Abstract

We show that a planar aperiodic lattice, mimicking the appearance of a sunflower, supports photonic bandgaps for weak dielectric contrast. The pattern’s high orientational order and spatially uniform modal pitch yields an isotropic Fourier space. A 2D structure of cylinders (ϵ=2) in air possesses a wide 21% TM bandgap, versus 5.6% for a sixfold lattice or 14% for a 12-fold fractal tiling. The isotropic gap frequencies imply flat bands, and thus application in nonlinear optics and low threshold lasers, where a reduced group velocity in all directions may be desired.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).
  2. M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
    [CrossRef] [PubMed]
  3. W. Steurer and D. Sutter-Widmer, J. Phys. D 40, R229 (2007).
    [CrossRef]
  4. M. Senechal, Not. Am. Math. Soc. 53, 886 (2006).
  5. C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
    [CrossRef]
  6. C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
    [CrossRef]
  7. M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
    [CrossRef] [PubMed]
  8. K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, J. Mod. Opt. 50, 2095 (2003).
  9. R. V. Moody, D. Postnikoff, and N. Strungaru, Ann. Henri Poincare 7, 711 (2006).
    [CrossRef]
  10. F. Zolla, D. Felbacq, and B. Guizal, Opt. Commun. 148, 6 (1998).
    [CrossRef]
  11. G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
    [CrossRef]
  12. A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, Opt. Lett. 33, 2716 (2008).
    [CrossRef] [PubMed]
  13. H. Vogel, Math. Biosci. 44, 179 (1979).
    [CrossRef]
  14. J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
    [CrossRef]
  15. B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
    [CrossRef]
  16. M. Soljačić and J. D. Joannopoulos, Nature Mater. 3, 211 (2004).
    [CrossRef]

2008 (3)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef] [PubMed]

A. Agrawal, N. Kejalakshmy, J. Chen, B. M. A. Rahman, and K. T. V. Grattan, Opt. Lett. 33, 2716 (2008).
[CrossRef] [PubMed]

2007 (1)

W. Steurer and D. Sutter-Widmer, J. Phys. D 40, R229 (2007).
[CrossRef]

2006 (3)

M. Senechal, Not. Am. Math. Soc. 53, 886 (2006).

R. V. Moody, D. Postnikoff, and N. Strungaru, Ann. Henri Poincare 7, 711 (2006).
[CrossRef]

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

2004 (2)

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
[CrossRef]

M. Soljačić and J. D. Joannopoulos, Nature Mater. 3, 211 (2004).
[CrossRef]

2003 (2)

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, J. Mod. Opt. 50, 2095 (2003).

2000 (2)

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
[CrossRef] [PubMed]

1999 (1)

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

1998 (1)

F. Zolla, D. Felbacq, and B. Guizal, Opt. Commun. 148, 6 (1998).
[CrossRef]

1979 (1)

H. Vogel, Math. Biosci. 44, 179 (1979).
[CrossRef]

Agrawal, A.

Band, S.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

Baumberg, J. J.

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
[CrossRef] [PubMed]

Beckett, D. H.

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

Busch, K.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
[CrossRef]

Chaikin, P. M.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef] [PubMed]

Charlton, M. D. B.

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
[CrossRef] [PubMed]

Chelnokov, A.

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, J. Mod. Opt. 50, 2095 (2003).

Chen, J.

Cheng, B.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
[CrossRef]

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

Cox, S. J.

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

David, S.

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, J. Mod. Opt. 50, 2095 (2003).

Felbacq, D.

F. Zolla, D. Felbacq, and B. Guizal, Opt. Commun. 148, 6 (1998).
[CrossRef]

Generowicz, J. M.

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

Grattan, K. T. V.

Guizal, B.

F. Zolla, D. Felbacq, and B. Guizal, Opt. Commun. 148, 6 (1998).
[CrossRef]

Hagmann, F.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
[CrossRef]

Hiett, B. P.

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

Jeong, H.-C.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef] [PubMed]

Jin, C.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
[CrossRef]

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

M. Soljačić and J. D. Joannopoulos, Nature Mater. 3, 211 (2004).
[CrossRef]

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Kejalakshmy, N.

Kingerl, K.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
[CrossRef]

Lee, T.

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

Li, Z.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
[CrossRef]

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

Lourtioz, J. M.

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, J. Mod. Opt. 50, 2095 (2003).

Man, B.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
[CrossRef]

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Mingaleev, S. F.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
[CrossRef]

Molinari, M.

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

Moody, R. V.

R. V. Moody, D. Postnikoff, and N. Strungaru, Ann. Henri Poincare 7, 711 (2006).
[CrossRef]

Netti, M. C.

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
[CrossRef] [PubMed]

Parker, G. J.

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
[CrossRef] [PubMed]

Postnikoff, D.

R. V. Moody, D. Postnikoff, and N. Strungaru, Ann. Henri Poincare 7, 711 (2006).
[CrossRef]

Rahman, B. M. A.

Rechtsman, M. C.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef] [PubMed]

Senechal, M.

M. Senechal, Not. Am. Math. Soc. 53, 886 (2006).

Soljacic, M.

M. Soljačić and J. D. Joannopoulos, Nature Mater. 3, 211 (2004).
[CrossRef]

Steinhardt, P. J.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef] [PubMed]

Steurer, W.

W. Steurer and D. Sutter-Widmer, J. Phys. D 40, R229 (2007).
[CrossRef]

Strungaru, N.

R. V. Moody, D. Postnikoff, and N. Strungaru, Ann. Henri Poincare 7, 711 (2006).
[CrossRef]

Sun, B.

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

Sutter-Widmer, D.

W. Steurer and D. Sutter-Widmer, J. Phys. D 40, R229 (2007).
[CrossRef]

Thomas, K. S.

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

Torquato, S.

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef] [PubMed]

Vogel, H.

H. Vogel, Math. Biosci. 44, 179 (1979).
[CrossRef]

Wang, K.

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, J. Mod. Opt. 50, 2095 (2003).

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Zarbakhsh, J.

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
[CrossRef]

Zhang, D.

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
[CrossRef]

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

Zolla, F.

F. Zolla, D. Felbacq, and B. Guizal, Opt. Commun. 148, 6 (1998).
[CrossRef]

Zoorob, M. E.

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
[CrossRef] [PubMed]

Ann. Henri Poincare (1)

R. V. Moody, D. Postnikoff, and N. Strungaru, Ann. Henri Poincare 7, 711 (2006).
[CrossRef]

Appl. Phys. Lett. (2)

J. Zarbakhsh, F. Hagmann, S. F. Mingaleev, K. Busch, and K. Kingerl, Appl. Phys. Lett. 84, 4687 (2004).
[CrossRef]

C. Jin, B. Cheng, B. Man, Z. Li, D. Zhang, S. Band, and B. Sun, Appl. Phys. Lett. 75, 1848 (1999).
[CrossRef]

J. Mater. Sci. Mater. Electron. (1)

B. P. Hiett, D. H. Beckett, S. J. Cox, J. M. Generowicz, M. Molinari, and K. S. Thomas, J. Mater. Sci. Mater. Electron. 14, 413 (2003).
[CrossRef]

J. Mod. Opt. (1)

K. Wang, S. David, A. Chelnokov, and J. M. Lourtioz, J. Mod. Opt. 50, 2095 (2003).

J. Phys. D (1)

W. Steurer and D. Sutter-Widmer, J. Phys. D 40, R229 (2007).
[CrossRef]

Math. Biosci. (1)

H. Vogel, Math. Biosci. 44, 179 (1979).
[CrossRef]

Nature (1)

M. E. Zoorob, M. D. B. Charlton, G. J. Parker, J. J. Baumberg, and M. C. Netti, Nature 404, 740 (2000).
[CrossRef] [PubMed]

Nature Mater. (1)

M. Soljačić and J. D. Joannopoulos, Nature Mater. 3, 211 (2004).
[CrossRef]

Not. Am. Math. Soc. (1)

M. Senechal, Not. Am. Math. Soc. 53, 886 (2006).

Opt. Commun. (1)

F. Zolla, D. Felbacq, and B. Guizal, Opt. Commun. 148, 6 (1998).
[CrossRef]

Opt. Lett. (1)

Philos. Trans. R. Soc. London Ser. A (1)

G. J. Parker, M. D. B. Charlton, M. E. Zoorob, J. J. Baumberg, M. C. Netti, and T. Lee, Philos. Trans. R. Soc. London Ser. A 364, 189 (2006).
[CrossRef]

Phys. Rev. B (1)

C. Jin, B. Cheng, B. Man, Z. Li, and D. Zhang, Phys. Rev. B 61, 10762 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

M. C. Rechtsman, H.-C. Jeong, P. M. Chaikin, S. Torquato, and P. J. Steinhardt, Phys. Rev. Lett. 101, 073902 (2008).
[CrossRef] [PubMed]

Other (1)

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Molding the Flow of Light, 2nd ed. (Princeton U. Press, 2008).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

(a) SEM image of a 500-point sunflower. (b) Delaunay triangulation of (a), highlighting the parastichies. (c) Calculated and (d) experimental diffraction patterns of the fabricated sample. The faint fourfold symmetry in (d) is an artefact due to the test patterns’ square tiling.

Fig. 2
Fig. 2

Calculated diffraction patterns of the sunflower for (a) a 50 a 2 section and (b) a 100 a 2 section, both offset r 0 = 100 a along the positive x axis. Darker shades indicate greater magnitudes, and the dc component is suppressed to improve contrast.

Fig. 3
Fig. 3

Transmitted intensity ( | E z | 2 , logscale) as a function of in-plane angle and normalized frequency ( ω a 2 π c = a λ ) for (a) hexagonal, (b) dodecagonal, (c) Stampfli-inflated, and (d) sunflower lattices. Darker shades indicate lower transmission. (e)–(h) Corresponding diffraction patterns showing only the strongest f n . Symbol size is proportional to | f n | .

Equations (3)

Equations on this page are rendered with MathJax. Learn more.

x ( n ) = A cos ( n Ψ ) n ,
y ( n ) = A sin ( n Ψ ) n ,
ρ ( r ) = n f n ( k n ) exp ( i k n r ) ,

Metrics