Abstract

We define the concept of an impedance matrix for three-dimensional (3D) photonic and metamaterial structures relative to a reference medium and show that it satisfies a matrix generalization of the basic algebraic properties of the wave impedance between homogeneous media. This definition of the impedance matrix is motivated by the structure of the Fresnel reflection and transmission matrices at the interface between the media. In the derivation of the Fresnel scattering matrices, the field in each medium is expressed by a Bloch mode expansion, with field matching at the interface being undertaken in a least-squares manner by exploiting a biorthogonality relation between primal and adjoint Bloch modes. A semi-analytic technique, based on the impedance matrix, is developed for modeling the scattering of light by 3D periodic photonic and metamaterial structures. The advantages (in design and intuition) of the formalism are demonstrated through two applications.

© 2013 Optical Society of America

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  1. A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).
  2. X. Wei, A. J. Wachters, and H. P. Urbach, “Finite-element model for three-dimensional optical scattering problems,” J. Opt. Soc. Am. A 24, 866–881 (2007).
    [CrossRef]
  3. K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comput. Phys. 231, 6969–6989 (2012).
    [CrossRef]
  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]
  5. K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
    [CrossRef]
  6. L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).
  7. R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).
  8. F. J. Lawrence, C. Martijn de Sterke, L. C. Botten, R. C. McPhedran, and K. B. Dossou, “Modeling photonic crystal interfaces and stacks: impedance-based approaches,” Adv. Opt. Photon. (to be published).
  9. S. Boscolo, C. Conti, M. Midrio, and C. Someda, “Numerical analysis of propagation and impedance matching in 2d photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 304–310 (2002).
    [CrossRef]
  10. J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
    [CrossRef]
  11. R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
    [CrossRef]
  12. B. Momeni, A. A. Eftekhar, and A. Adibi, “Effective impedance model for analysis of reflection at the interfaces of photonic crystals,” Opt. Lett. 32, 778–780 (2007).
    [CrossRef]
  13. B. Momeni, M. Badieirostami, and A. Adibi, “Accurate and efficient techniques for the analysis of reflection at the interfaces of three-dimensional photonic crystals,” J. Opt. Soc. Am. B 24, 2957–2963 (2007).
    [CrossRef]
  14. A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).
  15. M. Mazilu and K. Dholakia, “Optical impedance of metallic nano-structures,” Opt. Express 14, 7709–7722 (2006).
    [CrossRef]
  16. J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
    [CrossRef]
  17. F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. Martijn de Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 121114 (2008).
    [CrossRef]
  18. F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
    [CrossRef]
  19. A. K. Cousins and S. C. Gottschalk, “Application of the impedance formalism to diffraction gratings with multiple coating layers,” Appl. Opt. 29, 4268–4271 (1990).
    [CrossRef]
  20. 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]
  21. D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
    [CrossRef]
  22. L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
    [CrossRef]
  23. Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
    [CrossRef]
  24. K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
    [CrossRef]
  25. D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
    [CrossRef]
  26. W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
    [CrossRef]
  27. J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
    [CrossRef]
  28. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).
  29. A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
    [CrossRef]
  30. K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
    [CrossRef]
  31. R. Petit, ed., Electromagnetic Theory of Gratings, vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).
  32. 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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
    [CrossRef]
  33. L. C. Botten, N.-A. P. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
    [CrossRef]
  34. 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]
  35. L. C. Botten and R. C. McPhedran, “Completeness and modal expansion methods in diffraction theory,” Opt. Acta 32, 1479–1488 (1985).
    [CrossRef]
  36. T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
    [CrossRef]
  37. T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
    [CrossRef]
  38. G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
    [CrossRef]
  39. P. Kužel and J. Petzelt, “Time-resolved terahertz transmission spectroscopy of dielectrics,” Ferroelectrics 239, 79–86 (2000).
    [CrossRef]
  40. D. Chemicals, “ http://www.dow.com/cyclotene/solution/highfreq.htm ”.
  41. H. R. Philipp, “Silicon dioxide (SiO2) glass,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985), pp. 749–763.

2012

K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comput. Phys. 231, 6969–6989 (2012).
[CrossRef]

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
[CrossRef]

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).

2011

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

2010

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

2009

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
[CrossRef]

2008

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. Martijn de Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 121114 (2008).
[CrossRef]

2007

2006

M. Mazilu and K. Dholakia, “Optical impedance of metallic nano-structures,” Opt. Express 14, 7709–7722 (2006).
[CrossRef]

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

2005

K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
[CrossRef]

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
[CrossRef]

2004

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

2003

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
[CrossRef]

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

2002

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

S. Boscolo, C. Conti, M. Midrio, and C. Someda, “Numerical analysis of propagation and impedance matching in 2d photonic crystal waveguides with finite length,” J. Lightwave Technol. 20, 304–310 (2002).
[CrossRef]

2001

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

2000

1999

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

1996

1990

1985

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

1981

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]

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

Adibi, A.

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]

Andryieuski, A.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).

Asatryan, A. A.

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, N.-A. P. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

Badieirostami, M.

Bartal, G.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Biswas, R.

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

Boscolo, S.

Botten, L. C.

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
[CrossRef]

K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comput. Phys. 231, 6969–6989 (2012).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. Martijn de Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 121114 (2008).
[CrossRef]

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, N.-A. P. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[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]

F. J. Lawrence, C. Martijn de Sterke, L. C. Botten, R. C. McPhedran, and K. B. Dossou, “Modeling photonic crystal interfaces and stacks: impedance-based approaches,” Adv. Opt. Photon. (to be published).

Brekhovskikh, L. M.

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).

Byrne, M. A.

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
[CrossRef]

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

Collin, R. E.

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).

Conti, C.

Cousins, A. K.

Craig, M. S.

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]

Culshaw, I. S.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

de Sterke, C. M.

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, N.-A. P. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[CrossRef]

Dholakia, K.

Dolling, G.

Dossou, K.

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
[CrossRef]

Dossou, K. B.

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
[CrossRef]

K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comput. Phys. 231, 6969–6989 (2012).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. Martijn de Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 121114 (2008).
[CrossRef]

F. J. Lawrence, C. Martijn de Sterke, L. C. Botten, R. C. McPhedran, and K. B. Dossou, “Modeling photonic crystal interfaces and stacks: impedance-based approaches,” Adv. Opt. Photon. (to be published).

Eftekhar, A. A.

Enkrich, C.

Fontaine, M.

K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
[CrossRef]

Gaylord, T. K.

Genov, D. A.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Gomyo, A.

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
[CrossRef]

Gottschalk, S. C.

Ha, S.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).

Ho, K. M.

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

Ishikawa, A.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Kivshar, Y. S.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).

Kužel, P.

P. Kužel and J. Petzelt, “Time-resolved terahertz transmission spectroscopy of dielectrics,” Ferroelectrics 239, 79–86 (2000).
[CrossRef]

Lalanne, P.

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

Langtry, T. N.

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Lavrinenko, A. V.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).

Lawrence, F. J.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. Martijn de Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 121114 (2008).
[CrossRef]

F. J. Lawrence, C. Martijn de Sterke, L. C. Botten, R. C. McPhedran, and K. B. Dossou, “Modeling photonic crystal interfaces and stacks: impedance-based approaches,” Adv. Opt. Photon. (to be published).

Lederer, F.

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

Li, L.

Li, Z. Y.

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

Li, Z.-Y.

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Lin, L.-L.

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

Linden, S.

Markoš, P.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Martijn de Sterke, C.

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. Martijn de Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 121114 (2008).
[CrossRef]

F. J. Lawrence, C. Martijn de Sterke, L. C. Botten, R. C. McPhedran, and K. B. Dossou, “Modeling photonic crystal interfaces and stacks: impedance-based approaches,” Adv. Opt. Photon. (to be published).

Mazilu, M.

McPhedran, R. C.

K. B. Dossou, L. C. Botten, A. A. Asatryan, B. C. P. Sturmberg, M. A. Byrne, C. G. Poulton, R. C. McPhedran, and C. M. de Sterke, “Modal formulation for diffraction by absorbing photonic crystal slabs,” J. Opt. Soc. Am. A 29, 817–831 (2012).
[CrossRef]

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

L. C. Botten, N.-A. P. Nicorovici, A. A. Asatryan, R. C. McPhedran, C. M. de Sterke, and P. A. Robinson, “Formulation for electromagnetic scattering and propagation through grating stacks of metallic and dielectric cylinders for photonic crystal calculations. Part II. Properties and implementation,” J. Opt. Soc. Am. A 17, 2177–2190 (2000).
[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]

F. J. Lawrence, C. Martijn de Sterke, L. C. Botten, R. C. McPhedran, and K. B. Dossou, “Modeling photonic crystal interfaces and stacks: impedance-based approaches,” Adv. Opt. Photon. (to be published).

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Menzel, C.

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

Midrio, M.

Moharam, M. G.

Momeni, B.

Nicorovici, N. A.

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Nicorovici, N.-A. P.

Paul, T.

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

Perrins, W. T.

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

Petzelt, J.

P. Kužel and J. Petzelt, “Time-resolved terahertz transmission spectroscopy of dielectrics,” Ferroelectrics 239, 79–86 (2000).
[CrossRef]

Philipp, H. R.

H. R. Philipp, “Silicon dioxide (SiO2) glass,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985), pp. 749–763.

Poulton, C. G.

Robinson, P. A.

Rockstuhl, C.

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

Sauvan, C.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

Schultz, S.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Shirane, M.

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
[CrossRef]

Smigaj, W.

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

Smith, D. R.

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Someda, C.

Soukoulis, C. M.

G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, “Cut-wire pairs and plate pairs as magnetic atoms for optical metamaterials,” Opt. Lett. 30, 3198–3200 (2005).
[CrossRef]

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

Sturmberg, B. C. P.

Sukhorukov, A. A.

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

Tokushima, M.

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
[CrossRef]

Ulin-Avila, E.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Urbach, H. P.

Ushida, J.

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
[CrossRef]

Valentine, J.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Wachters, A. J.

Wegener, M.

Wei, X.

White, T. P.

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Whittaker, D. M.

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

Yamada, H.

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
[CrossRef]

Yang, J.

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

Zentgraf, T.

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Zhang, S.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Zhang, X.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
[CrossRef]

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Zhou, J. F.

Appl. Opt.

Appl. Phys. Lett.

R. Biswas, Z. Y. Li, and K. M. Ho, “Impedance of photonic crystals and photonic crystal waveguides,” Appl. Phys. Lett. 84, 1254–1256 (2004).
[CrossRef]

J. Yang, C. Sauvan, T. Paul, C. Rockstuhl, F. Lederer, and P. Lalanne, “Retrieving the effective parameters of metamaterials from the single interface scattering problem,” Appl. Phys. Lett. 97, 061102 (2010).
[CrossRef]

F. J. Lawrence, L. C. Botten, K. B. Dossou, and C. Martijn de Sterke, “Antireflection coatings for two-dimensional photonic crystals using a rigorous impedance definition,” Appl. Phys. Lett. 93, 121114 (2008).
[CrossRef]

Comput. Methods Appl. Mech. Eng.

K. Dossou and M. Fontaine, “A high order isoparametric finite element method for the computation of waveguide modes,” Comput. Methods Appl. Mech. Eng. 194, 837–858 (2005).
[CrossRef]

Ferroelectrics

P. Kužel and J. Petzelt, “Time-resolved terahertz transmission spectroscopy of dielectrics,” Ferroelectrics 239, 79–86 (2000).
[CrossRef]

J. Comput. Phys.

K. Dossou, M. A. Byrne, and L. C. Botten, “Finite element computation of grating scattering matrices and application to photonic crystal band calculations,” J. Comput. Phys. 219, 120–143 (2006).
[CrossRef]

K. B. Dossou and L. C. Botten, “A combined three-dimensional finite element and scattering matrix method for the analysis of plane wave diffraction by bi-periodic, multilayered structures,” J. Comput. Phys. 231, 6969–6989 (2012).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Opt. Soc. Am. B

Metamaterials

W. T. Perrins and R. C. McPhedran, “Metamaterials and the homogenization of composite materials,” Metamaterials 4, 24–31 (2010).
[CrossRef]

Nature

J. Valentine, S. Zhang, T. Zentgraf, E. Ulin-Avila, D. A. Genov, G. Bartal, and X. Zhang, “Three-dimensional optical metamaterial with a negative refractive index,” Nature 455, 376–379 (2008).
[CrossRef]

Opt. Acta

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

Opt. Express

Opt. Lett.

Phys. Rev. A

F. J. Lawrence, L. C. Botten, K. B. Dossou, C. Martijn de Sterke, and R. C. McPhedran, “Impedance of square and triangular lattice photonic crystals,” Phys. Rev. A 80, 023826 (2009).
[CrossRef]

Phys. Rev. B

D. R. Smith, S. Schultz, P. Markoš, and C. M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002).
[CrossRef]

T. Paul, C. Menzel, W. Śmigaj, C. Rockstuhl, P. Lalanne, and F. Lederer, “Reflection and transmission of light at periodic layered metamaterial films,” Phys. Rev. B 84, 115142 (2011).
[CrossRef]

A. Andryieuski, S. Ha, A. A. Sukhorukov, Y. S. Kivshar, and A. V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012).

D. M. Whittaker and I. S. Culshaw, “Scattering-matrix treatment of patterned multilayer photonic structures,” Phys. Rev. B 60, 2610–2618 (1999).
[CrossRef]

J. Ushida, M. Tokushima, M. Shirane, A. Gomyo, and H. Yamada, “Immittance matching for multidimensional open-system photonic crystals,” Phys. Rev. B 68, 155115 (2003).
[CrossRef]

Phys. Rev. E

L. C. Botten, N. A. Nicorovici, R. C. McPhedran, C. M. de Sterke, and A. A. Asatryan, “Photonic band structure calculations using scattering matrices,” Phys. Rev. E 64, 046603 (2001).
[CrossRef]

Z.-Y. Li and L.-L. Lin, “Photonic band structures solved by a plane-wave-based transfer-matrix method,” Phys. Rev. E 67, 046607 (2003).
[CrossRef]

T. P. White, L. C. Botten, C. M. de Sterke, R. C. McPhedran, A. A. Asatryan, and T. N. Langtry, “Bloch mode scattering matrix methods for modeling extended photonic crystal structures. II. Applications,” Phys. Rev. E 70, 056607 (2004).
[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 photonic crystal structures. I. Theory,” Phys. Rev. E 70, 056606 (2004).
[CrossRef]

Phys. Rev. Lett.

A. Ishikawa, S. Zhang, D. A. Genov, G. Bartal, and X. Zhang, “Deep subwavelength terahertz waveguides using gap magnetic plasmon,” Phys. Rev. Lett. 102, 043904 (2009).
[CrossRef]

Other

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton University, 2008).

R. Petit, ed., Electromagnetic Theory of Gratings, vol. 22 of Topics in Current Physics (Springer-Verlag, 1980).

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

L. M. Brekhovskikh, Waves in Layered Media (Academic, 1960).

R. E. Collin, Field Theory of Guided Waves, 2nd ed. (IEEE, 1991).

F. J. Lawrence, C. Martijn de Sterke, L. C. Botten, R. C. McPhedran, and K. B. Dossou, “Modeling photonic crystal interfaces and stacks: impedance-based approaches,” Adv. Opt. Photon. (to be published).

D. Chemicals, “ http://www.dow.com/cyclotene/solution/highfreq.htm ”.

H. R. Philipp, “Silicon dioxide (SiO2) glass,” in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, 1985), pp. 749–763.

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Figures (14)

Fig. 1.
Fig. 1.

(a) Layered metamaterials. (b) Unit cell. (c) Up–down symmetric unit cell. The vectors e1, e2, and e3 are the lattice vectors.

Fig. 2.
Fig. 2.

Unit cell of layered metamaterials: it consists of a stack of N z-invariant slabs (or layers) L1,,LN. The symbols Π and Π denote, respectively, the top and bottom interfaces of the unit cell. The reflection and transmission matrices for incidence over the upper interface are denoted R and T, while R and T are the reflection and transmission matrices for incidence over the lower interface.

Fig. 3.
Fig. 3.

Upper and lower panels illustrate incidence from a semi-infinite medium (1) into a semi-infinite medium (2) and vice versa.

Fig. 4.
Fig. 4.

Illustration of the field representation for reflection and transmission through a slab with two interfaces. The matrix P is the modal propagation matrix through the slab; when the slab is a stack of L identical metamaterial layers, we have P=ΛL.

Fig. 5.
Fig. 5.

Graphics (a) and (b) show, respectively, the geometry of the THz metamaterials analyzed in [29] and the optical metamaterials analyzed in [27].

Fig. 6.
Fig. 6.

THz metamaterials: Fresnel reflectance curves obtained using, respectively, a full numerical calculation (continuous curves) and the impedance matrix technique (dotted curves).

Fig. 7.
Fig. 7.

THz metamaterials: absolute value of the Fresnel reflection and transmission coefficients |rn| and |tn| in the expansions Eq. (61) for normal incidence by an x-polarized plane wave at the frequency 0.4 THz. The modal expansions of the reflected and transmitted fields are dominated, respectively, by the contributions of the x-polarized specular plane wave order and the least evanescent Bloch mode of the metamaterials.

Fig. 8.
Fig. 8.

THz metamaterials: the solid red curves and the dashed blue curves are, respectively, the real and imaginary parts of the effective index neff and effective impedance Zeff. The shaded rectangle indicates the frequency band where the real part of the effective magnetic permeability μeff is negative (see Fig. 9).

Fig. 9.
Fig. 9.

THz metamaterials: the solid red curves and the dashed blue curve are, respectively, the real and imaginary parts of the effective dielectric constant εeff=neff/Zeff and effective magnetic permeability μeff=neffZeff. The shaded rectangle indicates the frequency band [0.76 THz, 1.08 THz] where Re(μeff)<0.

Fig. 10.
Fig. 10.

THz metamaterials: transmittance over three layers (same notations as in Fig. 6). The full numerical calculation is based on Eq. (17).

Fig. 11.
Fig. 11.

Fresnel reflectance curves of the metal–dielectric optical metamaterials. The continuous and dashed curves are obtained using, respectively, a full numerical calculation [based on Eq. (60)] and the impedance matrix technique. The impedance matrix technique provides a well-converged result when Ptrunc=50.

Fig. 12.
Fig. 12.

Transmittance curves over three layers of the metal–dielectric optical metamaterials (same notations as in Fig. 11). The full numerical calculation is based on Eq. (17).

Fig. 13.
Fig. 13.

Optical metamaterials: absolute value of the Fresnel reflection and transmission coefficients |rn| and |tn| in the expansions Eq. (61) for normal incidence by an x-polarized plane wave at the wavelength λ=1600nm. Unlike the case in Fig. 7, here many modes have significant contributions to the modal expansions.

Fig. 14.
Fig. 14.

Optical metamaterials: transmittance curves over L=3 and L=10 layers. The continuous blue curve refers to the solution given by the full numerical calculation [using Eq. (17)]. The green-dashed curve and the red-dotted curve represent results obtained using the asymptotic expressions Eq. (63), in which the values of the Fresnel coefficients ρij and τij are computed from the scalar impedance model (the green-dashed curve) or extracted from the coefficient of scattering matrices constructed from impedance matrices with a high truncation number—here Ptrunc=200 (red-dotted curve).

Equations (63)

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×(1μ(r)×En(r))k02ε(r)En(r)=0,
Z0Hn=ik0μ×En,
E(r)=p,q[(χp,qTE)12(fp,qTE+fp,qTE+)Ep,qTE+(χp,qTM)12(fp,qTM+fp,qTM+)Ep,qTM]eikp,q·(rr0),
E(r)=p,q[(χp,qTE)12(fp,qTE+fp,qTE+)Ep,qTE+(χp,qTM)12(fp,qTM+fp,qTM+)Ep,qTM]eikp,q·(rr0).
(z×H(r))=p,q[(χp,qTE)12(fp,qTEfp,qTE+)Ep,qTE+(χp,qTM)12(fp,qTMfp,qTM+)Ep,qTM]eikp,q·(rr0),
(z×H(r))=p,q[(χp,qTE)12(fp,qTEfp,qTE+)Ep,qTE+(χp,qTM)12(fp,qTMfp,qTM+)Ep,qTM]eikp,q·(rr0).
αp=α0+2πpdx,βq=β0+2πqdyandγp,q=(k02εbαp2βq2)1/2,p,qZ,
Ep,qTE=1αp2+βq2[βqαp],Ep,qzTE=0
Ep,qTM=1αp2+βq2[αpβq],Ep,qzTM=αp2+βq2γp,q,ifγp,q0.
χp,qTE=γp,qk0andχp,qTM=εbk0γp,q.
[ff+]=T[ff+].
fn=κnfn,fn+=κnfn+withκn=eikn·e3,
T[fnfn+]=κn[fnfn+].
T=FLF1withF=[FFF+F+],L=[Λ00Λ],
Q=[0QQ0]
T=[IR0T]1[T0RI]=[TRT1RRT1T1RT1],
R[1,i+1]=R[1,i]+T[1,i]Ri+1(IR[1,i]Ri+1)1T[1,i],T[1,i+1]=Ti+1(IR[1,i]Ri+1)1T[1,i],R[1,i+1]=Ri+1+Ti+1R[1,i](IRi+1R[1,i])1Ti+1,T[1,i+1]=T[1,i](IRi+1R[1,i])1Ti+1,
ΩEn·(ez×Hm)dA=δnmandΩEn·(ez×Hm)dA=δnm.
En(x,y,z)=(En,(x,y,z),En,z(x,y,z)),
Hn(x,y,z)=(Hn,(x,y,z),Hn,z(x,y,z)).
P:[Ex(x,y,z)Ey(x,y,z)Ez(x,y,z)][Ex(x,y,z)Ey(x,y,z)Ez(x,y,z)].
0=P(×(1μ×En)k02εEn),
=×(1μ×P(En))k02εP(En).
E=n(cnEn+cn+En+),
H=n(cnHn+cn+Hn+).
En=(En,Enz)=(En+,Enz+),
Hn=(Hn,Hnz)=(Hn+,Hnz+).
Q:[Ex(x,y,z)Ey(x,y,z)Ez(x,y,z)][Ex(x,y,z)Ey(x,y,z)Ez(x,y,z)].
E=n(cn+cn+)En,,
H=n(cncn+)Hn,.
n(c1n+c1n+)E1n=nc2nE2n,
n(c1nc1n+)H1n=nc2nH2n.
(c1m+c1m+)=nJ12;mn(α0,β0)c2n,mN,
nK21;mn(α0,β0)(c1nc1n+)=c2m,mN,
J12;mn(α0,β0)=ΩE2n·(ez×H1m)dA,
K21;mn(α0,β0)=ΩE2m·(ez×H1n)dA=J12;nm(α0,β0).
{c1+c1+=J12(α0,β0)c2,J12(α0,β0)T(c1c1+)=c2,
c1+c1+=J12(α0,β0)J12(α0,β0)T(c1c1+),
c1+=(J12(α0,β0)J12(α0,β0)T+I)1(J12(α0,β0)J12(α0,β0)TI)c1.
c2=2J12(α0,β0)T(J12(α0,β0)J12(α0,β0)T+I)1c1.
R12=(J12(α0,β0)J12(α0,β0)T+I)1(J12(α0,β0)J12(α0,β0)TI),
T12=2J12(α0,β0)T(J12(α0,β0)J12(α0,β0)T+I)1.
R21=(I+J12(α0,β0)TJ12(α0,β0))1(IJ12(α0,β0)TJ12(α0,β0)),
T21=2J12(α0,β0)(I+J12(α0,β0)TJ12(α0,β0))1.
R=R12+T21ΛLR23ΛL(IR21ΛLR23ΛL)1T12,
T=T23ΛL(IR21ΛLR23ΛL)1T12,
Z2Z1Z2+Z1=Z2/Z11Z2/Z1+1.
Z=J12(α0,β0)J12(α0,β0)T
Jij(α0,β0)=Ji0(α0,β0)J0j(α0,β0).
(Ji0(α0,β0)J0j(α0,β0))nm=p=1Ji0;np(α0,β0)J0j;pm(α0,β0),
=p=1(ΩE0;p(r)·(ez×Hi;n(r))dA)(ΩEj;m(r)·(ez×H0;p(r))dA),
=p=1{Ω(ΩEj;m(r)·(ez×H0;p(r))dA)E0;p(r)·(ez×Hi;n(r))dA},
=Ω{p=1(ΩEj;m(r)·(ez×H0;p(r))dA)E0;p(r)}·(ez×Hi;n(r))dA
Ej;m,(r)=p=1(ΩEj;m(r)·(ez×H0;p(r))dA)E0;p,(r),
(Ji0(α0,β0)J0j(α0,β0))nm=ΩEj;m(r)·(ez×Hi;n(r))dA=Jij;nm(α0,β0)=(Jij(α0,β0))nm
Ji0(α0,β0)=(J0i(α0,β0))1,
R12=J1221J122+1=Z2/Z11Z2/Z1+1=1J1221+J122=R21,
T12=2J12J122+1=2Z2/Z1Z2/Z1+1=T21.
εm(ω)=1ωP2ω(ω+iγ)
R12=F+F1andT12=F1,
Erefl=nrnE1,nandEtrans=ntnE2,n,
J32(α0,β0)=J31(α0,β0)J12(α0,β0)=(J13(α0,β0))1J12(α0,β0).
R=ρ12+ρ23τ122Λ12L1ρ21ρ23Λ12L,T=τ12τ23Λ1L1ρ21ρ23Λ12L.

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