Abstract

The shift of optical cavity modes oscillates due to the time-varying refractive index under Kerr nonlinearity. To reveal its impact on bistability, we present a time-average approach that includes this dynamic shift characteristic in the expression of nonlinear transmission. It predicts the monotonous decrease of the maximum transmission with the increase of the frequency detuning. Also, the frequency detuning to start bistability should be corrected as about 2. Our analyses are supported by the simulations based on the finite-difference time-domain technique.

© 2006 Optical Society of America

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References

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  1. E. Centeno and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
    [CrossRef]
  2. S. Mingaleev and Y. Kivshar, "Nonlinear transmission and light localization in photonic crystal waveguides," J. Opt. Soc. Am. B 19, 2241-2249 (2002).
    [CrossRef]
  3. M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
    [CrossRef]
  4. M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
    [CrossRef]
  5. H. Lee, and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
    [CrossRef]
  6. M. Soljacic, E. Lidorikis, M. Ibanescu, S. G. Johnson, J. Joannopoulos, and Y. Fink, "Optical bistability and cutoff solitons in photonic bandgap fibers," Opt. Express 12, 1518-1527 (2004).
    [CrossRef] [PubMed]
  7. V. R. Almeida and M. Lipson, "Optical bistability on a silicon chip," Opt. Lett. 29, 2387-2389 (2004).
    [CrossRef] [PubMed]
  8. G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, "Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures," Opt. Express 13, 9623-9628 (2005).
    [CrossRef] [PubMed]
  9. G. P. Agrawal, Nonlinear Fiber Optics & Applications of Nonlinear Fiber Optics (Elsevier Science, USA, 2001), Chap. 2.
  10. S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122 (2005).
    [CrossRef]
  11. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995), Chap. 2.
  12. X. S. Lin, W. Q. Wu, H. Zhou, K. F. Zhou, and S. Lan, "Enhancement of unidirectional transmission through the coupling of nonlinear photonic crystal defects," Opt. Express 14, 2429-2439 (2006).
    [CrossRef] [PubMed]

2006 (1)

2005 (2)

G. Priem, P. Dumon, W. Bogaerts, D. Van Thourhout, G. Morthier, and R. Baets, "Optical bistability and pulsating behaviour in Silicon-On-Insulator ring resonator structures," Opt. Express 13, 9623-9628 (2005).
[CrossRef] [PubMed]

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122 (2005).
[CrossRef]

2004 (2)

2003 (2)

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

H. Lee, and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

2002 (2)

S. Mingaleev and Y. Kivshar, "Nonlinear transmission and light localization in photonic crystal waveguides," J. Opt. Soc. Am. B 19, 2241-2249 (2002).
[CrossRef]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

2000 (1)

E. Centeno and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

Agrawal, G. P.

H. Lee, and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

Almeida, V. R.

Baets, R.

Bogaerts, W.

Centeno, E.

E. Centeno and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

Chen, J. D.

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122 (2005).
[CrossRef]

Chen, X. W.

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122 (2005).
[CrossRef]

Dumon, P.

Fan, S.

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

Felbacq, D.

E. Centeno and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

Fink, Y.

M. Soljacic, E. Lidorikis, M. Ibanescu, S. G. Johnson, J. Joannopoulos, and Y. Fink, "Optical bistability and cutoff solitons in photonic bandgap fibers," Opt. Express 12, 1518-1527 (2004).
[CrossRef] [PubMed]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Ibanescu, M.

M. Soljacic, E. Lidorikis, M. Ibanescu, S. G. Johnson, J. Joannopoulos, and Y. Fink, "Optical bistability and cutoff solitons in photonic bandgap fibers," Opt. Express 12, 1518-1527 (2004).
[CrossRef] [PubMed]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Joannopoulos, J.

Joannopoulos, J. D.

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Johnson, S. G.

M. Soljacic, E. Lidorikis, M. Ibanescu, S. G. Johnson, J. Joannopoulos, and Y. Fink, "Optical bistability and cutoff solitons in photonic bandgap fibers," Opt. Express 12, 1518-1527 (2004).
[CrossRef] [PubMed]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Kivshar, Y.

Lan, S.

X. S. Lin, W. Q. Wu, H. Zhou, K. F. Zhou, and S. Lan, "Enhancement of unidirectional transmission through the coupling of nonlinear photonic crystal defects," Opt. Express 14, 2429-2439 (2006).
[CrossRef] [PubMed]

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122 (2005).
[CrossRef]

Lee, H.

H. Lee, and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

Lidorikis, E.

Lin, X. S.

X. S. Lin, W. Q. Wu, H. Zhou, K. F. Zhou, and S. Lan, "Enhancement of unidirectional transmission through the coupling of nonlinear photonic crystal defects," Opt. Express 14, 2429-2439 (2006).
[CrossRef] [PubMed]

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122 (2005).
[CrossRef]

Lipson, M.

Mingaleev, S.

Morthier, G.

Priem, G.

Soljacic, M.

M. Soljacic, E. Lidorikis, M. Ibanescu, S. G. Johnson, J. Joannopoulos, and Y. Fink, "Optical bistability and cutoff solitons in photonic bandgap fibers," Opt. Express 12, 1518-1527 (2004).
[CrossRef] [PubMed]

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Van Thourhout, D.

Wu, W. Q.

Yanik, M. F.

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

Zhou, H.

Zhou, K. F.

Appl. Phys. Lett. (1)

M. F. Yanik, S. Fan, and M. Soljacic, "High-contrast all-optical bistable switching in photonic crystal microcavities," Appl. Phys. Lett. 83, 2739-2741 (2003).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. Lee, and G. P. Agrawal, "Nonlinear switching of optical pulses in fiber Bragg gratings," IEEE J. Quantum Electron. 39, 508-515 (2003).
[CrossRef]

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

Opt. Express (3)

Opt. Lett. (1)

Phys. Rev. B (2)

E. Centeno and D. Felbacq, "Optical bistability in finite-size nonlinear bidimensional photonic crystals doped by a microcavity," Phys. Rev. B 62, R7683-R7686 (2000).
[CrossRef]

S. Lan, X. W. Chen, J. D. Chen, and X. S. Lin, "Physical origin of the ultrafast response of nonlinear photonic crystal atoms to the excitation of ultrashort pulses," Phys. Rev. B 71, 125122 (2005).
[CrossRef]

Phys. Rev. E (1)

M. Soljacic, M. Ibanescu, S. G. Johnson, Y. Fink, and J. D. Joannopoulos, "Optimal bistable switching in nonlinear photonic crystals," Phys. Rev. E 66, 055601 (2002).
[CrossRef]

Other (2)

G. P. Agrawal, Nonlinear Fiber Optics & Applications of Nonlinear Fiber Optics (Elsevier Science, USA, 2001), Chap. 2.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton University Press, Princeton, NJ, 1995), Chap. 2.

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Figures (3)

Fig. 1.
Fig. 1.

Theoretical transmissions of the nonlinear optical resonator when the dynamic shift character is considered (a); is not considered (b). The values shown in the top right corners are the corresponding frequency detuning.

Fig. 2.
Fig. 2.

(a) PC defect structure used in this paper. (b) Defect mode lineshape obtained by using the FDTD simulations. A Lorentzian lineshape (ηγ 2)/[γ 2 +(ω-ω 0)2] is also given for comparison.

Fig. 3.
Fig. 3.

Nonlinear transmissions of the PC defect structure based on FDTD simulation. The solid circles denote the stable transmissions when the input is the superposition of CW and Gaussian pulse. The empty circles denote the stable transmissions when the input is the CW only. The values shown in the top right corner are the corresponding frequency detuning.

Equations (8)

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1 n 0 2 ( r ) 2 E 0 ( r ) = ( ω c ) 2 E 0 ( r ) ,
1 n 0 2 ( r ) · 1 1 + 2 c ε 0 n 2 ( r ) E 0 2 ( r ) cos 2 ( ωt ) 2 E 0 ( r ) = ( ω + δω c ) 2 E 0 ( r ) ,
δω ω = cos 2 ( ωt ) c ε 0 2 n 2 ( r ) n 0 2 ( r ) E 0 4 ( r ) d r ε 0 n 0 2 ( r ) E 0 2 ( r ) d r .
p 0 = 2 γ out Q · [ ( ε 0 n 0 2 ( r ) E 0 2 ( r ) d r ) 2 c ε 0 2 n 2 ( r ) n 0 2 ( r ) E 0 2 ( r ) d r ] ,
δω = 2 γp out p 0 cos 2 ( ωt ) ,
T ¯ = p out p in = ω π 0 π ω η γ 2 γ 2 + ( ω ω 0 ) 2 · [ 2 cos 2 ( ωt ) ] d t = η ( 1 δ 2 ) ,
T ¯ = p out p in = ω π 0 π ω η γ 2 γ 2 + [ ω ( ω 0 γ p out p in ) ] 2 · [ 2 cos 2 ( ωt ) ] d t = η 1 + [ δ ( p out p 0 ) 2 ] .
T ¯ = p out p in = ω π 0 π ω η 1 + [ δ 2 ( p out p 0 ) cos 2 ( ωt ) ] 2 · [ 2 cos 2 ( ωt ) ] d t .

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