Abstract

It is well known that defects, such as holes, inside an infinite photonic crystal can sustain localized resonant modes whose frequencies fall within a forbidden band. Here we prove that finite, defect-free photonic crystals behave as mirrorless resonant cavities for frequencies within but near the edges of an allowed band, regardless of the shape of their outer boundary. The resonant modes are extended, surface-avoiding (nearly-Dirichlet) states that may lie inside or outside the light cone. Independent of the dimensionality, quality factors and finesses are on the order of, respectively, (L/λ)3 and L/λ, where λ is the vacuum wavelength and L >> λ is a typical size of the crystal. Similar topological modes exist in conventional Fabry-Pérot resonators, and in plasmonic media at frequencies just above those at which the refractive index vanishes.

© 2014 Optical Society of America

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References

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  1. A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  7. M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  21. Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express 19(22), 22191–22197 (2011).
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    [CrossRef] [PubMed]
  23. K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev. 58(7), 2172–2176 (2011).
    [CrossRef]

2011 (2)

K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev. 58(7), 2172–2176 (2011).
[CrossRef]

Q. Quan, I. B. Burgess, S. K. Y. Tang, D. L. Floyd, and M. Loncar, “High-Q, low index-contrast polymeric photonic crystal nanobeam cavities,” Opt. Express 19(22), 22191–22197 (2011).
[CrossRef] [PubMed]

2006 (3)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. 97(12), 124301 (2006).
[CrossRef] [PubMed]

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006).
[CrossRef]

2005 (2)

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72(4), 045126 (2005).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71(19), 195122 (2005).
[CrossRef]

2004 (1)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

2003 (2)

S. T. Thurman and G. M. Morris, “Controlling the spectral response in guided-mode resonance filter design,” Appl. Opt. 42(16), 3225–3233 (2003).
[CrossRef] [PubMed]

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

2002 (1)

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
[CrossRef]

2001 (1)

1999 (1)

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

1994 (1)

R. Merlin, “Raman scattering by surface-avoiding acoustic phonons in semi-infinite superlattices,” Philos. Mag. B 70(3), 761–766 (1994).
[CrossRef]

1993 (2)

S. S. Wang and R. Magnusson, “Theory and applications of guided-mode resonance filters,” Appl. Opt. 32(14), 2606–2613 (1993).
[CrossRef] [PubMed]

For a discussion of the analogous problem of a particle in a finite one-dimensional periodic potential, see:D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61(12), 1118–1124 (1993).

1987 (2)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

1961 (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[CrossRef]

1946 (1)

E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev. 69(11), 681 (1946).

1944 (1)

H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev. 65(11), 307–318 (1944).
[CrossRef]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik 330(3), 377–445 (1908).
[CrossRef]

Akahane, Y.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Allard, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
[CrossRef]

Asano, T.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Burgess, I. B.

Charbonneau-Lefort, M.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
[CrossRef]

Chutinan, A.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

Eckhause, T. A.

M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. 97(12), 124301 (2006).
[CrossRef] [PubMed]

Feshbach, H.

H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev. 65(11), 307–318 (1944).
[CrossRef]

Floyd, D. L.

Fox, A. G.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[CrossRef]

Garcia-Vidal, F. J.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Goldman, R. S.

M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. 97(12), 124301 (2006).
[CrossRef] [PubMed]

Green, A. A.

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71(19), 195122 (2005).
[CrossRef]

Ilchenko, V. S.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

Imada, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

Istrate, E.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71(19), 195122 (2005).
[CrossRef]

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
[CrossRef]

Joannopoulos, J. D.

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

Li, T.

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[CrossRef]

Loncar, M.

Magnusson, R.

Martín-Moreno, L.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Martorell, J.

For a discussion of the analogous problem of a particle in a finite one-dimensional periodic potential, see:D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61(12), 1118–1124 (1993).

Matsko, A. B.

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

Mazumder, P.

K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev. 58(7), 2172–2176 (2011).
[CrossRef]

Merlin, R.

M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. 97(12), 124301 (2006).
[CrossRef] [PubMed]

R. Merlin, “Raman scattering by surface-avoiding acoustic phonons in semi-infinite superlattices,” Philos. Mag. B 70(3), 761–766 (1994).
[CrossRef]

Mie, G.

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik 330(3), 377–445 (1908).
[CrossRef]

Morris, G. M.

Murata, M.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

Nair, S. V.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72(4), 045126 (2005).
[CrossRef]

Noda, S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

Pendry, J. B.

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Poon, J.

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
[CrossRef]

Purcell, E. M.

E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev. 69(11), 681 (1946).

Quan, Q.

Reason, M.

M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. 97(12), 124301 (2006).
[CrossRef] [PubMed]

Ruda, H. E.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72(4), 045126 (2005).
[CrossRef]

Sargent, E. H.

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71(19), 195122 (2005).
[CrossRef]

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
[CrossRef]

Sasaki, G.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

Song, B. S.

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Song, K.

K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev. 58(7), 2172–2176 (2011).
[CrossRef]

Sprung, D. W. L.

For a discussion of the analogous problem of a particle in a finite one-dimensional periodic potential, see:D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61(12), 1118–1124 (1993).

Tang, S. K. Y.

Thurman, S. T.

Tokuda, T.

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

Trigo, M.

M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. 97(12), 124301 (2006).
[CrossRef] [PubMed]

Wang, S. S.

Wu, H.

For a discussion of the analogous problem of a particle in a finite one-dimensional periodic potential, see:D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61(12), 1118–1124 (1993).

Xu, T.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72(4), 045126 (2005).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

Yang, S.

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72(4), 045126 (2005).
[CrossRef]

Am. J. Phys. (1)

For a discussion of the analogous problem of a particle in a finite one-dimensional periodic potential, see:D. W. L. Sprung, H. Wu, and J. Martorell, “Scattering by a finite periodic potential,” Am. J. Phys. 61(12), 1118–1124 (1993).

Annalen der Physik (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Annalen der Physik 330(3), 377–445 (1908).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

M. Imada, S. Noda, A. Chutinan, T. Tokuda, M. Murata, and G. Sasaki, “Coherent two-dimensional lasing action in surface-emitting laser with triangular-lattice photonic crystal structure,” Appl. Phys. Lett. 75(3), 316–318 (1999).
[CrossRef]

Bell Syst. Tech. J. (1)

A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell Syst. Tech. J. 40(2), 453–488 (1961).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

A. B. Matsko and V. S. Ilchenko, “Optical resonators with whispering-gallery modes-part I: basics,” IEEE J. Sel. Top. Quantum Electron. 12(1), 3–14 (2006).
[CrossRef]

IEEE Trans. Electron. Dev. (1)

K. Song and P. Mazumder, “Dynamic terahertz spoof surface plasmon-polariton switch based on resonance and absorption,” IEEE Trans. Electron. Dev. 58(7), 2172–2176 (2011).
[CrossRef]

Nature (1)

Y. Akahane, T. Asano, B. S. Song, and S. Noda, “High-Q photonic nanocavity in a two-dimensional photonic crystal,” Nature 425(6961), 944–947 (2003).
[CrossRef] [PubMed]

Opt. Express (2)

Philos. Mag. B (1)

R. Merlin, “Raman scattering by surface-avoiding acoustic phonons in semi-infinite superlattices,” Philos. Mag. B 70(3), 761–766 (1994).
[CrossRef]

Phys. Rev. (2)

H. Feshbach, “On the Perturbation of Boundary Conditions,” Phys. Rev. 65(11), 307–318 (1944).
[CrossRef]

E. M. Purcell, “Spontaneous Emission Probabilities at Radio Frequencies,” Phys. Rev. 69(11), 681 (1946).

Phys. Rev. B (3)

T. Xu, S. Yang, S. V. Nair, and H. E. Ruda, “Confined modes in finite-size photonic crystals,” Phys. Rev. B 72(4), 045126 (2005).
[CrossRef]

M. Charbonneau-Lefort, E. Istrate, M. Allard, J. Poon, and E. H. Sargent, “Photonic crystal heterostructures: Waveguiding phenomena and methods of solution in an envelope function picture,” Phys. Rev. B 65(12), 125318 (2002).
[CrossRef]

E. Istrate, A. A. Green, and E. H. Sargent, “Behavior of light at photonic crystal interfaces,” Phys. Rev. B 71(19), 195122 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

M. Trigo, T. A. Eckhause, M. Reason, R. S. Goldman, and R. Merlin, “Observation of surface-avoiding waves: a new class of extended states in periodic media,” Phys. Rev. Lett. 97(12), 124301 (2006).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58(20), 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58(23), 2486–2489 (1987).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

E. Istrate and E. H. Sargent, “Photonic crystal heterostructures and interfaces,” Rev. Mod. Phys. 78(2), 455–481 (2006).
[CrossRef]

Science (1)

J. B. Pendry, L. Martín-Moreno, and F. J. Garcia-Vidal, “Mimicking surface plasmons with structured surfaces,” Science 305(5685), 847–848 (2004).
[CrossRef] [PubMed]

Other (1)

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (University Press, Cambridge, 1958), p. 198.

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Figures (4)

Fig. 1
Fig. 1

Schematic diagram showing the finite PC and the length scales, L and λ, of the problem. C Ξ is a mathematical curve that tightly encloses the PC and Ξ is the interior region it defines.

Fig. 2
Fig. 2

Results for a multilayer stack (one dimensional PC) of length D and period d whose unit cell consists of two layers, of thicknesses d A /d=0.37 and d B /d=0.63 , and corresponding refractive indices n A =1.5 and n B =3.9 . (a) Transmission as a function of frequency; ω G 2.36c/d . (b) The square of the field for the first three modes. (c) Transmission data for different PC lengths.

Fig. 3
Fig. 3

(a) PC consisting of a square lattice of rods. The radius of the rods is r, the lattice parameter is a and ε1 (ε2) is the permittivity of the host (rods). Absorption spectra for a finite PC whose boundary is defined by a circle and a bow tie are shown respectively in (b) and (c); a plane wave with electric field oriented parallel to the rod axes impinges on the finite crystals from the left. Data for ε1 = 1, ε2 = 12 × (1 + 0.0001i) and r/a = 0.2. Contour plots are for the two resonances closest to the band edge at ωG.

Fig. 4
Fig. 4

(a) Absorption data for a plasmonic sphere of radius R; k 0 = ω 0 /c . (b) Contour plots of the electric field intensity for the two lowest eigenmodes. Results are for a plane perpendicular to the wavevector of the incident plane wave. (c) Absorption as a function of ωRn(ω) for various radii. Parameters are ε 0 =3 , ε =2 and γ/ ω 0 = 10 4 .

Equations (17)

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ε 1 (r) 2 Ψ+ k 2 Ψ=0 ,
Ψ= p A p H p (1) (kρ) e ipϕ ,
1 ε(r) r 2 G k (r, r )+ k 2 G k (r, r )=δ(r r )
η(r) C Ξ [ i θ( s ) G k (r, r )+ G k (r, r ) ] η( r )d s
G k (r, r )= m χ m (r) χ m ( r ) k 2 k m 2 ,
η χ t i mt χ m k 2 k m 2 [ C Ξ θ 1 χ m ηds ] .
K t = k t + G tt 2 k t +... η t = χ t + mt k m k t G tm k t 2 k m 2 χ m +...
G tm =i( k t / k m ) C Ξ θ 1 χ m χ t ds .
j C j ( K j 2 κ ˜ 2 ) η j =0
Ψ= q A q u q (r) e iq.r
Ψ(r) u 0 (r) E q (r)
G tm i k t k m C Ξ u 0 2 (r) E t D E m D θ(s) ds
Ω t ω t + c 2 2 ω t C Ξ u 0 2 (r)| E t D | 2 ( ln u 0 ik e ρ )n ds .
Ψ 2 πρ p A p e i[ kρ+p(ϕπ/2)π/4 ] ,
Ω t ω t i c 2 v G 2 n Ξ ω t 2 C Ξ | χ t | 2 ( e ρ n) ds
ε= ε + ω 0 2 ( ε 0 ε ) ω 0 2 ω 2 +iωγ ,
τ R ~ L v G (1R)

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