Abstract

Modal methods often used to model lamellar gratings that include infinitely or highly conducting metallic parts encounter numerical instabilities in some situations. In this paper, the origin of these numerical instabilities is determined, and then a stable algorithm solving this problem is proposed. In order to complete this analysis, the different geometries that can be handled without numerical instabilities are clearly defined. Numerical tests of the exact modal method implemented with the proposed solution are also presented. A test of convergence shows the efficiency of the method while the comparison with the fictitious sources method shows its accuracy.

© 2008 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. 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]
  2. 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]
  3. L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction grating,” Opt. Acta 28, 1103-1106 (1981).
    [Crossref]
  4. M. G. Moharam and T. K. Gaylord, “Rigorous coupled-waves analysis of metallic surface-relief grating,” J. Opt. Soc. Am. A 3, 1780-1787 (1986).
    [Crossref]
  5. T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
    [Crossref]
  6. S. Enoch, M. Nevière, E. Popov, and R. Reinisch, “Enhanced light transmission by hole arrays,” J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
    [Crossref]
  7. S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
    [Crossref] [PubMed]
  8. P. Andrew and W. L. Barnes, “Molecular fluorescence above metallic gratings,” Phys. Rev. B 64, 125405 (2001).
    [Crossref]
  9. J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83, 30 (2003).
    [Crossref]
  10. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
    [Crossref]
  11. M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
    [Crossref] [PubMed]
  12. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
    [Crossref] [PubMed]
  13. L. Li, “Formulation and comparison of two recursive matrix algorithms for modeling layered diffraction gratings,” J. Opt. Soc. Am. A 13, 1024-1035 (1996).
    [Crossref]
  14. S. Campbell, L. C. Botten, C. Martijn 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]
  15. L. Li, “A modal analysis of lamellar diffraction gratings in conical mountings,” J. Mod. Opt. 40, 553-573 (1993).
    [Crossref]
  16. B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
    [Crossref]
  17. Z.-Y. Li and K.-M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
    [Crossref]
  18. G. Tayeb and S. Enoch, “Combined fictitious sources-scattering matrix method,” J. Opt. Soc. Am. A 21, 1417-1423 (2004).
    [Crossref]
  19. C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, 1990).
  20. G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Special issue on Generalized Multipole Techniques (GMT) of Applied Computational Electromagnetics Society Journal 9, 90-100 (1994).
  21. D. Maystre, M. Saillard, and G. Tayeb, Scattering (Academic, 2001).
  22. D. Kaklamani and H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44, 48-64 (2002).
    [Crossref]
  23. G. Benelli, S. Enoch, and G. Tayeb, “Modelling of a single object embedded in a layered medium,” J. Mod. Opt. 54, 871-879 (2007).
    [Crossref]

2007 (2)

S. Campbell, L. C. Botten, C. Martijn 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]

G. Benelli, S. Enoch, and G. Tayeb, “Modelling of a single object embedded in a layered medium,” J. Mod. Opt. 54, 871-879 (2007).
[Crossref]

2006 (1)

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

2004 (1)

2003 (3)

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

Z.-Y. Li and K.-M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
[Crossref]

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83, 30 (2003).
[Crossref]

2002 (3)

D. Kaklamani and H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44, 48-64 (2002).
[Crossref]

S. Enoch, M. Nevière, E. Popov, and R. Reinisch, “Enhanced light transmission by hole arrays,” J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

2001 (3)

P. Andrew and W. L. Barnes, “Molecular fluorescence above metallic gratings,” Phys. Rev. B 64, 125405 (2001).
[Crossref]

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

1998 (1)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[Crossref]

1996 (1)

1993 (1)

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

1986 (1)

1981 (3)

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]

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

Adams, J. L.

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]

Anastassiu, H.

D. Kaklamani and H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44, 48-64 (2002).
[Crossref]

Andrew, P.

P. Andrew and W. L. Barnes, “Molecular fluorescence above metallic gratings,” Phys. Rev. B 64, 125405 (2001).
[Crossref]

Andrewartha, J. R.

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]

Barnes, W. L.

P. Andrew and W. L. Barnes, “Molecular fluorescence above metallic gratings,” Phys. Rev. B 64, 125405 (2001).
[Crossref]

Benelli, G.

G. Benelli, S. Enoch, and G. Tayeb, “Modelling of a single object embedded in a layered medium,” J. Mod. Opt. 54, 871-879 (2007).
[Crossref]

Botten, L. C.

S. Campbell, L. C. Botten, C. Martijn 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, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction grating,” Opt. Acta 28, 1103-1106 (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, J. L. Adams, and J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413-428 (1981).
[Crossref]

Campbell, S.

S. Campbell, L. C. Botten, C. Martijn 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]

Carminati, R.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Chen, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Craig, M. S.

L. C. Botten, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction grating,” 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]

de Dood, M.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

De Sterke, C. Martijn

S. Campbell, L. C. Botten, C. Martijn 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]

De Wilde, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Ebbesen, T. W.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[Crossref]

Enoch, S.

G. Benelli, S. Enoch, and G. Tayeb, “Modelling of a single object embedded in a layered medium,” J. Mod. Opt. 54, 871-879 (2007).
[Crossref]

G. Tayeb and S. Enoch, “Combined fictitious sources-scattering matrix method,” J. Opt. Soc. Am. A 21, 1417-1423 (2004).
[Crossref]

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

S. Enoch, M. Nevière, E. Popov, and R. Reinisch, “Enhanced light transmission by hole arrays,” J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Formanek, F.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Gaylord, T. K.

Ghaemi, H. F.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[Crossref]

Gilderdale, D. J.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

Gralak, B.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83, 30 (2003).
[Crossref]

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

Greffet, J.-J.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Guérin, N.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Hafner, C.

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, 1990).

Hajnal, J. V.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

Ho, K.-M.

Z.-Y. Li and K.-M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
[Crossref]

Joulain, K.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Kaklamani, D.

D. Kaklamani and H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44, 48-64 (2002).
[Crossref]

Kalkman, J.

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83, 30 (2003).
[Crossref]

Larkman, D. J.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

Lemoine, P.-A.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Lezec, H. J.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[Crossref]

Li, L.

Li, Z.-Y.

Z.-Y. Li and K.-M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
[Crossref]

Maystre, D.

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

D. Maystre, M. Saillard, and G. Tayeb, Scattering (Academic, 2001).

McPhedran, R. C.

S. Campbell, L. C. Botten, C. Martijn 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, M. S. Craig, and R. C. McPhedran, “Highly conducting lamellar diffraction grating,” 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]

Moharam, M. G.

Mulet, J.-P.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Nevière, M.

S. Enoch, M. Nevière, E. Popov, and R. Reinisch, “Enhanced light transmission by hole arrays,” J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

Pendry, J. B.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

Polman, A.

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83, 30 (2003).
[Crossref]

Popov, E.

S. Enoch, M. Nevière, E. Popov, and R. Reinisch, “Enhanced light transmission by hole arrays,” J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

Reinisch, R.

S. Enoch, M. Nevière, E. Popov, and R. Reinisch, “Enhanced light transmission by hole arrays,” J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

Sabouroux, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Saillard, M.

D. Maystre, M. Saillard, and G. Tayeb, Scattering (Academic, 2001).

Schultz, S.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

Shelby, R. A.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

Smith, D. R.

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

Strohhöfer, C.

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83, 30 (2003).
[Crossref]

Tayeb, G.

G. Benelli, S. Enoch, and G. Tayeb, “Modelling of a single object embedded in a layered medium,” J. Mod. Opt. 54, 871-879 (2007).
[Crossref]

G. Tayeb and S. Enoch, “Combined fictitious sources-scattering matrix method,” J. Opt. Soc. Am. A 21, 1417-1423 (2004).
[Crossref]

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

D. Maystre, M. Saillard, and G. Tayeb, Scattering (Academic, 2001).

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Special issue on Generalized Multipole Techniques (GMT) of Applied Computational Electromagnetics Society Journal 9, 90-100 (1994).

Thio, T.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[Crossref]

Vincent, P.

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Wiltshire, M. C.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

Wolff, P. A.

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[Crossref]

Young, I. R.

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

Appl. Phys. Lett. (1)

J. Kalkman, C. Strohhöfer, B. Gralak, and A. Polman, “Surface plasmon polariton modified emission of erbium in a metallodielectric grating,” Appl. Phys. Lett. 83, 30 (2003).
[Crossref]

IEEE Antennas Propag. Mag. (1)

D. Kaklamani and H. Anastassiu, “Aspects of the method of auxiliary sources (MAS) in computational electromagnetics,” IEEE Antennas Propag. Mag. 44, 48-64 (2002).
[Crossref]

J. Mod. Opt. (2)

G. Benelli, S. Enoch, and G. Tayeb, “Modelling of a single object embedded in a layered medium,” J. Mod. Opt. 54, 871-879 (2007).
[Crossref]

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

J. Opt. A, Pure Appl. Opt. (1)

S. Enoch, M. Nevière, E. Popov, and R. Reinisch, “Enhanced light transmission by hole arrays,” J. Opt. A, Pure Appl. Opt. 4, S83-S87 (2002).
[Crossref]

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

Nature (London) (2)

T. W. Ebbesen, H. J. Lezec, H. F. Ghaemi, T. Thio, and P. A. Wolff, “Extraordinary optical transmission through subwavelength hole arrays,” Nature (London) 391, 667-669 (1998).
[Crossref]

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature (London) 444, 740-743 (2006).
[Crossref]

Opt. Acta (3)

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]

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

Phys. Rev. B (2)

P. Andrew and W. L. Barnes, “Molecular fluorescence above metallic gratings,” Phys. Rev. B 64, 125405 (2001).
[Crossref]

Z.-Y. Li and K.-M. Ho, “Analytic modal solution to light propagation through layer-by-layer metallic photonic crystals,” Phys. Rev. B 67, 165104 (2003).
[Crossref]

Phys. Rev. E (1)

B. Gralak, M. de Dood, G. Tayeb, S. Enoch, and D. Maystre, “Theoretical study of photonic band gaps in woodpile crystals,” Phys. Rev. E 67, 066601 (2003).
[Crossref]

Phys. Rev. Lett. (1)

S. Enoch, G. Tayeb, P. Sabouroux, N. Guérin, and P. Vincent, “A metamaterial for directive emission,” Phys. Rev. Lett. 89, 213902 (2002).
[Crossref] [PubMed]

Science (2)

M. C. K. Wiltshire, J. B. Pendry, I. R. Young, D. J. Larkman, D. J. Gilderdale, and J. V. Hajnal, “Microstructured magnetic materials for RF flux guides in magnetic resonance imaging,” Science 291, 849-851 (2001).
[Crossref] [PubMed]

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
[Crossref] [PubMed]

Waves Random Complex Media (1)

S. Campbell, L. C. Botten, C. Martijn 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 (3)

C. Hafner, The Generalized Multipole Technique for Computational Electromagnetics (Artech House, 1990).

G. Tayeb, “The method of fictitious sources applied to diffraction gratings,” Special issue on Generalized Multipole Techniques (GMT) of Applied Computational Electromagnetics Society Journal 9, 90-100 (1994).

D. Maystre, M. Saillard, and G. Tayeb, Scattering (Academic, 2001).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Lamellar grating made of three layers.

Fig. 2
Fig. 2

Layer made of two rods per unit cell. First rod has width a and dielectic constant ϵ a . Second rod (shaded domain) has width d a and is made of infinitely conducting metal; thickness of the layer is h.

Fig. 3
Fig. 3

Stack of a homogenous layer and the layer represented on Fig. 2. The interface delimiting these two layers is represented by the dashed line at x 3 = h .

Fig. 4
Fig. 4

Structure containing two adjacent layers with infinitely conducting rods. The bottom layer is made of two rods per unit cell: the first rod has width b and dielectic constant ϵ b , the second rod (shaded domain) has width d b and is made of infinitely conducting metal; thickness of this layer is h b .

Fig. 5
Fig. 5

Structure containing two adjacent layers with infinitely conducting rods. Each layer contains four different rods: the two dielectric rods have widths c 1 = c 1 c 1 and c 2 = c 2 c 2 and dielectric constants ϵ c 1 and ϵ c 2 ( c = a for the top layer and c = b for the bottom layer). The other two rods (shaded domain) are made of infinitely conducting metal.

Fig. 6
Fig. 6

Structure that cannot be modeled using a numerical stacking algorithm, since a 1 < b 1 < a 1 < b 1 .

Fig. 7
Fig. 7

Structure under consideration. The spatial period is d = 20.0 . The four layers have widths a = 18.0 , b = 15.0 , c = 18.0 , e = 13.0 , and thicknesses h a = 4.0 , h b = 2.0 , h c = 3.0 , h e = 2.0 .

Fig. 8
Fig. 8

Convergence of the main reflected order (left) and of the total reflectivity (right) when the number of modes is increasing.

Fig. 9
Fig. 9

Total reflectivity as a function of the wavelength (from 2.0 to 3.0) for two different numbers of modes.

Fig. 10
Fig. 10

Discretized objects (rectangular cell and the F). Black points and open circles fictitious sources represent.

Fig. 11
Fig. 11

Maps of log 10 E 1 —the electric field along the periodicity direction—using the modal method (left) and the fictitious-sources method (right).

Equations (122)

Equations on this page are rendered with MathJax. Learn more.

[ ω 2 ϵ 1 × μ 0 1 × ] E ω = 0 ,
H ω = ( ω μ 0 ) 1 × E ω .
E ω = ( ω ϵ ) 1 × H ω , H ω = ( ω μ 0 ) 1 × E ω .
Ψ = { 1 in dielectric materials 0 in infinitely conducting metal } .
E ω = Ψ ( ω ϵ ) 1 × H ω , H ω = ( ω μ 0 ) 1 × E ω .
ϵ ( x + d ) = ϵ ( x ) = ϵ ( x 1 , x 3 ) , x R 3 .
Ψ a ( x 1 ) = { 1 0 x 1 + p d a 0 a < x 1 + p d < d } , p Z ,
E ω = Ψ a ( ω ϵ a ) 1 × H ω , H ω = ( ω μ 0 ) 1 × E ω .
G ω ( x 1 + d , x 2 , x 3 ) = exp [ i k 1 d ] G ω ( x 1 , x 2 , x 3 ) ,
x 1 , k 1 , x 2 , x 3 R
G ω ( x 1 , x 2 , x 3 ) = G ̂ ω ( x 1 , k 2 , x 3 ) exp [ i k 2 x 2 ] ,
x 1 , x 2 , k 2 , x 3 R .
F = [ F ( 1 ) F ( 2 ) ] , F ( j ) = [ E ̂ ω , j H ̂ ω , j ] , j = 1 , 2 ,
F ( x 3 ) = n N F a , n ( x 3 ) Φ a , n ,
F ( x 3 ) = p Z F 0 , p ( x 3 ) Φ 0 , p ,
E = P E F , P E = [ 1 0 0 0 0 0 1 0 ] ,
H = P H F , P H = [ 0 1 0 0 0 0 0 1 ] .
E a , n ( x 3 ) = P E F a , n ( x 3 ) , H a , n ( x 3 ) = P H F a , n ( x 3 ) , n N ,
E 0 , p ( x 3 ) = P E F 0 , p ( x 3 ) , H 0 , p ( x 3 ) = P H F 0 , p ( x 3 ) , p Z .
G 0 ( x 3 ) = [ , G a , 1 ( x 3 ) , G a , 0 ( x 3 ) , G a , 1 ( x 3 ) , , G a , p ( x 3 ) , ] ,
G a ( x 3 ) = [ G a , 0 ( x 3 ) , G a , 1 ( x 3 ) , , G a , n ( x 3 ) , ] , G = E , H .
E ( x 1 , h ) = E ( x 1 , h ) Ψ a ( x 1 ) = E ( x 1 , h + ) ,
H ( x 1 , h ) Ψ a ( x 1 ) = H ( x 1 , h + ) Ψ a ( x 1 ) , 0 x 1 d .
n N Φ a , n ( 1 ) E a , n ( h ) = Ψ a n N Φ a , n ( 1 ) E a , n ( h ) = p Z ϕ 0 , p E 0 , p ( h + ) ,
Ψ a n N Φ a , n ( 2 ) H a , n ( h ) = Ψ a p Z ϕ 0 , p H 0 , p ( h + ) , 0 x 1 d .
W 0 , a ( 1 ) E a ( h ) = E 0 ( h + ) ,
W 0 , a ( 2 ) H a ( h ) = U a H 0 ( h + ) H a ( h ) = W a , 0 ( 2 ) H 0 ( h + ) ,
[ W 0 , a ( j ) ] p , n = [ 0 , d ] d x 1 ϕ 0 , p ( x 1 ) ¯ Φ a , n ( j ) ( x 1 ) ,
p Z , n N , j = 1 , 2 ,
[ W a , 0 ( j ) ] n , p = [ 0 , d ] d x 1 Φ a , n ( j ) ( x 1 ) ϕ 0 , p ( x 1 ) ,
n N , p Z , j = 1 , 2 ,
[ U a ] p , q = [ 0 , d ] d x 1 ϕ 0 , p ( x 1 ) ¯ Ψ a ( x 1 ) ϕ 0 , q ( x 1 ) I ,
p Z , q Z .
n N [ W a , 0 ( j ) ] n , q Φ a , n ( j ) = Ψ a ϕ 0 , q I ϕ 0 , q I , q Z .
p Z n N [ W 0 , a ( j ) ] p , n [ W a , 0 ( j ) ] n , q ϕ 0 , p = Ψ a ϕ 0 , q I ϕ 0 , q I , q Z ,
W 0 , a ( j ) W a , 0 ( j ) = U a , j = 1 , 2 .
m N p Z [ W a , 0 ( j ) ] m , p [ W 0 , a ( j ) ] p , n Φ a , m ( j ) = Φ a , n ( j ) , n N ,
W a , 0 ( j ) W 0 , a ( j ) = I a , j = 1 , 2 ,
[ I a ] m , n = { I , m = n 0 , m n } , m , n N .
K = [ 1 0 0 0 ] , [ I + K ] 1 = [ 2 0 0 1 ] 1 = [ 1 2 0 0 1 ] .
[ E a ( h ) E a ( 0 ) ] = Z a [ H a ( h ) H a ( 0 ) ] , Z a = [ Z a ( 11 ) Z a ( 12 ) Z a ( 21 ) Z a ( 22 ) ] .
[ E a , n ( h ) E a , n ( 0 ) ] = Z a , n [ H a , n ( h ) H a , n ( 0 ) ] , Z a , n = [ Z a , n ( 11 ) Z a , n ( 12 ) Z a , n ( 21 ) Z a , n ( 22 ) ] n N .
[ E 0 ( h 0 + h ) E 0 ( h ) ] = Z 0 [ H 0 ( h 0 + h ) H 0 ( h ) ] , Z 0 = [ Z 0 ( 11 ) Z 0 ( 12 ) Z 0 ( 21 ) Z 0 ( 22 ) ] .
[ E 0 ( h ) E a ( 0 ) ] = Z ̃ a [ H 0 ( h ) H a ( 0 ) ] , Z ̃ a = [ W 0 , a ( 1 ) Z a ( 11 ) W a , 0 ( 2 ) W 0 , a ( 1 ) Z a ( 12 ) Z a ( 21 ) W a , 0 ( 2 ) Z a ( 22 ) ] .
[ E 0 ( h 0 + h ) E a ( 0 ) ] = Z 0 a [ H 0 ( h 0 + h ) H a ( 0 ) ] .
Z 0 ( 22 ) W 0 , a ( 1 ) Z a ( 11 ) W a , 0 ( 2 ) .
Ψ b ( x 1 ) = { 1 0 x 1 + p d b 0 b < x 1 + p d < d } , p Z .
[ E b ( 0 ) E b ( h b ) ] = Z b [ H b ( 0 ) H b ( h b ) ] , Z b = [ Z b ( 11 ) Z b ( 12 ) Z b ( 21 ) Z b ( 22 ) ] .
b a Ψ a Ψ b = Ψ a ;
W b , a ( 1 ) E a ( 0 + ) = E b ( 0 ) ,
H a ( 0 + ) = W a , b ( 2 ) H b ( 0 ) ,
[ W a , b ( j ) ] m , n = [ W b , a ( j ) ] n , m = [ 0 , d ] d x 1 Φ a , m ( j ) ( x 1 ) Φ b , n ( j ) ( x 1 ) ,
m , n N , j = 1 , 2 .
[ E a ( h ) E b ( 0 ) ] = Z ̂ a [ H a ( h ) H b ( 0 ) ] , Z ̂ a = [ Z a ( 11 ) Z a ( 12 ) W a , b ( 2 ) W b , a ( 1 ) Z a ( 21 ) W b , a ( 1 ) Z a ( 22 ) W a , b ( 2 ) ] .
W b , a ( 1 ) Z a ( 22 ) W a , b ( 2 ) Z b ( 11 ) .
Z ̃ ̂ a = [ W 0 , a ( 1 ) Z a ( 11 ) W a , 0 ( 2 ) W 0 , a ( 1 ) Z a ( 12 ) W a , b ( 2 ) W b , a ( 1 ) Z a ( 21 ) W a , 0 ( 2 ) W b , a ( 1 ) Z a ( 22 ) W a , b ( 2 ) ] .
a b Ψ a Ψ b = Ψ b ;
E a ( 0 + ) = W a , b ( 1 ) E b ( 0 ) ,
W b , a ( 2 ) H a ( 0 + ) = H b ( 0 ) ,
[ E a ( 0 ) E b ( h b ) ] = Z ̃ b [ H b ( 0 ) H b ( h b ) ] ,
Z ̃ b = [ W b , a ( 1 ) Z b ( 11 ) W a , b ( 2 ) W b , a ( 1 ) Z b ( 12 ) Z a ( 21 ) W a , b ( 2 ) Z a ( 22 ) ] .
Ψ a = Ψ a 1 + Ψ a 2 + + Ψ a q ,
Ψ a j = { 1 a j x 1 + p d a j 0 a j < x 1 + p d < a j + 1 } , p Z ,
a = ( a 1 , a 1 , a 2 , a 2 , , a q , a q , a q + 1 ) a q + 1 = d ,
Ψ b = Ψ b 1 + Ψ b 2 + + Ψ b l ,
Ψ a j Ψ b k = Ψ a j or Ψ a j Ψ b k = Ψ b k .
Ψ a j Ψ b k Ψ a j , Ψ a j Ψ b k Ψ b k , Ψ a j Ψ b k 0 .
R 2 d x 1 d x 2 G ω ( x ) 2 < , x 3 R , G ω = E ω , H ω .
[ F 0 ( G ω ) ] ( x 1 , k 2 , x 3 ) = 1 2 π R d x 2 exp ( i k 2 x 2 ) G ω ( x 1 , x 2 , x 3 ) ,
[ F d ( G ω ) ] ( x 1 , k 1 , x 2 , x 3 ) = p Z G ω ( x 1 + p d , x 2 , x 3 ) exp ( i k 1 p d ) ,
G ω ( x 1 , x 2 , x 3 ) = d 2 π ( π d , π d ) d k 1 [ F d ( G ω ) ] ( x 1 , k 1 , x 2 , x 3 ) ,
[ 0 , d ] d x 1 G ̂ ω ( x 1 , k 1 , k 2 , x 3 ) 2 < , k 1 , k 2 , x 3 R ,
G ̂ ω ( x 1 + d , k 1 , k 2 , x 3 ) = exp ( i k 1 d ) G ̂ ω ( x 1 , k 1 , k 2 , x 3 ) ,
x 1 , k 1 , k 2 , x 3 R .
E ̂ ω = Ψ a ( ω ϵ a ) 1 k 2 × H ̂ ω , H ̂ ω = ( ω μ 0 ) 1 k 2 × E ̂ ω ,
E ̂ ω , 1 ( x 1 , k 1 , k 2 , x 3 ) = E ̂ ω , 2 ( x 1 , k 1 , k 2 , x 3 ) = 0 ,
a x 1 d , x 3 = 0 , h ;
E ̂ ω , 2 ( x 1 , k 1 , k 2 , x 3 ) = E ̂ ω , 3 ( x 1 , k 1 , k 2 , x 3 ) = 0 ,
0 x 3 h , x 1 = 0 , a .
( 3 2 + L a ) E ̂ ω , j = 0 , L a = ω 2 ϵ a μ 0 k 2 2 + 1 2 , j = 2 , 3 ,
L a ϕ a , n = λ a , n ϕ a , n , n N .
ϕ a , n : x 1 2 a Ψ a ( x 1 ) sin ( n π x 1 a ) ,
λ a , n = ω 2 ϵ a μ 0 k 2 2 ( n π a ) 2 .
E ̂ ω , j ( x 3 ) = n N ϕ a , n [ E ̂ ω , j ( a , n ) ( 0 ) cos ( λ a , n x 3 ) + ( 3 E ̂ ω , j ( a , n ) ) ( 0 ) sin ( λ a , n x 3 ) λ a , n ] , j = 2 , 3 ,
E ̂ ω , j ( a , n ) ( x 3 ) = [ 0 , d ] d x 1 ϕ a , n ( x 1 ) E ̂ ω , j ( x 1 , x 3 ) ,
( 3 E ̂ ω , j ( a , n ) ) ( x 3 ) = [ 0 , d ] d x 1 ϕ a , n ( x 1 ) ( 3 E ̂ ω , j ) ( x 1 , x 3 ) ,
j = 2 , 3 ,
L a = ω 2 ϵ a μ 0 k 2 2 + 1 2
L a ϕ a , n = λ a , n ϕ a , n , n N .
ϕ a , 0 : x 1 1 a Ψ a ( x 1 ) ,
ϕ a , n : x 1 2 a Ψ a ( x 1 ) cos ( n π x 1 a ) , n N \ { 0 } .
Φ a , n = [ Φ a , n ( 1 ) 0 0 Φ a , n ( 2 ) ] , Λ a , n = [ λ a , n I 0 0 λ a , n I ] , n N ,
Φ a , n ( 1 ) = [ ϕ a , n 0 0 ϕ a , n ] , Φ a , n ( 2 ) = [ ϕ a , n 0 0 ϕ a , n ] ,
n N , I = [ 1 0 0 1 ] .
F ( x 3 ) = n N Φ a , n [ cos ( Λ a , n x 3 ) F a , n ( 0 ) + sin ( Λ a , n x 3 ) Λ a , n ( 3 F a , n ) ( 0 ) ] ,
F a , n ( x 3 ) = [ 0 , d ] d x 1 Φ a , n ( x 1 ) F ( x 1 , x 3 ) ,
( 3 F a , n ) ( x 3 ) = [ 0 , d ] d x 1 Φ a , n ( x 1 ) ( 3 F ) ( x 1 , x 3 ) ,
3 F = M a F , M a = [ i k 2 σ a 1 1 σ a + σ a 1 1 2 σ a + k 2 2 σ a 1 i k 2 σ a 1 1 ] ,
σ a = ω [ 0 μ 0 ϵ a 0 ] .
( 3 F a , n ) ( 0 ) = M a , n F a , n ( 0 ) , p N ,
M a , n = [ i k 2 ( n π a ) J σ a 1 [ ω 2 ϵ a μ 0 ( n π a ) 2 ] σ a 1 ( ω 2 ϵ 1 μ 0 k 2 2 ) σ a 1 i k 2 ( n π a ) J σ a 1 ] ,
J = [ 1 0 0 1 ] .
F ( x 3 ) = n N Φ a , n [ cos ( Λ a , n x 3 ) + sin ( Λ a , n x 3 ) Λ a , n M a , n ] F a , n ( 0 ) .
F a , n ( h ) = T a , n F a , n ( 0 ) , T a , n = cos ( Λ a , n h ) + sin ( Λ a , n h ) Λ a , n M a , n , n N ,
[ F a , n ( 1 ) ( h ) F a , n ( 1 ) ( 0 ) ] = R a , n [ F a , n ( 2 ) ( h ) F a , n ( 2 ) ( 0 ) ] ,
R a , n = [ R a , n ( 11 ) R a , n ( 12 ) R a , n ( 21 ) R a , n ( 22 ) ] n N .
R a , n ( 11 ) = [ ω 2 ϵ a μ 0 k 2 2 ] 1 [ λ a , n cos ( λ a , n h ) sin ( λ a , n h ) σ a + i k 2 p π a J ] ,
R a , n ( 12 ) = + [ ω 2 ϵ a μ 0 k 2 2 ] 1 λ a , n sin ( λ a , n h ) σ a ,
R a , n ( 21 ) = [ ω 2 ϵ a μ 0 k 2 2 ] 1 λ a , n sin ( λ a , n h ) σ a ,
R a , n ( 22 ) = [ ω 2 ϵ a μ 0 k 2 2 ] 1 [ λ a , n cos ( λ a , n h ) sin ( λ a , n h ) σ a i k 2 p π a J ] .
F ( x 3 + h ) = p N Φ 0 , p [ cos ( Λ 0 , p x 3 ) + sin ( Λ 0 , p x 3 ) Λ 0 , p M 0 , p ] F 0 , p ( h ) ,
ϕ 0 , p : x 1 1 d exp [ i ( k 1 + p 2 π d ) x 1 ] , p Z ,
λ 0 , p = ω 2 ϵ 0 μ 0 k 2 2 ( k 1 + p 2 π d ) 2 ,
M 0 , p = [ k 2 ( k 1 + p 2 π d ) σ 0 1 [ ω 2 ϵ 0 μ 0 ( k 1 + p 2 π d ) 2 ] σ 0 1 ( ω 2 ϵ 0 μ 0 k 2 2 ) σ 0 1 k 2 ( k 1 + p 2 π d ) σ 0 1 ]
σ 0 = ω [ 0 μ 0 ϵ 0 0 ] .
F 0 , p ( x 3 ) = [ 0 , d ] d x 1 Φ 0 , p ( x 1 ) ¯ F ( x 1 , x 3 )
[ F 0 , p ( 1 ) ( h 0 + h ) F 0 , p ( 1 ) ( h ) ] = R 0 , p [ F 0 , p ( 2 ) ( h 0 + h ) F 0 , p ( 2 ) ( h ) ] ,
R 0 , p = [ R 0 , p ( 11 ) R 0 , p ( 12 ) R 0 , p ( 21 ) R 0 , p ( 22 ) ] , p Z ,
R 0 , p ( 11 ) = [ ω 2 ϵ 0 μ 0 k 2 2 ] 1 [ λ 0 , p cos ( λ 0 , p h 0 ) sin ( λ 0 , p h 0 ) σ 0 k 2 ( k 1 + p 2 π d ) I ] ,
R 0 , p ( 12 ) = + [ ω 2 ϵ 0 μ 0 k 2 2 ] 1 λ 0 , p sin ( λ 0 , p h 0 ) σ 0 ,
R 0 , p ( 21 ) = [ ω 2 ϵ 0 μ 0 k 2 2 ] 1 λ 0 , p sin ( λ 0 , p h 0 ) σ 0 ,
R 0 , p ( 22 ) = [ ω 2 ϵ 0 μ 0 k 2 2 ] 1 [ λ 0 , p cos ( λ 0 , p h 0 ) sin ( λ 0 , p h 0 ) σ 0 + k 2 ( k 1 + p 2 π d ) I ] .

Metrics