Abstract

Defects in photonic crystals are local regions in which the translational symmetry is broken. The same definition can be applied to photonic quasicrystals except in this case the symmetry is the 2π/n rotational symmetry, where n is the rotational fold number. In this context, if no such defects are present, the structure is called “defect-free”. Even though photonic quasicrystal patterns can be defect-free, localized modes can still exist in such structures. These modes resemble those of a central potential that suggests that localization in photonic quasicrystals are actually “extended” modes of the rotational symmetry. A possible connection is suggested between these localized modes and short-range dependence of the photonic band gap (PBG). Such a connection implies a tight-binding description of PBG formation of photonic quasicrystals ‒ making them more similar to electronic semiconductors than regular photonic crystals. Physical coupling to these defect-free localized modes is demonstrated.

© 2007 Optical Society of America

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References

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  1. E. Yablonovitch, "Inhibited spontaneous emission in solid-state physics and electronics," Phys. Rev. Lett. 58, 2059 (1987).
    [CrossRef] [PubMed]
  2. S. John, "Strong localization of photons in certain disordered dielectric superlattices," Phys. Rev. Lett. 58, 2486 (1987).
    [CrossRef] [PubMed]
  3. A. Shinya, S. Mitsugi, T. Tanabe, M. Notomi, I. Yokohama, H. Takara, and S. Kawanishi, "All-optical flip-flop circuit composed of coupled two-port resonant tunneling filter in two-dimensional photonic crystal slab," Opt. Express 14, 1230-1235 (2006).
    [CrossRef] [PubMed]
  4. D. Shectman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range orientational order and no translation symmetry," Phys. Rev. Lett. 53, 1951 (1984).
    [CrossRef]
  5. Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
    [CrossRef]
  6. S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091 (1999).
    [CrossRef]
  7. Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
    [CrossRef]
  8. K. Wang, "Light wave states in two-dimensional quasiperiodic media," Phys. Rev. B 73, 235122 (2006).
    [CrossRef]
  9. A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
    [CrossRef] [PubMed]
  10. R. C. Gauthier and K. Mnaymneh, "FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states," Opt. Commun. 264, 78 (2006).
    [CrossRef]
  11. A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021 (2006).
    [CrossRef] [PubMed]
  12. R. C. Gauthier and K. Mnaymneh, "Design of photonic band gap structures through a dual-beam multiple exposure technique," Opt. Laser Technol. 36, 625 (2004).
    [CrossRef]
  13. R. C. Gauthier and K. Mnaymneh, "Photonic band gap properties of 12-fold quasicrystal determined through FDTD analysis," Opt. Express 13, 1985 (2005).
    [CrossRef] [PubMed]
  14. R. C. Gauthier and A. Ivanov, "Production of quasi-crystal template patterns using a dual beam multiple exposure technique," Opt. Express 12, 990 (2004).
    [CrossRef] [PubMed]
  15. A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
    [CrossRef]
  16. J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New Jersey, 1999).

2006

2005

R. C. Gauthier and K. Mnaymneh, "Photonic band gap properties of 12-fold quasicrystal determined through FDTD analysis," Opt. Express 13, 1985 (2005).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

2004

R. C. Gauthier and A. Ivanov, "Production of quasi-crystal template patterns using a dual beam multiple exposure technique," Opt. Express 12, 990 (2004).
[CrossRef] [PubMed]

R. C. Gauthier and K. Mnaymneh, "Design of photonic band gap structures through a dual-beam multiple exposure technique," Opt. Laser Technol. 36, 625 (2004).
[CrossRef]

2003

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

2001

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

1999

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091 (1999).
[CrossRef]

1998

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

1987

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

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

1984

D. Shectman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range orientational order and no translation symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Asatryan, A. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

Blech, I.

D. Shectman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range orientational order and no translation symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Botten, L. C.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

Busch, K.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

Cahn, J. W.

D. Shectman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range orientational order and no translation symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Capolino, F.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Chan, C. T.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091 (1999).
[CrossRef]

Chan, C.T.

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

Chan, Y. S.

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

Cheng, B.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Cheng, S. S. M.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091 (1999).
[CrossRef]

de Streke, M.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

Della-Villa, A.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Enoch, S.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Galdi, V.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Gauthier, R. C.

R. C. Gauthier and K. Mnaymneh, "FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states," Opt. Commun. 264, 78 (2006).
[CrossRef]

R. C. Gauthier and K. Mnaymneh, "Photonic band gap properties of 12-fold quasicrystal determined through FDTD analysis," Opt. Express 13, 1985 (2005).
[CrossRef] [PubMed]

R. C. Gauthier and K. Mnaymneh, "Design of photonic band gap structures through a dual-beam multiple exposure technique," Opt. Laser Technol. 36, 625 (2004).
[CrossRef]

R. C. Gauthier and A. Ivanov, "Production of quasi-crystal template patterns using a dual beam multiple exposure technique," Opt. Express 12, 990 (2004).
[CrossRef] [PubMed]

Gratias, D.

D. Shectman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range orientational order and no translation symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Hu, X.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Ivanov, A.

John, S.

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

Kawanishi, S.

Li, L. M.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091 (1999).
[CrossRef]

Liu, Z. Y.

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

McPhedran, R. C.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

Mitsugi, S.

Mnaymneh, K.

R. C. Gauthier and K. Mnaymneh, "FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states," Opt. Commun. 264, 78 (2006).
[CrossRef]

R. C. Gauthier and K. Mnaymneh, "Photonic band gap properties of 12-fold quasicrystal determined through FDTD analysis," Opt. Express 13, 1985 (2005).
[CrossRef] [PubMed]

R. C. Gauthier and K. Mnaymneh, "Design of photonic band gap structures through a dual-beam multiple exposure technique," Opt. Laser Technol. 36, 625 (2004).
[CrossRef]

Nicorovici, N. A.

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

Notomi, M.

Pierro, V.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Shectman, D.

D. Shectman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range orientational order and no translation symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Shinya, A.

Takara, H.

Tanabe, T.

Tayeb, G.

A. Della-Villa, S. Enoch, G. Tayeb, F. Capolino, V. Pierro, and V. Galdi, and, "Localized Modes in Photonic Quasicrystals with Penrose-Type Lattice," Opt. Express 14, 10021 (2006).
[CrossRef] [PubMed]

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

Wang, K.

K. Wang, "Light wave states in two-dimensional quasiperiodic media," Phys. Rev. B 73, 235122 (2006).
[CrossRef]

Wang, Y.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Xu, X.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Yablonovitch, E.

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

Yokohama, I.

Zhang, D.

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

Zhang, Z. Q.

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091 (1999).
[CrossRef]

Opt. Commun.

R. C. Gauthier and K. Mnaymneh, "FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states," Opt. Commun. 264, 78 (2006).
[CrossRef]

Opt. Express

Opt. Laser Technol.

R. C. Gauthier and K. Mnaymneh, "Design of photonic band gap structures through a dual-beam multiple exposure technique," Opt. Laser Technol. 36, 625 (2004).
[CrossRef]

Phys. Rev. B

S. S. M. Cheng, L. M. Li, C. T. Chan, and Z. Q. Zhang, "Defect and transmission properties of two-dimensional quasiperiodic photonic band-gap systems," Phys. Rev. B 59, 4091 (1999).
[CrossRef]

Y. Wang, X. Hu, X. Xu, B. Cheng, and D. Zhang, "Localized modes in defect-free dodecagonal quasiperiodic photonic crystals," Phys. Rev. B 68, 165106 (2003).
[CrossRef]

K. Wang, "Light wave states in two-dimensional quasiperiodic media," Phys. Rev. B 73, 235122 (2006).
[CrossRef]

Phys. Rev. E

A. A. Asatryan, K. Busch, R. C. McPhedran, L. C. Botten, M. de Streke, and N. A. Nicorovici, "Two-dimensional Green’s function and local density of states in photonic crystals consisting of a finite number of cylinders of infinte length," Phys. Rev. E 63, 46612 (2001).
[CrossRef]

Phys. Rev. Lett.

A. Della-Villa, S. Enoch, G. Tayeb, V. Pierro, V. Galdi, and F. Capolino, "Band gap formation and multiple scattering in photonic quasicrystals with a Penrose-type lattice," Phys. Rev. Lett. 94, 183903 (2005).
[CrossRef] [PubMed]

D. Shectman, I. Blech, D. Gratias, and J. W. Cahn, "Metallic phase with long-range orientational order and no translation symmetry," Phys. Rev. Lett. 53, 1951 (1984).
[CrossRef]

Y. S. Chan, C.T. Chan, and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 956 (1998).
[CrossRef]

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

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

Other

J. D. Jackson, Classical Electrodynamics (John Wiley & Sons, Inc., New Jersey, 1999).

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Figures (9)

Fig. 1.
Fig. 1.

Six plane families generating a 12-fold PhQ. The intensity threshold correlates the density, size and shape of the scatterers. (a) With an intensity threshold set to 66% of the total intensity, a sparse distribution of almost circular scatterers appears, while (b) if the intensity threshold is set to 50%, a denser distribution of various shaped scatterers appears.

Fig. 2.
Fig. 2.

Comparison of transmission data for 10 μm (upper plot) and 3 μm (lower plot). As the width changed, defect states appeared in the smaller width PhQ.

Fig. 3.
Fig. 3.

Localized optical modes in a 12-fold PhQ. The number of negative lobes (represented in red and a negative sign) or the number of positive lobes (represented in blue and a positive sign) equals the m number. The wavelength at which these modes occur are shown underneath each respective mode.

Fig. 4.
Fig. 4.

Localized modes in 8-fold PhQs when the dielectric scatterer shapes are different from circular.

Fig. 5.
Fig. 5.

Localized modes in 10-fold PhQs when the dielectric scatterer shapes are different from circular.

Fig. 6.
Fig. 6.

Waveguide insertion shifts resonant wavelength of localized mode.

Fig. 7.
Fig. 7.

(a) Transmission spectra of PhQ compared with straight waveguide. Losses are high to design roughness, but notice the large drop around the resonant wavelength for the localized mode. (b) Higher resolution spectra scan for localized mode.

Fig. 8.
Fig. 8.

(a) The physical picture of the 12-fold PhQ and the waveguide setup. (b) and (c) Intensity pictures of the scattering at off-dip wavelength values. The arrows point to weak scattering in the area of the center of the pattern. (d) Intensity picture of the localized mode shown exactly where the dip occurs. The dashed box represents the outline of quasicrystal pattern.

Fig. 9.
Fig. 9.

(a) Electric field and (b) power simulations of the dip wavelength.

Equations (3)

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H ̂ G ( r , ω ) = δ ( r r 0 )
ρ ( r 0 , ω ) = Im G ( r = r 0 , ω ) .
ψ ( r ) r ψ = l , m r l , 0 l , 0 R z ( ϕ ) l , m l , m | ψ

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