Abstract

The photonic band gap (PBG) properties of two classes of two-dimensional (2-D) triangular lattice fabricated by holographic lithography are investigated numerically. The effect of intensity threshold on the filling ratio and then the shape of “atoms”, and the corresponding photonic gap are comprehensively studied. Our results show that the recording geometry for a given 2-D triangular lattice is not unique, and this fact gives us more freedom in choosing proper recording geometry to obtain larger bandgaps.

© 2004 Optical Society of America

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References

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  • |

  1. E. Yablonovitch, �??Inhibited spontaneous emission in solid-state physics and electronics,�?? Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  2. J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).
  3. C. M. Anderson and K. P. Giapis, �??Larger two-dimensional photonic band gaps,�?? Phys. Rev. Lett. 77, 2949-2952 (1996).
    [CrossRef] [PubMed]
  4. D. L. Bullock, C. Shih, R. S. Margulies, �??Photonic band structure investigation of two-dimensional Bragg reflector mirrors for semiconductor laser mode control,�?? J. Opt. Soc. Am. B 10, 399-403 (1993).
    [CrossRef]
  5. J. C. Knight, T. A. Birks, P. St. J. Russell, D. M. Atkin, �??All-silica single-mode fiber with photonic crystal cladding,�?? Opt. Lett. 21, 1547-1549 (1996).
    [CrossRef] [PubMed]
  6. A. R. Parker, R. C. McPhedran, D. R. McKenzie, L. C. Botten, N. P. Nicorvici, �??Photonic engineering: Aphrodite's iridescence,�?? Nature 409, 36-37 (2001).
    [CrossRef] [PubMed]
  7. M. Campbell, D. N. Sharp, M. T. Harrison, R. G. Denning, A. J. Turberfield, �??Fabrication of photonic crystals for the visible spectrum by holographic lithography, �?? Nature 404, 53-56 (2000).
    [CrossRef] [PubMed]
  8. X. L. Yang, L. Z. Cai, Y. R. Wang, Q. Liu, �??Interference of four umbrellalike beams by a diffractive beam splitter for fabrication of two-dimensional square and trigonal lattices,�?? Opt. Lett. 28, 453-455 (2003).
    [CrossRef] [PubMed]
  9. L. Z. Cai, X. L. Yang, Y. R. Wang, �??Formation of a microfiber bundle by interference of three noncoplanar beams,�?? Opt. Lett. 26, 1858-1860 (2001).
    [CrossRef]
  10. A. Shishido, Ivan B. Diviliansky, I. C. Khoo, T. S. Mayer, �??Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,�?? Appl. Phys. Lett. 79, 3332-3334 (2001).
    [CrossRef]
  11. S. G. Johnson and J. D. Joannopoulos, �??Block-iterative frequency-domain methods for Maxwell's equations in a planewave basis,�?? Opt. Express 8, 173-190 (2001), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-8-3-173</a>
    [CrossRef] [PubMed]
  12. Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).
  13. A. J. Ward and J. B. Pendry, �??Calculating photonic Green's functions using a nonorthogonal finite-difference time-domain method,�?? Phys. Rev. B 58, 7252-7259 (1998).
    [CrossRef]
  14. J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185-200 (1994).
    [CrossRef]
  15. V. Berger, O. Gauthier-Lafaye, E. Costard, �??Photonic band gaps and holography,�?? J. Appl. Phys. 82, 60-64 (1997).
    [CrossRef]
  16. D. Cassagne, C. Jouanin, D. Bertho, �??Hexagonal photonic-band-gap structures,�?? Phys. Rev. B 53, 7134-7142 (1996).
    [CrossRef]
  17. P. R. Villeneuve, M. Piche, �??Photonic band gaps in two-dimensional square and hexagonal lattices,�?? Phys. Rev. B 46, 4969-4972 (1992).
    [CrossRef]
  18. X. L. Yang, L. Z. Cai, and Q. Liu, "Theoretical bandgap modeling of two-dimensional triangular photonic crystals formed by interference technique of three-noncoplanar beams," Opt. Express 11, 1050-1055 (2003), <a href= "http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1050">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-9-1050</a>
    [CrossRef] [PubMed]

Appl. Phys. Lett.

A. Shishido, Ivan B. Diviliansky, I. C. Khoo, T. S. Mayer, �??Direct Fabrication of Two-Dimensional Titania Arrays Using Interference Photolithography,�?? Appl. Phys. Lett. 79, 3332-3334 (2001).
[CrossRef]

J. Appl. Phys.

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

J. Comput. Phys.

J. P. Berenger, �??A perfectly matched layer for the absorption of electromagnetic waves,�?? J. Comput. Phys. 114, 185-200 (1994).
[CrossRef]

J. Opt. Soc. Am. B

Nature

A. R. Parker, R. C. McPhedran, D. R. McKenzie, L. C. Botten, N. P. Nicorvici, �??Photonic engineering: Aphrodite's iridescence,�?? Nature 409, 36-37 (2001).
[CrossRef] [PubMed]

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

Opt. Express

Opt. Lett.

Phys. Rev. B

D. Cassagne, C. Jouanin, D. Bertho, �??Hexagonal photonic-band-gap structures,�?? Phys. Rev. B 53, 7134-7142 (1996).
[CrossRef]

P. R. Villeneuve, M. Piche, �??Photonic band gaps in two-dimensional square and hexagonal lattices,�?? Phys. Rev. B 46, 4969-4972 (1992).
[CrossRef]

A. J. Ward and J. B. Pendry, �??Calculating photonic Green's functions using a nonorthogonal finite-difference time-domain method,�?? Phys. Rev. B 58, 7252-7259 (1998).
[CrossRef]

Phys. Rev. Lett.

C. M. Anderson and K. P. Giapis, �??Larger two-dimensional photonic band gaps,�?? Phys. Rev. Lett. 77, 2949-2952 (1996).
[CrossRef] [PubMed]

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

Other

J. D. Joannopoulos, R. D. Meade, J. N. Winn, Photonic Crystals (Princeton University Press, Princeton, 1995).

Richard Brent, Algorithms for minimization without derivatives (Prentice-Hall, 1973; republished by Dover in paperback, 2002).

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Figures (7)

Fig. 1.
Fig. 1.

(a) (I) 2-D triangular photonic lattice fabricated by four umbrellalike beam interference technique; (II) The first Brillouin zone with the symmetry points indicated.(b) The relation between threshold intensity and the filling ratio of dielectric material, where line (I) is for titania and line (II) is for GaAs.

Fig. 2.
Fig. 2.

Calculated results for TE polarization. (a)Gap to midgap frequency ratio (Δω/ω 0) as a function of titania FR; (b) Optimized photonic band gap structure with FR=60.4%.

Fig. 3.
Fig. 3.

Calculated titania PhC transmission spectra (a and d) and directional photonic band diagrams (b and c) for TE and TM polarizations, respectively.

Fig. 4.
Fig. 4.

Calculated optimized photonic band structure for TE polarization with the dielectric constant ratio 13.6:1. (a) The FR of GaAs corresponding to the maximum E3–4 is 49.4%; (b) The FR of GaAs corresponding to the maximum E4–5 is 7.1%.

Fig. 5.
Fig. 5.

(a)Gap map for the inverted GaAs structure, where arrows indicate these maximum TM Gap-midgap ratios; (b)TM Gap-midgap ratios as a function of intensity threshold.

Fig. 6.
Fig. 6.

The photonic band structure in the optimized case that the intensity threshold is 0.900. The solid curves are for the TE polarization, and the dotted curves are for the TM.

Fig. 7.
Fig. 7.

Gap map of full PBG for the inverted GaAs structure

Equations (2)

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I 0 ( x , y ) = cos 2 [ π 3 a ( 2 x ) ] + cos 2 [ π 3 a ( x + 3 y ) ] ,
I 0 = 3 + cos ( 4 π y 3 a ) + cos [ 2 π a ( x + y 3 ) ] + cos [ 2 π a ( x y 3 ) ] ,

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