Abstract

We investigate the enhancement of nonlinearity in one-dimensional (1D) photonic crystals (PCs) with Kerr nonlinearity by numerical Z-scan experiments based on the finite-difference time-domain technique. Focused Gaussian beams with well-defined waists and Rayleigh lengths necessary for Z-scan experiments are generated through a conjugated manipulation of the Gaussian beams propagating in free space. The Z-scan measurements used for bulk materials are naturally extended to 1D PCs after incorporating the frequency- and power-density-dependent reflections into their linear and nonlinear absorptions. The closed- and open-aperture Z-scan traces for the 1D PCs are obtained and a symmetric method is employed to modify the asymmetric closed-aperture traces. The nonlinearity enhancement factors at different frequencies in the first and second bands are derived numerically and analytically. A good agreement is found between the numerical and analytical results in the case of weak nonlinearity. Moreover, the dependences of the enhancement factor on the incident power density for different frequencies in the 1D PCs are extracted and they are found to be much different from those in bulk materials. It is revealed that the variation of the group velocity with increasing power density is responsible for the power-density dependence of the enhancement factor. It indicates that in practice one must deliberately choose the working frequency and power density of PC-based devices in order to achieve a maximum enhancement of nonlinearity.

© 2008 Optical Society of America

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2007 (2)

H. Y. Liu, S. Lan, L. J. Wu, Q. Guo, W. Hu, S. H. Liu, X. S. Lin, and A. V. Gopal, “Self-induced Anderson localization and optical limiting in photonic crystal coupled cavity waveguides with Kerr nonlinearity,” Appl. Phys. Lett. 90, 213507 (2007).
[CrossRef]

J. Hwang and J. W. Wu, “Investigation on dispersive properties of photonic crystals for employment of Z-scan method,” Proc. SPIE 6480, 648019 (2007).
[CrossRef]

2006 (2)

2005 (2)

J. Hwang and J. W. Wu, “Determination of optical Kerr nonlinearity of a photonic bandgap structure by Z-scan measurement,” Opt. Lett. 30, 875-877 (2005).
[CrossRef] [PubMed]

Y. H. Liu, X. Y. Hu, D. X. Zhang, B. Y. Cheng, D. Z. Zhang, and Q. B. Meng, “Subpicosecond optical switching in polystyrene opal,” Appl. Phys. Lett. 86, 151102 (2005).
[CrossRef]

2004 (3)

2003 (3)

Z. B. Liu, J. G. Tian, W. P. Zang, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, “Influence of nonlinear absorption on Z-scan measurements of nonlinear refraction,” Chin. Phys. Lett. 20, 509-512 (2003).
[CrossRef]

L. J. Wu, M. Mazilu, and T. F. Krauss, “Beam steering in planar-photonic crystals: from superprism to supercollimator,” J. Lightwave Technol. 21, 561-566 (2003).
[CrossRef]

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518-2520 (2003).
[CrossRef]

2002 (2)

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, “Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines,” Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

M. Soljacic, S. G. Johnson, S. Fan, M. Ibanescu, E. Ippen, and J. D. Joannopoulos, “Photonic-crystal slow-light enhancement of nonlinear phase sensitivity,” J. Opt. Soc. Am. B 19, 2052-2059 (2002).
[CrossRef]

2001 (1)

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, “Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses,” J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

2000 (3)

A. Haché and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089-4091 (2000).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

1999 (2)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals: toward microscale lightwave circuits,” J. Lightwave Technol. 17, 2032-2038 (1999).
[CrossRef]

1998 (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[CrossRef]

1997 (1)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

1996 (1)

1994 (1)

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

1990 (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[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]

1966 (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966). In this paper, a commercial software developed by Rsoft Design Group (http://www.rsoftdesign.com) is used for FDTD simulations.
[CrossRef]

Appl. Phys. Lett. (7)

M. Loncǎr, D. Nedeljkovic, T. Doll, J. Vuckovic, A. Scherer, and T. P. Pearsall, “Waveguiding in planar photonic crystals,” Appl. Phys. Lett. 77, 1937-1939 (2000).
[CrossRef]

Y. Sugimoto, S. Lan, S. Nishikawa, N. Ikeda, H. Ishikawa, and K. Asakawa, “Design and fabrication of impurity band-based photonic crystal waveguides for optical delay lines,” Appl. Phys. Lett. 81, 1946-1948 (2002).
[CrossRef]

S. Lan, A. V. Gopal, K. Kanamoto, and H. Ishikawa, “Ultrafast response of photonic crystal atoms with Kerr nonlinearity to ultrashort optical pulses,” Appl. Phys. Lett. 84, 5124-5126 (2004).
[CrossRef]

Y. H. Liu, X. Y. Hu, D. X. Zhang, B. Y. Cheng, D. Z. Zhang, and Q. B. Meng, “Subpicosecond optical switching in polystyrene opal,” Appl. Phys. Lett. 86, 151102 (2005).
[CrossRef]

A. Haché and M. Bourgeois, “Ultrafast all-optical switching in a silicon-based photonic crystal,” Appl. Phys. Lett. 77, 4089-4091 (2000).
[CrossRef]

X. Y. Hu, Q. Zhang, Y. H. Liu, B. Y. Cheng, and D. Z. Zhang, “Ultrafast three-dimensional tunable photonic crystal,” Appl. Phys. Lett. 83, 2518-2520 (2003).
[CrossRef]

H. Y. Liu, S. Lan, L. J. Wu, Q. Guo, W. Hu, S. H. Liu, X. S. Lin, and A. V. Gopal, “Self-induced Anderson localization and optical limiting in photonic crystal coupled cavity waveguides with Kerr nonlinearity,” Appl. Phys. Lett. 90, 213507 (2007).
[CrossRef]

Chin. Phys. Lett. (1)

Z. B. Liu, J. G. Tian, W. P. Zang, W. Y. Zhou, C. P. Zhang, and G. Y. Zhang, “Influence of nonlinear absorption on Z-scan measurements of nonlinear refraction,” Chin. Phys. Lett. 20, 509-512 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

M. Sheik-Bahae, A. A. Said, T. Wei, D. J. Hagan, and E. W. Van Stryland, “Sensitive measurement of optical nonlinearities using a single beam,” IEEE J. Quantum Electron. 26, 760-769 (1990).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. Yee, “Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media,” IEEE Trans. Antennas Propag. 14, 302-307 (1966). In this paper, a commercial software developed by Rsoft Design Group (http://www.rsoftdesign.com) is used for FDTD simulations.
[CrossRef]

J. Appl. Phys. (1)

S. Lan, S. Nishikawa, H. Ishikawa, and O. Wada, “Design of impurity band-based photonic crystal waveguides and delay lines for ultrashort optical pulses,” J. Appl. Phys. 90, 4321-4327 (2001).
[CrossRef]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (3)

Nature (2)

J. S. Foresi, P. R. Villeneuve, J. Ferrera, E. R. Thoen, G. Steinmeyer, S. Fan, J. D. Joannopoulos, L. C. Kimerling, H. I. Smith, and E. P. Ippen, “Photonic crystals: putting a new twist on light,” Nature 386, 143-149 (1997).
[CrossRef]

S. Noda, A. Chutinan, and M. Imada, “Trapping and emission of photons by a single defect in a photonic bandgap structure,” Nature 407, 608-610 (2000).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. B (1)

H. Kosaka, T. Kawashima, A. Tomita, M. Notomi, T. Tamamura, T. Sato, and S. Kawakami, “Superprism phenomena in photonic crystals,” Phys. Rev. B 58, R10096-R10099 (1998).
[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]

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

M. Scalora, J. P. Dowling, C. M. Bowden, and M. J. Bloemer, “Optical limiting and switching of ultrashort pulses in nonlinear photonic band gap materials,” Phys. Rev. Lett. 73, 1368-1371 (1994).
[CrossRef] [PubMed]

Proc. SPIE (1)

J. Hwang and J. W. Wu, “Investigation on dispersive properties of photonic crystals for employment of Z-scan method,” Proc. SPIE 6480, 648019 (2007).
[CrossRef]

Science (1)

O. Painter, R. K. Lee, A. Scherer, A. Yariv, J. D. O'Brien, P. D. Dapkus, and I. Kim, “Two-dimensional photonic band-gap defect mode laser,” Science 284, 1819-1821 (1999).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Linear transmission spectrum for the 1D PC studied in this paper. The frequencies at which the enhancement of nonlinearity is investigated are indicated by arrows.

Fig. 2
Fig. 2

Electric field distribution of the focused Gaussian beams generated by a conjugated technique.

Fig. 3
Fig. 3

Z-scan traces obtained by the division method and the symmetric method for the (a) bulk material ( TiO 2 film) and (b) 1D PC.

Fig. 4
Fig. 4

Closed-aperture Z-scan traces for the frequencies in the first (a) and (b) second bands. The insets show the details of the Z-scan traces obtained at ω D and ω I where the transmission valleys appear.

Fig. 5
Fig. 5

Open-aperture Z-scan traces for the frequencies in the (a) first and (b) second bands.

Fig. 6
Fig. 6

Comparison of the nonlinearity enhancement factors at different frequencies derived numerically (filled circles and triangles) and analytically (open squares). The linear transmission spectrum is also shown by a dashed curve for reference (the transmittance has been multiplied by a factor of 10).

Fig. 7
Fig. 7

Frequency spectra for (a) transmission, (b) transmission phase and its derivative, and (c) group velocity of the 1D PC calculated by the transfer matrix method.

Fig. 8
Fig. 8

Dependence of the peak-to-valley difference in transmittance on the incident power density for the bulk material ( TiO 2 film) and the 1D PC at ω B .

Fig. 9
Fig. 9

Dependence of the nonlinearity enhancement factor on the incident power density for the 1D PC at different frequencies ( ω A , ω B , and ω C ) in the first band.

Fig. 10
Fig. 10

Locations of different frequencies ( ω A , ω B , and ω C ) in the first band relative to the group velocity minimum.

Tables (1)

Tables Icon

Table 1 Values of ν g , σ, α 0 , and L eff for the 1D PC Used in the Calculation of η a

Equations (14)

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n ( I ) = n 0 + n 2 I ,
α ( I ) = α 0 + β I ,
α ( ω , I ) = α 0 ( ω ) + β ( ω ) I ,
Δ ϕ 0 Δ T p v .
Δ ϕ 0 = k n 2 I 0 L eff bulk ,
η = Δ T p v PC Δ T p v bulk ,
Δ ϕ 0 = δ k L eff ,
δ k δ ω ( d ω d k ) 1 = δ ω ν g ,
δ ω = σ δ n n ω .
Δ ϕ 0 σ δ n n ω ν g L eff .
δ n n = ε 0 c n 2 E 0 2 ,
Δ ϕ 0 ε 0 c n 2 E 0 2 ω ν g σ L eff .
η = Δ ϕ 0 PC Δ ϕ 0 bulk = E 0 PC 2 E 0 bulk 2 ( ν g bulk ν g PC ) σ PC L eff PC σ bulk L eff bulk ( ν g bulk ν g PC ) 2 σ PC L eff PC σ bulk L eff bulk .
ν g PC = ( σ bulk L eff bulk σ PC L eff PC η ) 1 2 ν g bulk .

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