Abstract

We have investigated numerically the photonic bandgap (PBG) and spectral properties of two-dimensional (2-D) square structures fabricated by holographic lithography. Based on the block-iterative frequency-domain method and on the nonorthogonal finite-difference time-domain method, we have calculated band structure as a function of intensity threshold and shown that the PBG of 2-D titania arrays opens only for TM polarization and that directional PBGs can be obtained simultaneously for TE- and TM-polarized waves. In addition, we have shown that symmetry reduction of the atom introduced by holographic lithography can lift the band degeneracies and create an absolute PBG for a square lattice of dielectric rods in air.

© 2004 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2003 (1)

2002 (2)

X. L. Yang and L. Z. Cai, “Wave design of the interference of three noncoplanar beams for microfiber fabrication,” Opt. Commun. 208, 293–297 (2002).
[CrossRef]

L. Z. Cai, X. L. Yang, and Y. R. Wang, “All fourteen Bravais lattices can be formed by interference of four noncoplanar beams,” Opt. Lett. 27, 900–902 (2002).
[CrossRef]

2001 (3)

2000 (1)

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 (2)

B. D’Urso, O. Painter, J. D. O’Brien, T. Tombrello, A. Yariv, and A. Scherer, “Modal reflectivity in finite-depth two-dimensional photonic-crystal microcavities,” J. Opt. Soc. Am. B 15, 1155–1159 (1998).
[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]

1997 (1)

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

1996 (2)

1994 (1)

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

1993 (1)

1992 (1)

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

1987 (2)

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

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Atkin, D. M.

Berenger, J. P.

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

Berger, V.

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

Bertho, D.

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

Birks, T. A.

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]

Cassagne, D.

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

Costard, E.

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

D’Urso, B.

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]

Diviliansky, I. B.

A. Shishido, I. B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Gauthier-Lafaye, O.

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

Joannopoulos, J. D.

John, S.

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, S. G.

Jouanin, C.

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

Khoo, I. C.

A. Shishido, I. B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Knight, J. C.

Liu, Q.

Margulies, R. S.

Mayer, T. S.

A. Shishido, I. B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

O’Brien, J. D.

Painter, O.

Pendry, J. B.

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]

Piche, M.

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

Russell, P. St. J.

Scherer, A.

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.

Shishido, A.

A. Shishido, I. B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

Tombrello, T.

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 and hexagonal lattices,” Phys. Rev. B 46, 4969–4972 (1992).
[CrossRef]

Wang, Y. R.

Ward, A. J.

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]

Yablonovitch, E.

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

Yang, X. L.

Yariv, A.

Appl. Phys. Lett. (1)

A. Shishido, I. B. Diviliansky, I. C. Khoo, and T. S. Mayer, “Direct fabrication of two-dimensional titania arrays using interference photolithography,” Appl. Phys. Lett. 79, 3332–3334 (2001).
[CrossRef]

J. Appl. Phys. (1)

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

J. Comput. Phys. (1)

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 (2)

Nature (1)

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. Commun. (1)

X. L. Yang and L. Z. Cai, “Wave design of the interference of three noncoplanar beams for microfiber fabrication,” Opt. Commun. 208, 293–297 (2002).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Rev. B (3)

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]

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

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

Phys. Rev. Lett. (2)

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

S. John, “Strong localization of photons in certain disordered dielectric superlattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Other (1)

R. Brent, Algorithms for Minimization without Derivatives (Prentice-Hall, Englewood Cliffs, N.J., 1973; reprinted by Dover, New York, 2002).

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Figures (6)

Fig. 1
Fig. 1

(a) 2-D square photonic lattice fabricated by interference of three noncoplanar beams. (b) First Brillouin zone, with the symmetry points indicated.

Fig. 2
Fig. 2

TM gap map of the 2-D square titania arrays.

Fig. 3
Fig. 3

Gap map of the Γ–X directional PBG for 2-D titania arrays.

Fig. 4
Fig. 4

(a) Γ–X directional photonic band diagrams and (b) calculated transmission spectra for TE and TM polarizations, where It=4.54.

Fig. 5
Fig. 5

Gap map of full PBG for the inverted GaAs structure.

Fig. 6
Fig. 6

PBG for the optimal case that the intensity threshold is 2.0; a 6.11% full PBG appears’ between E1 and M3.

Equations (1)

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I0(x, y)=3+cos2πax+cos2πay+cos2πa(x-y),

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