Abstract

In this paper, we derive the couple-mode equations for third-order nonlinear effects in photonic crystal waveguides by employing the modal theory. These nonlinear interactions include self-phase modulation, cross-phase modulation and degenerate four-wave mixing. The equations similar to that in nonlinear fiber optics could be expanded and applied for third-order nonlinear processes in other periodic waveguides. Based on the equations, we systematically analyze the group-velocity dispersion, optical propagation loss, effective interaction area, slow light enhanced factor and phase mismatch for a slow light engineered silicon photonic crystal waveguide. Considering the two-photon and free-carrier absorptions, the wavelength conversion efficiencies in two low-dispersion regions are numerically simulated by utilizing finite difference method. Finally, we investigate the influence of slow light enhanced multiple four-wave-mixing process on the conversion efficiency.

© 2012 OSA

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    [CrossRef] [PubMed]
  3. C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  14. M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express17, 18340–18353 (2009).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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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] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  39. L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
    [CrossRef]
  40. S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
  44. Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009).
    [CrossRef]
  45. X. Liu, “Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers,” Phys. Rev. A77, 043818 (2008).
    [CrossRef]

2012 (2)

S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J.4, 224–233 (2012).
[CrossRef]

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

2011 (7)

J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express19, 4458–4463 (2011).
[CrossRef] [PubMed]

I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express19, 6093–6099 (2011).
[CrossRef] [PubMed]

B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett.36, 1728–1730 (2011).
[CrossRef] [PubMed]

P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett.36, 2629–2631 (2011).
[CrossRef] [PubMed]

C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett.36, 3413–3415 (2011).
[CrossRef] [PubMed]

N. Matsuda, T. Kato, K. Harada, H. Takesue, E. Kuramochi, H. Taniyama, and M. Notomi, “Slow light enhanced optical nonlinearity in a silicon photonic crystal coupled-resonator optical waveguide,” Opt. Express19, 19861–19874 (2011).
[CrossRef] [PubMed]

I. Cestier, A. Willinger, P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, and G. Eisenstein, “Efficient parametric interactions in a low loss GaInP photonic crystal waveguide,” Opt. Lett.36, 3936–3938 (2011).
[CrossRef] [PubMed]

2010 (13)

C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express18, 6831–6840 (2010).
[CrossRef] [PubMed]

V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett.35, 1440–1442 (2010).
[CrossRef] [PubMed]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010).
[CrossRef] [PubMed]

M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express18, 21024–21029 (2010).
[CrossRef] [PubMed]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B81, 155115 (2010).
[CrossRef]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308 (2010).
[CrossRef]

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 257–266 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A81, 023820 (2010).
[CrossRef]

2009 (7)

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009).
[CrossRef]

Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009).
[CrossRef]

A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express17, 552–557 (2009).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009).
[CrossRef] [PubMed]

K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express17, 7206–7216 (2009).
[CrossRef] [PubMed]

M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express17, 18340–18353 (2009).
[CrossRef] [PubMed]

2008 (6)

X. Liu, “Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers,” Phys. Rev. A77, 043818 (2008).
[CrossRef]

J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett.93, 251105 (2008).
[CrossRef]

E. Centeno and C. Ciracì, “Theory of backward second-harmonic localization in nonlinear left-handed media,” Phys. Rev. B78, 235101 (2008).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics2, 465–473 (2008).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B77, 115124 (2008).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express16, 6227–6232 (2008).
[CrossRef] [PubMed]

2007 (3)

2005 (1)

S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005).
[CrossRef] [PubMed]

2004 (1)

M. Soljacic and J. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3, 211–220 (2004).
[CrossRef] [PubMed]

2003 (1)

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E68, 065601–065601 (2003).
[CrossRef]

2001 (1)

Agrawal, G.

L. Yin and G. Agrawal, “Impact of two-photon absorption on self-phase modulation in silicon waveguides,” Opt. Lett.32, 2031–2033 (2007).
[CrossRef] [PubMed]

G. Agrawal, Nonlinear Fiber Optics and Applications of Nonlinear Fiber Optics, 4th ed. (Elsevier Science, New York, 2007).

Asakawa, K.

Baba, T.

T. Baba, “Slow light in photonic crystals,” Nat. Photonics2, 465–473 (2008).
[CrossRef]

Bao, C.

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

Baron, A.

Beggs, D.

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

Boucaud, P.

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308 (2010).
[CrossRef]

Bräuer, A.

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E68, 065601–065601 (2003).
[CrossRef]

Cassan, E.

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

Centeno, E.

E. Centeno and C. Ciracì, “Theory of backward second-harmonic localization in nonlinear left-handed media,” Phys. Rev. B78, 235101 (2008).
[CrossRef]

Cestier, I.

Checoury, X.

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308 (2010).
[CrossRef]

Chen, L.

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

Ciracì, C.

E. Centeno and C. Ciracì, “Theory of backward second-harmonic localization in nonlinear left-handed media,” Phys. Rev. B78, 235101 (2008).
[CrossRef]

Clark, A.

Colman, P.

S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J.4, 224–233 (2012).
[CrossRef]

I. Cestier, A. Willinger, P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, and G. Eisenstein, “Efficient parametric interactions in a low loss GaInP photonic crystal waveguide,” Opt. Lett.36, 3936–3938 (2011).
[CrossRef] [PubMed]

I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express19, 6093–6099 (2011).
[CrossRef] [PubMed]

P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett.36, 2629–2631 (2011).
[CrossRef] [PubMed]

V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett.35, 1440–1442 (2010).
[CrossRef] [PubMed]

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009).
[CrossRef]

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009).
[CrossRef]

Combrie, S.

S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J.4, 224–233 (2012).
[CrossRef]

M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express18, 21024–21029 (2010).
[CrossRef] [PubMed]

Combrié, S.

P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett.36, 2629–2631 (2011).
[CrossRef] [PubMed]

I. Cestier, A. Willinger, P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, and G. Eisenstein, “Efficient parametric interactions in a low loss GaInP photonic crystal waveguide,” Opt. Lett.36, 3936–3938 (2011).
[CrossRef] [PubMed]

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett.35, 1440–1442 (2010).
[CrossRef] [PubMed]

Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009).
[CrossRef]

A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express17, 552–557 (2009).
[CrossRef] [PubMed]

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009).
[CrossRef]

CombriÚ, S.

Corcoran, B.

B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett.36, 1728–1730 (2011).
[CrossRef] [PubMed]

C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express18, 6831–6840 (2010).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009).
[CrossRef] [PubMed]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

De Rossi, A.

S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J.4, 224–233 (2012).
[CrossRef]

I. Cestier, A. Willinger, P. Colman, S. Combrié, G. Lehoucq, A. De Rossi, and G. Eisenstein, “Efficient parametric interactions in a low loss GaInP photonic crystal waveguide,” Opt. Lett.36, 3936–3938 (2011).
[CrossRef] [PubMed]

P. Colman, I. Cestier, A. Willinger, S. Combrié, G. Lehoucq, G. Eisenstein, and A. De Rossi, “Observation of parametric gain due to four-wave mixing in dispersion engineered GaInP photonic crystal waveguides,” Opt. Lett.36, 2629–2631 (2011).
[CrossRef] [PubMed]

I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express19, 6093–6099 (2011).
[CrossRef] [PubMed]

M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express18, 21024–21029 (2010).
[CrossRef] [PubMed]

V. Eckhouse, I. Cestier, G. Eisenstein, S. Combrié, P. Colman, A. De Rossi, M. Santagiustina, C. Someda, and G. Vadalà, “Highly efficient four wave mixing in GaInP photonic crystal waveguides,” Opt. Lett.35, 1440–1442 (2010).
[CrossRef] [PubMed]

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009).
[CrossRef]

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009).
[CrossRef]

A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express17, 552–557 (2009).
[CrossRef] [PubMed]

Delaye, P.

Dubreuil, N.

Ebnali-Heidari, M.

Eckhouse, V.

Eggleton, B.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

Eggleton, B. J.

Eisenstein, G.

Etrich, C.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A81, 023820 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B77, 115124 (2008).
[CrossRef]

Frey, R.

Gao, D.

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

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Grillet, C.

C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett.36, 3413–3415 (2011).
[CrossRef] [PubMed]

C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express18, 6831–6840 (2010).
[CrossRef] [PubMed]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express17, 18340–18353 (2009).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009).
[CrossRef] [PubMed]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

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X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308 (2010).
[CrossRef]

Harada, K.

Hou, J.

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

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S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005).
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Husko, C.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009).
[CrossRef]

Ikeda, N.

Iliew, R.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A81, 023820 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B77, 115124 (2008).
[CrossRef]

Inoue, K.

Joannopoulos, J.

Johnson, S.

Kato, T.

Kivshar, Y.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A81, 023820 (2010).
[CrossRef]

Krauss, T.

C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett.36, 3413–3415 (2011).
[CrossRef] [PubMed]

J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express19, 4458–4463 (2011).
[CrossRef] [PubMed]

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

L. O’Faolain, T. White, D. O’Brien, X. Yuan, M. Settle, and T. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express15, 13129–13138 (2007).
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T. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys.40, 2666–2670 (2007).
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[CrossRef]

Kwong, D.-L.

Lalanne, P.

Lederer, F.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A81, 023820 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B77, 115124 (2008).
[CrossRef]

Lehoucq, G.

Li, J.

Liu, X.

X. Liu, “Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers,” Phys. Rev. A77, 043818 (2008).
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Love, J.

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

Lu, L.

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B81, 155115 (2010).
[CrossRef]

Marshall, G.

Matsuda, N.

Mazoyer, S.

McMillan, J.

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 257–266 (2010).
[CrossRef]

J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett.93, 251105 (2008).
[CrossRef]

McMillan, J. F.

Melloni, A.

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

Michaelis, D.

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E68, 065601–065601 (2003).
[CrossRef]

Mock, A.

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B81, 155115 (2010).
[CrossRef]

Monat, C.

B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett.36, 1728–1730 (2011).
[CrossRef] [PubMed]

C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett.36, 3413–3415 (2011).
[CrossRef] [PubMed]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express18, 6831–6840 (2010).
[CrossRef] [PubMed]

M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express17, 18340–18353 (2009).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009).
[CrossRef] [PubMed]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

Moravvej-Farshi, M.

Morichetti, F.

Moss, D.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

Moss, D. J.

Notomi, M.

O’Brien, D.

O’Brien, J.

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B81, 155115 (2010).
[CrossRef]

O’Faolain, L.

B. Corcoran, M. D. Pelusi, C. Monat, J. Li, L. O’Faolain, T. F. Krauss, and B. J. Eggleton, “Ultracompact 160Gbaud all-optical demultiplexing exploiting slow light in an engineered silicon photonic crystal waveguide,” Opt. Lett.36, 1728–1730 (2011).
[CrossRef] [PubMed]

J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express19, 4458–4463 (2011).
[CrossRef] [PubMed]

C. Xiong, C. Monat, A. Clark, C. Grillet, G. Marshall, M. Steel, J. Li, L. O’Faolain, T. Krauss, J. Rarity, and B. J. Eggleton, “Slow-light enhanced correlated photon pair generation in a silicon photonic crystal waveguide,” Opt. Lett.36, 3413–3415 (2011).
[CrossRef] [PubMed]

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009).
[CrossRef] [PubMed]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express16, 6227–6232 (2008).
[CrossRef] [PubMed]

L. O’Faolain, T. White, D. O’Brien, X. Yuan, M. Settle, and T. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express15, 13129–13138 (2007).
[CrossRef]

Oda, H.

Panoiu, N.

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 257–266 (2010).
[CrossRef]

Pelusi, M.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

Pelusi, M. D.

Pertsch, T.

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A81, 023820 (2010).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B77, 115124 (2008).
[CrossRef]

Peschel, U.

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E68, 065601–065601 (2003).
[CrossRef]

Pudo, D.

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

Ramunno, L.

S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005).
[CrossRef] [PubMed]

Rarity, J.

Rey, I.

Roosen, G.

Roy, S.

S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J.4, 224–233 (2012).
[CrossRef]

Ryasnyanskiy, A.

Sagnes, I.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

Santagiustina, M.

Schulz, S.

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

Settle, M.

Sipe, J.

S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005).
[CrossRef] [PubMed]

Snyder, A.

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

Soljacic, M.

M. Soljacic and J. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3, 211–220 (2004).
[CrossRef] [PubMed]

Someda, C.

Spasenovic, M.

Steel, M.

Takesue, H.

Taniyama, H.

Tran, Q.

Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009).
[CrossRef]

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009).
[CrossRef]

Vadala, G.

Vadalà, G.

Vy Tran, Q.

Wächter, C.

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E68, 065601–065601 (2003).
[CrossRef]

White, T.

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

L. O’Faolain, T. White, D. O’Brien, X. Yuan, M. Settle, and T. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express15, 13129–13138 (2007).
[CrossRef]

White, T. P.

Willinger, A.

Wong, C.

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 257–266 (2010).
[CrossRef]

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett.93, 251105 (2008).
[CrossRef]

Wong, C. W.

Wu, H.

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

Xiong, C.

Yariv, A.

A. Yariv, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, USA, 2007).

Yin, L.

Young, J.

S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005).
[CrossRef] [PubMed]

Yu, M.

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010).
[CrossRef] [PubMed]

J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett.93, 251105 (2008).
[CrossRef]

Yuan, X.

Zhang, X.

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

Appl. Phys. Lett. (3)

J. McMillan, M. Yu, D. Kwong, and C. Wong, “Observation of spontaneous Raman scattering in silicon slow-light photonic crystal waveguides,” Appl. Phys. Lett.93, 251105 (2008).
[CrossRef]

S. Combrié, Q. Tran, A. De Rossi, C. Husko, and P. Colman, “High quality GaInP nonlinear photonic crystals with minimized nonlinear absorption,” Appl. Phys. Lett.95, 221108 (2009).
[CrossRef]

Q. Tran, S. Combrié, P. Colman, and A. De Rossi, “Photonic crystal membrane waveguides with low insertion losses,” Appl. Phys. Lett.95, 061105 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

C. Monat, B. Corcoran, D. Pudo, M. Ebnali-Heidari, C. Grillet, M. Pelusi, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, “Slow light enhanced nonlinear optics in silicon photonic crystal waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 344–356 (2010).
[CrossRef]

N. Panoiu, J. McMillan, and C. Wong, “Theoretical analysis of pulse dynamics in silicon photonic crystal wire waveguides,” IEEE J. Sel. Top. Quantum Electron.16, 257–266 (2010).
[CrossRef]

IEEE Photon. J. (1)

S. Roy, M. Santagiustina, P. Colman, S. Combrie, and A. De Rossi, “Modeling the Dispersion of the Nonlinearity in Slow Mode Photonic Crystal Waveguides,” IEEE Photon. J.4, 224–233 (2012).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. Bao, J. Hou, H. Wu, E. Cassan, L. Chen, D. Gao, and X. Zhang, “Flat band slow light with high coupling efficiency in one-dimensional grating waveguides,” IEEE Photon. Technol. Lett.24, 7–9 (2012).
[CrossRef]

J. Opt. (1)

S. Schulz, L. O’Faolain, D. Beggs, T. White, A. Melloni, and T. Krauss, “Dispersion engineered slow light in photonic crystals: a comparison,” J. Opt.12, 104004 (2010).
[CrossRef]

J. Phys. D: Appl. Phys. (1)

T. Krauss, “Slow light in photonic crystal waveguides,” J. Phys. D: Appl. Phys.40, 2666–2670 (2007).
[CrossRef]

Nat. Mater. (1)

M. Soljacic and J. Joannopoulos, “Enhancement of nonlinear effects using photonic crystals,” Nat. Mater.3, 211–220 (2004).
[CrossRef] [PubMed]

Nat. Photonics (3)

B. Corcoran, C. Monat, C. Grillet, D. Moss, B. Eggleton, T. White, L. O’Faolain, and T. Krauss, “Green light emission in silicon through slow-light enhanced third-harmonic generation in photonic-crystal waveguides,” Nat. Photonics3, 206–210 (2009).
[CrossRef]

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. Wong, and A. De Rossi, “Temporal solitons and pulse compression in photonic crystal waveguides,” Nat. Photonics4, 862–868 (2010).
[CrossRef]

T. Baba, “Slow light in photonic crystals,” Nat. Photonics2, 465–473 (2008).
[CrossRef]

Opt. Express (15)

L. O’Faolain, T. White, D. O’Brien, X. Yuan, M. Settle, and T. Krauss, “Dependence of extrinsic loss on group velocity in photonic crystal waveguides,” Opt. Express15, 13129–13138 (2007).
[CrossRef]

J. Li, T. P. White, L. O’Faolain, A. Gomez-Iglesias, and T. F. Krauss, “Systematic design of flat band slow light in photonic crystal waveguides,” Opt. Express16, 6227–6232 (2008).
[CrossRef] [PubMed]

A. Baron, A. Ryasnyanskiy, N. Dubreuil, P. Delaye, Q. Vy Tran, S. Combrié, A. De Rossi, R. Frey, and G. Roosen, “Light localization induced enhancement of third order nonlinearities in a GaAs photonic crystal waveguide,” Opt. Express17, 552–557 (2009).
[CrossRef] [PubMed]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, “Slow light enhancement of nonlinear effects in silicon engineered photonic crystal waveguides,” Opt. Express17, 2944–2953 (2009).
[CrossRef] [PubMed]

K. Inoue, H. Oda, N. Ikeda, and K. Asakawa, “Enhanced third-order nonlinear effects in slow-light photonic-crystal slab waveguides of line-defect,” Opt. Express17, 7206–7216 (2009).
[CrossRef] [PubMed]

M. Ebnali-Heidari, C. Monat, C. Grillet, and M. Moravvej-Farshi, “A proposal for enhancing four-wave mixing in slow light engineered photonic crystal waveguides and its application to optical regeneration,” Opt. Express17, 18340–18353 (2009).
[CrossRef] [PubMed]

C. Monat, C. Grillet, B. Corcoran, D. J. Moss, B. J. Eggleton, T. P. White, and T. F. Krauss, “Investigation of phase matching for third-harmonic generation in silicon slow light photonic crystal waveguides using Fourier optics,” Opt. Express18, 6831–6840 (2010).
[CrossRef] [PubMed]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, “Observation of four-wave mixing in slow-light silicon photonic crystal waveguides,” Opt. Express18, 15484–15497 (2010).
[CrossRef] [PubMed]

M. Santagiustina, C. Someda, G. Vadala, S. Combrie, and A. De Rossi, “Theory of slow light enhanced four-wave mixing in photonic crystal waveguides,” Opt. Express18, 21024–21029 (2010).
[CrossRef] [PubMed]

C. Monat, M. Ebnali-Heidari, C. Grillet, B. Corcoran, B. Eggleton, T. White, L. O’Faolain, J. Li, and T. Krauss, “Four-wave mixing in slow light engineered silicon photonic crystal waveguides,” Opt. Express18, 22915–22927 (2010).
[CrossRef] [PubMed]

L. O’Faolain, S. Schulz, D. Beggs, T. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. Hugonin, P. Lalanne, and T. Krauss, “Loss engineered slow light waveguides,” Opt. Express18, 27627–27638 (2010).
[CrossRef]

J. Li, L. O’Faolain, I. Rey, and T. Krauss, “Four-wave mixing in photonic crystal waveguides: slow light enhancement and limitations,” Opt. Express19, 4458–4463 (2011).
[CrossRef] [PubMed]

I. Cestier, A. Willinger, V. Eckhouse, G. Eisenstein, S. CombriÚ, P. Colman, G. Lehoucq, and A. De Rossi, “Time domain switching/demultiplexing using four wave mixing in GaInP photonic crystal waveguides,” Opt. Express19, 6093–6099 (2011).
[CrossRef] [PubMed]

S. Johnson and J. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Express8, 173–190 (2001).
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N. Matsuda, T. Kato, K. Harada, H. Takesue, E. Kuramochi, H. Taniyama, and M. Notomi, “Slow light enhanced optical nonlinearity in a silicon photonic crystal coupled-resonator optical waveguide,” Opt. Express19, 19861–19874 (2011).
[CrossRef] [PubMed]

Opt. Lett. (6)

Phys. Rev. A (2)

R. Iliew, C. Etrich, T. Pertsch, F. Lederer, and Y. Kivshar, “Huge enhancement of backward second-harmonic generation with slow light in photonic crystals,” Phys. Rev. A81, 023820 (2010).
[CrossRef]

X. Liu, “Theory and experiments for multiple four-wave-mixing processes with multifrequency pumps in optical fibers,” Phys. Rev. A77, 043818 (2008).
[CrossRef]

Phys. Rev. B (4)

E. Centeno and C. Ciracì, “Theory of backward second-harmonic localization in nonlinear left-handed media,” Phys. Rev. B78, 235101 (2008).
[CrossRef]

R. Iliew, C. Etrich, T. Pertsch, and F. Lederer, “Slow-light enhanced collinear second-harmonic generation in two-dimensional photonic crystals,” Phys. Rev. B77, 115124 (2008).
[CrossRef]

A. Mock, L. Lu, and J. O’Brien, “Space group theory and Fourier space analysis of two-dimensional photonic crystal waveguides,” Phys. Rev. B81, 155115 (2010).
[CrossRef]

X. Checoury, Z. Han, and P. Boucaud, “Stimulated Raman scattering in silicon photonic crystal waveguides under continuous excitation,” Phys. Rev. B82, 041308 (2010).
[CrossRef]

Phys. Rev. E (1)

D. Michaelis, U. Peschel, C. Wächter, and A. Bräuer, “Reciprocity theorem and perturbation theory for photonic crystal waveguides,” Phys. Rev. E68, 065601–065601 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

S. Hughes, L. Ramunno, J. Young, and J. Sipe, “Extrinsic optical scattering loss in photonic crystal waveguides: role of fabrication disorder and photon group velocity,” Phys. Rev. Lett.94, 033903 (2005).
[CrossRef] [PubMed]

Other (3)

A. Yariv, Photonics: Optical Electronics in Modern Communications, 6th ed. (Oxford University Press, USA, 2007).

A. Snyder and J. Love, Optical Waveguide Theory (Springer, 1983).

G. Agrawal, Nonlinear Fiber Optics and Applications of Nonlinear Fiber Optics, 4th ed. (Elsevier Science, New York, 2007).

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Figures (11)

Fig. 1
Fig. 1

Schematic of the modified W1 PCW (the modified geometry will be shown in next figure). Green region indicates the electric-field energy density of TE mode with group index 98. The rectangle outline depicts the real unit cell Ω. The y–z plane boundary of the rectangle is the area for the calculation of time-average modal Poynting vector.

Fig. 2
Fig. 2

(a) Dispersion curves of two TE-like line-defect bands in slow light engineered Si PCW. The engineered PCW is obtained by symmetrically displacing the first rows of holes about the waveguide axis. The displacements relative to the unmodified lattice (red lines) are given by s, where shifts toward the waveguide centre are defined to be positive. (b) Group index and propagation loss as a function of the wavelength. (c) β1 and GVD β2 as a function of the wavelength.

Fig. 3
Fig. 3

Electric field profiles of Bloch waves in X–Y and Y–Z plane for (a) wavelength 1565 nm with group index 5.11, (b) wavelength 1600 nm with group index 27.72, (c) wavelength 1615 nm with group index 118.14. (d) Effective interaction areas as a function of the pump wavelength λp.

Fig. 4
Fig. 4

(a) Slow light enhanced factors τ and (b) phase mismatch Δk as a function of the pump wavelength λp.

Fig. 5
Fig. 5

The conversion efficiency as a function of the pump wavelength λp (a) for fixed waveguide length L = 300 μm and different pump power Pp with or without TPA and FCA, (b) for fixed pump power Pp = 0.3 W, Ps = 1 mW and different device lengthes with or without TPA and FCA.

Fig. 6
Fig. 6

(a) The XPM effective interaction areas as a function of Δλ. (b)The XPM enhanced factors as a function of Δλ, where Δλ = λpλs

Fig. 7
Fig. 7

The conversion efficiency as a function of the waveguide length L with Δλ = 1 nm and 3 nm for (a)λp = 1604 nm and λp = 1599 nm, (c) λp = 1579 nm and λp = 1574 nm. The conversion efficiency as a function of the Δλ for (b)λp = 1604 nm and λp = 1599 nm with L = 400 μm, (d) λp = 1579 nm and λp = 1574 nm with L = 1 mm. Where Pp = 0.1W and Ps = 1 mW are used.

Fig. 8
Fig. 8

Scheme of multiple FWM process.

Fig. 9
Fig. 9

Optical pulse evolutions along the waveguide with 100 ps pulse input pump with peak power 100 mW and 1 mW CW input signal, (a) pump wave power Pp, (b) signal wave power Ps, (c) idler wave power Pi, (d) idler wave power Pj.

Fig. 10
Fig. 10

(a) Output signal power gain for various coupled input peak pump powers. (b) Output pump and two idler pulses with input pump peak power 100 mW and signal power 1mW.

Fig. 11
Fig. 11

(a) The peak powers and (b) conversion efficiencies of two idler waves as a function of the input signal power with pump peak power 100 mW.

Equations (16)

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E 1 = e p , s , i ( x , y , z ) exp [ i k p , s , i ( ω ) x ] , H 1 = h p , s , i ( x , y , z ) exp [ i k p , s , i ( ω ) x ] .
E 2 = 1 2 a p ( x , t ) e p ( x , y , a ) exp ( i k p x ) exp ( i ω p t ) + 1 2 a s ( x , t ) e s ( x , y , z ) exp ( i k s x ) exp ( i ω s t ) + 1 2 a i ( x , t ) e i ( x , y , z ) exp ( i k i x ) exp ( i ω i t ) + c . c . , H 2 = 1 2 a p ( x , t ) h p ( x , y , z ) exp ( i k p x ) exp ( i ω p t ) + 1 2 a s ( x , t ) h s ( x , y , z ) exp ( i k s x ) exp ( i ω s t ) + 1 2 a i ( x , t ) h i ( x , y , z ) exp ( i k i x ) exp ( i ω i t ) + c . c .
S ( E 2 × H 1 * + E 1 * × H 2 ) d A = i ω V E 1 * P N L d V .
P N L = 1 2 [ P p exp ( i k p x i ω p t ) + P s exp ( i k s x i ω s t ) + P i exp ( i k i x i ω i t ) ] + c . c .
P p = 3 ε 0 4 χ [ | a p e p | 2 a p e p + 2 ( | a s e s | 2 + | a i e i | 2 ) a p e p + 2 a p * a s a i e p * e s e i exp [ i ( k s + k i 2 k p ) x ] , P s = 3 ε 0 4 χ [ | a s e s | 2 a s e s + 2 ( | a p e p | 2 + | a i e i | 2 ) a s e s + a i * a p a p e i * e p e p exp [ i ( k s + k i 2 k p ) x ] , P i = 3 ε 0 4 χ [ | a i e i | 2 a i e i + 2 ( | a p e p | 2 + | a s e s | 2 ) a i e i + a s * a p a p e s * e p e p exp [ i ( k s + k i 2 k p ) x ] .
a ˜ p ( x , ω ) x i [ k p ( ω ) k p ] a ˜ p ( x , ω ) = i ω 2 a S p exp [ i ( k p ( ω ) k p ) ] Ω e p * P N L exp [ i k p ( ω ) x ] d V , a ˜ s ( x , ω ) x i [ k s ( ω ) k s ] a ˜ s ( x , ω ) = i ω 2 a S s exp [ i ( k s ( ω ) k s ) ] Ω e s * P N L exp [ i k s ( ω ) x ] d V , a ˜ i ( x , ω ) x i [ k i ( ω ) k i ] a ˜ i ( x , ω ) = i ω 2 a S i exp [ i ( k i ( ω ) k i ) ] Ω e i * P N L exp [ i k i ( ω ) x ] d V .
k p , s , i ( ω ) = k p , s , i + ( ω ω p , s , i ) β p , s , i 1 + 1 2 ( ω ω p , s , i ) 2 β p , s , i 2 +
a p ( x , t ) x + β p 1 a p ( x , t ) t + i 2 β p 2 a p 2 ( x , t ) t 2 = [ | a p | 2 a p Ω χ e p * | e p | 2 e p d V + 2 | a s | 2 a p Ω χ e p * | e s | 2 e p d V + 2 | a i | 2 a p Ω χ e p * | e i | 2 e p d V + 2 a p * a s a i exp ( i Δ k x ) Ω χ e p * e p * e s e i d V ] i ω p 2 a S p 3 ε 0 8 , a s ( x , t ) x + β s 1 a s ( x , t ) t + i 2 β s 2 a s 2 ( x , t ) t 2 = [ | a s | 2 a s Ω χ e s * | e s | 2 e s d V + 2 | a p | 2 a s Ω χ e s * | e p | 2 e s d V + 2 | a i | 2 a s Ω χ e s * | e i | 2 e s d V + a i * a p a p exp ( i Δ k x ) Ω χ e s * e i * e p e p d V ] i ω s 2 a S s 3 ε 0 8 , a i ( x , t ) x + β i 1 a i ( x , t ) t + i 2 β i 2 a i 2 ( x , t ) t 2 = [ | a i | 2 a i Ω χ e i * | e i | 2 e i d V + 2 | a p | 2 a i Ω χ e i * | e p | 2 e i d V + 2 | a s | 2 a i Ω χ e i * | e s | 2 e i d V + a s * a p a p exp ( i Δ k x ) Ω χ e i * e s * e p e p d V ] i ω i 2 a S i 3 ε 0 8 .
A p ( x , t ) x + β p 1 A p ( x , t ) t + i 2 β p 2 A p 2 ( x , t ) t 2 = i n 2 ω p c [ ( m p ) 2 | A P | 2 A P f p p p p + 2 m p m s | A s | 2 A P f p s p s + 2 m p m i | A i | 2 A P f p i p i + 2 m p m s m i A p * A s A i exp ( i Δ k x ) f p p s i ] , A s ( x , t ) x + β s 1 A s ( x , t ) t + i 2 β s 2 A s 2 ( x , t ) t 2 = i n 2 ω s c [ ( m s ) 2 | A s | 2 A s f s s s s + 2 m s m p | A p | 2 A s f s p s p + 2 m s m i | A i | 2 A s f s i s i + m p m s m i A i * A p A p exp ( i Δ k x ) f s i p p ] , A i ( x , t ) x + β i 1 A i ( x , t ) t + i 2 β i 2 A i 2 ( x , t ) t 2 = i n 2 ω i c [ ( m i ) 2 | A i | 2 A i f i i i i + 2 m p m i | A p | 2 A i f i p i p + 2 m s m i | A s | 2 A i f i s i s + m p m s m i A s * A p A p exp ( i Δ k x ) f i s p p ] .
f 1234 = a ( ε n 1 ε n 2 ε n 3 ε n 4 ) 1 / 2 ( χ 1111 ( 3 ) ) 1 Ω χ e 1 * e 2 * e 3 e 4 d V [ Ω ε 1 ( r ) | e 1 | 2 d V Ω ε 2 ( r ) | e 2 | 2 d V Ω ε 3 ( r ) | e 3 | 2 d V Ω ε 4 ( r ) | e 4 | 2 d V ] 1 / 2 .
τ 1234 = m 1 m 2 m 3 m 4 .
A p x + β p 1 A p t + i 2 β p 2 A p 2 t 2 = i [ γ p p p p | A P | 2 A p + 2 γ p p s s | A s | 2 A p + 2 γ p p i i | A i | 2 A p + 2 γ p p s i A p * A s A i exp ( i Δ k x ) ] , A s x + β s 1 A s t + i 2 β s 2 A s 2 t 2 = i [ γ s s s s | A s | 2 A s + 2 γ p p s s | A p | 2 A s + 2 γ s s i i | A i | 2 A s + γ p p s i A i * A p A p exp ( i Δ k x ) ] , A i x + β i 1 A i t + i 2 β i 2 A i 2 t 2 = i [ γ i i i i | A i | 2 A i + 2 γ p p i i | A p | 2 A i + 2 γ s s i i | A s | 2 A i + γ p p s i A s * A p A p exp ( i Δ k x ) ] .
A eff = 1 a ( Ω | e 1 | 2 d V Ω | e 2 | 2 d V Ω | e 3 | 2 d V Ω | e 4 | 2 d V ) 1 / 2 Ω e 1 * e 2 * e 3 e 4 d V .
A p x + β p 1 A p t + i 2 β p 2 A p 2 t 2 = 1 2 ( α l p + α f c p + ζ p p p p | A p | 2 + 2 ζ p p s s | A s | 2 + 2 ζ p p i i | A i | 2 ) A p + i [ δ n f c p A p + γ p p p p | A p | 2 A P + 2 γ p p s s | A s | 2 A P + 2 γ p p i i | A i | 2 A p + 2 γ p p s i A p * A s A i exp ( i Δ k x ) ] , A s x + β s 1 A s t + i 2 β s 2 A s 2 t 2 = 1 2 ( α l s + α f c s + ζ s s s s | A s | 2 + 2 ζ p p s s | A p | 2 + 2 ζ s s i i | A i | 2 ) A s + i [ δ n f c s A s + γ s s s s | A s | 2 A s + 2 γ p p s s | A p | 2 A s + 2 γ s s i i | A i | 2 A s + γ p p s i A i * A p A p exp ( i Δ k x ) ] , A i x + β i 1 A i t + i 2 β i 2 A i 2 t 2 = 1 2 ( α l i + α f c i + ζ i i i i | A i | 2 + 2 ζ p p i i | A p | 2 + 2 ζ s s i i | A s | 2 ) A i + i [ δ n f c i A i + γ i i i i | A i | 2 A i + 2 γ p p i i | A p | 2 A i + 2 γ s s i i | A s | 2 A i + γ p p s i A s * A p A p exp ( i Δ k x ) ] .
σ q = 1.45 × 10 9 τ p p p p ( λ q / 1550 ) 2 μ m 2 , k q c = 1.35 × 10 9 τ p p p p ( λ q / 1550 ) 2 μ m 3 , N c ( t ) t = β T P A 2 h v p ( A eff p p p p ) 2 | A p | 4 N c ( t ) τ c .
A p x + β p 1 A p t + i 2 β p 2 A p 2 t 2 = 1 2 ( α l p + α f c p + ζ p p p p | A p | 2 + 2 ζ p p s s | A s | 2 ) A p + i [ δ n f c p A p + γ p p p p | A p | 2 A P + 2 γ p p s s | A s | 2 A P + 2 γ p p s i A p * A s A i exp ( i Δ k p p s i x ) + γ s s p j A j * A s A s exp ( i Δ k s s p j x ) + 2 γ s p i j A s * A i A j exp ( i Δ k s p i j x ) ] , A s x + β s 1 A s t + i 2 β s 2 A s 2 t 2 = 1 2 ( α l s + α f c s + ζ s s s s | A s | 2 + 2 ζ p p s s | A p | 2 ) A s + i [ δ n f c s A s + γ s s s s | A s | 2 A s + 2 γ p p s s | A p | 2 A s + 2 γ s s p j A s * A p A j exp ( i Δ k s s p j x ) + γ p p s i A i * A p A p exp ( i Δ k p p s i x ) + 2 γ s p i j A p * A i A j exp ( i Δ k s p i j x ) ] , A i x + β i 1 A i t + i 2 β i 2 A i 2 t 2 = 1 2 ( α l i + α f c i + 2 ζ p p i i | A p | 2 + 2 ζ s s i i | A s | 2 ) A i + i [ δ n f c i A i + 2 γ p p i i | A p | 2 A i + 2 γ s s i i | A s | 2 A i + γ p p s i A s * A p A p exp ( i Δ k p p s i x ) + 2 γ s p i j A j * A s A p exp ( i Δ k s p i j x ) ] . A j x + β j 1 A j t + i 2 β j 2 A j 2 t 2 = 1 2 ( α l i + α f c j + 2 ζ p p j j | A p | 2 + 2 ζ s s j j | A s | 2 ) A j + i [ δ n f c j A j + 2 γ p p j j | A p | 2 A j + 2 γ s s j j | A s | 2 A j + γ s s p j A s * A s A p exp ( i Δ k s s p j x ) + 2 γ s p i j A i * A s A p exp ( i Δ k s p i j x ) ] .

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