Abstract

We consider lamellar gratings made of dielectric or lossy materials used in classical diffraction mounts. We show how the modal diffraction formulation may be generalized to deal with slanted lamellar gratings and illustrate the accuracy and versatility of the new method through study of highly slanted gratings in a homogenization limit. We also comment on the completeness of the eigenmode basis and present tests enabling this completeness to be verified numerically.

© 2008 Optical Society of America

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  1. M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).
  2. E. Popov, M. Neviere, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
    [CrossRef]
  3. N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express 15, 11427-11432 (2007).
    [CrossRef] [PubMed]
  4. J. Chandezon, M. T. Dupuis, G. Cornet, and D. Maystre, “Multicoated gratings: a differential formalism applicable in the entire optical region,” J. Opt. Soc. Am. 72, 839-846 (1982).
    [CrossRef]
  5. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811-818 (1981).
    [CrossRef]
  6. L. Li, “Oblique-coordinate system-based Chandezon method for modeling one-dimensional periodic, multilayer, inhomogeneous, anisotropic gratings,” J. Opt. Soc. Am. A 16, 2521-2531 (1999).
    [CrossRef]
  7. T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073-1080 (1997).
    [CrossRef]
  8. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
    [CrossRef]
  9. P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
    [CrossRef]
  10. J. Y. Suratteau, M. Cadilhac, and R. Petit, “On the numerical study of deep dielectric lamellar gratings,” J. Opt. (Paris) 14, 273-288 (1983).
    [CrossRef]
  11. L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
    [CrossRef]
  12. L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
    [CrossRef]
  13. A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511-538 (1987).
    [CrossRef]
  14. L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479-1488 (1985).
    [CrossRef]
  15. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
    [CrossRef]
  16. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 16, 2581-2591 (1993).
    [CrossRef]
  17. J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
    [CrossRef]
  18. S. Kaushik, “Vector Fresnel equations and Airy formula for one-dimensional multilayer and surface relief gratings,” J. Opt. Soc. Am. A 14, 596-609 (1997).
    [CrossRef]
  19. S. Campbell, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
    [CrossRef]
  20. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for the problem of extraordinary light transmission through subwavelength holes,” EPL 80, 24002 (2007).
    [CrossRef]
  21. B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
    [CrossRef]
  22. G. Granet, J. Chandezon, and O. Coudert, “Extension of the C method to nonhomogeneous media: application to nonhomogeneous layers with parallel modulated faces and to inclined lamellar gratings,” J. Opt. Soc. Am. A 14, 1576-1582 (1997).
    [CrossRef]
  23. β2 here is the square of the modes propagation constant, which we denote μ2.
  24. D. J. Bergman, “The dielectric constant of a composite material--A problem in classical physics,” Phys. Rep. 43, 377-407 (1978).
    [CrossRef]
  25. G. W. Milton, “Bounds using the analytic method,” in The Theory of Composites (Cambridge U. Press, 2002), pp. 569-589.
    [CrossRef]
  26. L. C. Botten, R. C. McPhedran, and G. W. Milton, “Perfectly Conducting Lamellar Gratings: Babinets Principle and Circuit Models,” J. Mod. Opt. 42, 2453-2473 (1995).
    [CrossRef]
  27. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
    [CrossRef]
  28. H. Hoffman, “Relative convergence in mode-matching solutions of mircrostrip problems,” Electron. Lett. 10, 126-127 (1974).
    [CrossRef]
  29. T. Itoh and R. Mittra, “Relative convergence phenomenon arising in the solution of diffraction from strip grating on a dielectric slab,” Proc. IEEE 59, 1363-1365 (1971).
    [CrossRef]
  30. J. P. Plumey, B. Guizal, and J. Chandezon, “Coordinate transformation method as applied to asymmetric gratings with vertical facets,” J. Opt. Soc. Am. A 14, 610-617 (1997).
    [CrossRef]
  31. O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133-1135 (2001).
    [CrossRef] [PubMed]
  32. O. Toader and S. John, “Square spiral photonic crystals: Robust architecture for microfabrication of materials with large three-dimensional photonic band gaps,” Phys. Rev. E 66, 016610 (2002).
    [CrossRef]
  33. D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
    [CrossRef]
  34. A. Dirks and H. Leamy, “Columnar microstructure in vapor-deposited thin-films,” Thin Solid Films 47, 219-233 (1977).
    [CrossRef]

2007 (5)

S. Campbell, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for the problem of extraordinary light transmission through subwavelength holes,” EPL 80, 24002 (2007).
[CrossRef]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[CrossRef]

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

N. Bonod, E. Popov, L. Li, and B. Chernov, “Unidirectional excitation of surface plasmons by slanted gratings,” Opt. Express 15, 11427-11432 (2007).
[CrossRef] [PubMed]

2004 (1)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

2003 (1)

M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).

2002 (2)

E. Popov, M. Neviere, B. Gralak, and G. Tayeb, “Staircase approximation validity for arbitrary-shaped gratings,” J. Opt. Soc. Am. A 19, 33-42 (2002).
[CrossRef]

O. Toader and S. John, “Square spiral photonic crystals: Robust architecture for microfabrication of materials with large three-dimensional photonic band gaps,” Phys. Rev. E 66, 016610 (2002).
[CrossRef]

2001 (1)

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133-1135 (2001).
[CrossRef] [PubMed]

1999 (1)

1997 (4)

1995 (1)

L. C. Botten, R. C. McPhedran, and G. W. Milton, “Perfectly Conducting Lamellar Gratings: Babinets Principle and Circuit Models,” J. Mod. Opt. 42, 2453-2473 (1995).
[CrossRef]

1994 (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

1993 (2)

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 16, 2581-2591 (1993).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

1987 (1)

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511-538 (1987).
[CrossRef]

1985 (1)

L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479-1488 (1985).
[CrossRef]

1983 (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, “On the numerical study of deep dielectric lamellar gratings,” J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

1982 (2)

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

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

1981 (4)

M. G. Moharam and T. K. Gaylord, “Rigorous coupled-wave analysis of planar-grating diffraction,” J. Opt. Soc. Am. 71, 811-818 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

1978 (1)

D. J. Bergman, “The dielectric constant of a composite material--A problem in classical physics,” Phys. Rep. 43, 377-407 (1978).
[CrossRef]

1977 (1)

A. Dirks and H. Leamy, “Columnar microstructure in vapor-deposited thin-films,” Thin Solid Films 47, 219-233 (1977).
[CrossRef]

1974 (1)

H. Hoffman, “Relative convergence in mode-matching solutions of mircrostrip problems,” Electron. Lett. 10, 126-127 (1974).
[CrossRef]

1971 (1)

T. Itoh and R. Mittra, “Relative convergence phenomenon arising in the solution of diffraction from strip grating on a dielectric slab,” Proc. IEEE 59, 1363-1365 (1971).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Asatryan, A. A.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Bergman, D. J.

D. J. Bergman, “The dielectric constant of a composite material--A problem in classical physics,” Phys. Rep. 43, 377-407 (1978).
[CrossRef]

Bonod, N.

Botten, L. C.

S. Campbell, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, R. C. McPhedran, and G. W. Milton, “Perfectly Conducting Lamellar Gratings: Babinets Principle and Circuit Models,” J. Mod. Opt. 42, 2453-2473 (1995).
[CrossRef]

L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479-1488 (1985).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

Bur, J.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Cadilhac, M.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “On the numerical study of deep dielectric lamellar gratings,” J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

Campbell, S.

S. Campbell, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

Chandezon, J.

Chang, A.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Chernov, B.

Cornet, G.

Coudert, O.

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

Davidson, M. P.

M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).

de Sterke, C. M.

S. Campbell, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Dirks, A.

A. Dirks and H. Leamy, “Columnar microstructure in vapor-deposited thin-films,” Thin Solid Films 47, 219-233 (1977).
[CrossRef]

Dupuis, M. T.

Gaylord, T. K.

Gorkunov, M.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for the problem of extraordinary light transmission through subwavelength holes,” EPL 80, 24002 (2007).
[CrossRef]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[CrossRef]

Gralak, B.

Granet, G.

Guizal, B.

Harris, J. B.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073-1080 (1997).
[CrossRef]

Hoffman, H.

H. Hoffman, “Relative convergence in mode-matching solutions of mircrostrip problems,” Electron. Lett. 10, 126-127 (1974).
[CrossRef]

Itoh, T.

T. Itoh and R. Mittra, “Relative convergence phenomenon arising in the solution of diffraction from strip grating on a dielectric slab,” Proc. IEEE 59, 1363-1365 (1971).
[CrossRef]

John, S.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

O. Toader and S. John, “Square spiral photonic crystals: Robust architecture for microfabrication of materials with large three-dimensional photonic band gaps,” Phys. Rev. E 66, 016610 (2002).
[CrossRef]

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133-1135 (2001).
[CrossRef] [PubMed]

Kaushik, S.

Langtry, T. N.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Leamy, H.

A. Dirks and H. Leamy, “Columnar microstructure in vapor-deposited thin-films,” Thin Solid Films 47, 219-233 (1977).
[CrossRef]

Li, L.

Lin, S.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Lu, T.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Maystre, D.

McPhedran, R. C.

S. Campbell, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, R. C. McPhedran, and G. W. Milton, “Perfectly Conducting Lamellar Gratings: Babinets Principle and Circuit Models,” J. Mod. Opt. 42, 2453-2473 (1995).
[CrossRef]

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511-538 (1987).
[CrossRef]

L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479-1488 (1985).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

Miller, J. M.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Milton, G. W.

L. C. Botten, R. C. McPhedran, and G. W. Milton, “Perfectly Conducting Lamellar Gratings: Babinets Principle and Circuit Models,” J. Mod. Opt. 42, 2453-2473 (1995).
[CrossRef]

G. W. Milton, “Bounds using the analytic method,” in The Theory of Composites (Cambridge U. Press, 2002), pp. 569-589.
[CrossRef]

Mittra, R.

T. Itoh and R. Mittra, “Relative convergence phenomenon arising in the solution of diffraction from strip grating on a dielectric slab,” Proc. IEEE 59, 1363-1365 (1971).
[CrossRef]

Moharam, M. G.

Neviere, M.

Noponen, E.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Petit, R.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “On the numerical study of deep dielectric lamellar gratings,” J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

Plumey, J. P.

Podivilov, E.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for the problem of extraordinary light transmission through subwavelength holes,” EPL 80, 24002 (2007).
[CrossRef]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[CrossRef]

Popov, E.

Preist, T. W.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073-1080 (1997).
[CrossRef]

Roberts, A.

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511-538 (1987).
[CrossRef]

Sambles, J. R.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073-1080 (1997).
[CrossRef]

Sanda, P. N.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Sheng, P.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Stepleman, R. S.

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Sturman, B.

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for the problem of extraordinary light transmission through subwavelength holes,” EPL 80, 24002 (2007).
[CrossRef]

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[CrossRef]

Suratteau, J. Y.

J. Y. Suratteau, M. Cadilhac, and R. Petit, “On the numerical study of deep dielectric lamellar gratings,” J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

Taghizadeh, M. R.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Tayeb, G.

Toader, O.

O. Toader and S. John, “Square spiral photonic crystals: Robust architecture for microfabrication of materials with large three-dimensional photonic band gaps,” Phys. Rev. E 66, 016610 (2002).
[CrossRef]

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133-1135 (2001).
[CrossRef] [PubMed]

Turunen, J.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Vasara, A.

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Wang, R.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Wanstall, N. P.

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073-1080 (1997).
[CrossRef]

White, T. P.

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Yang, Z.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Ye, D.

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Electron. Lett. (1)

H. Hoffman, “Relative convergence in mode-matching solutions of mircrostrip problems,” Electron. Lett. 10, 126-127 (1974).
[CrossRef]

EPL (1)

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for the problem of extraordinary light transmission through subwavelength holes,” EPL 80, 24002 (2007).
[CrossRef]

J. Mod. Opt. (5)

L. C. Botten, R. C. McPhedran, and G. W. Milton, “Perfectly Conducting Lamellar Gratings: Babinets Principle and Circuit Models,” J. Mod. Opt. 42, 2453-2473 (1995).
[CrossRef]

T. W. Preist, J. B. Harris, N. P. Wanstall, and J. R. Sambles, “Optical response of blazed and overhanging gratings using oblique Chandezon transformations,” J. Mod. Opt. 44, 1073-1080 (1997).
[CrossRef]

A. Roberts and R. C. McPhedran, “Power losses in highly conducting lamellar gratings,” J. Mod. Opt. 34, 511-538 (1987).
[CrossRef]

L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
[CrossRef]

M. P. Davidson, “A modal model for diffraction gratings,” J. Mod. Opt. 50, 1817-1834 (2003).

J. Opt. (Paris) (1)

J. Y. Suratteau, M. Cadilhac, and R. Petit, “On the numerical study of deep dielectric lamellar gratings,” J. Opt. (Paris) 14, 273-288 (1983).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (6)

J. Phys. D (1)

D. Ye, Z. Yang, A. Chang, J. Bur, S. Lin, T. Lu, R. Wang, and S. John, “Experimental realization of a well-controlled 3D silicon spiral photonic crystal,” J. Phys. D 40, 2624-2628 (2007).
[CrossRef]

Opt. Acta (4)

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087-1102 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, and J. L. Adams, “Highly conducting lamellar diffraction gratings,” Opt. Acta 28, 1103-1106 (1981).
[CrossRef]

L. C. Botten, M. S. Craig, R. C. McPhedran, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[CrossRef]

L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479-1488 (1985).
[CrossRef]

Opt. Commun. (1)

J. M. Miller, J. Turunen, E. Noponen, A. Vasara, and M. R. Taghizadeh, “Rigorous modal theory for multiply grooved lamellar gratings,” Opt. Commun. 111, 526-535 (1994).
[CrossRef]

Opt. Express (1)

Phys. Rep. (1)

D. J. Bergman, “The dielectric constant of a composite material--A problem in classical physics,” Phys. Rep. 43, 377-407 (1978).
[CrossRef]

Phys. Rev. B (2)

B. Sturman, E. Podivilov, and M. Gorkunov, “Eigenmodes for metal-dielectric light-transmitting nanostructures,” Phys. Rev. B 76, 125104 (2007).
[CrossRef]

P. Sheng, R. S. Stepleman, and P. N. Sanda, “Exact eigenfunctions for square wave gratings--application to diffraction and surface-plasmon calculations,” Phys. Rev. B 26, 2907-2916 (1982).
[CrossRef]

Phys. Rev. E (2)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, “Bloch mode scattering matrix methods for modeling extended PC structures,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

O. Toader and S. John, “Square spiral photonic crystals: Robust architecture for microfabrication of materials with large three-dimensional photonic band gaps,” Phys. Rev. E 66, 016610 (2002).
[CrossRef]

Proc. IEEE (1)

T. Itoh and R. Mittra, “Relative convergence phenomenon arising in the solution of diffraction from strip grating on a dielectric slab,” Proc. IEEE 59, 1363-1365 (1971).
[CrossRef]

Science (1)

O. Toader and S. John, “Proposed square spiral microfabrication architecture for large three-dimensional photonic band gap crystals,” Science 292, 1133-1135 (2001).
[CrossRef] [PubMed]

Thin Solid Films (1)

A. Dirks and H. Leamy, “Columnar microstructure in vapor-deposited thin-films,” Thin Solid Films 47, 219-233 (1977).
[CrossRef]

Waves Random Complex Media (1)

S. Campbell, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Fresnel formulation for multi-element lamellar diffraction gratings in conical mountings,” Waves Random Complex Media 17, 455-475 (2007).
[CrossRef]

Other (2)

β2 here is the square of the modes propagation constant, which we denote μ2.

G. W. Milton, “Bounds using the analytic method,” in The Theory of Composites (Cambridge U. Press, 2002), pp. 569-589.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Geometry of the slanted lamellar grating. The subscripted ϵ terms are dielectric constants, θ is the slant angle, ϕ is the incident angle, d is the period, f is the fill fraction, and h is the thickness of the slanted lamellar grating.

Fig. 2
Fig. 2

One vertical period of the slanted lamellar grating is shown. Vectors c 1 ± denote modal amplitudes of basis 1 in region 1, and vectors c 2 ± denote modal amplitudes of basis 2 in region 2. R 12 and T 12 denote Fresnel reflection and transmission matrices.

Fig. 3
Fig. 3

Slanted lamellar grating is formed from a stack of padded cells. The Fresnel matrices R 12 , T 12 , R 21 , and T 21 are for the interface between air and a semi-infinite lamellar grating.

Fig. 4
Fig. 4

Schematic periods of slanted lamellar gratings for (a) no slant, (b) θ = 45 ° , (c) θ = 63.4 ° , (d) θ = 85.2 ° .

Fig. 5
Fig. 5

Energy reflected from a lamellar grating under normal E z incidence versus slant angle for f = 0.5 , d = 1 , h = 0.5 , ϵ 1 = 1 , ϵ 2 = 25 ( n 2 = 5 ) , λ = 1.1 and the grating suspended in air. Angles where tan θ increases by 1 are shown as vertical lines, from tan ( 85.2 ° ) = 12 up to tan ( 87.7 ° ) = 26 .

Fig. 6
Fig. 6

Energy reflected from a lamellar grating under normal H z (solid curve) and E z (dashed curve) incidence versus slant angle θ, with f = 0.5 , d = 1 , h = 0.5 , ϵ 1 = 1 , ϵ 2 = 44.9757 + 2.9524 i ( n 2 = 0.22 + i 6.71 ) , λ = 1.1 and the grating suspended in air.

Fig. 7
Fig. 7

Energy absorbed by a lamellar grating under normal H z incidence as a function of slant angle θ and height h, where f = 0.5 , d = 1 , ϵ 1 = 1 , ϵ 2 = 44.9757 + 2.9524 i ( n 2 = 0.22 + i 6.71 ) , λ = 1.1 and the grating suspended in air. The maximum absorptance (lighter fringes) is 0.517, and the minimum is 0.017.

Tables (3)

Tables Icon

Table 1 Truncation Order a as a Function of ϵ

Tables Icon

Table 2 Eigenvalue Behavior as ϵ Approaches Zero

Tables Icon

Table 3 Comparison of Efficiencies Calculated Using the DMM and the C Method a

Equations (111)

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cos ( β m c ) cos ( γ m g ) 1 2 ( β m ϵ 2 γ m ϵ 1 + γ m ϵ 1 β m ϵ 2 ) sin ( β m c ) sin ( γ m g ) = cos ( α 0 d ) ,
γ m 2 = β m 2 + k 0 2 ( ϵ 2 ϵ 1 ) ,
u m ( x + d ) = u m ( x ) exp ( i α 0 d ) , α 0 = 2 π sin ( ϕ ) λ ,
w p ± ( x , y ) = 1 d exp ( i α p x ) exp ( ± i χ p y ) ,
α p = α 0 + 2 π p d ,
χ p = { ( k 0 2 α p 2 ) for α p k 0 i ( α p 2 k 0 2 ) for α p > k 0 } .
u m ( x + d ) ¯ = u m ( x ) ¯ exp ( i α 0 d ) .
d u l ( d ) ¯ u m ( x ) d x = δ l m ,
P p l = 1 d d exp ( i α p x ) u l ( x ) d x .
1 d exp ( i α p x ) = m P p m ¯ u m ( x ) ,
1 d d exp ( i ( α p α q ) x ) d x = δ p q = m P p m ¯ P q m ,
P P H = I .
f m ( x ) + [ k 0 2 ϵ ( x ) μ m 2 ] f m ( x ) = 0 ,
d u n A ( x ) u m ( x ) d x = δ n m ,
K p l = 1 d d exp ( i α p x ) u l A ( x ) d x .
δ p q = m K p m P q m .
A p l = 1 d d exp ( i α p x ) ϵ ( x ) u l ( x ) d x ,
δ p q = m K p m A q m .
E z = m μ m 1 2 [ c m exp ( i μ m y ) + c m + exp ( i μ m y ) ] u m ( x ) ,
K x = m 1 k 0 μ m 1 2 [ c m exp ( i μ m y ) c m + exp ( i μ m y ) ] u m ( x ) ,
u 1 m ( x ) = u 2 m ( x δ x ) ,
u 1 m A ( x ) = u 2 m A ( x δ x ) .
m μ m 1 2 ( c 1 m + c 1 m + ) d u 1 m ( x ) u 1 l A ( x ) d x = m μ m 1 2 ( c 2 m + c 2 m + ) d u 2 m ( x ) u 1 l A ( x ) d x ,
μ 1 2 ( c 1 + c 1 + ) = K μ 1 2 ( c 2 + c 2 + ) ,
K l m = d u 1 l A ( x ) u 2 m ( x ) d x .
J μ 1 2 ( c 1 c 1 + ) = μ 1 2 ( c 2 c 2 + ) ,
J l m = d u 2 l A ( x ) u 1 m ( x ) d x .
J ( α 0 ) = K T ( α 0 ) .
R 12 = ( A B + I ) 1 ( A B I ) ,
T 12 = 2 B ( I + A B ) 1 ,
A = μ 1 2 K ( α 0 ) μ 1 2 ,
B = μ 1 2 K T ( α 0 ) μ 1 2 .
R 21 = ( I + B A ) 1 ( I B A ) ,
T 21 = 2 A ( I + B A ) 1 .
R 21 T ( α 0 ) = R 21 ( α 0 ) ,
R 12 T ( α 0 ) = R 12 ( α 0 ) ,
T 21 T ( α 0 ) = T 21 ( α 0 ) .
u 1 m ( x ) = u 2 m ( x δ x ) ,
= u 2 m ( x ) u 2 m ( x ) δ x ,
K ( α 0 ) = I V ( α 0 ) δ x ,
V l m = d [ d d x u 2 l ( α 0 , x ) ] u 2 m ( α 0 , x ) d x .
A = I + M A δ x M A = μ 1 2 V ( α 0 ) μ 1 2 ,
B = I + M B δ x M B = μ 1 2 V T ( α 0 ) μ 1 2 .
T interface = [ T 12 R 21 T 21 1 R 12 R 21 T 21 1 T 21 1 R 12 T 21 1 ] .
T interface = I + [ M a M s M s M a ] δ x + O ( δ x 2 ) ,
M s = 1 2 ( M B + M A ) ,
M a = 1 2 ( M B M A ) .
T pad = [ Q 0 0 Q 1 ] ( I + [ M a M s M s M a ] δ x ) [ Q 0 0 Q 1 ] ,
T pad = I + M δ y ,
M = i [ μ 0 0 μ ] + tan θ [ M a M s M s M a ] .
T N = ( I + M δ y ) N .
T = lim N ( I + M δ y ) N = lim N ( I + h N M ) N = exp ( h M ) .
T = X exp ( h Λ ) X 1 .
X = [ F F F + F + ] ,
T = [ F F F + F + ] [ exp ( h Λ ) 0 0 exp ( h Λ ) ] [ F F F + F + ] 1 .
T = [ F F F + F + ] [ exp ( h Λ ) 0 0 exp ( h Λ ) ] [ F 1 0 0 F + 1 ] × [ I R R I ] [ ( I R R ) 1 0 0 ( I R R ) 1 ] ,
T = [ T R T 1 R R T 1 T 1 R T 1 ] ,
R = ( R P R P ) ( I R P R P ) 1 ,
T = ( I R R ) P ( I R P R P ) 1 ,
P = F exp ( h Λ ) F 1 ,
P = F + exp ( h Λ ) F + 1 .
R = ( R P R P ) ( I R P R P ) 1 ,
T = ( I R R ) P ( I R P R P ) 1 .
R 23 = R ,
T 23 = T ,
R 32 = R ,
T 32 = T .
E z = p χ p 1 2 [ e p exp ( i χ p y ) + e p + exp ( i χ p y ) ] ( 1 d ) exp ( i α p x ) ,
K x = 1 k 0 p χ p 1 2 [ e p exp ( i χ p y ) + e p + exp ( i χ p y ) ] ( 1 d ) exp ( i α p x ) ,
Y μ 1 2 ( c + c + ) = χ 1 2 ( e + e + ) ,
Y q m = d ( 1 d ) exp ( i α q x ) u m ( x ) d x .
μ 1 2 ( c c + ) = Z T χ 1 2 ( e e + ) ,
Z p m = d ( 1 d ) exp ( i α p x ) u m A ( x ) d x .
R 12 = ( G H + I ) 1 ( G H I ) ,
T 12 = 2 H ( I + G H ) 1 ,
G = χ 1 2 Y μ 1 2 ,
H = μ 1 2 Z T χ 1 2 .
R 21 = ( I + H G ) 1 ( I H G ) ,
T 21 = 2 G ( I + H G ) 1 .
R 14 = R 13 + T 31 R 34 ( I R 31 R 34 ) 1 T 13 ,
T 14 = T 34 ( I R 31 R 34 ) 1 T 13 ,
R 13 = R 12 + T 21 R 23 ( I R 21 R 23 ) 1 T 12 ,
T 13 = T 23 ( I R 21 R 23 ) 1 T 12 ,
R 31 = R 32 + T 23 R 21 ( I R 23 R 21 ) 1 T 32 ,
T 31 = T 21 ( I R 23 R 21 ) 1 T 32 .
d 1 ϵ ( x ) u n A ( x ) u m ( x ) d x = δ n m .
K z = m μ m 1 2 [ c m exp ( i μ m y ) c m + exp ( i μ m y ) ] u m ( x ) ,
E x = m 1 k 0 μ m 1 2 [ c m exp ( i μ m y ) + c m + exp ( i β m y ) ] u m ( x ) ϵ ( x ) .
μ 1 2 ( c 1 c 1 + ) = K μ 1 2 ( c 2 c 2 + ) ,
K l m = d 1 ϵ 1 ( x ) u 1 l A ( x ) u 2 m ( x ) d x .
J μ 1 2 ( c 1 + c 1 + ) = μ 1 2 ( c 2 + c 2 ) ,
J l m = d 1 ϵ 1 ( x ) u 2 l A ( x ) u 1 m ( x ) d x .
R 12 = ( I + A B ) 1 ( I A B ) ,
T 12 = 2 B ( I + A B ) 1 ,
R 21 = ( I + B A ) 1 ( B A I ) ,
T 21 = 2 A ( I + B A ) 1 .
u 2 m ( x ) = u 1 m ( x + δ x ) ,
= u 1 m ( x ) + u 1 m ( x ) δ x ,
K ( α 0 ) = I + V ( α 0 ) δ x ,
V ( α 0 ) = d 1 ϵ 1 ( x ) u 1 l ( α 0 , x ) [ d d x u 1 m ( α 0 , x ) ] d x .
T interface = [ I + M a δ x M s δ x M s δ x I + M a δ x ] ,
M A = μ 1 2 V ( α 0 ) μ 1 2 ,
M B = μ 1 2 V T ( α 0 ) μ 1 2 ,
M a = 1 2 ( M B M A ) ,
M s = 1 2 ( M B + M A ) .
K z = p χ p 1 2 [ f p exp ( i χ p y ) f p + exp ( i χ p y ) ] ( 1 d ) exp ( i α p x ) ,
E x = 1 k 0 p χ p 1 2 [ f p exp ( i χ p y ) + f p + exp ( i χ p y ) ] ( 1 d ) exp ( i α p x ) .
Y χ 1 2 ( f f + ) = μ 1 2 ( c c + ) ,
Y m p = d ( 1 d ) exp ( i α p x ) [ 1 ϵ 1 ( x ) ] u 1 m A ( x ) d x .
χ 1 2 ( f + f + ) = Z T μ 1 2 ( c + c + ) ,
Z m q = d 1 d 1 ϵ 1 ( x ) u 1 m ( x ) exp ( i α q x ) d x .

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