Abstract

Through analysis of the dispersion relation in a photonic bandgap structure, the effective optical Kerr nonlinearity that determines a Z-scan profile particularly near the stop-band edges, is derived. Near and inside the stop band, the nonlinear optical phase change that originates from an off-resonant response is converted into a change in nonlinear optical intensity through Bragg reflection. The Z-scan measurement of a cholesteric liquid-crystal photonic bandgap structure confirmed that off-resonant Kerr nonlinearity is responsible for the characteristic open-aperture Z-scan profiles near the stop-band edges.

© 2005 Optical Society of America

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  1. M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
    [Crossref]
  2. D. V. Petrov, J. Opt. Soc. Am. B 13, 1491 (1996).
    [Crossref]
  3. G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
    [Crossref] [PubMed]
  4. J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
    [Crossref]
  5. S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
    [Crossref]
  6. X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
    [Crossref]
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    [Crossref] [PubMed]
  8. A. Haché and M. Bourgeois, Appl. Phys. Lett. 77, 4089 (2000).
    [Crossref]
  9. P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, Oxford, 1993).
  10. I. I. Tahan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
    [Crossref]
  11. K. Y. Wong and A. F. Garito, Phys. Rev. A 34, 5051 (1986).
    [Crossref] [PubMed]

2004 (2)

J. Hwang, N. Y. Ha, H. J. Chang, B. Park, and J. W. Wu, Opt. Lett. 29, 2644 (2004).
[Crossref] [PubMed]

S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[Crossref]

2003 (1)

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
[Crossref]

2000 (1)

A. Haché and M. Bourgeois, Appl. Phys. Lett. 77, 4089 (2000).
[Crossref]

1997 (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[Crossref]

1996 (2)

D. V. Petrov, J. Opt. Soc. Am. B 13, 1491 (1996).
[Crossref]

I. I. Tahan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[Crossref]

1995 (1)

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

1986 (1)

K. Y. Wong and A. F. Garito, Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

Bourgeois, M.

A. Haché and M. Bourgeois, Appl. Phys. Lett. 77, 4089 (2000).
[Crossref]

Boyd, R. W.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Chang, H. J.

Cheng, B.

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
[Crossref]

De Gennes, P. G.

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, Oxford, 1993).

Fan, S.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[Crossref]

Fischer, G. L.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Garito, A. F.

K. Y. Wong and A. F. Garito, Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

Gehr, R. J.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Gopal, A. V.

S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[Crossref]

Ha, N. Y.

Haché, A.

A. Haché and M. Bourgeois, Appl. Phys. Lett. 77, 4089 (2000).
[Crossref]

Hagan, D. J.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Hu, X.

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
[Crossref]

Hwang, J.

Ishikawa, H.

S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[Crossref]

Jenekhe, S. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Joannopoulos, J. D.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[Crossref]

Kanamoto, K.

S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[Crossref]

Lan, S.

S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[Crossref]

Liu, Y.

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
[Crossref]

Osaheni, J. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Park, B.

Petrov, D. V.

Prost, J.

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, Oxford, 1993).

Said, A. A.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Sheik-Bahae, M.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Sipe, J. E.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Tahan, I. I.

I. I. Tahan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[Crossref]

Van Stryland, E. W.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Villeneuve, P. R.

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[Crossref]

Watson, G. H.

I. I. Tahan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[Crossref]

Wei, T.

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

Weller-Brophy, L. A.

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Wong, K. Y.

K. Y. Wong and A. F. Garito, Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

Wu, J. W.

Zhang, D.

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
[Crossref]

Zhang, Q.

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
[Crossref]

Appl. Phys. Lett. (3)

S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, Appl. Phys. Lett. 84, 5124 (2004).
[Crossref]

X. Hu, Q. Zhang, Y. Liu, B. Cheng, and D. Zhang, Appl. Phys. Lett. 83, 2518 (2003).
[Crossref]

A. Haché and M. Bourgeois, Appl. Phys. Lett. 77, 4089 (2000).
[Crossref]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, IEEE J. Quantum Electron. 26, 760 (1990).
[Crossref]

J. Opt. Soc. Am. B (1)

Nature (1)

J. D. Joannopoulos, P. R. Villeneuve, and S. Fan, Nature 386, 143 (1997).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

K. Y. Wong and A. F. Garito, Phys. Rev. A 34, 5051 (1986).
[Crossref] [PubMed]

Phys. Rev. B (1)

I. I. Tahan and G. H. Watson, Phys. Rev. B 54, 7593 (1996).
[Crossref]

Phys. Rev. Lett. (1)

G. L. Fischer, R. W. Boyd, R. J. Gehr, S. A. Jenekhe, J. A. Osaheni, J. E. Sipe, and L. A. Weller-Brophy, Phys. Rev. Lett. 74, 1871 (1995).
[Crossref] [PubMed]

Other (1)

P. G. De Gennes and J. Prost, The Physics of Liquid Crystals (Clarendon, Oxford, 1993).

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Figures (3)

Fig. 1
Fig. 1

Theoretically simulated (a) reflection spectrum, (b) F ( ω ) , and (c) Δ k ( I ) of a PC structure. Solid (dashed) curves, real (imaginary) parts of (b) F ( ω ) and (c) Δ k ( I ) .

Fig. 2
Fig. 2

Simulated Z-scan profiles with open aperture near the PBG as functions of normalized frequency and normalized distance z z 0 .

Fig. 3
Fig. 3

Open aperture Z-scan measurement data of a one-dimensional CLC PC. Symbols, experimental data; waves, theoretical fit. Inset, transmission spectrum of the CLC sample.

Tables (1)

Tables Icon

Table 1 Parameters in Dispersion Relation Eq. (5) for a Three-Dimensional fcc PC and a One-Dimensional CLC PC

Equations (5)

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d Δ ϕ d z = Re [ Δ k ( I ) ] ,
d I d z = α I Im [ Δ k ( I ) ] I .
k = G 2 ± F ( ω ) + Δ k ( I ) ,
F ( ω ) = ( G 2 4 ) + ϵ 0 L ( ω c ) 2 [ G 2 ϵ 0 L ( ω c ) 2 + ( U G L ) 2 ( ω c ) 4 ] 1 2 ,
Δ k ( I ) = 1 2 F ( ω ) { ε 0 NL ( ω c ) 2 G ε 0 NL ( ω c ) + 2 U G L U G NL ( ω c ) 3 2 [ G 2 ε 0 L + ( U G L ) 2 ( ω c ) 2 ] 1 2 } I .

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