Abstract

By treating the quasicrystal structure as an irrational cut of a higher-dimensional periodic structure, we present a coupled-wave theory, which provides an efficient method to analyze the electromagnetic propagation through a photonic quasicrystal. Computation algorithms for simple one- and two-dimensional birefringent photonic quasicrystals are implemented to study the characteristic diffractions of the aperiodic structures. Our numerical results are in good agreement with results obtained by the finite-difference time-domain (FDTD) method.

© 2012 Optical Society of America

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  1. D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
    [CrossRef]
  2. G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Two-dimensional photonic quasicrystals by single beam computer-generated holography,” Opt. Express 16, 5164–5170 (2008).
    [CrossRef]
  3. J. B. Yeo, S. D. Yun, N. H. Kim, and H. Y. Lee, “Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography,” J. Vac. Sci. Technol. B 27, 1886–1889 (2009).
    [CrossRef]
  4. Y. Yang, S. Zhang, and G. P. Wang, “Fabrication of two-dimensional metallodielectric quasicrystals by single-beam holography,” Appl. Phys. Lett. 88, 251104 (2006).
    [CrossRef]
  5. M. Guo, Z. Xu, and X. Wang, “Photofabrication of two-dimensional quasi-crystal patterns on UV-curable molecular Azo glass films,” Langmuir 24, 2740–2745 (2008).
    [CrossRef]
  6. M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
    [CrossRef]
  7. J.-M. Dubois, “The applied physics of quasicrystals,” Phys. Scr. T 1993(T49A), 17–23 (1993).
    [CrossRef]
  8. S. P. Gorkhali, J. Qi, and G. P. Crawford, “Electrically switchable mesoscale Penrose quasicrystal structure,” Appl. Phys. Lett. 86, 011110 (2005).
    [CrossRef]
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    [CrossRef]
  10. L. P. Lee and R. Szema, “Photonic systems inspirations from biological optics for advanced photonic systems,” Science 310, 1148–1150 (2005).
    [CrossRef]
  11. J. Huang, X. Wang, and Z. L. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
    [CrossRef]
  12. S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
    [CrossRef]
  13. R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264, 78–88 (2006).
    [CrossRef]
  14. R. C. Gauthier and K. Mnaymueh, “Photonic band gap properties of 12-fold quasi-crystal determined through FDTD analysis,” Opt. Express 13, 1985–1998 (2005).
    [CrossRef]
  15. M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
    [CrossRef]
  16. A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
    [CrossRef]
  17. A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, “Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem,” Phys. Rev. B 77, 104201 (2008).
    [CrossRef]
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    [CrossRef]
  20. M. G. Moharam, E. B. Grann, D. A. Pommet, and T. K. Gaylord, “Formulation for stable and efficient implementation of the rigorous coupled-wave analysis of binary gratings,” J. Opt. Soc. Am. A 12, 1068–1076 (1995).
    [CrossRef]
  21. I. L. Ho, Y. C. Chang, C. H. Huang, and W. Y. Li, “A detailed derivation of rigorous coupled wave algorithms for three-dimensional periodic liquid-crystal microstructures,” Liq. Cryst. 38, 241–252 (2011).
    [CrossRef]
  22. K. Rokushima and J. Yamakita, “Analysis of anisotropic dielectric gratings,” J. Opt. Soc. Am. 73, 901–908 (1983).
    [CrossRef]

2011

I. L. Ho, Y. C. Chang, C. H. Huang, and W. Y. Li, “A detailed derivation of rigorous coupled wave algorithms for three-dimensional periodic liquid-crystal microstructures,” Liq. Cryst. 38, 241–252 (2011).
[CrossRef]

2009

J. B. Yeo, S. D. Yun, N. H. Kim, and H. Y. Lee, “Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography,” J. Vac. Sci. Technol. B 27, 1886–1889 (2009).
[CrossRef]

2008

M. Guo, Z. Xu, and X. Wang, “Photofabrication of two-dimensional quasi-crystal patterns on UV-curable molecular Azo glass films,” Langmuir 24, 2740–2745 (2008).
[CrossRef]

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, “Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem,” Phys. Rev. B 77, 104201 (2008).
[CrossRef]

G. Zito, B. Piccirillo, E. Santamato, A. Marino, V. Tkachenko, and G. Abbate, “Two-dimensional photonic quasicrystals by single beam computer-generated holography,” Opt. Express 16, 5164–5170 (2008).
[CrossRef]

2007

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

2006

Y. Yang, S. Zhang, and G. P. Wang, “Fabrication of two-dimensional metallodielectric quasicrystals by single-beam holography,” Appl. Phys. Lett. 88, 251104 (2006).
[CrossRef]

J. Huang, X. Wang, and Z. L. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264, 78–88 (2006).
[CrossRef]

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

2005

S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
[CrossRef]

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

S. P. Gorkhali, G. P. Crawford, and J. Qi, “Electrically switchable two-dimensional Penrose quasi-crystal,” Mol. Cryst. Liq. Cryst. 433, 297–308 (2005).
[CrossRef]

L. P. Lee and R. Szema, “Photonic systems inspirations from biological optics for advanced photonic systems,” Science 310, 1148–1150 (2005).
[CrossRef]

R. C. Gauthier and K. Mnaymueh, “Photonic band gap properties of 12-fold quasi-crystal determined through FDTD analysis,” Opt. Express 13, 1985–1998 (2005).
[CrossRef]

2000

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

1995

1993

J.-M. Dubois, “The applied physics of quasicrystals,” Phys. Scr. T 1993(T49A), 17–23 (1993).
[CrossRef]

1990

J. F. Sadoc1 and R. Mosseri, “A new method to generate quasicrystalline structures: examples in 2D tilings,” J. Phys. 51, 205–221 (1990).
[CrossRef]

1983

Abbate, G.

Abram, R. A.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

Avniel, Y.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, “Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem,” Phys. Rev. B 77, 104201 (2008).
[CrossRef]

Bogart, G. R.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Bogart, K. H. A.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Brand, S.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

Capolino, F.

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

Chang, Y. C.

I. L. Ho, Y. C. Chang, C. H. Huang, and W. Y. Li, “A detailed derivation of rigorous coupled wave algorithms for three-dimensional periodic liquid-crystal microstructures,” Liq. Cryst. 38, 241–252 (2011).
[CrossRef]

Charlton, M. D. B.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Crawford, G. P.

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

S. P. Gorkhali, G. P. Crawford, and J. Qi, “Electrically switchable two-dimensional Penrose quasi-crystal,” Mol. Cryst. Liq. Cryst. 433, 297–308 (2005).
[CrossRef]

DeLa Rue, R. M.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

Dubois, J.-M.

J.-M. Dubois, “The applied physics of quasicrystals,” Phys. Scr. T 1993(T49A), 17–23 (1993).
[CrossRef]

Enoch, S.

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

Galdi, V.

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

Gauthier, R. C.

R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264, 78–88 (2006).
[CrossRef]

R. C. Gauthier and K. Mnaymueh, “Photonic band gap properties of 12-fold quasi-crystal determined through FDTD analysis,” Opt. Express 13, 1985–1998 (2005).
[CrossRef]

Gaylord, T. K.

Gorkhali, S. P.

S. P. Gorkhali, G. P. Crawford, and J. Qi, “Electrically switchable two-dimensional Penrose quasi-crystal,” Mol. Cryst. Liq. Cryst. 433, 297–308 (2005).
[CrossRef]

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

Grann, E. B.

Guo, M.

M. Guo, Z. Xu, and X. Wang, “Photofabrication of two-dimensional quasi-crystal patterns on UV-curable molecular Azo glass films,” Langmuir 24, 2740–2745 (2008).
[CrossRef]

Ho, I. L.

I. L. Ho, Y. C. Chang, C. H. Huang, and W. Y. Li, “A detailed derivation of rigorous coupled wave algorithms for three-dimensional periodic liquid-crystal microstructures,” Liq. Cryst. 38, 241–252 (2011).
[CrossRef]

Huang, C. H.

I. L. Ho, Y. C. Chang, C. H. Huang, and W. Y. Li, “A detailed derivation of rigorous coupled wave algorithms for three-dimensional periodic liquid-crystal microstructures,” Liq. Cryst. 38, 241–252 (2011).
[CrossRef]

Huang, J.

J. Huang, X. Wang, and Z. L. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

Hwang, I. K.

S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
[CrossRef]

Janot, C.

C. Janot, Quasicrystals: a Primer (Oxford University, 1997).

Jeon, S.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Johnson, H. T.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Johnson, S. G.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, “Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem,” Phys. Rev. B 77, 104201 (2008).
[CrossRef]

Kaliteevski, M. A.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

Kim, N. H.

J. B. Yeo, S. D. Yun, N. H. Kim, and H. Y. Lee, “Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography,” J. Vac. Sci. Technol. B 27, 1886–1889 (2009).
[CrossRef]

Kim, S. H.

S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
[CrossRef]

Kim, S. K.

S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
[CrossRef]

Krauss, T. F.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

Lee, H. Y.

J. B. Yeo, S. D. Yun, N. H. Kim, and H. Y. Lee, “Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography,” J. Vac. Sci. Technol. B 27, 1886–1889 (2009).
[CrossRef]

Lee, J. H.

S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
[CrossRef]

Lee, L. P.

L. P. Lee and R. Szema, “Photonic systems inspirations from biological optics for advanced photonic systems,” Science 310, 1148–1150 (2005).
[CrossRef]

Lee, T.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Lee, Y. H.

S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
[CrossRef]

Li, W. Y.

I. L. Ho, Y. C. Chang, C. H. Huang, and W. Y. Li, “A detailed derivation of rigorous coupled wave algorithms for three-dimensional periodic liquid-crystal microstructures,” Liq. Cryst. 38, 241–252 (2011).
[CrossRef]

Liao, H.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Marino, A.

McCauley, A. P.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, “Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem,” Phys. Rev. B 77, 104201 (2008).
[CrossRef]

Millar, P.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

Mnaymneh, K.

R. C. Gauthier and K. Mnaymneh, “FDTD analysis of 12-fold photonic quasi-crystal central pattern localized states,” Opt. Commun. 264, 78–88 (2006).
[CrossRef]

Mnaymueh, K.

Moharam, M. G.

Mosseri, R.

J. F. Sadoc1 and R. Mosseri, “A new method to generate quasicrystalline structures: examples in 2D tilings,” J. Phys. 51, 205–221 (1990).
[CrossRef]

Piccirillo, B.

Pierro, V.

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

Pommet, D. A.

Qi, J.

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

S. P. Gorkhali, G. P. Crawford, and J. Qi, “Electrically switchable two-dimensional Penrose quasi-crystal,” Mol. Cryst. Liq. Cryst. 433, 297–308 (2005).
[CrossRef]

Rodriguez, A. W.

A. W. Rodriguez, A. P. McCauley, Y. Avniel, and S. G. Johnson, “Computation and visualization of photonic quasicrystal spectra via Bloch’s theorem,” Phys. Rev. B 77, 104201 (2008).
[CrossRef]

Rogers, J. A.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Rokushima, K.

Sadoc1, J. F.

J. F. Sadoc1 and R. Mosseri, “A new method to generate quasicrystalline structures: examples in 2D tilings,” J. Phys. 51, 205–221 (1990).
[CrossRef]

Santamato, E.

Shir, D.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Szema, R.

L. P. Lee and R. Szema, “Photonic systems inspirations from biological optics for advanced photonic systems,” Science 310, 1148–1150 (2005).
[CrossRef]

Tayeb, G.

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

Tkachenko, V.

Villa, A. D.

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

Wang, G. P.

Y. Yang, S. Zhang, and G. P. Wang, “Fabrication of two-dimensional metallodielectric quasicrystals by single-beam holography,” Appl. Phys. Lett. 88, 251104 (2006).
[CrossRef]

Wang, X.

M. Guo, Z. Xu, and X. Wang, “Photofabrication of two-dimensional quasi-crystal patterns on UV-curable molecular Azo glass films,” Langmuir 24, 2740–2745 (2008).
[CrossRef]

J. Huang, X. Wang, and Z. L. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

Wang, Z. L.

J. Huang, X. Wang, and Z. L. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

Xiao, D.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

Xu, Z.

M. Guo, Z. Xu, and X. Wang, “Photofabrication of two-dimensional quasi-crystal patterns on UV-curable molecular Azo glass films,” Langmuir 24, 2740–2745 (2008).
[CrossRef]

Yamakita, J.

Yang, Y.

Y. Yang, S. Zhang, and G. P. Wang, “Fabrication of two-dimensional metallodielectric quasicrystals by single-beam holography,” Appl. Phys. Lett. 88, 251104 (2006).
[CrossRef]

Yeo, J. B.

J. B. Yeo, S. D. Yun, N. H. Kim, and H. Y. Lee, “Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography,” J. Vac. Sci. Technol. B 27, 1886–1889 (2009).
[CrossRef]

Yun, S. D.

J. B. Yeo, S. D. Yun, N. H. Kim, and H. Y. Lee, “Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography,” J. Vac. Sci. Technol. B 27, 1886–1889 (2009).
[CrossRef]

Zhang, S.

Y. Yang, S. Zhang, and G. P. Wang, “Fabrication of two-dimensional metallodielectric quasicrystals by single-beam holography,” Appl. Phys. Lett. 88, 251104 (2006).
[CrossRef]

Zito, G.

Zoorob, M. E.

M. D. B. Charlton, M. E. Zoorob, and T. Lee, “Photonic quasi-crystal LEDs: design, modelling, and optimisation,” Proc. SPIE 6486, 64860R (2007).
[CrossRef]

Appl. Phys. Lett.

Y. Yang, S. Zhang, and G. P. Wang, “Fabrication of two-dimensional metallodielectric quasicrystals by single-beam holography,” Appl. Phys. Lett. 88, 251104 (2006).
[CrossRef]

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

S. K. Kim, J. H. Lee, S. H. Kim, I. K. Hwang, and Y. H. Lee, Appl. Phys. Lett. 86, 031101 (2005).
[CrossRef]

IEEE Antennas Wireless Propag. Lett.

A. D. Villa, V. Galdi, F. Capolino, V. Pierro, S. Enoch, and G. Tayeb, “A comparative study of representative categories of EBG dielectric quasi-crystals,” IEEE Antennas Wireless Propag. Lett. 5, 331–334 (2006).
[CrossRef]

J. Mod. Opt.

M. A. Kaliteevski, S. Brand, R. A. Abram, T. F. Krauss, R. M. DeLa Rue, and P. Millar, “Two-dimensional Penrose-tiled photonic quasicrystals; diffraction of light and fractal density of modes,” J. Mod. Opt. 47, 1771–1778 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

J. Phys.

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[CrossRef]

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J. B. Yeo, S. D. Yun, N. H. Kim, and H. Y. Lee, “Fabrication of Si-based two-dimensional photonic quasicrystals by using multiple-exposure holographic lithography,” J. Vac. Sci. Technol. B 27, 1886–1889 (2009).
[CrossRef]

Langmuir

M. Guo, Z. Xu, and X. Wang, “Photofabrication of two-dimensional quasi-crystal patterns on UV-curable molecular Azo glass films,” Langmuir 24, 2740–2745 (2008).
[CrossRef]

Liq. Cryst.

I. L. Ho, Y. C. Chang, C. H. Huang, and W. Y. Li, “A detailed derivation of rigorous coupled wave algorithms for three-dimensional periodic liquid-crystal microstructures,” Liq. Cryst. 38, 241–252 (2011).
[CrossRef]

Mol. Cryst. Liq. Cryst.

S. P. Gorkhali, G. P. Crawford, and J. Qi, “Electrically switchable two-dimensional Penrose quasi-crystal,” Mol. Cryst. Liq. Cryst. 433, 297–308 (2005).
[CrossRef]

Nano Lett.

D. Shir, H. Liao, S. Jeon, D. Xiao, H. T. Johnson, G. R. Bogart, K. H. A. Bogart, and J. A. Rogers, “Three-dimensional nanostructures formed by single step, two-photon exposures through elastomeric penrose quasicrystal phase masks,” Nano Lett. 8, 2236–2244 (2008).
[CrossRef]

J. Huang, X. Wang, and Z. L. Wang, “Controlled replication of butterfly wings for achieving tunable photonic properties,” Nano Lett. 6, 2325–2331 (2006).
[CrossRef]

Opt. Commun.

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[CrossRef]

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[CrossRef]

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[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Quasiperiodic pattern generated from higher-dimensional space. (a) The cut method to derive a Fibonacci sequence from a 2D periodic structure ρ(x1x1+x2x2), (b) the corresponding projection operation in the 2D reciprocal space, a bubble chart with bubble size |ρ(k)| at positions k=k1k1+k2k2, (c) the generated quasiperiodic structure as a result of cutting along the x dimension in (a), (d) the Fourier amplitudes of the quasiperiodic structure |ρ(kx)|, shown by projecting |ρ(k)| on the kx line in (b).

Fig. 2.
Fig. 2.

(a) Density distribution function ρ(x,y) of the octagonal quasicrystal structure in (x,y) dimensions obtained by an irrational cut of the 4D (x1,x2,y1,y2) tesseract lattice. (b) Fourier amplitudes, |ρ(kx,ky)| in (kx,ky) dimensions, obtained by the projection from the 4D reciprocal lattice of the tesseract structure, in which the magnitudes are revealed by the size of the bubbles. The truncation of the number of Fourier components, Ng, is set at 500. (c) Density distribution function ρ(x,y) obtained by the straightforward Fourier transform of ρ(kx,ky) in (b).

Fig. 3.
Fig. 3.

Transmitted field |E| through a 1D quasicrystal structure as a function of diffraction angle θemit. (a) RCWA result with Ng=10, (b) RCWA result with Ng=50, and (c) RCWA result with Ng=500; (d) FDTD result for a finite-length 1D quasicrystal.

Fig. 4.
Fig. 4.

Transmitted field |E| through a 2D octagonal quasicrystal layer calculated by RCWA for three incident angles: (a) θ=0, (b) θ=π/4, and (c) θ=π/3. The thickness of the layer is Zt=2μm, and the number of Fourier components used is Ng=60.

Fig. 5.
Fig. 5.

(a) Transmitted field |E| through a 2D octagonal quasicrystal supercell calculated by the FDTD method for three incident angles: (a) θ=0, (b) θ=π/4, and (c) θ=π/3. The thickness of the layer is Zt=2μm, and the 2D supercell used has the dimension length×width=31μm×31μm.

Fig. 6.
Fig. 6.

(a) Transmission amplitude |T| as a function of incident angle θ for three diffraction orders: [ghuv]=[0000], [1000], and [0100]; (b) reflection amplitude |R| as a function of incident angle θ for three diffraction orders: [ghuv]=[0000], [1000], and [0100]. The labels T0000 (R0000), T1000 (R1000), and T0100 (R0100) denote the upper, middle, and lower curves, respectively. The thickness of the layer is Zt=5μm.

Equations (29)

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ρ(r)=ρlat(r)ρs(r)ρlat(rr)ρs(r)dr,
ρlat(r)=x1,x2Nδ(rx1x1x2x2),
ρs(r)={1w,ifw2xw2andΔ2xcΔ20,otherwise.
FT[ρ(r)]=FT[ρlat(r)ρg(r)]=FT[ρlat(r)]×FT[ρg(r)].
ρwδ(k)=FT[ρwδ(r)]=FT[ρlat(r)ρs,wδ(r)]=FT[ρlat(r)]×FT[ρs,wδ(r)]=k1,k2Nδ(kk1k1k2k2)×2sinΔ2kxckxc.
ρwδ(kx)=k1,k2N2sinΔ2kxckxcδ(kk1k1k2k2)dkxc=k1,k2Nδ[(k·k^x)k^xk1(k1·k^x)k^xk2(k2·k^x)k^x]×2sinΔ2kxckxcδ[(k·k^xc)k^xck1(k1·k^xc)k^xck2(k2·k^xc)k^xc]dkxck1,k2Nδ(kxGxk1k2)2sinΔ2Gxck1k2Gxck1k2,
Gxk1k2=2πak1τ+k22+τ,Gxck1k2=2πak1τ+k22+τ,
[k^xk^xc]=[cosαsinαsinαcosα][k^1k^2].
[xyxcyc]=Ω[x1x2y1y2]12[2011021111201102][x1x2y1y2],
xmuvεmuvexp[2πi(x1+mx2+uy1+vy2)/a]=muvxεmuvexp[i2πa(muv)Ω1Ω(x1x2y1y2)T]=muvxεmuvexp[i2πa(muv)ΩT(xyxcyc)T]=muvxεmuvexp[i(kTΩc1Tx+kTΩc2Ty+kTΩc3Txc+kTΩc4Tyc)]g[muv]ikx,gεgexp[i(kx,gx+ky,gy+kxc,gxc+kyc,gyc)],
ε=[εxxεxyεxzεyxεyyεyzεzxεzyεzz],
εxx=no2+(ne2no2)sin2θocos2ϕo,εxy=εyx=(ne2no2)sin2θosinϕocosϕo,εxz=εzx=(ne2no2)sinθocosθocosϕo,εyy=no2+(ne2no2)sin2θosin2ϕo,εyz=εzy=(ne2no2)sinθocosθosinϕo,εzz=no2+(ne2no2)cos2θo,
εij(x1,x2,y1,y2;z)=g[muv]εij,g(z)exp(i(x1+mx2+uy1+vy2)2πa),
εij,g(z)=(1a)40a0a0a0aεg(x1,x2,y1,y2;z)×exp(i(x1+mx2+uy1+vy2)2πa)dx1dx2dy1dy2,
εij(x,y;z)=g[muv]εij,g(z)exp[ikx,gx+iky,gy].
Y0E(x,y;z)=geg(z)exp[ik0(nx,gx+ny,gy)],
Z0H(x,y;z)=ghg(z)exp[ik0(nx,gx+ny,gy)],
nx,g=nIsinθcosϕ+kx,g/k0,
ny,g=nIsinθsinϕ+ky,g/k0.
[eg(z)hg(z)]=[Eg+Hg+]eik0ξgz+[EgHg]eik0ξgz.
ft/z=iGqcft,
Gqc=[n˜xε˜zz1ε˜zxn˜xε˜zz1n˜x1n˜xε˜zz1ε˜zyn˜xε˜zz1n˜yε˜xzε˜zz1ε˜zxε˜xx+n˜yn˜yε˜xzε˜zz1n˜xε˜xzε˜zz1ε˜zyε˜xyn˜yn˜xε˜xzε˜zz1n˜yn˜yε˜zz1ε˜zxn˜yε˜zz1n˜xn˜yε˜zz1ε˜zyn˜yε˜zz1n˜y+1ε˜yzε˜zz1ε˜zx+ε˜yx+n˜xn˜yε˜yzε˜zz1n˜xε˜yzε˜zz1ε˜zy+ε˜yyn˜xn˜xε˜yzε˜zz1n˜y],
Tqc1GqcTqc=Kqc
ft(z)=TqceiKqcz[ft+ft].
[ft,III+ft,III]=TIIIMqc(TI)1[ft,I+ft,I]W1[ft,I+ft,I],
Mqc=TqceiKqcZqc(Tqc)1
[ft,I+ft,I]=[W1W2W3W4][ft,III+0],
ft,III+=W11ft,I+,
ft,I=W3W11ft,I+.

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