Abstract

We examine the effects of multiphoton absorption, free carriers, and disorder-induced linear scattering in slow-light photonic crystal waveguides. We derive an analytic formulation for self-phase modulation including the group velocity scaling of the nonlinear phase shift in materials limited by three-photon absorption as a representative nonlinear process. We investigate the role of free carriers and derive an approximate critical intensity at which these effects begin to strongly modify the optical field. This critical intensity is employed to determine an optimal group index for the self-phase modulation in the slow-light devices. These observations are confirmed with numerical modeling.

© 2011 Optical Society of America

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  1. T. Baba, Nat. Photon. 2, 465 (2008).
    [CrossRef]
  2. M. Soljačić and J. D. Joannopoulos, Nat. Mater. 3, 211(2004).
    [CrossRef] [PubMed]
  3. P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
    [CrossRef]
  4. J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Opt. Express 18, 15484 (2010).
    [CrossRef] [PubMed]
  5. H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
    [CrossRef]
  6. J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Appl. Phys. Lett. 93, 251105 (2008).
    [CrossRef]
  7. B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
    [CrossRef]
  8. C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, Opt. Express 17, 2944 (2009).
    [CrossRef] [PubMed]
  9. C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
    [CrossRef]
  10. A. D. Bristow, N. Rotenberg, and H. van Driel, Appl. Phys. Lett. 90, 191104 (2007).
    [CrossRef]
  11. X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, Nat. Photon. 4, 557 (2010).
    [CrossRef]
  12. L. Yin and G. Agrawal, Opt. Lett. 32, 2031 (2007).
    [CrossRef] [PubMed]
  13. N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
    [CrossRef]
  14. L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, Opt. Express 18, 27627 (2010).
    [CrossRef]
  15. M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
    [CrossRef] [PubMed]
  16. C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
    [CrossRef]

2010 (5)

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Opt. Express 18, 15484 (2010).
[CrossRef] [PubMed]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, Nat. Photon. 4, 557 (2010).
[CrossRef]

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

L. O’Faolain, S. A. Schulz, D. M. Beggs, T. P. White, M. Spasenović, L. Kuipers, F. Morichetti, A. Melloni, S. Mazoyer, J. P. Hugonin, P. Lalanne, and T. F. Krauss, Opt. Express 18, 27627 (2010).
[CrossRef]

2009 (5)

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
[CrossRef]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, Opt. Express 17, 2944 (2009).
[CrossRef] [PubMed]

C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
[CrossRef]

2008 (3)

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Appl. Phys. Lett. 93, 251105 (2008).
[CrossRef]

T. Baba, Nat. Photon. 2, 465 (2008).
[CrossRef]

2007 (2)

L. Yin and G. Agrawal, Opt. Lett. 32, 2031 (2007).
[CrossRef] [PubMed]

A. D. Bristow, N. Rotenberg, and H. van Driel, Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

2004 (1)

M. Soljačić and J. D. Joannopoulos, Nat. Mater. 3, 211(2004).
[CrossRef] [PubMed]

Agrawal, G.

Asakawa, K.

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

Baba, T.

T. Baba, Nat. Photon. 2, 465 (2008).
[CrossRef]

Beggs, D. M.

Bristow, A. D.

A. D. Bristow, N. Rotenberg, and H. van Driel, Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Colman, P.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

Combrié, S.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
[CrossRef]

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

Corcoran, B.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
[CrossRef]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, Opt. Express 17, 2944 (2009).
[CrossRef] [PubMed]

De Rossi, A.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
[CrossRef]

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

Ebnali-Heidari, M.

Eggleton, B.

Eggleton, B. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
[CrossRef]

Gabet, R.

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

Green, W. M. J.

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, Nat. Photon. 4, 557 (2010).
[CrossRef]

Grillet, C.

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, Opt. Express 17, 2944 (2009).
[CrossRef] [PubMed]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
[CrossRef]

Hughes, S.

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

Hugonin, J. P.

Husko, C.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

Ikeda, N.

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

Inoue, K.

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

Jaouen, Y.

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

Joannopoulos, J. D.

M. Soljačić and J. D. Joannopoulos, Nat. Mater. 3, 211(2004).
[CrossRef] [PubMed]

Krauss, T. F.

Kuipers, L.

Kwong, D.-L.

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Opt. Express 18, 15484 (2010).
[CrossRef] [PubMed]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Appl. Phys. Lett. 93, 251105 (2008).
[CrossRef]

Lalanne, P.

Liu, X.

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, Nat. Photon. 4, 557 (2010).
[CrossRef]

Mazoyer, S.

McMillan, J. F.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Opt. Express 18, 15484 (2010).
[CrossRef] [PubMed]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Appl. Phys. Lett. 93, 251105 (2008).
[CrossRef]

Melloni, A.

Monat, C.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
[CrossRef]

C. Monat, B. Corcoran, M. Ebnali-Heidari, C. Grillet, B. Eggleton, T. White, L. O’Faolain, and T. F. Krauss, Opt. Express 17, 2944 (2009).
[CrossRef] [PubMed]

Morichetti, F.

Moss, D. J.

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
[CrossRef]

O’Faolain, L.

Oda, H.

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

Osgood, R. M.

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, Nat. Photon. 4, 557 (2010).
[CrossRef]

Panoiu, N. C.

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

Patterson, M.

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

Raineri, F.

C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

Rotenberg, N.

A. D. Bristow, N. Rotenberg, and H. van Driel, Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Sagnes, I.

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

Schulz, S. A.

Soljacic, M.

M. Soljačić and J. D. Joannopoulos, Nat. Mater. 3, 211(2004).
[CrossRef] [PubMed]

Spasenovic, M.

Sugimoto, Y.

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

Tran, Q.

C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
[CrossRef]

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

van Driel, H.

A. D. Bristow, N. Rotenberg, and H. van Driel, Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

Vlasov, Y. A.

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, Nat. Photon. 4, 557 (2010).
[CrossRef]

White, T.

White, T. P.

Wong, C. W.

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Opt. Express 18, 15484 (2010).
[CrossRef] [PubMed]

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

C. Husko, S. Combrié, Q. Tran, F. Raineri, C. W. Wong, and A. De Rossi, Opt. Express 17, 22442 (2009).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Appl. Phys. Lett. 93, 251105 (2008).
[CrossRef]

Yamanaka, A.

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

Yin, L.

Yu, M.

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Opt. Express 18, 15484 (2010).
[CrossRef] [PubMed]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Appl. Phys. Lett. 93, 251105 (2008).
[CrossRef]

Appl. Phys. Lett. (4)

H. Oda, K. Inoue, A. Yamanaka, N. Ikeda, Y. Sugimoto, and K. Asakawa, Appl. Phys. Lett. 93, 051114 (2008).
[CrossRef]

J. F. McMillan, M. Yu, D.-L. Kwong, and C. W. Wong, Appl. Phys. Lett. 93, 251105 (2008).
[CrossRef]

A. D. Bristow, N. Rotenberg, and H. van Driel, Appl. Phys. Lett. 90, 191104 (2007).
[CrossRef]

C. Husko, A. De Rossi, S. Combrié, Q. Tran, F. Raineri, and C. W. Wong, Appl. Phys. Lett. 94, 021111 (2009).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (1)

N. C. Panoiu, J. F. McMillan, and C. W. Wong, IEEE J. Sel. Top. Quantum Electron. 16, 257 (2010).
[CrossRef]

Nat. Mater. (1)

M. Soljačić and J. D. Joannopoulos, Nat. Mater. 3, 211(2004).
[CrossRef] [PubMed]

Nat. Photon. (4)

P. Colman, C. Husko, S. Combrié, I. Sagnes, C. W. Wong, and A. De Rossi, Nat. Photon. 4, 862 (2010).
[CrossRef]

X. Liu, R. M. Osgood, Y. A. Vlasov, and W. M. J. Green, Nat. Photon. 4, 557 (2010).
[CrossRef]

B. Corcoran, C. Monat, C. Grillet, D. J. Moss, B. J. Eggleton, T. P. White, L. O’Faolain, and T. F. Krauss, Nat. Photon. 3, 206 (2009).
[CrossRef]

T. Baba, Nat. Photon. 2, 465 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (1)

Phys. Rev. Lett. (1)

M. Patterson, S. Hughes, S. Combrié, Q. Tran, A. De Rossi, R. Gabet, and Y. Jaouen, Phys. Rev. Lett. 102, 253903(2009).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

(a) Phase shift, ϕ, as a function of I o for (i) GaInP and (ii) Si. The straight dashed lines are α only, while the curves shows ϕ impacted by TPA or ThPA. (b) Effective lengths, L eff ( TPA ) [ Si ] and L eff ( ThPA ) [ GaInP ] versus I o with L eff ( lin. ) shown (dashed black line) as reference.

Fig. 2
Fig. 2

(a) Critical intensity as a function of n g . Boxed regions indicated recent slow-light SPM measurement ranges [8, 9]. (b) Estimated minimum critical length L crit , min to achieve ϕ = π as a function of n g .

Fig. 3
Fig. 3

(a) Phase shift (π) versus delay for P = 4 W at n g = 20 for GaInP, solid, and Si, dashed. (b) Spectra corresponding to panel (a). (c) Phase shift ϕ versus n g for a fixed length ( L = 1 mm ) and variable scattering loss, α eff in GaInP. The critical intensity I c (Fig. 2), γ eff , and α eff scale with group index. The peak in the various curves demonstrate an optimal n g shifting toward smaller n g for increasing α eff . (d) Same as (c), except with α eff = 10 dB / cm and variable L.

Equations (4)

Equations on this page are rendered with MathJax. Learn more.

E z = α 2 E + i k 0 n 2 | E | 2 E α 3 2 | E | 4 E + ( i k o d n d N σ 2 ) N c E ,
I out ( L , I 0 , t ) = I 0 e α L [ 1 + 2 α 3 I 0 2 L 3 eff ] 1 / 2 ,
ϕ ( L , I 0 , t ) = k o n 2 α α 3 arctan [ γ ] , γ = 2 L 3 eff I o α α 3 1 + exp ( α L ) [ 1 + 2 α 3 I o 2 L 3 eff ] 1 / 2 ,
I c = n 2 3 ω α 3 T 0 d n d N 1 ( n g n 0 ) .

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