Abstract

We investigate the phonon-polariton band gaps in periodic and quasi-periodic (Fibonacci-type) multilayers made up of both positive (SiO2) and negative refractive index materials (metamaterials) following the Fibonacci sequence in the terahertz region. The behavior of the polaritonic band gaps as a function of the multilayer period is investigated systematically. Our theoretical model makes use of a transfer matrix approach to simplify the algebra involved and to set up analytical phonon-polariton dispersion relations (bulk and surface modes). We also present a quantitative analysis of the results, pointing out the distribution of the allowed polaritonic bandwidths for high Fibonacci generations, which gives good insight about their localization and power laws.

© 2009 Optical Society of America

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    [CrossRef]
  2. S. A. Ramakrishna, “Negative refraction at visible frequencies,” Rep. Prog. Phys. 68, 449-521 (2005).
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  3. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
    [CrossRef] [PubMed]
  4. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292, 77-79 (2001).
    [CrossRef] [PubMed]
  5. M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696-10705 (2000).
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  6. P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
    [CrossRef] [PubMed]
  7. E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
    [CrossRef] [PubMed]
  8. A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
    [CrossRef] [PubMed]
  9. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
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  10. S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486-2489 (1987).
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  11. S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50, 1764-1769 (1994).
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  12. Y. Zeng, X. Chen, and W. Lu, “Modified spontaneous emission from a two-dimensional photonic crystal,” Phys. Rev. E 70, 047601-047603 (2004).
    [CrossRef]
  13. A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
    [CrossRef] [PubMed]
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  15. M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000).
    [CrossRef]
  16. N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127-12133 (1998).
    [CrossRef]
  17. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
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  23. E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225-337 (2003).
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  24. E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).
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  26. D. Mittleman, Sensing with Terahertz Radiation (Springer-Verlag, 2003).
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    [CrossRef]
  28. M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76, 165117 (2007).
    [CrossRef]
  29. T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
    [CrossRef]
  30. H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523-534 (1973).
    [CrossRef]
  31. T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
    [CrossRef] [PubMed]
  32. M. K. Farr, J. G. Taylor, and S. K. Sinha, “Lattice dynamics of GaSb,” Phys. Rev. B 11, 1587-1594 (1975).
    [CrossRef]
  33. M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: mapping and escape,” Phys. Rev. Lett. 50, 1870-1872 (1983).
    [CrossRef]
  34. M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020-1033 (1987).
    [CrossRef]
  35. I. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611-036614 (2002).
    [CrossRef]
  36. L. Wu, S. He, and L. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103-235108 (2003).
    [CrossRef]
  37. N. C. Panoiu, R. M. Osgood, Jr., S. Zhang, and S. R. J. Brueck, “Zero-n−bandgap in photonic crystal superlattices,” J. Opt. Soc. Am. B 23, 506-513 (2006).
    [CrossRef]
  38. E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69, 397-442 (2006).
    [CrossRef]
  39. P. Hawrylak and J. J. Quinn, “Critical plasmons of a quasiperiodic semiconductor superlattice,” Phys. Rev. Lett. 57, 380-383 (1986).
    [CrossRef] [PubMed]
  40. K. L. Yeh, T. Hornung, J. C. Vaughan, and K. A. Nelson, in Ultrafast Phenomena XV, P.Corkum, D.M.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2007).

2007

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76, 165117 (2007).
[CrossRef]

2006

2005

S. A. Ramakrishna, “Negative refraction at visible frequencies,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

2004

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Y. Zeng, X. Chen, and W. Lu, “Modified spontaneous emission from a two-dimensional photonic crystal,” Phys. Rev. E 70, 047601-047603 (2004).
[CrossRef]

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
[CrossRef]

2003

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225-337 (2003).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

L. Wu, S. He, and L. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103-235108 (2003).
[CrossRef]

2002

G. M. Turner, M. C. Beard, and C. A. Schmuttenmaer, “Carrier localization and cooling in dye-sensitized nanocrystalline titanium dioxide,” J. Phys. Chem. B 106, 11716-11719 (2002).
[CrossRef]

I. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611-036614 (2002).
[CrossRef]

2001

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

2000

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000).
[CrossRef]

1998

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

1997

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

1996

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

1994

S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50, 1764-1769 (1994).
[CrossRef] [PubMed]

1993

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233, 67-135 (1993).
[CrossRef]

1987

M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020-1033 (1987).
[CrossRef]

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

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

1986

P. Hawrylak and J. J. Quinn, “Critical plasmons of a quasiperiodic semiconductor superlattice,” Phys. Rev. Lett. 57, 380-383 (1986).
[CrossRef] [PubMed]

1983

M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: mapping and escape,” Phys. Rev. Lett. 50, 1870-1872 (1983).
[CrossRef]

1975

M. K. Farr, J. G. Taylor, and S. K. Sinha, “Lattice dynamics of GaSb,” Phys. Rev. B 11, 1587-1594 (1975).
[CrossRef]

1974

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817-926 (1974).
[CrossRef]

1973

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523-534 (1973).
[CrossRef]

1968

V. G. Veselago, “The electrodynamics of substance with simultaneously negative values of ϵand μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

1964

R. Loudon, “Raman effect in crystals,” Adv. Phys. 13, 423 (1964).
[CrossRef]

Albuquerque, E. L.

M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76, 165117 (2007).
[CrossRef]

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225-337 (2003).
[CrossRef]

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233, 67-135 (1993).
[CrossRef]

E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).

Anand, S.

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Aydin, K.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Basov, D. N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Beard, M. C.

G. M. Turner, M. C. Beard, and C. A. Schmuttenmaer, “Carrier localization and cooling in dye-sensitized nanocrystalline titanium dioxide,” J. Phys. Chem. B 106, 11716-11719 (2002).
[CrossRef]

Berrier, A.

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Brueck, S. R. J.

Burstein, E.

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817-926 (1974).
[CrossRef]

Busch, K.

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Chen, J. C.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Chen, X.

Y. Zeng, X. Chen, and W. Lu, “Modified spontaneous emission from a two-dimensional photonic crystal,” Phys. Rev. E 70, 047601-047603 (2004).
[CrossRef]

Cottam, M. G.

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225-337 (2003).
[CrossRef]

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233, 67-135 (1993).
[CrossRef]

E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).

Cubukcu, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Dawson, P.

T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
[CrossRef]

de Medeiros, F. F.

M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76, 165117 (2007).
[CrossRef]

Derov, J. S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

Falge, H. J.

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523-534 (1973).
[CrossRef]

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Fang, N.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Farr, M. K.

M. K. Farr, J. G. Taylor, and S. K. Sinha, “Lattice dynamics of GaSb,” Phys. Rev. B 11, 1587-1594 (1975).
[CrossRef]

Feurer, T.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

Foteinopoulou, S.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Hawrylak, P.

P. Hawrylak and J. J. Quinn, “Critical plasmons of a quasiperiodic semiconductor superlattice,” Phys. Rev. Lett. 57, 380-383 (1986).
[CrossRef] [PubMed]

He, S.

L. Wu, S. He, and L. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103-235108 (2003).
[CrossRef]

Hein, G.

T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
[CrossRef]

Hornung, T.

K. L. Yeh, T. Hornung, J. C. Vaughan, and K. A. Nelson, in Ultrafast Phenomena XV, P.Corkum, D.M.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2007).

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

S. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Kluwer, 2002).

John, S.

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50, 1764-1769 (1994).
[CrossRef] [PubMed]

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

Johnson, S.

S. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Kluwer, 2002).

Kadanoff, L. P.

M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: mapping and escape,” Phys. Rev. Lett. 50, 1870-1872 (1983).
[CrossRef]

Kleine-Ostmann, T.

T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
[CrossRef]

Koch, M.

T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
[CrossRef]

Kohmoto, M.

M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020-1033 (1987).
[CrossRef]

M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: mapping and escape,” Phys. Rev. Lett. 50, 1870-1872 (1983).
[CrossRef]

Kosaka, H.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000).
[CrossRef]

Kurland, I.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Loudon, R.

R. Loudon, “Raman effect in crystals,” Adv. Phys. 13, 423 (1964).
[CrossRef]

Lu, W.

Y. Zeng, X. Chen, and W. Lu, “Modified spontaneous emission from a two-dimensional photonic crystal,” Phys. Rev. E 70, 047601-047603 (2004).
[CrossRef]

Lu, W. T.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

Maciá, E.

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69, 397-442 (2006).
[CrossRef]

Mauriz, P. W.

M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76, 165117 (2007).
[CrossRef]

Mekis, A.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Mills, D. L.

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817-926 (1974).
[CrossRef]

Mittleman, D.

D. Mittleman, Sensing with Terahertz Radiation (Springer-Verlag, 2003).

Modinos, A.

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

Mulot, M.

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Nefedov, I. S.

I. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611-036614 (2002).
[CrossRef]

Nelson, K. A.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

K. L. Yeh, T. Hornung, J. C. Vaughan, and K. A. Nelson, in Ultrafast Phenomena XV, P.Corkum, D.M.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2007).

Notomi, M.

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

Osgood, R. M.

Otto, A.

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523-534 (1973).
[CrossRef]

Ozbay, E.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Padilla, W. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Panoiu, N. C.

Parimi, P. V.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

Pendry, J. B.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

Pierz, K.

T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
[CrossRef]

Quang, T.

S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50, 1764-1769 (1994).
[CrossRef] [PubMed]

Qui, M.

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Quinn, J. J.

P. Hawrylak and J. J. Quinn, “Critical plasmons of a quasiperiodic semiconductor superlattice,” Phys. Rev. Lett. 57, 380-383 (1986).
[CrossRef] [PubMed]

Ramakrishna, S. A.

S. A. Ramakrishna, “Negative refraction at visible frequencies,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

Schmuttenmaer, C. A.

G. M. Turner, M. C. Beard, and C. A. Schmuttenmaer, “Carrier localization and cooling in dye-sensitized nanocrystalline titanium dioxide,” J. Phys. Chem. B 106, 11716-11719 (2002).
[CrossRef]

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]

Shen, L.

L. Wu, S. He, and L. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103-235108 (2003).
[CrossRef]

Sinha, S. K.

M. K. Farr, J. G. Taylor, and S. K. Sinha, “Lattice dynamics of GaSb,” Phys. Rev. B 11, 1587-1594 (1975).
[CrossRef]

Smith, D. R.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[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]

Sokoloff, J.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

Soukoulis, C. M.

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Sridhar, S.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

Statz, E. R.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

Stefanou, N.

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

Stoyanov, N. S.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

Sutherland, B.

M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020-1033 (1987).
[CrossRef]

Swillo, M.

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Talneau, A.

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Tang, C.

M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020-1033 (1987).
[CrossRef]

M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: mapping and escape,” Phys. Rev. Lett. 50, 1870-1872 (1983).
[CrossRef]

Taylor, J. G.

M. K. Farr, J. G. Taylor, and S. K. Sinha, “Lattice dynamics of GaSb,” Phys. Rev. B 11, 1587-1594 (1975).
[CrossRef]

Thylén, L.

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

Tokushima, M.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000).
[CrossRef]

Tomita, A.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000).
[CrossRef]

Tretyakov, S. A.

I. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611-036614 (2002).
[CrossRef]

Turner, G. M.

G. M. Turner, M. C. Beard, and C. A. Schmuttenmaer, “Carrier localization and cooling in dye-sensitized nanocrystalline titanium dioxide,” J. Phys. Chem. B 106, 11716-11719 (2002).
[CrossRef]

Vasconcelos, M. S.

M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76, 165117 (2007).
[CrossRef]

Vaughan, J. C.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

K. L. Yeh, T. Hornung, J. C. Vaughan, and K. A. Nelson, in Ultrafast Phenomena XV, P.Corkum, D.M.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2007).

Veselago, V. G.

V. G. Veselago, “The electrodynamics of substance with simultaneously negative values of ϵand μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Vier, D. C.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

Vodo, P.

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

Ward, D. W.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

Wu, L.

L. Wu, S. He, and L. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103-235108 (2003).
[CrossRef]

Yablonovitch, E.

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

Yamada, H.

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000).
[CrossRef]

Yeh, K. L.

K. L. Yeh, T. Hornung, J. C. Vaughan, and K. A. Nelson, in Ultrafast Phenomena XV, P.Corkum, D.M.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2007).

Yen, T. J.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Zeng, Y.

Y. Zeng, X. Chen, and W. Lu, “Modified spontaneous emission from a two-dimensional photonic crystal,” Phys. Rev. E 70, 047601-047603 (2004).
[CrossRef]

Zhang, S.

Zhang, X.

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Adv. Phys.

R. Loudon, “Raman effect in crystals,” Adv. Phys. 13, 423 (1964).
[CrossRef]

Annu. Rev. Mater. Res.

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Res. 37, 317-350 (2007).
[CrossRef]

Appl. Phys. Lett.

T. Kleine-Ostmann, P. Dawson, K. Pierz, G. Hein, and M. Koch, “Room-temperature operation of an electrically driven terahertz modulator,” Appl. Phys. Lett. 84, 3555-3557 (2004).
[CrossRef]

M. Tokushima, H. Kosaka, A. Tomita, and H. Yamada, “Lightwave propagation through a 120° sharply bent single-line-defect photonic crystal waveguide,” Appl. Phys. Lett. 76, 952-954 (2000).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. Chem. B

G. M. Turner, M. C. Beard, and C. A. Schmuttenmaer, “Carrier localization and cooling in dye-sensitized nanocrystalline titanium dioxide,” J. Phys. Chem. B 106, 11716-11719 (2002).
[CrossRef]

Nature

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

E. Cubukcu, K. Aydin, E. Ozbay, S. Foteinopoulou, and C. M. Soukoulis, “Electromagnetic waves: negative refraction by photonic crystals,” Nature 423, 604-605 (2003).
[CrossRef] [PubMed]

Phys. Rep.

E. L. Albuquerque and M. G. Cottam, “Superlattice plasmon-polaritons,” Phys. Rep. 233, 67-135 (1993).
[CrossRef]

E. L. Albuquerque and M. G. Cottam, “Theory of elementary excitations in quasiperiodic structures,” Phys. Rep. 376, 225-337 (2003).
[CrossRef]

Phys. Rev. A

S. John and T. Quang, “Spontaneous emission near the edge of a photonic band gap,” Phys. Rev. A 50, 1764-1769 (1994).
[CrossRef] [PubMed]

Phys. Rev. B

M. Notomi, “Theory of light propagation in strongly modulated photonic crystals: refraction like behavior in the vicinity of the photonic band gap,” Phys. Rev. B 62, 10696-10705 (2000).
[CrossRef]

N. Stefanou and A. Modinos, “Impurity bands in photonic insulators,” Phys. Rev. B 57, 12127-12133 (1998).
[CrossRef]

M. S. Vasconcelos, P. W. Mauriz, F. F. de Medeiros, and E. L. Albuquerque, “Photonic band gaps in quasiperiodic photonic crystals with negative refractive index,” Phys. Rev. B 76, 165117 (2007).
[CrossRef]

M. K. Farr, J. G. Taylor, and S. K. Sinha, “Lattice dynamics of GaSb,” Phys. Rev. B 11, 1587-1594 (1975).
[CrossRef]

M. Kohmoto, B. Sutherland, and C. Tang, “Critical wave functions and a Cantor-set spectrum of a one-dimensional quasicrystal model,” Phys. Rev. B 35, 1020-1033 (1987).
[CrossRef]

L. Wu, S. He, and L. Shen, “Band structure for a one-dimensional photonic crystal containing left-handed materials,” Phys. Rev. B 67, 235103-235108 (2003).
[CrossRef]

Phys. Rev. E

I. S. Nefedov and S. A. Tretyakov, “Photonic band gap structure containing metamaterial with negative permittivity and permeability,” Phys. Rev. E 66, 036611-036614 (2002).
[CrossRef]

K. Busch and S. John, “Photonic band gap formation in certain self-organizing systems,” Phys. Rev. E 58, 3896-3908 (1998).
[CrossRef]

Y. Zeng, X. Chen, and W. Lu, “Modified spontaneous emission from a two-dimensional photonic crystal,” Phys. Rev. E 70, 047601-047603 (2004).
[CrossRef]

Phys. Rev. Lett.

A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos, “High transmission through sharp bends in photonic crystal waveguides,” Phys. Rev. Lett. 77, 3787-3790 (1996).
[CrossRef] [PubMed]

P. V. Parimi, W. T. Lu, P. Vodo, J. Sokoloff, J. S. Derov, and S. Sridhar, “Negative refraction and left-handed electromagnetism in microwave photonic crystals,” Phys. Rev. Lett. 92, 127401-127404 (2004).
[CrossRef] [PubMed]

A. Berrier, M. Mulot, M. Swillo, M. Qui, L. Thylén, A. Talneau, and S. Anand, “Negative refraction at infrared wavelengths in a two-dimensional photonic crystal,” Phys. Rev. Lett. 93, 073902-073905 (2004).
[CrossRef] [PubMed]

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

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

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966-3969 (2000).
[CrossRef] [PubMed]

M. Kohmoto, L. P. Kadanoff, and C. Tang, “Localization problem in one dimension: mapping and escape,” Phys. Rev. Lett. 50, 1870-1872 (1983).
[CrossRef]

P. Hawrylak and J. J. Quinn, “Critical plasmons of a quasiperiodic semiconductor superlattice,” Phys. Rev. Lett. 57, 380-383 (1986).
[CrossRef] [PubMed]

Phys. Status Solidi B

H. J. Falge and A. Otto, “Dispersion of phonon-like surface polaritons on α-quartz observed by attenuated total reflection,” Phys. Status Solidi B 56, 523-534 (1973).
[CrossRef]

Rep. Prog. Phys.

D. L. Mills and E. Burstein, “Polaritons: the electromagnetic modes of media,” Rep. Prog. Phys. 37, 817-926 (1974).
[CrossRef]

S. A. Ramakrishna, “Negative refraction at visible frequencies,” Rep. Prog. Phys. 68, 449-521 (2005).
[CrossRef]

E. Maciá, “The role of aperiodic order in science and technology,” Rep. Prog. Phys. 69, 397-442 (2006).
[CrossRef]

Science

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

T. J. Yen, W. J. Padilla, N. Fang, D. C. Vier, D. R. Smith, J. B. Pendry, D. N. Basov, and X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494-1496 (2004).
[CrossRef] [PubMed]

Sov. Phys. Usp.

V. G. Veselago, “The electrodynamics of substance with simultaneously negative values of ϵand μ,” Sov. Phys. Usp. 10, 509-514 (1968).
[CrossRef]

Other

S. Johnson and J. D. Joannopoulos, Photonic Crystals: The Road from Theory to Practice (Kluwer, 2002).

Polaritons, E.Burstein and F.de Martini, eds. (Pergamon, 1974).

E. L. Albuquerque and M. G. Cottam, Polaritons in Periodic and Quasiperiodic Structures (Elsevier, 2004).

D. Mittleman, Sensing with Terahertz Radiation (Springer-Verlag, 2003).

K. L. Yeh, T. Hornung, J. C. Vaughan, and K. A. Nelson, in Ultrafast Phenomena XV, P.Corkum, D.M.Jonas, R.J. D.Miller, and A.M.Weiner, eds. (Springer, 2007).

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Figures (4)

Fig. 1
Fig. 1

(a) Polaritonic band gap frequency spectrum, as given by Eq. (16) (bulk modes) and Eq. (17) (surface modes) for a periodic photonic crystal. The shaded areas represent the bulk mode regions, while the surface modes are represented by full lines. The dashed line represents the light line in the vacuum, while the chain-dotted line is the light line in layer B ( SiO 2 ) . (b) Zoom of (a) for the region 17.34 ω 26.01   THz and 0.0 k x d A 0.25 .

Fig. 2
Fig. 2

Polaritonic band gap frequency spectrum considering a fourth generation Fibonacci quasi-periodic photonic crystal. The shaded areas represent the bulk mode regions, while the surface modes are represented by full lines. The dashed line represents the light line in the vacuum, while the chain-dotted line is the light line in layer B ( SiO 2 ) .

Fig. 3
Fig. 3

Polaritonic band gap spectrum against the dimensionless Bloch wave vector Q L for the thickness ratio d B / d A = 3.90 , considering the fifth generation of the Fibonacci quasi-periodic polaritonic superlattice. (b) Projected polaritonic band structure plotted as a function of the reduced in-plane wave vector K x = k x L / 2 π . The shaded and the white areas correspond to the passbands and to the stop bands of the structure, respectively.

Fig. 4
Fig. 4

(a) Bandwidth distribution of the phonon polaritons, for k x d A = 0.5 , as a function of the generation number n. (b) Log–log representation of the total width of the allowed regions Δ versus the Fibonacci number F n , for three different values of the dimensionless in-plane wave vector k x d A .

Equations (30)

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

ϵ A = ϵ ω L 2 ω 2 ω T 2 ω 2 ,
μ A = 1 F ω 2 ω 2 ω 0 2 ,
E j ( x , y , z ) = ( E x j , 0 , E z j ) exp ( i k x x i ω t ) ,
H j ( x , y , z ) = ( 0 , H y j , 0 ) exp ( i k x x i ω t ) .
E x j ( z ) = α 1 j n   exp ( k z j z ) + α 2 j n   exp ( k z j z ) ,
E z j ( z ) = ( i k x / k z j ) [ α 1 j n   exp ( k z j z ) α 2 j n   exp ( k z j z ) ] ,
H y j ( z ) = [ i ω ϵ 0 ϵ A ( ω ) / k z j ] [ α 1 j n   exp ( k z j z ) α 2 j n   exp ( k z j z ) ] ,
k z j = { [ k x 2 ϵ j μ j ω 2 / c 2 ] 1 / 2 if   k x > ( ϵ j μ j ) 1 / 2 ( ω / c ) , i [ ϵ j μ j ω 2 / c 2 k x 2 ] 1 / 2 if   k x < ( ϵ j μ j ) 1 / 2 ( ω / c ) . }
| α j n = [ α 1 j n α 2 j n ] ,
M A | α A n = N B | α B n ,
M B | α B n = N A | α A n + 1 ,
M j = ( f j f ¯ j f j / ( Z j   cos   θ j ) f ¯ j / ( Z j   cos   θ j ) ) ,
N j = ( 1 1 1 / ( Z j   cos   θ j ) 1 / ( Z j   cos   θ j ) ) .
f j = exp ( k z j d j ) ,     f ¯ j = 1 / f j .
| α A n + 1 = T | α A n ,
cos ( Q L ) = ( 1 / 2 ) Tr ( T ) .
T 11 + λ T 12 = exp ( β L ) = T 21 λ 1 + T 22 ,
T S n + 1 = T S n 1 T S n
Tr ( T S n + 1 ) = Tr ( T S n ) Tr ( T S n 1 ) Tr ( T S n 2 ) .
x n = ( 1 / 2 ) Tr ( T S n ) ,
x n + 1 = 2 x n x n 1 x n 2 .
x 1 = ( f A f ¯ B + f ¯ A f B ) ( r 1 + r 1 1 ) / 4 ,
x 0 = [ f B ( 1 + r 1 ) + f ¯ B ( 1 + r 1 ) ] / 4 ,
x 1 = [ f A ( 1 + r 1 1 ) + f ¯ A ( 1 + r 1 1 ) ] / 4 ,
r n + 1 = F ( r n )
r 1 = ( x 1 , y 1 , z 1 ) .
x n + 1 = y n ,     y n + 1 = z n ,     z n + 1 = 2 y n z n x n .
I = x n 2 + y n 2 + z n 2 2 x n y n z n 1.
I = { ( f A 2 f ¯ B 2 + f ¯ A 2 f B 2 ) [ ( r 1 + r 1 1 ) 2 2 ( r 1 + r 1 1 ) ] + ( f B 2 + f ¯ B 2 f A 2 f ¯ A 2 ) [ r 1 ( 2 + r 1 ) r 1 1 ( 2 + r 1 1 ) ] + 2 ( r 1 + r 1 1 ) 2 } / 32 1.
η ¯ n = ( F n 1 η A d A + F n 2 η B d B ) / L = 0 ,

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