Abstract

In holographic fabrication of photonic crystals the shape and size of the dielectric columns or particles (“atoms”) are determined by the isointensity surfaces of the interference field. Therefore their photonic band gap (PBG) properties are closely related to their fabrication design. As an example, we have investigated the PBGs of a kind of holographically formed two-dimensional (2-D) square lattice with pincushion columns rotated by 45°, and shown that this structure has complete PBGs in a wide range of dielectric contrast comparable to or even larger than those of the same lattice with square columns reported before. The optical design for making this structure is also given. This work may demonstrate the unique feature and advantages of photonic crystals made by holographic method and provide a guideline for their design and experimental fabrication.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |

  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. R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannapoulos, �??Existence of a photonic band gap in two- dimensions ,�?? Appl. Phys. Lett. 61, 495-497 (1992)
    [CrossRef]
  4. P. R. Villeneuve, M. Piche, �??Phonic band gaps in two-dimensional square and hexagonal lattices,�?? Phys. Rev. B 46, 4969-4972 (1992).
    [CrossRef]
  5. P. R. Villeneuve, M. Piche, �??Photonic band gaps in two-dimensional square lattices: Square and circular lattices,�?? Phys. Rev. B 46, 4973-4975 (1992).
    [CrossRef]
  6. 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]
  7. C. M. Anderson and K. P. Giapis, �??Larger two-dimensional photonic band gaps,�?? Phys. Rev. Lett. 77, 2949-2952 (1996).
    [CrossRef] [PubMed]
  8. 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]
  9. S. Y. Lin, G. Arjavalingam, W. M. Robertson, �??Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,�?? J. Mod. Opt. 41, 385-393 (1994)
    [CrossRef]
  10. Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Large absolute band gaps in two-dimensional anisotropic photonic crystals,�?? Phys. Rev. Lett. 77, 2574-2977 (1998).
    [CrossRef]
  11. M. Qiu, S. He, �??Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,�?? J. Opt. Soc. Am. A 17, 1027-1030 (2000).
    [CrossRef]
  12. M. Agio, L. C. Andreanm, �??Complete photonic band gap in a two-dimensional chessboard lattice,�?? Phys. Rev. B 61, 15519-15522 (2000).
    [CrossRef]
  13. 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]
  14. L. Z. Cai, X. L. Yang, Y .R. Wang, �??Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,�?? J. Opt. Soc. Am. A 19, 2238-2244 (2002).
    [CrossRef]
  15. Y.A. Vlasov, X. Z. Bo, J. C. Sturm, D. J. Norris, �??On-chip natural assembly of silicon photonic bangap crystals,�?? Nature 414, 289-293 (2001).
    [CrossRef] [PubMed]
  16. X. L. Yang, L. Z. Cai, 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).
    [CrossRef] [PubMed]
  17. X. L. Yang, L. Z. Cai, Q. Liu, H. K. Liu, �??Theoretical bandgap modeling of two-dimensional square photonic crystals fabricated by interference technique of three-noncoplanar beams,�?? J. Opt. Soc. Am. B 21, 1050-1055 (2004).
    [CrossRef]
  18. M. Leung, Y. F. Liu, �??Photon band structures: The plane-wave method,�?? Phys. Rev. B 41, 10188-10190 (1990).
    [CrossRef]
  19. T. Kondo, S. Matsuo, S. Juodkazis, H. Misawa, �??Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,�?? Appl. Phys. Lett. 79, 725-727 (2001).
    [CrossRef]

Appl. Phys. Lett.

R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannapoulos, �??Existence of a photonic band gap in two- dimensions ,�?? Appl. Phys. Lett. 61, 495-497 (1992)
[CrossRef]

T. Kondo, S. Matsuo, S. Juodkazis, H. Misawa, �??Femtosecond laser interference technique with diffractive beam splitter for fabrication of three-dimensional photonic crystals,�?? Appl. Phys. Lett. 79, 725-727 (2001).
[CrossRef]

J. Mod. Opt.

S. Y. Lin, G. Arjavalingam, W. M. Robertson, �??Investigation of absolute photonic band-gaps in 2-dimensional dielectric structures,�?? J. Mod. Opt. 41, 385-393 (1994)
[CrossRef]

J. Opt. Soc. Am. A

M. Qiu, S. He, �??Optimal design of two-dimensional photonic crystal of square lattice with large complete two-dimensional bandgap,�?? J. Opt. Soc. Am. A 17, 1027-1030 (2000).
[CrossRef]

L. Z. Cai, X. L. Yang, Y .R. Wang, �??Formation of three-dimensional periodic microstructures by interference of four noncoplanar beams,�?? J. Opt. Soc. Am. A 19, 2238-2244 (2002).
[CrossRef]

J. Opt. Soc. Am. B

Nature

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]

Nautre

Y.A. Vlasov, X. Z. Bo, J. C. Sturm, D. J. Norris, �??On-chip natural assembly of silicon photonic bangap crystals,�?? Nature 414, 289-293 (2001).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Phys. Rev. B

M. Leung, Y. F. Liu, �??Photon band structures: The plane-wave method,�?? Phys. Rev. B 41, 10188-10190 (1990).
[CrossRef]

M. Agio, L. C. Andreanm, �??Complete photonic band gap in a two-dimensional chessboard lattice,�?? Phys. Rev. B 61, 15519-15522 (2000).
[CrossRef]

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

P. R. Villeneuve, M. Piche, �??Photonic band gaps in two-dimensional square lattices: Square and circular lattices,�?? Phys. Rev. B 46, 4973-4975 (1992).
[CrossRef]

Phys. Rev. Lett.

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

Z. Y. Li, B. Y. Gu, G. Z. Yang, �??Large absolute band gaps in two-dimensional anisotropic photonic crystals,�?? Phys. Rev. Lett. 77, 2574-2977 (1998).
[CrossRef]

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

Other

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

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 (9)

Fig. 1.
Fig. 1.

The relation between threshold intensity I t and the filling ratio f of dielectric material when c=0.31, where line (I) is for the normal structure and line (II) for the inverse structure.

Fig. 2.
Fig. 2.

Variation of the shape and size of the cross section of dielectric columns with different I t when c=0.31. (a) I t=1.26, f=0.278; (b) I t=1.32, f=0.314; (c) I t=1.37, f=0.347; (d) I t=1.40, f=0.375.

Fig. 3.
Fig. 3.

Variation of relative band gap with intensity threshold I t for the inverse structure in the case of c=0.31 and ε=8.9.

Fig. 4.
Fig. 4.

Gap map for the inversed structure when c=0.31 and ε=8.9.

Fig. 5.
Fig. 5.

The photonic band structure in the optimized case I t=1.37 when c=0.31 and ε=8.9. The solid curves are for the p polarization, and the dotted curves are for the s polarization.

Fig. 6.
Fig. 6.

Different column shape and size of inverse structure of ε=8.9 yielding maximum relative PBG for different values of c. (a) c=0.1, I t=1.78, f=0.443; (b) c=0.2, I t=1.59, f=0.407; (c) c=0.3, I t=1.39, f=0.354; (d) c=0.4, I t=1.20, f=0.293.

Fig. 7.
Fig. 7.

Relation between the value of c and the corresponding maximum relative band width when ε=8.9.

Fig. 8.
Fig. 8.

Optimized column shapes yielding maximum PBG for different ε. (a) ε=12, c=0.24, I t=1.52, f=0.395; (b) ε=16, c=0.20, I t=1.60, f=0.414.

Fig. 9.
Fig. 9.

Optical design of holographic fabrication of the structure expressed by Eq. (1).

Tables (1)

Tables Icon

Table 1 Comparison of maximum PBGs for two structures with different dielectric constants

Equations (15)

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

I ( x , y ) = 2 + cos ( 2 π a x ) + cos ( 2 π a y ) + { cos [ 2 π a ( x + y ) ] + cos [ 2 π a ( x y ) ] } ,
K 1 = K ( sin θ 2 , sin θ 2 , cos θ ) , K 2 = K ( sin θ 2 , sin θ 2 , cos θ ) ,
K 3 = K ( sin θ 2 , sin θ 2 , cos θ ) , K 4 = K ( sin θ 2 , sin θ 2 , cos θ ) ,
I ( x , y ) = 4 + 2 { ( e 14 + e 23 ) cos ( 2 π x a ) + ( e 12 + e 34 ) cos ( 2 π y a )
+ e 13 cos [ 2 π ( x + y ) a ] + e 24 cos [ 2 π ( x y ) a ] ,
e 13 = e 24 = c ( e 14 + e 23 ) = c ( e 12 + e 34 ) .
e 1 = ( l , m , n ) , e 2 = ( l , m , n ) , e 3 = ( p , q , r ) , e 4 = ( p , q , r ) ,
l 2 + m 2 + n 2 = 1 , p 2 + q 2 + r 2 = 1 ,
( l + m ) sin θ + 2 n cos θ = 0 , ( p + q ) sin θ 2 r cos θ = 0 ,
lp + nr + ( 1 + 2 c ) mq ( 1 2 c ) = 0 , m 2 + q 2 2 mq / ( 1 2 c ) = 1 .
e 1 = ( 0.76253 , 0.42730 , 0.48575 ) , e 2 = ( 0.76253 , 0.42730 , 0.48575 ) ,
e 3 = ( 0.91602 , 0.31839 , 0.24398 ) , e 4 = ( 0.91602 , 0.31839 , 0.24398 ) .
I ( x , y ) = 2.864 { 1.397 + cos ( 2 π x a ) + cos ( 2 π y a )
+ 0.31 cos [ 2 π ( x + y ) a ] + 0.31 cos [ 2 π ( x y ) a ] } .
I t = 2.864 ( I t 0.603 ) .

Metrics