Abstract

The propagation characteristics of a subwavelength plasmonic crystal are studied based on its complex Bloch band structure. Photonic crystal bands are generated with an alternative 2D Finite Element Method formulation in which the Bloch wave problem is reduced to a quadratic eigenvalue system for the Bloch wavevector amplitude k. This method constitutes an efficient and convenient alternative to nonlinear search methods normally employed in the calculation of photonic bands when dispersive materials are involved. The method yields complex wavevector Bloch modes that determine the wave-scattering characteristics of finite crystals. This is evidenced in a comparison between the band structure of the square-lattice plasmonic crystal and scattering transfer-functions from a corresponding finite crystal slab. We report on a wave interference effect that leads to transmission resonances similar to Fano resonances, as well as on the isotropy of the crystal’s negative index band. Our results indicate that effective propagation constants obtained from scattering simulations may not always be directly related to individual crystal Bloch bands.

© 2007 Optical Society of America

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References

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princteon University Press, 1995).
  2. K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (New York: Springer, 2001).
  3. S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001).
    [CrossRef] [PubMed]
  4. B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, "Aplication of finite element methods to photonic crystal modelling," IEE Proc - Sci. Meas. Technol. 149, 293-296 (2002).
    [CrossRef]
  5. G. Shvets and Y. A. Urzhumov, "Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances," Phys. Rev. Lett. 93, 243902 (2004).
    [CrossRef]
  6. G. Shvets and Y. Urzhumov, "Electric and magnetic properties of sub-wavelength plasmonic crystals," J. Opt. A: Pure Appl. Opt. 7, S23-S31 (2005).
    [CrossRef]
  7. C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
    [CrossRef] [PubMed]
  8. A. Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM J. Numer. Anal. 10, 674-689 (1973).
    [CrossRef]
  9. A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
    [CrossRef]
  10. E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
    [CrossRef]
  11. [Online]. Available: http://www.comsol.com
  12. U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
    [CrossRef]
  13. J. Jin, The Finite Element Method in Electromagnetics, (2nd ed. Wiley, 2002).
  14. F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001).
    [CrossRef]
  15. G. Shvets and Y. Urzhumov, "Negative index meta-materials based on two-dimensional metallic structures," J. Opt. A: Pure Appl. Opt. 8, S122-S130 (2006).
    [CrossRef]
  16. D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
    [CrossRef]
  17. K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
    [CrossRef]
  18. H. G. Winful, "The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities," New J. Phys., Phys. 8, 101 (2006).
    [CrossRef]

2006 (2)

G. Shvets and Y. Urzhumov, "Negative index meta-materials based on two-dimensional metallic structures," J. Opt. A: Pure Appl. Opt. 8, S122-S130 (2006).
[CrossRef]

H. G. Winful, "The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities," New J. Phys., Phys. 8, 101 (2006).
[CrossRef]

2005 (5)

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

G. Shvets and Y. Urzhumov, "Electric and magnetic properties of sub-wavelength plasmonic crystals," J. Opt. A: Pure Appl. Opt. 7, S23-S31 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
[CrossRef]

2004 (2)

G. Shvets and Y. A. Urzhumov, "Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances," Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

2002 (1)

B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, "Aplication of finite element methods to photonic crystal modelling," IEE Proc - Sci. Meas. Technol. 149, 293-296 (2002).
[CrossRef]

2001 (2)

1973 (1)

A. Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM J. Numer. Anal. 10, 674-689 (1973).
[CrossRef]

1961 (1)

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Bienstman, P.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Burger, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Enkrich, C.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Fan, S.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Fano, U.

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Green, A. A.

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Hiett, B. P.

B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, "Aplication of finite element methods to photonic crystal modelling," IEE Proc - Sci. Meas. Technol. 149, 293-296 (2002).
[CrossRef]

Huang, K. C.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Istrate, E.

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Jiang, X.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Joannopoulos, J. D.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

S. G. Johnson and J. D. Joannopoulos, "Block-iterative frequency-domain methods for maxwell’s equations in a planewave basis," Opt. Express 8, 173-190 (2001).
[CrossRef] [PubMed]

Johnson, S. G.

Koschny, T.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Lidorikis, E.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Linden, S.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Meerbergen, K.

F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001).
[CrossRef]

Nelson, K. A.

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

Poulton, C.

A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
[CrossRef]

Ruhe, A.

A. Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM J. Numer. Anal. 10, 674-689 (1973).
[CrossRef]

Sargent, E. H.

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Schmidt, F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Shvets, G.

G. Shvets and Y. Urzhumov, "Negative index meta-materials based on two-dimensional metallic structures," J. Opt. A: Pure Appl. Opt. 8, S122-S130 (2006).
[CrossRef]

G. Shvets and Y. Urzhumov, "Electric and magnetic properties of sub-wavelength plasmonic crystals," J. Opt. A: Pure Appl. Opt. 7, S23-S31 (2005).
[CrossRef]

G. Shvets and Y. A. Urzhumov, "Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances," Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

Smith, D. R.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Soukoulis, C. M.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Spence, A.

A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
[CrossRef]

Tisseur, F.

F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001).
[CrossRef]

Urzhumov, Y.

G. Shvets and Y. Urzhumov, "Negative index meta-materials based on two-dimensional metallic structures," J. Opt. A: Pure Appl. Opt. 8, S122-S130 (2006).
[CrossRef]

G. Shvets and Y. Urzhumov, "Electric and magnetic properties of sub-wavelength plasmonic crystals," J. Opt. A: Pure Appl. Opt. 7, S23-S31 (2005).
[CrossRef]

Urzhumov, Y. A.

G. Shvets and Y. A. Urzhumov, "Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances," Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

Vier, D. C.

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Wegener, M.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Winful, H. G.

H. G. Winful, "The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities," New J. Phys., Phys. 8, 101 (2006).
[CrossRef]

Zhou, J. F.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

Zschiedrich, L.

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

IEE Proc - Sci. Meas. Technol. (1)

B. P. Hiett, B. D. H., S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, "Aplication of finite element methods to photonic crystal modelling," IEE Proc - Sci. Meas. Technol. 149, 293-296 (2002).
[CrossRef]

J. Comput. Phys. (1)

A. Spence and C. Poulton, "Photonic band structure calculations using nonlinear eigenvalue techniques," J. Comput. Phys. 204, 65-81 (2005).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (2)

G. Shvets and Y. Urzhumov, "Electric and magnetic properties of sub-wavelength plasmonic crystals," J. Opt. A: Pure Appl. Opt. 7, S23-S31 (2005).
[CrossRef]

G. Shvets and Y. Urzhumov, "Negative index meta-materials based on two-dimensional metallic structures," J. Opt. A: Pure Appl. Opt. 8, S122-S130 (2006).
[CrossRef]

Opt. Express (1)

Phys. (1)

H. G. Winful, "The meaning of group delay in barrier tunnelling: a re-examination of superluminal group velocities," New J. Phys., Phys. 8, 101 (2006).
[CrossRef]

Phys. Rev. (1)

U. Fano, "Effects of configuration interaction on intensities and phase shifts," Phys. Rev. 124, 1866-1878 (1961).
[CrossRef]

Phys. Rev. B (2)

K. C. Huang, E. Lidorikis, X. Jiang, J. D. Joannopoulos, K. A. Nelson, P. Bienstman, and S. Fan, "Nature of lossy bloch states in polaritonic photonic crystals," Phys. Rev. B 69, 195111 (2004).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, "Behavior of light at photonic crystal interfaces," Phys. Rev. B 71, 195122 (2005).
[CrossRef]

Phys. Rev. E (1)

D. R. Smith, D. C. Vier, T. Koschny, and C. M. Soukoulis, "Electromagnetic parameter retrieval from inhomogeneous metamaterials," Phys. Rev. E 71, 036617 (2005).
[CrossRef]

Phys. Rev. Lett. (2)

C. Enkrich, M. Wegener, S. Linden, S. Burger, L. Zschiedrich, F. Schmidt, J. F. Zhou, T. Koschny, and C. M. Soukoulis, "Magnetic metamaterials at telecommunication and visible frequencies," Phys. Rev. Lett. 95, 203901 (2005).
[CrossRef] [PubMed]

G. Shvets and Y. A. Urzhumov, "Engineering the electromagnetic properties of periodic nanostructures using electrostatic resonances," Phys. Rev. Lett. 93, 243902 (2004).
[CrossRef]

SIAM J. Numer. Anal. (1)

A. Ruhe, "Algorithms for the nonlinear eigenvalue problem," SIAM J. Numer. Anal. 10, 674-689 (1973).
[CrossRef]

SIAM Rev. (1)

F. Tisseur and K. Meerbergen, "The quadratic eigenvalue problem," SIAM Rev. 43, 235-286 (2001).
[CrossRef]

Other (4)

J. Jin, The Finite Element Method in Electromagnetics, (2nd ed. Wiley, 2002).

[Online]. Available: http://www.comsol.com

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light. (Princteon University Press, 1995).

K. Sakoda, Optical Properties of Photonic Crystal, ser. Optical Sciences. (New York: Springer, 2001).

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Figures (7)

Fig. 1.
Fig. 1.

(a)Square-lattice plasmonic crystal band structure as function of kr . (b) Band structure as function of ki . Crosses indicate bands with ki =0; circles, ki ≠0; and triangles, kr π/a, ki ≠0

Fig. 2.
Fig. 2.

(a)Amplitude-squared transmission and reflection coefficients, (b) Band structure as function of kr and (c) Band structure as function of ki for a.

Fig. 3.
Fig. 3.

(a) Comparison between numerically calculated transmission coefficient amplitudes for 3- and 4-unit-cell crystal slabs (thin lines) and the analytical expression |t(Ω)|= A ·| exp(-ki (Ω)·N·a)+i·β· t FP(ki (Ω)·N·a)| (thick lines), with ki (Ω) and kr (Ω) taken from the purely real and imaginary Bloch bands respectively. (b)Amplitude and phase of the transmission transfer function for the 4-unit-cell crystal with ωc /ωp =0.001 (thick line) and ωc =0 (thin line).

Fig. 4.
Fig. 4.

(a) Band structure for (a)kr and (b)ki . Dots correspond to ωc =0, crosses to ωc /ωp =0.001.

Fig. 5.
Fig. 5.

Low-frequency, purely real bands of the square-lattice plasmonic crystal over the first Brillouin zone.

Fig. 6.
Fig. 6.

High-frequency, purely real bands of the square-lattice plasmonic crystal over the first Brillouin zone.

Fig. 7.
Fig. 7.

Bulk Plasmon band along Γ-M (0°) and Γ-X (45°) directions.

Equations (14)

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

· p ϕ + ω 2 c 2 q ϕ = 0 .
· ( p u ) i · ( p k u ) i k · p u k 2 pu = ω 2 c 2 qu
· ( p u ) = · [ p ( u ) w ] p u · w
· ( p k u ) w = · [ p k uw ] k · ( pu w )
· [ p ( u ) w ] p u · w i · ( p k uw ) +
i k · { pu w [ p ( u ) w ] } k 2 puw =
ω 2 c 2 quw
Γ p u · wd Γ + k 2 Γ puwd Γ +
ik { Γ pu k ̂ · wd Γ Γ p k ̂ · ( u ) wd Γ } =
= ω 2 c 2 Γ quwd Γ + δ Γ pw ( u i k u ) · n ̂ d δ Γ
[ A ω 2 c 2 D ] u = ik ( C B ) u + k 2 Cu .
n eff = ln Y ( ω ) i ω Δ c
X ( 1 r 2 + t 2 ) 2 t
Y X ± 1 X 2

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