Abstract

We construct fourteen complex periodic two-dimensional (2D) photonic structures with different structural symmetries by arranging the small portions of a 12-fold quasicrystal on square or hexagonal lattices. The corresponding reciprocal lattices confirm that all of them demonstrate the 12-fold-like characteristics due to the analogue short-range arrangements. We then investigate their photonic bandgap properties at different dielectric contrast levels (dielectric rods in air background). Our results suggest that all structures possess analogue transverse magnetic (TM) gaps in both Si and glass photonic crystals due to the similarity of their local geometries. However, the arrangements of the basic elements, total symmetries, and the coupling between the local and the lattice symmetries have greater impact on the glass photonic crystals, which show much larger deviation of gap sizes from different structures. Furthermore, we find that the minimal dielectric contrast to achieve the TM gap in the complex lattices (dielectric-in-air) can be as low as ε = 1.44, whereas the inverse structures may open a 2D complete gap in silicon nitride (ε = 4.1).

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References

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    [CrossRef]
  3. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
    [CrossRef] [PubMed]
  4. X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2008 (1)

Y. Yang, Q. Z. Li, and G. P. Wang, “Fabrication of periodic complex photonic crystals constructed with a portion of photonic quasicrystals by interference lithography,” Appl. Phys. Lett. 93(6), 061112 (2008).
[CrossRef]

2007 (1)

2006 (1)

2005 (1)

W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[CrossRef] [PubMed]

2003 (1)

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
[CrossRef]

2001 (2)

S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express 8(3), 173–190 (2001).
[CrossRef] [PubMed]

X. D. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63(8), 081105 (2001).
[CrossRef]

2000 (1)

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

1998 (1)

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[CrossRef]

1992 (1)

M. Baake, R. Klitzing, and M. Schlottmann, “Fractally shaped acceptance domains of quasiperiodic square-triangular tilings with dodecaonal symmetry,” Physica A 191(1-4), 554–558 (1992).
[CrossRef]

1986 (1)

P. Stampfli, “A dodecagonal quasiperiodic lattice in two dimensions,” Helv. Phys. Acta 59, 1260–1263 (1986).

Baake, M.

M. Baake, R. Klitzing, and M. Schlottmann, “Fractally shaped acceptance domains of quasiperiodic square-triangular tilings with dodecaonal symmetry,” Physica A 191(1-4), 554–558 (1992).
[CrossRef]

Baumberg, J. J.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Capolino, F.

Chaikin, P. M.

W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[CrossRef] [PubMed]

Chan, C. T.

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
[CrossRef]

X. D. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63(8), 081105 (2001).
[CrossRef]

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (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(5), 956–959 (1998).
[CrossRef]

Charlton, M. D. B.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Della Villa, A.

Enoch, S.

Galdi, V.

Joannopoulos, J. D.

Johnson, S. G.

Klitzing, R.

M. Baake, R. Klitzing, and M. Schlottmann, “Fractally shaped acceptance domains of quasiperiodic square-triangular tilings with dodecaonal symmetry,” Physica A 191(1-4), 554–558 (1992).
[CrossRef]

Li, Q. Z.

Y. Yang, Q. Z. Li, and G. P. Wang, “Fabrication of periodic complex photonic crystals constructed with a portion of photonic quasicrystals by interference lithography,” Appl. Phys. Lett. 93(6), 061112 (2008).
[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(5), 956–959 (1998).
[CrossRef]

Man, W. N.

W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[CrossRef] [PubMed]

Megens, M.

W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[CrossRef] [PubMed]

Netti, M. C.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Ng, C. Y.

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
[CrossRef]

Parker, G. J.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Pierro, V.

Schlottmann, M.

M. Baake, R. Klitzing, and M. Schlottmann, “Fractally shaped acceptance domains of quasiperiodic square-triangular tilings with dodecaonal symmetry,” Physica A 191(1-4), 554–558 (1992).
[CrossRef]

Sheng, P.

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
[CrossRef]

Stampfli, P.

P. Stampfli, “A dodecagonal quasiperiodic lattice in two dimensions,” Helv. Phys. Acta 59, 1260–1263 (1986).

Steinhardt, P. J.

W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[CrossRef] [PubMed]

Tam, W. Y.

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
[CrossRef]

Tayeb, G.

Wang, G. P.

Y. Yang, Q. Z. Li, and G. P. Wang, “Fabrication of periodic complex photonic crystals constructed with a portion of photonic quasicrystals by interference lithography,” Appl. Phys. Lett. 93(6), 061112 (2008).
[CrossRef]

Y. Yang and G. P. Wang, “Two-dimensional photonic crystals constructed with a portion of photonic quasicrystals,” Opt. Express 15(10), 5991–5996 (2007).
[CrossRef] [PubMed]

Wang, X.

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
[CrossRef]

Yang, Y.

Y. Yang, Q. Z. Li, and G. P. Wang, “Fabrication of periodic complex photonic crystals constructed with a portion of photonic quasicrystals by interference lithography,” Appl. Phys. Lett. 93(6), 061112 (2008).
[CrossRef]

Y. Yang and G. P. Wang, “Two-dimensional photonic crystals constructed with a portion of photonic quasicrystals,” Opt. Express 15(10), 5991–5996 (2007).
[CrossRef] [PubMed]

Zhang, X. D.

X. D. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63(8), 081105 (2001).
[CrossRef]

Zhang, Z. Q.

X. D. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63(8), 081105 (2001).
[CrossRef]

Zoorob, M. E.

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

Adv. Mater. (1)

X. Wang, C. Y. Ng, W. Y. Tam, C. T. Chan, and P. Sheng, “Large-area two-dimensional mesoscale quasi-crystals,” Adv. Mater. 15(18), 1526–1528 (2003).
[CrossRef]

Appl. Phys. Lett. (1)

Y. Yang, Q. Z. Li, and G. P. Wang, “Fabrication of periodic complex photonic crystals constructed with a portion of photonic quasicrystals by interference lithography,” Appl. Phys. Lett. 93(6), 061112 (2008).
[CrossRef]

Helv. Phys. Acta (1)

P. Stampfli, “A dodecagonal quasiperiodic lattice in two dimensions,” Helv. Phys. Acta 59, 1260–1263 (1986).

Nature (2)

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, “Complete photonic bandgaps in 12-fold symmetric quasicrystals,” Nature 404(6779), 740–743 (2000).
[CrossRef] [PubMed]

W. N. Man, M. Megens, P. J. Steinhardt, and P. M. Chaikin, “Experimental measurement of the photonic properties of icosahedral quasicrystals,” Nature 436(7053), 993–996 (2005).
[CrossRef] [PubMed]

Opt. Express (3)

Phys. Rev. B (1)

X. D. Zhang, Z. Q. Zhang, and C. T. Chan, “Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals,” Phys. Rev. B 63(8), 081105 (2001).
[CrossRef]

Phys. Rev. Lett. (1)

Y. S. Chan, C. T. Chan, and Z. Y. Liu, “Photonic band gaps in two dimensional photonic quasicrystals,” Phys. Rev. Lett. 80(5), 956–959 (1998).
[CrossRef]

Physica A (1)

M. Baake, R. Klitzing, and M. Schlottmann, “Fractally shaped acceptance domains of quasiperiodic square-triangular tilings with dodecaonal symmetry,” Physica A 191(1-4), 554–558 (1992).
[CrossRef]

Other (1)

J. D. Joannopoulos, S. G. Johnson, R. D. Meade, and J. N. Winn, Photonic crystals (Princeton University Press, 2008).

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Figures (3)

Fig. 1
Fig. 1

(a-c) 12-fold QCs. The shadowed areas illustrate portions with different sizes. (d-f) The corresponding Fourier transformation of the lattice points shown in the shadowed areas. (g, h) Illustration of the same portion as shown in the red shadowed area in (a). (i, j) Illustration of the same portion as shown in the green shadowed area in (b). (g) and (h) or (i) and (j) are rotated by 90° respectively.

Fig. 2
Fig. 2

Complex periodic 2D structures constructed with small portions of 12-fold QCs showed in Fig. 1. The constructed rules are showed by blue and red veins or points. The grey shadowed areas indicate the unit cells. Inset: the corresponding plane groups.

Fig. 3
Fig. 3

Reciprocal lattices of the complex 2D structures seen in Fig. 2(a-n), showing 12-fold-like features, in comparison with that of the square-triangular tiling 12-fold quasicrystal (o).

Tables (1)

Tables Icon

Table 1 TM bandgap properties in Si and glass photonic crystals with complex 2D structures.

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