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

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References

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    [Crossref] [PubMed]
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    [Crossref]
  4. P. R. Villeneuve and M. Piche, “Phonic band gaps in two-dimensional square and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2004 (1)

2003 (1)

2002 (1)

2001 (2)

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

T. Kondo, S. Matsuo, S. Juodkazis, and 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]

2000 (3)

M. Qiu and 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]

M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (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]

1998 (1)

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998).
[Crossref]

1996 (2)

1994 (1)

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

1993 (1)

1992 (3)

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992)
[Crossref]

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

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

1990 (1)

M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
[Crossref]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

Agio, M.

M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000).
[Crossref]

Anderson, C. M.

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

Andreanm, L. C.

M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000).
[Crossref]

Arjavalingam, G.

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

Atkin, D. M.

Birks, T. A.

Bo, X. Z.

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

Brommer, K. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992)
[Crossref]

Bullock, D. L.

Cai, L. Z.

Campbell, M.

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]

Denning, R. G.

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]

Giapis, K. P.

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

Gu, B. Y.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998).
[Crossref]

Harrison, M. T.

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]

He, S.

M. Qiu and 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]

Joannapoulos, J. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992)
[Crossref]

Joannopoulos, J. D.

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

Juodkazis, S.

T. Kondo, S. Matsuo, S. Juodkazis, and 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]

Knight, J. C.

Kondo, T.

T. Kondo, S. Matsuo, S. Juodkazis, and 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]

Leung, M.

M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
[Crossref]

Li, Z. Y.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998).
[Crossref]

Lin, S. Y.

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

Liu, H. K.

Liu, Q.

Liu, Y. F.

M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
[Crossref]

Margulies, R. S.

Matsuo, S.

T. Kondo, S. Matsuo, S. Juodkazis, and 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]

Meade, R. D.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992)
[Crossref]

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

Misawa, H.

T. Kondo, S. Matsuo, S. Juodkazis, and 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]

Norris, D. J.

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

Piche, M.

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

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

Qiu, M.

M. Qiu and 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]

Rappe, A. M.

R. D. Meade, K. D. Brommer, A. M. Rappe, and J. D. Joannapoulos, “Existence of a photonic band gap in two dimensions,” Appl. Phys. Lett. 61, 495–497 (1992)
[Crossref]

Robertson, W. M.

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

Russell, P. St. J.

Sharp, D. N.

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]

Shih, C.

Sturm, J. C.

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

Turberfield, A. J.

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]

Villeneuve, P. R.

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

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

Vlasov, Y.A.

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

Wang, Y. R.

Winn, J. N.

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

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

Yang, G. Z.

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998).
[Crossref]

Yang, X. L.

Appl. Phys. Lett. (2)

R. D. Meade, K. D. Brommer, A. M. Rappe, and 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, and 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. (1)

S. Y. Lin, G. Arjavalingam, and 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 (2)

M. Qiu and 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, and 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 (2)

Nature (2)

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

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]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. B (4)

M. Agio and L. C. Andreanm, “Complete photonic band gap in a two-dimensional chessboard lattice,” Phys. Rev. B 61, 15519–15522 (2000).
[Crossref]

M. Leung and Y. F. Liu, “Photon band structures: The plane-wave method,” Phys. Rev. B 41, 10188–10190 (1990).
[Crossref]

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

P. R. Villeneuve and 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. (3)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[Crossref] [PubMed]

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

Z. Y. Li, B. Y. Gu, and G. Z. Yang, “Large absolute band gaps in two-dimensional anisotropic photonic crystals,” Phys. Rev. Lett. 77, 2574–2977 (1998).
[Crossref]

Other (1)

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

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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 ) .

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