Abstract

Quasicrystals, realized in metal alloys, are a class of lattices exhibiting symmetries that fall outside the usual classification for periodic crystals. They do not have translational symmetry and yet the lattice points are well ordered. Furthermore, they exhibit higher rotational symmetry than periodic crystals. Because of the higher symmetry (more spherical), they are more optimal than periodic crystals in achieving complete photonic bandgaps in a new class of materials called photonic crystals in which the propagation of light in certain frequency ranges is forbidden. The potential of quasicrystals has been demonstrated in two dimensions for the infrared range and, recently, in three-dimensional icosahedral quasicrystals fabricated using a stereo lithography method for the microwave range and direct laser writing for the IR range. Here, we report the fabrication and optical characterization of icosahedral quasicrystals using a holographic lithography method for the visible range. The icosahedral pattern, generated using a novel 7-beam optical interference holography, is recorded on photoresists and holographic plates. Electron micrographs of the photoresist samples show clearly the symmetry of the icosahedral quasicrystals in the submicron range, while the holographic plate samples exhibit bandgaps in the angular-dependent transmission spectra in the visible range. Calculations of the bandgaps due to reflection planes inside the icosahedral quasicrystal show good agreement with the experimental results.

© 2007 Optical Society of America

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2006 (5)

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides and J. W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006).
[CrossRef] [PubMed]

Y. K. Pang, J. C. W. Lee, C. T. Ho and W. Y. Tam, "Realization of woodpile structure using optical interference holography," Opt. Express 14, 9013-9019 (2006).
[CrossRef]

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

R. Ma, J. Xu and W. Y. Tam, "Wide bandgap photonic structures in dichromate gelatin emulsions," Appl. Phys. Lett. 89, 081116 (2006).
[CrossRef]

2005 (1)

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

2004 (1)

M. Deuble, G. von Freymann, M. Wegener, S. Pereira, K. Busch and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nature Mater. 3, 444-447 (2004).
[CrossRef]

2003 (3)

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

Yu. V. Miklyaev, D. C. Meisel, A. Blanco and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography fabrication optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

2002 (2)

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer and C. López, "Optical study of the full photonic band gap in silicon inverse opals," Appl. Phys. Lett. 81, 4925-4927 (2002).
[CrossRef]

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

2001 (2)

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar and R. M. De La Rue, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter. 13, 10459-10470 (2001).
[CrossRef]

X. Zhang, Z. Q. Zhang and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, 081105/1-4 (2001).
[CrossRef]

2000 (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, 740-743 (2000).
[CrossRef] [PubMed]

C. J. Jin, B. Y. Cheng, B. Y. Man, Z. L. Li and D. Z. Zhang, "Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region," Phys. Rev. B 61, 10762-10767 (2000).
[CrossRef]

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

1998 (3)

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

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

J. E. G. J. Wijnhoven and W. L. Vos, "Preparation of photonic crystals made of air spheres in titania," Science 281, 802-804 (1998).
[CrossRef]

1997 (1)

N. Yamamoto, S. Noda and A. Sasaki, "New realization method for three-dimensional photonic crystal in the optical wavelength region: experimental consideration," Jpn. J. Appl. Phys. 36, 1907-1911 (1997).
[CrossRef]

1988 (1)

D. S. Rokhsar, D. C. Wright and N. D. Mermin, "Scale equivalence of quasicrystallographic space groups," Phys. Rev. B 37, 8145-8149 (1988).
[CrossRef]

1987 (2)

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]

1985 (1)

D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy and P. J. Steinhardt, "Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals," Phys. Rev. Lett. 54, 1520-1523 (1985).
[CrossRef] [PubMed]

1984 (1)

D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, "Metallic phase with long range orientataional order and no translational symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Abram, R. A.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar and R. M. De La Rue, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter. 13, 10459-10470 (2001).
[CrossRef]

Aizenberg, J.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

Bartal, G.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides and J. W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006).
[CrossRef] [PubMed]

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, 740-743 (2000).
[CrossRef] [PubMed]

Biswas, R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Blanco, A.

Yu. V. Miklyaev, D. C. Meisel, A. Blanco and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography fabrication optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer and C. López, "Optical study of the full photonic band gap in silicon inverse opals," Appl. Phys. Lett. 81, 4925-4927 (2002).
[CrossRef]

Blech, I.

D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, "Metallic phase with long range orientataional order and no translational symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Brand, S.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar and R. M. De La Rue, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter. 13, 10459-10470 (2001).
[CrossRef]

Bur, J.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Busch, K.

M. Deuble, G. von Freymann, M. Wegener, S. Pereira, K. Busch and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nature Mater. 3, 444-447 (2004).
[CrossRef]

Cademartiri, L.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

Cahn, J. W.

D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, "Metallic phase with long range orientataional order and no translational symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Campbell, M.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Chaikin, P. M.

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

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

Chan, C. T.

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

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

X. Zhang, Z. Q. Zhang and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, 081105/1-4 (2001).
[CrossRef]

Y. S. Chan, C. T. Chan and Z. Y. Liu, "Photonic band gaps in two dimensional photonic quasicrystals," Phys. Rev. Lett. 80, 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, 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, 740-743 (2000).
[CrossRef] [PubMed]

Chen, Y. L.

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

Cheng, B. Y.

C. J. Jin, B. Y. Cheng, B. Y. Man, Z. L. Li and D. Z. Zhang, "Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region," Phys. Rev. B 61, 10762-10767 (2000).
[CrossRef]

Christodoulides, D. N.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides and J. W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006).
[CrossRef] [PubMed]

De La Rue, R. M.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar and R. M. De La Rue, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter. 13, 10459-10470 (2001).
[CrossRef]

Denning, R. G.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Deuble, M.

M. Deuble, G. von Freymann, M. Wegener, S. Pereira, K. Busch and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nature Mater. 3, 444-447 (2004).
[CrossRef]

Fleischer, J. W.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides and J. W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006).
[CrossRef] [PubMed]

Fleming, J. G.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Freedman, B.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides and J. W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006).
[CrossRef] [PubMed]

Freymann, G. V.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

Gratias, D.

D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, "Metallic phase with long range orientataional order and no translational symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Harrison, M. T.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Hermatschweiler, M.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

Hetherington, D. L.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Ho, C. T.

Y. K. Pang, J. C. W. Lee, C. T. Ho and W. Y. Tam, "Realization of woodpile structure using optical interference holography," Opt. Express 14, 9013-9019 (2006).
[CrossRef]

Ho, K. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Ibisate, M.

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer and C. López, "Optical study of the full photonic band gap in silicon inverse opals," Appl. Phys. Lett. 81, 4925-4927 (2002).
[CrossRef]

Jin, C. J.

C. J. Jin, B. Y. Cheng, B. Y. Man, Z. L. Li and D. Z. Zhang, "Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region," Phys. Rev. B 61, 10762-10767 (2000).
[CrossRef]

John, S.

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

Kaliteevski, M. A.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar and R. M. De La Rue, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter. 13, 10459-10470 (2001).
[CrossRef]

Krauss, T. F.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar and R. M. De La Rue, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter. 13, 10459-10470 (2001).
[CrossRef]

Kurtz, S. R.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Ledermann, A.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

Lee, J. C. W.

Y. K. Pang, J. C. W. Lee, C. T. Ho and W. Y. Tam, "Realization of woodpile structure using optical interference holography," Opt. Express 14, 9013-9019 (2006).
[CrossRef]

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

Levine, D.

D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy and P. J. Steinhardt, "Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals," Phys. Rev. Lett. 54, 1520-1523 (1985).
[CrossRef] [PubMed]

Li, Z. L.

C. J. Jin, B. Y. Cheng, B. Y. Man, Z. L. Li and D. Z. Zhang, "Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region," Phys. Rev. B 61, 10762-10767 (2000).
[CrossRef]

Lifshitz, R.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides and J. W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006).
[CrossRef] [PubMed]

Lin, S. Y.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[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, 956-959 (1998).
[CrossRef]

López, C.

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer and C. López, "Optical study of the full photonic band gap in silicon inverse opals," Appl. Phys. Lett. 81, 4925-4927 (2002).
[CrossRef]

Lubensky, T. C.

D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy and P. J. Steinhardt, "Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals," Phys. Rev. Lett. 54, 1520-1523 (1985).
[CrossRef] [PubMed]

Ma, R.

R. Ma, J. Xu and W. Y. Tam, "Wide bandgap photonic structures in dichromate gelatin emulsions," Appl. Phys. Lett. 89, 081116 (2006).
[CrossRef]

Man, B. Y.

C. J. Jin, B. Y. Cheng, B. Y. Man, Z. L. Li and D. Z. Zhang, "Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region," Phys. Rev. B 61, 10762-10767 (2000).
[CrossRef]

Man, W.

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

Megens, M.

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

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

Meisel, D. C.

Yu. V. Miklyaev, D. C. Meisel, A. Blanco and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography fabrication optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

Mermin, N. D.

D. S. Rokhsar, D. C. Wright and N. D. Mermin, "Scale equivalence of quasicrystallographic space groups," Phys. Rev. B 37, 8145-8149 (1988).
[CrossRef]

Meseguer, F.

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer and C. López, "Optical study of the full photonic band gap in silicon inverse opals," Appl. Phys. Lett. 81, 4925-4927 (2002).
[CrossRef]

Miklyaev, Yu. V.

Yu. V. Miklyaev, D. C. Meisel, A. Blanco and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography fabrication optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

Millar, P.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, P. Millar and R. M. De La Rue, "Diffraction and transmission of light in low-refractive index Penrose-tiled photonic quasicrystals," J. Phys.: Condens. Matter. 13, 10459-10470 (2001).
[CrossRef]

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, 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. Mat. 15, 1526-1528 (2003).
[CrossRef]

Noda, S.

N. Yamamoto, S. Noda and A. Sasaki, "New realization method for three-dimensional photonic crystal in the optical wavelength region: experimental consideration," Jpn. J. Appl. Phys. 36, 1907-1911 (1997).
[CrossRef]

Ostlund, S.

D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy and P. J. Steinhardt, "Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals," Phys. Rev. Lett. 54, 1520-1523 (1985).
[CrossRef] [PubMed]

Ozin, G. A.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

Palacios-Lidón, E.

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer and C. López, "Optical study of the full photonic band gap in silicon inverse opals," Appl. Phys. Lett. 81, 4925-4927 (2002).
[CrossRef]

Pang, Y. K.

Y. K. Pang, J. C. W. Lee, C. T. Ho and W. Y. Tam, "Realization of woodpile structure using optical interference holography," Opt. Express 14, 9013-9019 (2006).
[CrossRef]

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (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, 740-743 (2000).
[CrossRef] [PubMed]

Pereira, S.

M. Deuble, G. von Freymann, M. Wegener, S. Pereira, K. Busch and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nature Mater. 3, 444-447 (2004).
[CrossRef]

Ramaswamy, S.

D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy and P. J. Steinhardt, "Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals," Phys. Rev. Lett. 54, 1520-1523 (1985).
[CrossRef] [PubMed]

Rokhsar, D. S.

D. S. Rokhsar, D. C. Wright and N. D. Mermin, "Scale equivalence of quasicrystallographic space groups," Phys. Rev. B 37, 8145-8149 (1988).
[CrossRef]

Russel, W. B.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

Sasaki, A.

N. Yamamoto, S. Noda and A. Sasaki, "New realization method for three-dimensional photonic crystal in the optical wavelength region: experimental consideration," Jpn. J. Appl. Phys. 36, 1907-1911 (1997).
[CrossRef]

Segev, M.

B. Freedman, G. Bartal, M. Segev, R. Lifshitz, D. N. Christodoulides and J. W. Fleischer, "Wave and defect dynamics in nonlinear photonic quasicrystals," Nature 440, 1166-1169 (2006).
[CrossRef] [PubMed]

Sharp, D. N.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

Shechtman, D.

D. Shechtman, I. Blech, D. Gratias and J. W. Cahn, "Metallic phase with long range orientataional order and no translational symmetry," Phys. Rev. Lett. 53, 1951-1953 (1984).
[CrossRef]

Sheng, P.

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

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

Sigalas, M. M.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Smith, B. K.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Soukoulis, C. M.

M. Deuble, G. von Freymann, M. Wegener, S. Pereira, K. Busch and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nature Mater. 3, 444-447 (2004).
[CrossRef]

Steinhardt, P. J.

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

D. Levine, T. C. Lubensky, S. Ostlund, S. Ramaswamy and P. J. Steinhardt, "Elasticity and Dislocations in Pentagonal and Icosahedral Quasicrystals," Phys. Rev. Lett. 54, 1520-1523 (1985).
[CrossRef] [PubMed]

Su, H. M.

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

Tam, W. Y.

Y. K. Pang, J. C. W. Lee, C. T. Ho and W. Y. Tam, "Realization of woodpile structure using optical interference holography," Opt. Express 14, 9013-9019 (2006).
[CrossRef]

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

R. Ma, J. Xu and W. Y. Tam, "Wide bandgap photonic structures in dichromate gelatin emulsions," Appl. Phys. Lett. 89, 081116 (2006).
[CrossRef]

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

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

Toninelli, C.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

Turberfield, A. J.

M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning and A. J. Turberfield, "Fabrication of photonic crystals for the visible spectrum by holographic lithography," Nature 404, 53-56 (2000).
[CrossRef] [PubMed]

von Freymann, G.

M. Deuble, G. von Freymann, M. Wegener, S. Pereira, K. Busch and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nature Mater. 3, 444-447 (2004).
[CrossRef]

Yu. V. Miklyaev, D. C. Meisel, A. Blanco and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography fabrication optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

Vos, W. L.

J. E. G. J. Wijnhoven and W. L. Vos, "Preparation of photonic crystals made of air spheres in titania," Science 281, 802-804 (1998).
[CrossRef]

Wang, H. Z.

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

Wang, X.

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

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

Wegener, M.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

M. Deuble, G. von Freymann, M. Wegener, S. Pereira, K. Busch and C. M. Soukoulis, "Direct laser writing of three-dimensional photonic-crystal templates for telecommunications," Nature Mater. 3, 444-447 (2004).
[CrossRef]

Wiersma, D. S.

A. Ledermann, L. Cademartiri, M. Hermatschweiler, C. Toninelli, G. A. Ozin, D. S. Wiersma, M. Wegener and G. V. Freymann, "Three-dimensional silicon inverse photonic quasicrystals for infrared wavelengths," Nature Mater. 5, 942-945 (2006).
[CrossRef]

Wijnhoven, J. E. G. J.

J. E. G. J. Wijnhoven and W. L. Vos, "Preparation of photonic crystals made of air spheres in titania," Science 281, 802-804 (1998).
[CrossRef]

Wiltzius, P.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

Wright, D. C.

D. S. Rokhsar, D. C. Wright and N. D. Mermin, "Scale equivalence of quasicrystallographic space groups," Phys. Rev. B 37, 8145-8149 (1988).
[CrossRef]

Xu, J.

R. Ma, J. Xu and W. Y. Tam, "Wide bandgap photonic structures in dichromate gelatin emulsions," Appl. Phys. Lett. 89, 081116 (2006).
[CrossRef]

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

Xu, J. F.

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

Yablonovitch, E.

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

Yamamoto, N.

N. Yamamoto, S. Noda and A. Sasaki, "New realization method for three-dimensional photonic crystal in the optical wavelength region: experimental consideration," Jpn. J. Appl. Phys. 36, 1907-1911 (1997).
[CrossRef]

Yang, S.

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

Zeng, Z. H.

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

Zhang, D. Z.

C. J. Jin, B. Y. Cheng, B. Y. Man, Z. L. Li and D. Z. Zhang, "Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region," Phys. Rev. B 61, 10762-10767 (2000).
[CrossRef]

Zhang, X.

X. Zhang, Z. Q. Zhang and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, 081105/1-4 (2001).
[CrossRef]

Zhang, Z. Q.

X. Zhang, Z. Q. Zhang and C. T. Chan, "Absolute photonic band gaps in 12-fold symmetric photonic quasicrystals," Phys. Rev. B 63, 081105/1-4 (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, 740-743 (2000).
[CrossRef] [PubMed]

Zubrzycki, W.

S. Y. Lin, J. G. Fleming, D. L. Hetherington, B. K. Smith, R. Biswas, K. M. Ho, M. M. Sigalas, W. Zubrzycki, S. R. Kurtz and J. Bur, "Three-dimensional photonic crystal operating at infrared wavelengths," Nature 394, 251-253 (1998).
[CrossRef]

Adv. Mat. (1)

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

Appl. Phys. Lett. (5)

X. Wang, J. Xu, J. C. W. Lee, Y. K. Pang, W. Y. Tam, C. T. Chan, and P. Sheng, "Realization of optical periodic quasicrystals using holographic lithography," Appl. Phys. Lett. 88, 051901 (2006).
[CrossRef]

Yu. V. Miklyaev, D. C. Meisel, A. Blanco and G. von Freymann, "Three-dimensional face-centered-cubic photonic crystal templates by laser holography fabrication optical characterization, and band-structure calculations," Appl. Phys. Lett. 82, 1284-1286 (2003).
[CrossRef]

X. Wang, J. F. Xu, H. M. Su, Z. H. Zeng, Y. L. Chen, H. Z. Wang, Y. K. Pang and W. Y. Tam, "Three-dimensional photonic crystals fabricated by visible light holographic lithography," Appl. Phys. Lett. 82, 2212-2214 (2003).
[CrossRef]

R. Ma, J. Xu and W. Y. Tam, "Wide bandgap photonic structures in dichromate gelatin emulsions," Appl. Phys. Lett. 89, 081116 (2006).
[CrossRef]

E. Palacios-Lidón, A. Blanco, M. Ibisate, F. Meseguer and C. López, "Optical study of the full photonic band gap in silicon inverse opals," Appl. Phys. Lett. 81, 4925-4927 (2002).
[CrossRef]

Chem. Mat. (1)

S. Yang, M. Megens, J. Aizenberg, P. Wiltzius, P. M. Chaikin and W. B. Russel, "Creating periodic three-dimensional structures by multibeam interference of visible laser," Chem. Mat. 14, 2831-2833 (2002).
[CrossRef]

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Figures (4)

Fig. 1.
Fig. 1.

Beam configuration for icosahedral quasicrystals. (a) 7-beam configuration for the icosahedral quasicrystal. (b) Icosahedral quasicrystal lattice (red) with φ= 63.4° and the simulated 5-fold (F), 3-fold (U), and 2-fold (P) symmetry projections using 70% intensity cutoff. (c) Actual 7-beam arrangement (for 5-fold symmetry) using a truncated pentagonal prism. (d) Setup for obtaining 2-fold and 3-fold symmetry projections on the surface of photoresist using a pair of prisms.

Fig. 2.
Fig. 2.

Electron micrographs of fabricated icosahedral quasicrystals using incidence angle φ = 63.4° in SU8. (a) Icosahedral quasicrystal lattice using φ = 63.4°. (b) SEM image of the 5-fold symmetry obtained using configuration in Fig. 1(b). (c) SEM image of the 3-fold symmetry obtained using configuration in Fig. 1(c) with prism angle 37.4°. (d) SEM image of the 2-fold symmetry obtained using configuration in Fig. 1(c) with prism angle 31.7°. The upper-left insets in (b-d) are simulated projections of the corresponding symmetries using a 40% intensity cutoff. The circle in (b) shows the 5-fold symmetry. The lower-right inset in (b) is the cross-sectional SEM image of the sample. The lower-left inset, size 60 μm x 80 μm, in (b) is a 5-fold normal reflection image. The scale bars are all 1 μm.

Fig. 3.
Fig. 3.

Electron micrographs of fabricated icosahedral quasicrystals using incidence angle φ = 53.2° in SU8. (a). Icosahedral quasicrystal lattice using φ = 53.2°. (b). SEM image of the 5-fold symmetry obtained using configuration in Fig. 1(b). (c). SEM image of the 3-fold symmetry obtained using configuration in Fig. 1(c) with prism angle 30°. (d). SEM image of the 2-fold symmetry obtained using configuration in Fig. 1(c) with prism angle 30°. The upper-left insets in (b-d) are simulated projections of the corresponding symmetries using a 40% intensity cutoff. The circle in (b) shows the 5-fold symmetry. The lower-right inset in (b) is the cross-sectional SEM image of the sample. The scale bars are all 1 μm.

Fig. 4.
Fig. 4.

Optical measurements of fabricated icosahedral quasicrystals using incidence angle cp = 55.7° in DCG gelatin. (a) Normal reflectance (green) and transmittance (red) of the DCG icosahedral quasicrystal sample using the setup in the top inset. Right inset is a diffraction pattern of icosahedral quasicrystal. Lower-left insets are photos of the normal reflection (top green) and transmission (bottom light purple) from diffuse white light. (b) and (c), Angular transmission spectra for rotation along and perpendicular to the 5-fold axis, respectively. The scale bar on the right is the transmittance. The color lines are bandgaps of reflection planes inside the icosahedral quasicrystal obtained by the reciprocal vectors ∆k0–6 (magenta), ∆k i–6(green), ∆k0–j (red), and ∆k jj (yellow), for i,j= 1–5.

Tables (1)

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Table 1. Structural parameters for the icosahedral quasicrystal. τ = (1 + √5)/2 is the Golden Mean.

Equations (5)

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[ a 0 = ( 0,1 , τ ) = OF ¯ a 1 = ( τ , 0,1 ) = OA ¯ a 2 = ( 1 , τ , 0 ) = OB ¯ a 3 = ( 1 , τ , 0 ) = OC ¯ a 4 = ( τ , 0,1 ) = OD ¯ a 5 = ( 0, 1 , τ ) = OE ¯ ]
[ q 0 = 2 a 0 = ( 0,2,2 τ ) q 1 = a 0 + a 1 = ( τ , 1,1 + τ ) q 2 = a 0 + a 2 = ( 1,1 + τ , τ ) q 3 = a 0 + a 3 = ( 1,1 + τ , τ ) q 4 = a 0 + a 4 = ( τ , 1,1 + τ ) q 5 = a 0 + a 5 = ( 0,0,2 τ ) ]
[ k 0 = ( 0,1 , τ ) k 1 = ( τ , 0, 1 ) k 2 = ( 1 , τ , 0 ) k 3 = ( 1 , τ , 0 ) k 4 = ( τ , 0 , 1 , ) k 5 = ( 0,1 , τ ) k 6 = ( 0, 1 , τ ) = k 0 ]
[ q 0 = k 0 k 6 q n = k 0 k n , n = 1 5 ]
I ( r ) = l , m = 0 6 E l e i k l r i δ l E m * e i k m r + i δ m ,

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