Abstract

A simple and efficient interference method for fabricating highly symmetric two-dimensional (2-D) quasi-periodic structures (QPSs) is theoretically and experimentally demonstrated. With a three-beam interference technique, one can fabricate a periodic 2-D structure having sixfold symmetry. When this structure is multiduplicated into other specific orientations its combination results in a QPS with multifold symmetry. By use of n exposures with a rotation angle of 60°/n, one can create a 2-D QPS with six n-fold symmetry. The QPS with a super high symmetry level, as high as 60-fold, is demonstrated for the first time to the best of our knowledge. The diffraction pattern of a QPS is consistent with the Fourier transform calculation. The fabricated structures should be useful for many applications, such as isotropic bandgap materials and extraction enhancement of light-emitting diodes.

© 2007 Optical Society of America

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References

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2006

N. D. Lai, J. H. Lin, Y. Y. Huang, and C. C. Hsu, "Fabrication of two- and three-dimensional quasi-periodic structures with 12-fold symmetry by interference technique," Opt. Express 14, 10746-10752 (2006).
[CrossRef] [PubMed]

W. Mao, G. Liang, H. Zou, R. Zhang, H. Wang, and Z. Zeng, "Design and fabrication of two-dimensional holographic photonic quasi crystals with high-order symmetries," J. Opt. Soc. Am. B 23, 2046-2050 (2006).
[CrossRef]

2005

N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, "Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique," Opt. Express 13, 9605-9611 (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]

S. P. Gorkhali, J. Qi, and G. P. Crawford, "Electrically switchable mesoscale Penrose quasicrystal structure," Appl. Phys. Lett. 86, 011110 (2005).
[CrossRef]

2003

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

S. Wong, V. Kitaev, and G. A. Ozin, "Colloidal crystal films: advances in universality and perfection," J. Am. Chem. Soc. 125, 15589-15598 (2003).
[CrossRef] [PubMed]

2002

M. Hase, H. Miyazaki, M. Egashira, N. Shinya, K. M. Kojima, and S.-I. Uchida, "Isotropic photonic band gap and anisotropic structures in transmission spectra of two-dimensional fivefold and eightfold symmetric quasiperiodic photonic crystals," Phys. Rev. B 66, 214205 (2002).
[CrossRef]

M. Straub and M. Gu, "Near-infrared photonic crystals with higher-order bandgaps generated by two-photon photopolymerization," Opt. Lett. 27, 1824-1826 (2002).
[CrossRef]

2001

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

2000

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. Jin, B. Cheng, B. Man, Z. Li, and D. 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]

S. David, A. Chelnokov, and J.-M. Lourtioz, "Wide angularly isotropic photonic bandgaps obtained from two-dimensional photonic crystals with Archimedean-like tilings," Opt. Lett. 25, 1001-1003 (2000).
[CrossRef]

1999

1998

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]

1997

V. Berger, O. Gauthier-Lafaye, and E. Costard, "Photonic band gaps and holography," J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

1994

C. S. Vikram, W. K. Witherow, and J. D. Trolinger, "Fringe contrast and phase effects in multi-colour holography," J. Mod. Opt. 41, 1531-1536 (1994).
[CrossRef]

1984

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

Adv. Mater.

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

Appl. Phys. Lett.

S. P. Gorkhali, J. Qi, and G. P. Crawford, "Electrically switchable mesoscale Penrose quasicrystal structure," Appl. Phys. Lett. 86, 011110 (2005).
[CrossRef]

J. Am. Chem. Soc.

S. Wong, V. Kitaev, and G. A. Ozin, "Colloidal crystal films: advances in universality and perfection," J. Am. Chem. Soc. 125, 15589-15598 (2003).
[CrossRef] [PubMed]

J. Appl. Phys.

V. Berger, O. Gauthier-Lafaye, and E. Costard, "Photonic band gaps and holography," J. Appl. Phys. 82, 60-64 (1997).
[CrossRef]

J. Mod. Opt.

C. S. Vikram, W. K. Witherow, and J. D. Trolinger, "Fringe contrast and phase effects in multi-colour holography," J. Mod. Opt. 41, 1531-1536 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Nature

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]

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]

Opt. Express

N. D. Lai, W. P. Liang, J. H. Lin, C. C. Hsu, and C. H. Lin, "Fabrication of two- and three-dimensional periodic structures by multi-exposure of two-beam interference technique," Opt. Express 13, 9605-9611 (2005).
[CrossRef] [PubMed]

N. D. Lai, J. H. Lin, Y. Y. Huang, and C. C. Hsu, "Fabrication of two- and three-dimensional quasi-periodic structures with 12-fold symmetry by interference technique," Opt. Express 14, 10746-10752 (2006).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev. B

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, "Two-dimensional dodecagonal and decagonal quasiperiodic photonic crystals in the microwave region," Phys. Rev. B 61, 10762-10767 (2000).
[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 (2001).
[CrossRef]

M. Hase, H. Miyazaki, M. Egashira, N. Shinya, K. M. Kojima, and S.-I. Uchida, "Isotropic photonic band gap and anisotropic structures in transmission spectra of two-dimensional fivefold and eightfold symmetric quasiperiodic photonic crystals," Phys. Rev. B 66, 214205 (2002).
[CrossRef]

Phys. Rev. Lett.

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

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

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]

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Figures (4)

Fig. 1
Fig. 1

(Color online) (a) Projection of a multisurface prism along the z axis. A 1 , A 2 , and A 3 denote the surfaces that correspond to three laser beams. (b) Three beams after they have passed through the prism, changed their directions, and overlapped at one point in the z axis in which a photoresist sample was placed for fabrication. The sample can be rotated by an angle α for multiple exposures.

Fig. 2
Fig. 2

(Color online) Illustration of (a) a hexagonal and (b) a combination of this hexagonal and its duplication oriented in another direction by an angle α. (c) Example of a 24-fold QPS obtained with a combination of four hexagonal structures. Isointensity distribution of (d) a hexagonal structure and (e) an 18-fold QPS. (f) Isointensity distribution of an 18-fold QPS obtained with randomly chosen rotation centers.

Fig. 3
Fig. 3

(Color online) (a) Scanning electron microscope images of an 18-fold symmetry structure. (b) Zoom of a particular area that shows the 18-fold symmetry level of the structure. (c) Corresponding experimental diffraction pattern. (d) Calculated Fourier transform of the 18-fold symmetry QPS shown in Fig. 2(f).

Fig. 4
Fig. 4

(Color online) Scanning electron microscope images and corresponding diffraction patterns of (a), (b) a 24-fold, (c), (d) a 36-fold, and (e), (f) a 60-fold QPS obtained with four, six, and ten exposures, respectively. Insets in (a), (c), and (e) represent zooms of particular areas of the corresponding structures.

Equations (42)

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60 ° / n
( λ = 442   nm )
2   cm
6   mm
A 1
A 2
R = 6 n ; α n = 60 ° / n ; t n = t 1 / n ,
α n
α n ( i ) = i * ( 60 ° / n )
i = 0 , 1 , 2 , n 1
t n
t 1
α n ( i )
α 4 = 15 °
I multi exposure = n I α n ( i ) ,
I α ( i )
α n ( i )
n = 10
θ = 15.7 °
1.1   μm
2   μm
65   ° C
95 ° C
3   min
( 442   nm )
15   μm
A 1
A 2
A 3
2   mW
t 1
7 min
α 3 ( i ) = 0 °
40 ° ( δ α 0.1 ° )
6   mm × 6   mm
α 4 = 15 °
α 6 = 10 °
α 10 = 6 °
β n
A 1
A 2
A 3

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