).
  30. E. N. Glytsis, T. K. Gaylord, “Antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief grating on a lossy substrate,” Appl. Opt. 27, 4288–4304 (1988).
    [CrossRef] [PubMed]
  31. T. Matsuda, Y. Okuno, “Diffraction efficiency of Fourier gratings,” Inst. Electron. Inform. Commun. Eng. Tech. Rept. AP88, 105 (1988).
  32. K. Hagiwara, M. Iida, H. Asakura, Y. Nagaoka, “Holographic Fourier gratings,” presented at the Institute of Electronics and Information Communication Engineers National Conference Record1987.
  33. N. F. Hartman, T. K. Gaylord. “Antireflection gold surface-relief gratings: experimental characteristics,” Appl. Opt. 27, 3738–3743 (1988).
    [CrossRef] [PubMed]
  34. R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
    [CrossRef]
  35. S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
    [CrossRef]
  36. M. Koshiba, M. Suzuki, “Application of the boundary-element method to waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 301–307 (1986).
    [CrossRef]
  37. H. A. Kalhor, A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,” J. Opt. Soc. Am. 61, 43–48 (1971).
    [CrossRef]
  38. D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
    [CrossRef]
  39. R. A. Depine, “Scattering of a wave at a periodic boundary: analytical expression for the surface impedance,” J. Opt. Soc. Am. A 5, 507–510 (1988).
    [CrossRef]

1988 (6)

M. K. Moaveni, “Diffraction characteristics of metallic reflection gratings,” Proc. Inst. Electr. Eng. Part J 135, 318–324 (1988).

L.-J. Stanković, S. Jovićević, “Modified least squares method with application to diffraction and eigenvalue problems,” Proc. Inst. Electr. Eng. Part H 135, 339–343 (1988).

T. Matsuda, Y. Okuno, “Diffraction efficiency of Fourier gratings,” Inst. Electron. Inform. Commun. Eng. Tech. Rept. AP88, 105 (1988).

R. A. Depine, “Scattering of a wave at a periodic boundary: analytical expression for the surface impedance,” J. Opt. Soc. Am. A 5, 507–510 (1988).
[CrossRef]

N. F. Hartman, T. K. Gaylord. “Antireflection gold surface-relief gratings: experimental characteristics,” Appl. Opt. 27, 3738–3743 (1988).
[CrossRef] [PubMed]

E. N. Glytsis, T. K. Gaylord, “Antireflection surface structure: dielectric layer(s) over a high spatial-frequency surface-relief grating on a lossy substrate,” Appl. Opt. 27, 4288–4304 (1988).
[CrossRef] [PubMed]

1987 (5)

1986 (5)

K. Yasuura, M. Murayama, “Numerical analysis of diffraction from a sinusoidal metal grating,” Trans. Inst. Commun. Eng. Jpn. J69-B, 198–205 (1986).

D. Agassi, T. F. George, “Convergent scheme for light scattering from an arbitrary deep metallic grating,” Phys. Rev. B 33, 2393–2400 (1986).
[CrossRef]

Y. Nakata, M. Koshiba, M. Suzuki, “Finite-element analysis of plane wave diffraction from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J69-C, 1503–1511 (1986).

M. Koshiba, M. Suzuki, “Application of the boundary-element method to waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 301–307 (1986).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Rigorous coupled-wave analysis of metallic surface-relief gratings,” J. Opt. Soc. Am. A 3, 1780–1787 (1986).
[CrossRef]

1985 (1)

M. Dahleh, R. Nevels, L. Tsang, “Plane-wave diffraction by a dielectric-coated corrugated surface,” J. Appl. Phys. 58, 646–650 (1985).
[CrossRef]

1984 (3)

1983 (6)

1982 (3)

1981 (1)

1980 (2)

1978 (3)

1975 (1)

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

1971 (1)

Agassi, D.

D. Agassi, T. F. George, “Convergent scheme for light scattering from an arbitrary deep metallic grating,” Phys. Rev. B 33, 2393–2400 (1986).
[CrossRef]

Asakura, H.

K. Hagiwara, M. Iida, H. Asakura, Y. Nagaoka, “Holographic Fourier gratings,” presented at the Institute of Electronics and Information Communication Engineers National Conference Record1987.

Bertoni, H. L.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Breidne, M.

Cadilhac, M.

Case, S. K.

Chandezon, J.

Chang, K. C.

Chuang, S. L.

Cornet, G.

Dahleh, M.

M. Dahleh, R. Nevels, L. Tsang, “Plane-wave diffraction by a dielectric-coated corrugated surface,” J. Appl. Phys. 58, 646–650 (1985).
[CrossRef]

Depine, R. A.

Dupuis, M. T.

Enger, R. C.

Fedosejevs, R.

Fukai, I.

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

Gavrielides, A.

Gaylord, T. K.

George, T. F.

D. Agassi, T. F. George, “Convergent scheme for light scattering from an arbitrary deep metallic grating,” Phys. Rev. B 33, 2393–2400 (1986).
[CrossRef]

Glytsis, E. N.

Hagiwara, K.

K. Hagiwara, M. Iida, H. Asakura, Y. Nagaoka, “Holographic Fourier gratings,” presented at the Institute of Electronics and Information Communication Engineers National Conference Record1987.

Hartman, N. F.

Iida, M.

K. Hagiwara, M. Iida, H. Asakura, Y. Nagaoka, “Holographic Fourier gratings,” presented at the Institute of Electronics and Information Communication Engineers National Conference Record1987.

Ilcisin, K. J.

Jovicevic, S.

L.-J. Stanković, S. Jovićević, “Modified least squares method with application to diffraction and eigenvalue problems,” Proc. Inst. Electr. Eng. Part H 135, 339–343 (1988).

Kagami, S.

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

Kalhor, H. A.

Kong, J. A.

Koshiba, M.

Y. Nakata, M. Koshiba, “Finite-element analysis of plane wave diffraction from metallic gratings with arbitrary complex permittivity,” Trans. Inst. Electron. Inform. Commun. Eng. J70-C, 1513–1522 (1987).

Y. Nakata, M. Koshiba, M. Suzuki, “Finite-element analysis of plane wave diffraction from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J69-C, 1503–1511 (1986).

M. Koshiba, M. Suzuki, “Application of the boundary-element method to waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 301–307 (1986).
[CrossRef]

Leskiw, D. M.

Loewen, E. G.

Mashev, L. B.

Matsuda, T.

Maystre, D.

Mei, K. K.

Moaveni, M. K.

M. K. Moaveni, “Diffraction characteristics of metallic reflection gratings,” Proc. Inst. Electr. Eng. Part J 135, 318–324 (1988).

Moharam, M. G.

Murayama, M.

K. Yasuura, M. Murayama, “Numerical analysis of diffraction from a sinusoidal metal grating,” Trans. Inst. Commun. Eng. Jpn. J69-B, 198–205 (1986).

Nagaoka, Y.

K. Hagiwara, M. Iida, H. Asakura, Y. Nagaoka, “Holographic Fourier gratings,” presented at the Institute of Electronics and Information Communication Engineers National Conference Record1987.

Nakata, Y.

Y. Nakata, M. Koshiba, “Finite-element analysis of plane wave diffraction from metallic gratings with arbitrary complex permittivity,” Trans. Inst. Electron. Inform. Commun. Eng. J70-C, 1513–1522 (1987).

Y. Nakata, M. Koshiba, M. Suzuki, “Finite-element analysis of plane wave diffraction from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J69-C, 1503–1511 (1986).

Neureuther, A. R.

Nevels, R.

M. Dahleh, R. Nevels, L. Tsang, “Plane-wave diffraction by a dielectric-coated corrugated surface,” J. Appl. Phys. 58, 646–650 (1985).
[CrossRef]

Okuno, Y.

Peng, S. T.

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Peterson, P.

Petit, R.

Popov, E. K.

Rokushima, K.

J. Yamakita, K. Rokushima, “Scattering of plane waves from dielectric gratings with deep grooves,” Trans. Inst. Electron. Commun. Eng. Jpn. J66-B, 375–382 (1983).

Schwering, F.

Sincerbox, G. T.

Stah, V.

Stankovic, L.-J.

L.-J. Stanković, S. Jovićević, “Modified least squares method with application to diffraction and eigenvalue problems,” Proc. Inst. Electr. Eng. Part H 135, 339–343 (1988).

Suzuki, M.

Y. Nakata, M. Koshiba, M. Suzuki, “Finite-element analysis of plane wave diffraction from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J69-C, 1503–1511 (1986).

M. Koshiba, M. Suzuki, “Application of the boundary-element method to waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 301–307 (1986).
[CrossRef]

Tamir, T.

K. C. Chang, V. Stah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Tomita, M.

K. Yasuura, M. Tomita, “Numerical analysis of plane wave scattering from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J61-B, 662–669 (1978).

Tremain, D. E.

Tsang, L.

M. Dahleh, R. Nevels, L. Tsang, “Plane-wave diffraction by a dielectric-coated corrugated surface,” J. Appl. Phys. 58, 646–650 (1985).
[CrossRef]

van den Berg, P. M.

Werlich, H.

Whitman, G. M.

Wirgin, A.

Yamakita, J.

J. Yamakita, K. Rokushima, “Scattering of plane waves from dielectric gratings with deep grooves,” Trans. Inst. Electron. Commun. Eng. Jpn. J66-B, 375–382 (1983).

Yasuura, K.

K. Yasuura, M. Murayama, “Numerical analysis of diffraction from a sinusoidal metal grating,” Trans. Inst. Commun. Eng. Jpn. J69-B, 198–205 (1986).

K. Yasuura, M. Tomita, “Numerical analysis of plane wave scattering from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J61-B, 662–669 (1978).

Yung, B.

Appl. Opt. (8)

IEEE Trans. Microwave Theory Tech. (3)

S. Kagami, I. Fukai, “Application of boundary-element method to electromagnetic field problems,” IEEE Trans. Microwave Theory Tech. MTT-32, 455–461 (1984).
[CrossRef]

M. Koshiba, M. Suzuki, “Application of the boundary-element method to waveguide discontinuities,” IEEE Trans. Microwave Theory Tech. MTT-34, 301–307 (1986).
[CrossRef]

S. T. Peng, T. Tamir, H. L. Bertoni, “Theory of periodic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. MTT-23, 123–133 (1975).
[CrossRef]

Inst. Electron. Inform. Commun. Eng. Tech. Rept. (1)

T. Matsuda, Y. Okuno, “Diffraction efficiency of Fourier gratings,” Inst. Electron. Inform. Commun. Eng. Tech. Rept. AP88, 105 (1988).

J. Appl. Phys. (1)

M. Dahleh, R. Nevels, L. Tsang, “Plane-wave diffraction by a dielectric-coated corrugated surface,” J. Appl. Phys. 58, 646–650 (1985).
[CrossRef]

J. Opt. Soc. Am. (13)

D. E. Tremain, K. K. Mei, “Application of the unimoment method to scattering from periodic dielectric structures,” J. Opt. Soc. Am. 68, 775–783 (1978).
[CrossRef]

K. C. Chang, V. Stah, T. Tamir, “Scattering and guiding of waves by dielectric gratings with arbitrary profiles,” J. Opt. Soc. Am. 70, 804–813 (1980).
[CrossRef]

M. G. Moharam, T. K. Gaylord, “Diffraction analysis of dielectric surface-relief gratings,” J. Opt. Soc. Am. 72, 1385–1392 (1982).
[CrossRef]

G. M. Whitman, D. M. Leskiw, F. Schwering, “Rigorous theory of scattering by perfectly conducting periodic surfaces with trapezoidal height profile. TE and TM polarization,” J. Opt. Soc. Am. 70, 1495–1503 (1980).
[CrossRef]

P. M. van den Berg, “Reflection by a grating: Rayleigh methods,” J. Opt. Soc. Am. 71, 1224–1229 (1981).
[CrossRef]

J. Chandezon, M. T. Dupuis, G. Cornet, D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839–846 (1982).
[CrossRef]

M. Breidne, D. Maystre, “Variational theory of diffraction gratings and its application to the study of ghosts,” J. Opt. Soc. Am. 72, 499–506 (1982).
[CrossRef]

S. L. Chuang, J. A. Kong, “Wave scattering and guidance by dielectric waveguides with periodic surfaces,” J. Opt. Soc. Am. 73, 669–679 (1983).
[CrossRef]

R. Petit, M. Cadilhac, “Form of the electromagnetic field in the groove region of a perfectly conducting echelette grating,” J. Opt. Soc. Am. 73, 963–965 (1983).
[CrossRef]

A. Wirgin, “Scattering from sinusoidal gratings: an evaluation of the Kirchhoff approximation,” J. Opt. Soc. Am. 73, 1028–1041 (1983).
[CrossRef]

Y. Okuno, T. Matsuda, “Mode-matching method with a higher-order smoothing procedure for the numerical solution of diffraction by a grating,” J. Opt. Soc. Am. 73, 1305–1311 (1983).
[CrossRef]

H. A. Kalhor, A. R. Neureuther, “Numerical method for the analysis of diffraction gratings,” J. Opt. Soc. Am. 61, 43–48 (1971).
[CrossRef]

D. Maystre, “A new general integral theory for dielectric coated gratings,” J. Opt. Soc. Am. 68, 490–495 (1978).
[CrossRef]

J. Opt. Soc. Am. A (3)

Phys. Rev. B (1)

D. Agassi, T. F. George, “Convergent scheme for light scattering from an arbitrary deep metallic grating,” Phys. Rev. B 33, 2393–2400 (1986).
[CrossRef]

Proc. Inst. Electr. Eng. Part H (1)

L.-J. Stanković, S. Jovićević, “Modified least squares method with application to diffraction and eigenvalue problems,” Proc. Inst. Electr. Eng. Part H 135, 339–343 (1988).

Proc. Inst. Electr. Eng. Part J (1)

M. K. Moaveni, “Diffraction characteristics of metallic reflection gratings,” Proc. Inst. Electr. Eng. Part J 135, 318–324 (1988).

Trans. Inst. Commun. Eng. Jpn. (1)

K. Yasuura, M. Murayama, “Numerical analysis of diffraction from a sinusoidal metal grating,” Trans. Inst. Commun. Eng. Jpn. J69-B, 198–205 (1986).

Trans. Inst. Electron. Commun. Eng. Jpn. (3)

K. Yasuura, M. Tomita, “Numerical analysis of plane wave scattering from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J61-B, 662–669 (1978).

J. Yamakita, K. Rokushima, “Scattering of plane waves from dielectric gratings with deep grooves,” Trans. Inst. Electron. Commun. Eng. Jpn. J66-B, 375–382 (1983).

Y. Nakata, M. Koshiba, M. Suzuki, “Finite-element analysis of plane wave diffraction from dielectric gratings,” Trans. Inst. Electron. Commun. Eng. Jpn. J69-C, 1503–1511 (1986).

Trans. Inst. Electron. Inform. Commun. Eng. (1)

Y. Nakata, M. Koshiba, “Finite-element analysis of plane wave diffraction from metallic gratings with arbitrary complex permittivity,” Trans. Inst. Electron. Inform. Commun. Eng. J70-C, 1513–1522 (1987).

Other (2)

K. Hagiwara, M. Iida, H. Asakura, Y. Nagaoka, “Holographic Fourier gratings,” presented at the Institute of Electronics and Information Communication Engineers National Conference Record1987.

R. Petit, Electromagnetic Theory of Gratings (Springer-Verlag, Berlin, 1980).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Geometry of groove grating.

Fig. 2
Fig. 2

Two-dimensional region Ω surrounded by boundary Γ.

Fig. 3
Fig. 3

Quadratic line element with three nodes.

Fig. 4
Fig. 4

Holographic grating of complicated shape fabricated by IBM10 (1 = 1.0, 2 = 1.642 = 2.6896, d = 458 nm).

Fig. 5
Fig. 5

Quadratic triangular element with six nodes for the FEM.

Fig. 6
Fig. 6

Element division of the holographic grating for the FEM (2360 triangular elements, 4879 nodes).

Fig. 7
Fig. 7

Element division of the holographic grating for the boundary-element method (input region: 170 line elements, 340 nodes; output region: 106 line elements, 212 nodes).

Fig. 8
Fig. 8

Wavelength dependence of diffraction efficiency at the fixed incident angle θ = 36.86 deg.

Fig. 9
Fig. 9

Incident-angle dependence of diffraction efficiency at the fixed wavelength λ = 458 nm.

Fig. 10
Fig. 10

Wavelength dependence of diffraction efficiency under the Bragg condition 2d sin θ = λ.

Fig. 11
Fig. 11

Surface profile of Fourier gratings.

Fig. 12
Fig. 12

Incident-angle dependence of relative reflected powers for the Fourier grating.

Fig. 13
Fig. 13

Groove-depth dependence of relative reflected powers for the Fourier grating.

Equations (43)

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1 p ( 2 φ x 2 + 2 φ y 2 ) + k 0 2 q φ = 0 ,
k 0 = ω μ 0 0 ,
φ = E z , p = 1 , q = for TE waves ,
φ = H z , p = , q = 1 for TM waves .
φ | Γ 4 = ξ φ | Γ 3 ,
1 p φ x | Γ 4 = ξ 1 p φ x | Γ 3 ,
ξ = exp ( j 1 k 0 d sin ϑ ) .
φ i + Γ G i n φ d Γ = Γ φ n G i d Γ ,
G i = 1 4 j H 0 ( 2 ) ( k | r r i | ) ,
G i n = k 4 j H 1 ( 2 ) ( k | r r i | ) cos ρ .
k = k 0 .
ϑ i 2 π φ i + Γ G i n φ d Γ = Γ φ n G i d Γ ,
φ = { N } T { φ } e ,
φ n = { N } T { φ n } e ,
{ φ } e = [ φ 1 φ 2 φ 3 ] T ,
{ φ n } e = [ φ n | 1 φ n | 2 φ n | 3 ] T ,
{ N } = [ N 1 N 2 N 3 ] T .
[ H ] { φ } = [ G ] { φ n } ,
h i j = ϑ i 2 π δ i j + e Γ G i n N j d Γ ,
g i j = e Γ G i N j d Γ .
φ ( x , y = y i ) = δ i 1 2 exp ( j κ 10 y 1 ) f 10 ( x ) n = [ 1 j κ i n f i n ( x ) 0 d f i n * ( x ) 1 p i φ ( x , y ) n | y = y i d x ] ,
f i n ( x ) = ( p i d ) 1 / 2 exp ( j β n x ) , n = 0 , ± 1 , ± 2 , ,
β n = ( 1 ) 1 / 2 k 0 sin ϑ + 2 n π / d , n = 0 , ± 1 , ± 2 , ,
κ i n = ( k 0 2 i β n 2 ) 1 / 2 , Im ( κ i n ) 0 , n = 0 , ± 1 , ± 2 , .
{ φ } i = δ i 1 { f } 1 [ Z ] i { φ n } i ,
{ f } 1 = 2 exp ( j κ 10 y 1 ) { f 0 } 1 ,
[ Z ] i = n = [ 1 j κ i n p i { f n } i e e f i n * ( x ) { N } i T d x ] .
[ H ] ( I ) { φ } ( I ) = [ G ] ( I ) { φ n } ( I ) I = 1 , 2 ,
{ φ } 4 ( I ) = ξ { φ } 3 ( I ) , I = 1 , 2 ,
{ φ n } 4 ( I ) = ξ { φ n } 3 ( I ) , I = 1 , 2 ,
{ φ } 5 ( 1 ) = { φ } 5 ( 2 ) ,
{ φ n } 5 ( 1 ) = { φ n } 5 ( 2 ) for TE waves ,
{ φ n } 5 ( 1 ) = 1 2 { φ n } 5 ( 2 ) for TM waves .
P n r = κ 1 n κ 10 | 0 d g 1 n * ( x ) { N } 1 T d x { φ } 1 δ n 0 exp ( j κ 10 y 1 ) | 2 ,
P n t = κ 2 n κ 10 | 0 d g 2 n * ( x ) { N } 2 T d x { φ } 2 | 2 ,
g i n ( x ) = ( 1 p i d ) 1 / 2 exp ( j β n x ) , i = 1 , 2.
Z m = [ μ 0 / ( 0 2 ) ] 1 / 2 .
{ φ n } 5 ( 1 ) = j ω μ 0 Z m { φ } 5 ( 1 ) for TE waves ,
{ φ n } 5 ( 1 ) = j ω 0 Z m { φ } 5 ( 1 ) for TM waves .
{ φ } 5 ( 1 ) = 0 for TE waves .
{ φ n } 5 ( 1 ) = 0 for TM waves .
2 d sin ϑ = m λ for m th-order wave .
υ = η ( u ) = h [ sin ( 2 π u / d ) + γ sin ( 4 π u / d + δ ) ] ,

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