Abstract

We have studied the phenomenon of two-photon absorption in photonic nanowires doped with an ensemble of four-level nanoparticles. The photonic nanowire is made from two photonic crystals, A and B, where A is embedded in B. Photons are confined in the nanowire due to the band structure engineering of photonic crystals. It is considered that one of the resonance levels of nanoparticles lies near a photon-bound state of the nanowire. The two-photon absorption coefficient has been calculated using the time-dependent Schrödinger equation. It is found that there is an inhibition of two-photon absorption in the system. This is due to the strong coupling between nanoparticles and the nanowire.

© 2009 Optical Society of America

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  1. Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
    [CrossRef]
  2. M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300-1320 (2005).
    [CrossRef]
  3. B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
    [CrossRef]
  4. M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
    [CrossRef]
  5. N. A. Wolchover, F. Luan, A. K. George, J. C. Knight, and F. G. Omenetto, “High nonlinearity glass photonic crystal nanowires,” Opt. Express 15, 829-833 (2007).
    [CrossRef] [PubMed]
  6. N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287-2292 (1985).
    [CrossRef] [PubMed]
  7. D. Suter, The Physics of Laser-Atom Interaction (Cambridge U. Press, 1997), and references therein.
    [CrossRef]
  8. G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two-photon absorption,” Phys. Rev. Lett. 77, 1039-1942 (1996).
    [CrossRef] [PubMed]
  9. J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
    [CrossRef]
  10. D. McGloin, “Coherent effects in a driven Vee scheme,” J. Phys. B 36, 2861-2871 (2003).
    [CrossRef]
  11. J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
    [CrossRef]
  12. M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absorptive photon switching by quantum interference,” Phys. Rev. A 64, 041801 (2001).
    [CrossRef]
  13. W. Jiang, Q. Chen, Y. Zhang, and G.-C. Guo, “Optical pumping-assisted electromagnetically induced transparency,” Phys. Rev. A 73, 053804 (2006).
    [CrossRef]
  14. D. D. Yavuz, “All-optical femtosecond switch using two-photon absorption,” Phys. Rev. A 74, 053804 (2006).
    [CrossRef]
  15. G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
    [CrossRef]
  16. M. R. Singh, “Inhibition of two-photon absorption due to dipole-dipole interaction in nanoparticles,” Phys. Lett. A 372, 5083-5089 (2008).
    [CrossRef]
  17. J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. B. Meade, Photonic Crystals (Princeton U. Press, 2008).
  18. M. R. Singh, “A study of optoelectronics in photonic nanowires made from photonic crystals,” Appl. Phys. B: Lasers Opt. 93, 91-102 (2008).
    [CrossRef]
  19. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
    [CrossRef] [PubMed]
  20. V. I. Rupasov and M. R. Singh, “Quantum gap solitons and many-polariton-atom-bound states in dispersive medium and photonic bandgap,” Phys. Rev. Lett. 77, 338-341 (1996).
    [CrossRef] [PubMed]
  21. M. R. Singh and R. B. Lipson, “Optical switching in nonlinear photonic crystals lightly doped with nanostructures,” J. Phys. B 41, 015401 (2008).
    [CrossRef]
  22. D. Petrosyan and G. Kurizki, “Photon-photon correlations and entanglement in doped photonic crystals,” Phys. Rev. A 64, 23810 (2001).
    [CrossRef]
  23. P. Tran, “Optical limiting and switching of short pulses by use of a nonlinear photonic bandgap structure with a defect,” J. Opt. Soc. Am. B 14, 2589-2595 (1997).
    [CrossRef]
  24. M. R. Singh, “Dipole-dipole interaction in photonic-bandgap materials doped with nanoparticles,” Phys. Rev. A 75, 043809 (2007).
    [CrossRef]
  25. S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic bandgap material,” Phys. Rev. Lett. 76, 2484-2487 (1996).
    [CrossRef] [PubMed]
  26. M. R. Singh, “Transparency in nanophotonic quantum wires,” J. Phys. B 42, 065503 (2009).
    [CrossRef]
  27. P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455-503 (2000), and references therein.
    [CrossRef]
  28. S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43, 12772-12789 (1991).
    [CrossRef]
  29. M. R. Singh, Recent Research Activities in Chemical Physics: From Atomic Scale to Macroscale, E.Paspalakis and A.F.Terzis, eds. (Transworld Research Network, Trivandrum, 2007), chap. 5, pp. 101-165.
  30. E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency near a photonic band edge,” Phys. Rev. A 60, R33-R66 (1999).
    [CrossRef]
  31. D. G. Angelakis, E. Paspalakis, and P. L. Knight, “Coherent phenomena in photonic crystals,” Phys. Rev. A 64, 013801 (2001).
    [CrossRef]
  32. M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

2009 (1)

M. R. Singh, “Transparency in nanophotonic quantum wires,” J. Phys. B 42, 065503 (2009).
[CrossRef]

2008 (5)

M. R. Singh and R. B. Lipson, “Optical switching in nonlinear photonic crystals lightly doped with nanostructures,” J. Phys. B 41, 015401 (2008).
[CrossRef]

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
[CrossRef]

M. R. Singh, “Inhibition of two-photon absorption due to dipole-dipole interaction in nanoparticles,” Phys. Lett. A 372, 5083-5089 (2008).
[CrossRef]

M. R. Singh, “A study of optoelectronics in photonic nanowires made from photonic crystals,” Appl. Phys. B: Lasers Opt. 93, 91-102 (2008).
[CrossRef]

2007 (3)

N. A. Wolchover, F. Luan, A. K. George, J. C. Knight, and F. G. Omenetto, “High nonlinearity glass photonic crystal nanowires,” Opt. Express 15, 829-833 (2007).
[CrossRef] [PubMed]

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

M. R. Singh, “Dipole-dipole interaction in photonic-bandgap materials doped with nanoparticles,” Phys. Rev. A 75, 043809 (2007).
[CrossRef]

2006 (2)

W. Jiang, Q. Chen, Y. Zhang, and G.-C. Guo, “Optical pumping-assisted electromagnetically induced transparency,” Phys. Rev. A 73, 053804 (2006).
[CrossRef]

D. D. Yavuz, “All-optical femtosecond switch using two-photon absorption,” Phys. Rev. A 74, 053804 (2006).
[CrossRef]

2005 (3)

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[CrossRef]

M. A. Foster, A. C. Turner, M. Lipson, and A. L. Gaeta, “Nonlinear optics in photonic nanowires,” Opt. Express 16, 1300-1320 (2005).
[CrossRef]

2003 (1)

D. McGloin, “Coherent effects in a driven Vee scheme,” J. Phys. B 36, 2861-2871 (2003).
[CrossRef]

2001 (3)

M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absorptive photon switching by quantum interference,” Phys. Rev. A 64, 041801 (2001).
[CrossRef]

D. Petrosyan and G. Kurizki, “Photon-photon correlations and entanglement in doped photonic crystals,” Phys. Rev. A 64, 23810 (2001).
[CrossRef]

D. G. Angelakis, E. Paspalakis, and P. L. Knight, “Coherent phenomena in photonic crystals,” Phys. Rev. A 64, 013801 (2001).
[CrossRef]

2000 (2)

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455-503 (2000), and references therein.
[CrossRef]

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

1999 (1)

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency near a photonic band edge,” Phys. Rev. A 60, R33-R66 (1999).
[CrossRef]

1997 (1)

1996 (3)

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic bandgap material,” Phys. Rev. Lett. 76, 2484-2487 (1996).
[CrossRef] [PubMed]

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two-photon absorption,” Phys. Rev. Lett. 77, 1039-1942 (1996).
[CrossRef] [PubMed]

V. I. Rupasov and M. R. Singh, “Quantum gap solitons and many-polariton-atom-bound states in dispersive medium and photonic bandgap,” Phys. Rev. Lett. 77, 338-341 (1996).
[CrossRef] [PubMed]

1991 (1)

S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

1985 (1)

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

Agarwal, G. S.

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two-photon absorption,” Phys. Rev. Lett. 77, 1039-1942 (1996).
[CrossRef] [PubMed]

Angelakis, D. G.

D. G. Angelakis, E. Paspalakis, and P. L. Knight, “Coherent phenomena in photonic crystals,” Phys. Rev. A 64, 013801 (2001).
[CrossRef]

Bay, S.

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455-503 (2000), and references therein.
[CrossRef]

Bergman, K.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Biberman, A.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Chen, K. X.

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

Chen, Q.

W. Jiang, Q. Chen, Y. Zhang, and G.-C. Guo, “Optical pumping-assisted electromagnetically induced transparency,” Phys. Rev. A 73, 053804 (2006).
[CrossRef]

Chen, X.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Cheng, G.

J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[CrossRef]

Chou, C.-Y.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Dadap, J. I.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Du, D.

J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[CrossRef]

Foster, M. A.

Gaeta, A. L.

Gao, J. Y.

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

George, A. K.

Green, W. M. J.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Guo, G.-C.

W. Jiang, Q. Chen, Y. Zhang, and G.-C. Guo, “Optical pumping-assisted electromagnetically induced transparency,” Phys. Rev. A 73, 053804 (2006).
[CrossRef]

Guo, X.

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

Harshawardhan, W.

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two-photon absorption,” Phys. Rev. Lett. 77, 1039-1942 (1996).
[CrossRef] [PubMed]

Haus, H. A.

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

Hsieh, I-W.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Hu, X.

J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[CrossRef]

Hua, Z. Y.

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

Imoto, N.

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

Jiang, W.

W. Jiang, Q. Chen, Y. Zhang, and G.-C. Guo, “Optical pumping-assisted electromagnetically induced transparency,” Phys. Rev. A 73, 053804 (2006).
[CrossRef]

Jiang, Y.

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

Joannopoulos, J. D.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. B. Meade, Photonic Crystals (Princeton U. Press, 2008).

John, S.

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic bandgap material,” Phys. Rev. Lett. 76, 2484-2487 (1996).
[CrossRef] [PubMed]

S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

Johnson, S. G.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. B. Meade, Photonic Crystals (Princeton U. Press, 2008).

Knight, J. C.

Knight, P. L.

D. G. Angelakis, E. Paspalakis, and P. L. Knight, “Coherent phenomena in photonic crystals,” Phys. Rev. A 64, 013801 (2001).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency near a photonic band edge,” Phys. Rev. A 60, R33-R66 (1999).
[CrossRef]

Kurizki, G.

D. Petrosyan and G. Kurizki, “Photon-photon correlations and entanglement in doped photonic crystals,” Phys. Rev. A 64, 23810 (2001).
[CrossRef]

Kylstra, N. J.

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency near a photonic band edge,” Phys. Rev. A 60, R33-R66 (1999).
[CrossRef]

Lambropoulos, P.

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455-503 (2000), and references therein.
[CrossRef]

Lee, B. G.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Li, X.

J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[CrossRef]

Liphardt, J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Lipson, M.

Lipson, R. B.

M. R. Singh and R. B. Lipson, “Optical switching in nonlinear photonic crystals lightly doped with nanostructures,” J. Phys. B 41, 015401 (2008).
[CrossRef]

Liu, X.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Luan, F.

Mai, G. H.

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

McGloin, D.

D. McGloin, “Coherent effects in a driven Vee scheme,” J. Phys. B 36, 2861-2871 (2003).
[CrossRef]

Meade, R. B.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. B. Meade, Photonic Crystals (Princeton U. Press, 2008).

Nakayama, Y.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Nielsen, T. R.

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455-503 (2000), and references therein.
[CrossRef]

Nikolopoulos, G. M.

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455-503 (2000), and references therein.
[CrossRef]

Omenetto, F. G.

Onorato1, R. M.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Osgood, R. M.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Paspalakis, E.

D. G. Angelakis, E. Paspalakis, and P. L. Knight, “Coherent phenomena in photonic crystals,” Phys. Rev. A 64, 013801 (2001).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency near a photonic band edge,” Phys. Rev. A 60, R33-R66 (1999).
[CrossRef]

Pauzauskie, P. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Petrosyan, D.

D. Petrosyan and G. Kurizki, “Photon-photon correlations and entanglement in doped photonic crystals,” Phys. Rev. A 64, 23810 (2001).
[CrossRef]

Poulton, C. G.

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
[CrossRef]

Prill Sempere, L. N.

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
[CrossRef]

Quang, T.

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic bandgap material,” Phys. Rev. Lett. 76, 2484-2487 (1996).
[CrossRef] [PubMed]

Radenovic, A.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Rajiv, K.

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

Rickey, E. G.

M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absorptive photon switching by quantum interference,” Phys. Rev. A 64, 041801 (2001).
[CrossRef]

Rupasov, V. I.

V. I. Rupasov and M. R. Singh, “Quantum gap solitons and many-polariton-atom-bound states in dispersive medium and photonic bandgap,” Phys. Rev. Lett. 77, 338-341 (1996).
[CrossRef] [PubMed]

Russell, P. St. J.

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
[CrossRef]

Saykally, R. J.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Schmidt, M. A.

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
[CrossRef]

Scully, M. O.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

Sekaric, L.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Shen, J.

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

Singh, M. R.

M. R. Singh, “Transparency in nanophotonic quantum wires,” J. Phys. B 42, 065503 (2009).
[CrossRef]

M. R. Singh and R. B. Lipson, “Optical switching in nonlinear photonic crystals lightly doped with nanostructures,” J. Phys. B 41, 015401 (2008).
[CrossRef]

M. R. Singh, “Inhibition of two-photon absorption due to dipole-dipole interaction in nanoparticles,” Phys. Lett. A 372, 5083-5089 (2008).
[CrossRef]

M. R. Singh, “A study of optoelectronics in photonic nanowires made from photonic crystals,” Appl. Phys. B: Lasers Opt. 93, 91-102 (2008).
[CrossRef]

M. R. Singh, “Dipole-dipole interaction in photonic-bandgap materials doped with nanoparticles,” Phys. Rev. A 75, 043809 (2007).
[CrossRef]

V. I. Rupasov and M. R. Singh, “Quantum gap solitons and many-polariton-atom-bound states in dispersive medium and photonic bandgap,” Phys. Rev. Lett. 77, 338-341 (1996).
[CrossRef] [PubMed]

M. R. Singh, Recent Research Activities in Chemical Physics: From Atomic Scale to Macroscale, E.Paspalakis and A.F.Terzis, eds. (Transworld Research Network, Trivandrum, 2007), chap. 5, pp. 101-165.

Suter, D.

D. Suter, The Physics of Laser-Atom Interaction (Cambridge U. Press, 1997), and references therein.
[CrossRef]

Tangi, S. H.

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

Tran, P.

Turner, A. C.

Tyagi, H. K.

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
[CrossRef]

Vlasov, Y. A.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Wang, D.

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

Wang, J.

S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. B. Meade, Photonic Crystals (Princeton U. Press, 2008).

Wolchover, N. A.

Xia, F.

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

Yamamoto, Y.

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

Yan, M.

M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absorptive photon switching by quantum interference,” Phys. Rev. A 64, 041801 (2001).
[CrossRef]

Yang, P.

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Yang, S. U.

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

Yavuz, D. D.

D. D. Yavuz, “All-optical femtosecond switch using two-photon absorption,” Phys. Rev. A 74, 053804 (2006).
[CrossRef]

Zhang, Y.

W. Jiang, Q. Chen, Y. Zhang, and G.-C. Guo, “Optical pumping-assisted electromagnetically induced transparency,” Phys. Rev. A 73, 053804 (2006).
[CrossRef]

Zhang, Z. J.

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

Zhao, B.

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

Zhu, Y.

M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absorptive photon switching by quantum interference,” Phys. Rev. A 64, 041801 (2001).
[CrossRef]

Zou, J. J.

J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[CrossRef]

Zubairy, M. S.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

Appl. Phys. B: Lasers Opt. (2)

G. H. Mai, J. Shen, K. Rajiv, S. H. Tangi, Z. J. Zhang, and Z. Y. Hua, “Optimization of two-photon absorption enhancement in one-dimensional photonic crystals with defect states,” Appl. Phys. B: Lasers Opt. 80, 359-363 (2005).
[CrossRef]

M. R. Singh, “A study of optoelectronics in photonic nanowires made from photonic crystals,” Appl. Phys. B: Lasers Opt. 93, 91-102 (2008).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

B. G. Lee, X. Chen, A. Biberman, X. Liu, I-W. Hsieh, C.-Y. Chou, J. I. Dadap, F. Xia, W. M. J. Green, L. Sekaric, Y. A. Vlasov, R. M. Osgood, Jr., and K. Bergman, “Ultrahigh-bandwidth silicon photonic nanowire waveguides for on-chip networks,” IEEE Photonics Technol. Lett. 20, 398-400 (2008).
[CrossRef]

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

J. Phys. B (3)

M. R. Singh and R. B. Lipson, “Optical switching in nonlinear photonic crystals lightly doped with nanostructures,” J. Phys. B 41, 015401 (2008).
[CrossRef]

M. R. Singh, “Transparency in nanophotonic quantum wires,” J. Phys. B 42, 065503 (2009).
[CrossRef]

D. McGloin, “Coherent effects in a driven Vee scheme,” J. Phys. B 36, 2861-2871 (2003).
[CrossRef]

Nature (1)

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato1, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447, 1096-1101 (2007).
[CrossRef]

Opt. Express (2)

Phys. Lett. A (1)

M. R. Singh, “Inhibition of two-photon absorption due to dipole-dipole interaction in nanoparticles,” Phys. Lett. A 372, 5083-5089 (2008).
[CrossRef]

Phys. Rev. A (10)

J. J. Zou, X. Hu, G. Cheng, X. Li, and D. Du, “Inhibition of two-photon absorption in a three-level system with a pair of bichromatic fields,” Phys. Rev. A 72, 055802 (2005).
[CrossRef]

M. Yan, E. G. Rickey, and Y. Zhu, “Observation of absorptive photon switching by quantum interference,” Phys. Rev. A 64, 041801 (2001).
[CrossRef]

W. Jiang, Q. Chen, Y. Zhang, and G.-C. Guo, “Optical pumping-assisted electromagnetically induced transparency,” Phys. Rev. A 73, 053804 (2006).
[CrossRef]

D. D. Yavuz, “All-optical femtosecond switch using two-photon absorption,” Phys. Rev. A 74, 053804 (2006).
[CrossRef]

N. Imoto, H. A. Haus, and Y. Yamamoto, “Quantum nondemolition measurement of the photon number via the optical Kerr effect,” Phys. Rev. A 32, 2287-2292 (1985).
[CrossRef] [PubMed]

J. Y. Gao, S. U. Yang, D. Wang, X. Guo, K. X. Chen, Y. Jiang, and B. Zhao, “Electromagnetically induced inhibition of two-photon absorption in sodium vapor,” Phys. Rev. A 61, 023401 (2000).
[CrossRef]

D. Petrosyan and G. Kurizki, “Photon-photon correlations and entanglement in doped photonic crystals,” Phys. Rev. A 64, 23810 (2001).
[CrossRef]

M. R. Singh, “Dipole-dipole interaction in photonic-bandgap materials doped with nanoparticles,” Phys. Rev. A 75, 043809 (2007).
[CrossRef]

E. Paspalakis, N. J. Kylstra, and P. L. Knight, “Transparency near a photonic band edge,” Phys. Rev. A 60, R33-R66 (1999).
[CrossRef]

D. G. Angelakis, E. Paspalakis, and P. L. Knight, “Coherent phenomena in photonic crystals,” Phys. Rev. A 64, 013801 (2001).
[CrossRef]

Phys. Rev. B (2)

S. John and J. Wang, “Quantum optics of localized light in a photonic band gap,” Phys. Rev. B 43, 12772-12789 (1991).
[CrossRef]

M. A. Schmidt, L. N. Prill Sempere, H. K. Tyagi, C. G. Poulton, and P. St. J. Russell, “Waveguiding and plasmon resonances in two-dimensional photonic lattices of gold and silver nanowires,” Phys. Rev. B 77, 033417 (2008).
[CrossRef]

Phys. Rev. Lett. (4)

G. S. Agarwal and W. Harshawardhan, “Inhibition and enhancement of two-photon absorption,” Phys. Rev. Lett. 77, 1039-1942 (1996).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059-2062 (1987).
[CrossRef] [PubMed]

V. I. Rupasov and M. R. Singh, “Quantum gap solitons and many-polariton-atom-bound states in dispersive medium and photonic bandgap,” Phys. Rev. Lett. 77, 338-341 (1996).
[CrossRef] [PubMed]

S. John and T. Quang, “Resonant nonlinear dielectric response in a photonic bandgap material,” Phys. Rev. Lett. 76, 2484-2487 (1996).
[CrossRef] [PubMed]

Rep. Prog. Phys. (1)

P. Lambropoulos, G. M. Nikolopoulos, T. R. Nielsen, and S. Bay, “Fundamental quantum optics in structured reservoirs,” Rep. Prog. Phys. 63, 455-503 (2000), and references therein.
[CrossRef]

Other (4)

M. R. Singh, Recent Research Activities in Chemical Physics: From Atomic Scale to Macroscale, E.Paspalakis and A.F.Terzis, eds. (Transworld Research Network, Trivandrum, 2007), chap. 5, pp. 101-165.

M. O. Scully and M. S. Zubairy, Quantum Optics (Cambridge U. Press, 1997).

J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. B. Meade, Photonic Crystals (Princeton U. Press, 2008).

D. Suter, The Physics of Laser-Atom Interaction (Cambridge U. Press, 1997), and references therein.
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of the photonic nanowire is plotted (top view). The nanowire is made from A and B photonic crystals, where A is embeded B. The A photonic crystal is doped with noninteracting nanoparticles.

Fig. 2
Fig. 2

Schematic energy diagram for the photonic nanowire. Bandgaps for crystals A and B are denoted by ϵ a g and ϵ b g , respectively. Conduction and valence well depths are denoted as Δ ϵ c and Δ ϵ v , respectively. Photons have bound states within Δ ϵ c and Δ ϵ v .

Fig. 3
Fig. 3

Schematic diagram of a four-level nanoparticle where four levels are denoted by | a , | b , | c , and | d . A probe field monitors the transition a b , and the pump field is applied between levels | b and | c . It is considered that transition b d couples to one of the photonic bound states of the nanowire. This is called the DBP interaction.

Fig. 4
Fig. 4

Plot of TPA coefficient α α 0 versus the pump field detuning δ b c . Solid, dashed, and dash-dotted curves are plotted for δ a b = 0 , δ a b = 1 , and δ a b = + 1 , respectively. The DBP interaction is absent.

Fig. 5
Fig. 5

Three-dimension plot of TPA coefficient α α 0 versus probe field detuning δ a b and pump field detuning δ b c . The DBP interaction is absent.

Fig. 6
Fig. 6

Three-dimension plot of TPA coefficient α α 0 versus probe field detuning δ a b and two-photon detuning δ a c . The DBP interaction is absent.

Fig. 7
Fig. 7

Three-dimension plot of TPA coefficient α α 0 versus probe field detuning δ a b and two-photon detuning δ a c . The DBP interaction is present.

Fig. 8
Fig. 8

Three-dimension plot of TPA coefficient α α 0 versus two-photon detuning δ a c and DBP detuning δ m p . The DBP interaction is present.

Equations (29)

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H F = Ω a b σ a b + e i ( ε b a ε p ) t Ω b c σ b c + e i ( ε c b ε c ) t + h c ,
H DBP = i = b , c , d ε i a σ i a z + m , p , k z ε m p k z p m p k z + p m p k z m , p , k z ( ε m p k z μ b d 2 2 ϵ 0 2 A d z ) p m p k z σ b d + e i ( ε b d ε m p k z ) t + h c ,
F A ( ε m p k z ) = k z 2 + k m 2 + k p 2 ,
F A ( ε m p k z ) = 1 L a arccos [ ( n a s + 1 ) 2 4 n a s cos ( 4 ε m p k z n a s r a c ) ( n a s 1 ) 2 4 n a s ] ,
k m tan ( k m d x m π 2 ) = F A 2 ( ϵ b c ) F B 2 ( ϵ b c ) k m 2 ,
k p tan ( k p d y p π 2 ) = F A 2 ( ϵ b c ) F B 2 ( ϵ b c ) k p 2 ,
F B ( ε k ) = 1 L b arccos [ ( n b s + 1 ) 2 4 n b s cos ( 4 ε k n b s r b c ) ( n b s 1 ) 2 4 n b s ] ,
| ψ ( t ) = i = a , b , c c i ( t ) | φ i + m p k z c d k ( t ) | φ d k z m p ,
α = | c c ( t ) | 2 .
d c a d t = i ( μ a b E p 2 ) c b ,
d c b d t = [ ( i δ a b γ b ) c b + i ( μ a b E a 2 ) c a + i ( μ b c E c 2 ) c c + i m , p , k z g ( ε m p k z ) c d k z m p ] ,
d c c d t = ( i δ a b + i δ b c γ c ) c c + i ( μ b c E c 2 ) c b ,
d c d k z m p d t = i [ δ a b i ( ε m p k z ε b d ) γ d ] c d k z m p + i g * ( ε m p k z ) c b .
α = | Ω a Ω b [ γ c i ( δ a b + δ b c ) ] [ γ b i δ a b + i Ξ DBI + i Ξ c ] | 2 ,
Ξ c = | Ω c | 2 ( δ a b + δ b c ) + i γ c ,
Ξ DBI = m , p d ε m p k z ( ρ ( ε m p k z ) | g ( ε m p k z ) | 2 ( δ a b δ m p ) + i γ d ) .
δ a b = ε p ε b a ,
δ b c = ε c ε c b ,
δ m p = ε m p ε b d .
ρ ( ε m p k z ) = m p d z ς ( ε m p ) 2 π ε m p k z ε m p Θ ( ε m p k z ε n m ) ,
ς ( ε m p ) = F A ( ε m p ) ( n a s + 1 ) 2 ( 2 r a ) sin ( 4 ε m p n a s r a c ) L a sin [ F A ( ε m p ) L a ] ,
Ξ DBI = m , p Λ m p i γ d + ( δ a b δ m p ) ,
Λ m p = γ 0 ( 3 π 3 c 3 2 ε c b 2 A ) ς ( ε m p ) [ sin ( F A L a ) ] 1 2 ,
α = α 0 [ γ d 2 + ( δ a b + δ b c ) 2 ] [ ( γ b Γ DBI Γ c ) 2 + ( δ b a + Δ DBI + Δ c ) 2 ] ,
Δ c = | Ω c | 2 ( δ a b + δ b c ) ( δ a b + δ b c ) 2 + γ c 2 ,
Γ c = | Ω c | 2 γ c ( δ a b + δ b c ) 2 + γ c 2 .
Δ DBI = m , p Λ m p 2 [ ( δ a b δ m p ) 2 + γ d 2 ] 1 2 δ m p [ ( δ a b δ m p ) 2 + γ d 2 ] 1 2 ,
Γ DBI = m , p Λ m p 2 [ ( δ a b δ m p ) 2 + γ d 2 ] 1 2 + δ m p [ ( δ a b δ m p ) 2 + γ d 2 ] 1 2 .
α = α 0 [ γ d 2 + ( δ a b + δ b c ) 2 ] [ ( γ 2 Γ r ) 2 + ( δ b a + Δ r ) 2 ] .

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