Abstract

We consider spontaneous four-wave mixing in a microring resonator, presenting photon-pair generation rates and biphoton wave functions. We show how generation rates can be simply predicted from the performance of the device in the classical regime and that a wide variety of biphoton wave functions can be achieved by varying the pump pulse duration.

© 2010 Optical Society of America

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  1. J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, Opt. Express 14, 12388 (2006).
    [CrossRef] [PubMed]
  2. A. Hayat, P. Ginzburg, and M. Orenstein, Nat. Photonics 2, 238 (2008).
    [CrossRef]
  3. K.-I. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-I. Itabashi, Opt. Express 16, 20368 (2008).
    [CrossRef] [PubMed]
  4. S. Celemmen, K. Phan Huy, W. Bogaerts, R. G. Baets, Ph. Emplit, and S. Massar, Opt. Express 17, 16558 (2009).
    [CrossRef]
  5. X. Caillet, V. Berger, G. Leo, and S. Ducci, J. Mod. Opt. 56, 232 (2009).
    [CrossRef]
  6. A. Thomas, H. Herrmann, and W. Sohler, “Novel source of polarization entangled photon pairs using a PPLN waveguide with interlaced domains,” presented at ECIO 2010, Cambridge, UK, 7–9 April 2010.
  7. We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
    [CrossRef]
  8. J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
    [CrossRef]
  9. L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
    [CrossRef]
  10. Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, Opt. Lett. 32, 826 (2007).
    [CrossRef] [PubMed]
  11. Z. Yang and J. E. Sipe, Opt. Lett. 32, 3296 (2007).
    [CrossRef] [PubMed]
  12. P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, Opt. Lett. 25, 554 (2000).
    [CrossRef]
  13. Q. Lin, F. Yaman, and G. P. Agrawal, Phys. Rev. A 75, 023803 (2007).
    [CrossRef]
  14. E. Brainis, Phys. Rev. A 79, 023840 (2009).
    [CrossRef]
  15. A. Gondarenko, J. S. Levy, and M. Lipson, Opt. Express 17, 11366 (2009).
    [CrossRef] [PubMed]
  16. Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
    [CrossRef]
  17. Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
    [CrossRef]
  18. C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
    [CrossRef] [PubMed]
  19. Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).
    [CrossRef]

2010 (2)

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

2009 (5)

S. Celemmen, K. Phan Huy, W. Bogaerts, R. G. Baets, Ph. Emplit, and S. Massar, Opt. Express 17, 16558 (2009).
[CrossRef]

X. Caillet, V. Berger, G. Leo, and S. Ducci, J. Mod. Opt. 56, 232 (2009).
[CrossRef]

E. Brainis, Phys. Rev. A 79, 023840 (2009).
[CrossRef]

A. Gondarenko, J. S. Levy, and M. Lipson, Opt. Express 17, 11366 (2009).
[CrossRef] [PubMed]

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

2008 (4)

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

A. Hayat, P. Ginzburg, and M. Orenstein, Nat. Photonics 2, 238 (2008).
[CrossRef]

K.-I. Harada, H. Takesue, H. Fukuda, T. Tsuchizawa, T. Watanabe, K. Yamada, Y. Tokura, and S.-I. Itabashi, Opt. Express 16, 20368 (2008).
[CrossRef] [PubMed]

2007 (3)

2006 (1)

2004 (1)

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef] [PubMed]

2000 (1)

1991 (1)

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).
[CrossRef]

Absil, P. P.

Agrawal, G. P.

Q. Lin, F. Yaman, and G. P. Agrawal, Phys. Rev. A 75, 023803 (2007).
[CrossRef]

Aitchison, J. S.

Baets, R. G.

Berger, V.

X. Caillet, V. Berger, G. Leo, and S. Ducci, J. Mod. Opt. 56, 232 (2009).
[CrossRef]

Bogaerts, W.

Brainis, E.

E. Brainis, Phys. Rev. A 79, 023840 (2009).
[CrossRef]

Bristow, A. D.

Caillet, X.

X. Caillet, V. Berger, G. Leo, and S. Ducci, J. Mod. Opt. 56, 232 (2009).
[CrossRef]

Celemmen, S.

Chak, P.

Chen, Y.

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

Cho, P. S.

Chu, S.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Citrin, D. S.

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

Ducci, S.

X. Caillet, V. Berger, G. Leo, and S. Ducci, J. Mod. Opt. 56, 232 (2009).
[CrossRef]

Duchesne, D.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Eberly, J. H.

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef] [PubMed]

Emplit, Ph.

Feng, J.

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

Ferrera, M.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Foster, M. A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, Opt. Express 14, 12388 (2006).
[CrossRef] [PubMed]

Fukuda, H.

Gaeta, A. L.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

J. E. Sharping, K. F. Lee, M. A. Foster, A. C. Turner, B. S. Schmidt, M. Lipson, A. L. Gaeta, and P. Kumar, Opt. Express 14, 12388 (2006).
[CrossRef] [PubMed]

Ginzburg, P.

A. Hayat, P. Ginzburg, and M. Orenstein, Nat. Photonics 2, 238 (2008).
[CrossRef]

Gondarenko, A.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

A. Gondarenko, J. S. Levy, and M. Lipson, Opt. Express 17, 11366 (2009).
[CrossRef] [PubMed]

Harada, K.-I.

Hayat, A.

A. Hayat, P. Ginzburg, and M. Orenstein, Nat. Photonics 2, 238 (2008).
[CrossRef]

Herrmann, H.

A. Thomas, H. Herrmann, and W. Sohler, “Novel source of polarization entangled photon pairs using a PPLN waveguide with interlaced domains,” presented at ECIO 2010, Cambridge, UK, 7–9 April 2010.

Ho, P.-T.

Hryniewicz, J. V.

Itabashi, S.-I.

Iyer, R.

Joneckis, L. G.

Kim, Y. S.

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).
[CrossRef]

Kumar, P.

Law, C. K.

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef] [PubMed]

Lee, K. F.

Leo, G.

X. Caillet, V. Berger, G. Leo, and S. Ducci, J. Mod. Opt. 56, 232 (2009).
[CrossRef]

Levy, J. S.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

A. Gondarenko, J. S. Levy, and M. Lipson, Opt. Express 17, 11366 (2009).
[CrossRef] [PubMed]

Lin, Q.

Q. Lin, F. Yaman, and G. P. Agrawal, Phys. Rev. A 75, 023803 (2007).
[CrossRef]

Lipson, M.

Liscidini, M.

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Little, B. E.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

P. P. Absil, J. V. Hryniewicz, B. E. Little, P. S. Cho, R. A. Wilson, L. G. Joneckis, and P.-T. Ho, Opt. Lett. 25, 554 (2000).
[CrossRef]

Massar, S.

Morandotti, R.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Moss, D. J.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Noz, M. E.

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).
[CrossRef]

Orenstein, M.

A. Hayat, P. Ginzburg, and M. Orenstein, Nat. Photonics 2, 238 (2008).
[CrossRef]

Phan Huy, K.

Razzari, L.

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Schmidt, B. S.

Sharping, J. E.

Sipe, J. E.

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, Opt. Lett. 32, 826 (2007).
[CrossRef] [PubMed]

Z. Yang and J. E. Sipe, Opt. Lett. 32, 3296 (2007).
[CrossRef] [PubMed]

Smirl, A. L.

Sohler, W.

A. Thomas, H. Herrmann, and W. Sohler, “Novel source of polarization entangled photon pairs using a PPLN waveguide with interlaced domains,” presented at ECIO 2010, Cambridge, UK, 7–9 April 2010.

Summers, C. J.

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

Takesue, H.

Thomas, A.

A. Thomas, H. Herrmann, and W. Sohler, “Novel source of polarization entangled photon pairs using a PPLN waveguide with interlaced domains,” presented at ECIO 2010, Cambridge, UK, 7–9 April 2010.

Tokura, Y.

Tsuchizawa, T.

Turner, A. C.

Turner-Foster, A. C.

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

van Driel, H. M.

Watanabe, T.

Wilson, R. A.

Yamada, K.

Yaman, F.

Q. Lin, F. Yaman, and G. P. Agrawal, Phys. Rev. A 75, 023803 (2007).
[CrossRef]

Yang, Z.

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

Z. Yang and J. E. Sipe, Opt. Lett. 32, 3296 (2007).
[CrossRef] [PubMed]

Z. Yang, P. Chak, A. D. Bristow, H. M. van Driel, R. Iyer, J. S. Aitchison, A. L. Smirl, and J. E. Sipe, Opt. Lett. 32, 826 (2007).
[CrossRef] [PubMed]

Yu, J.

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

Zhou, Z.

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

J. Micro./Nanolith. MEMS MOEMS (1)

Y. Chen, J. Feng, Z. Zhou, J. Yu, C. J. Summers, and D. S. Citrin, J. Micro./Nanolith. MEMS MOEMS 8, 043060 (2009).
[CrossRef]

J. Mod. Opt. (1)

X. Caillet, V. Berger, G. Leo, and S. Ducci, J. Mod. Opt. 56, 232 (2009).
[CrossRef]

Nat. Photonics (4)

A. Hayat, P. Ginzburg, and M. Orenstein, Nat. Photonics 2, 238 (2008).
[CrossRef]

We consider a single-channel version of the ring structure used in M. Ferrera, L. Razzari, D. Duchesne, R. Morandotti, Z. Yang, M. Liscidini, J. E. Sipe, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 2, 737 (2008).
[CrossRef]

J. S. Levy, A. Gondarenko, M. A. Foster, A. C. Turner-Foster, A. L. Gaeta, and M. Lipson, Nat. Photonics 4, 37 (2010).
[CrossRef]

L. Razzari, D. Duchesne, M. Ferrera, R. Morandotti, S. Chu, B. E. Little, and D. J. Moss, Nat. Photonics 4, 41 (2010).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. A (3)

Q. Lin, F. Yaman, and G. P. Agrawal, Phys. Rev. A 75, 023803 (2007).
[CrossRef]

E. Brainis, Phys. Rev. A 79, 023840 (2009).
[CrossRef]

Z. Yang, M. Liscidini, and J. E. Sipe, Phys. Rev. A 77, 033808 (2008).
[CrossRef]

Phys. Rev. Lett. (1)

C. K. Law and J. H. Eberly, Phys. Rev. Lett. 92, 127903 (2004).
[CrossRef] [PubMed]

Other (2)

Y. S. Kim and M. E. Noz, Phase Space Picture of Quantum Mechanics (World Scientific, 1991).
[CrossRef]

A. Thomas, H. Herrmann, and W. Sohler, “Novel source of polarization entangled photon pairs using a PPLN waveguide with interlaced domains,” presented at ECIO 2010, Cambridge, UK, 7–9 April 2010.

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Figures (2)

Fig. 1
Fig. 1

(a) Schematic of a single-channel ring resonator. (b) Cross section of the channel.

Fig. 2
Fig. 2

Biphoton probability density for a pump pulse centered at N P = 310 with (a) T = 5 ns and (b) T = 5 ps . We consider N = 311 and N ¯ = 309 .

Equations (10)

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

H ^ ch = μ { ω μ d z ψ ^ μ ( z ) ψ ^ μ ( z ) + i 2 v μ d z ( ψ ^ μ ( z ) z ψ ^ μ ( z ) h . c . ) } , H ^ cp = 2 π μ { c μ b ^ μ ψ ^ μ ( 0 ) + h . c . } , H ^ R = μ ω μ b ^ μ b ^ μ μ 1 , μ 2 , μ 3 , μ 4 S μ 1 μ 2 μ 3 μ 4 b ^ μ 1 b ^ μ 2 b ^ μ 3 b ^ μ 4 ,
ϕ P ( k ) = exp ( ( k k N P ) 2 / ( 4 Δ k 2 ) ) / ( 2 π Δ k ) 1 / 2 ,
2 | β N N ¯ | 2 = 4 γ 2 2 ω N ω N ¯ | α | 4 Ω N P 2 Ω N Ω N ¯ v N P 4 v N v N ¯ L 2 π 2 J N N ¯ ,
J N N ¯ = d k 1 d k 2 | d k ϕ P ( k ) ϕ P ( Δ ω ( k 1 , k 2 ) / v N P k ) ( Δ Ω N P ( k ) ) 1 Δ Ω N ( k 1 ) Δ Ω N ¯ ( k 2 ) Δ Ω N P ( Δ ω ( k 1 , k 2 ) / v N P k ) | 2 ,
| β ϕ N N ¯ ( k 1 , k 2 ) / β N N ¯ | 2 = e ( Δ ω ( k 1 , k 2 ) / v N P 2 k N P ) 2 / ( 4 Δ k 2 ) J N N ¯ Ω N P 4 | Δ Ω N ( k 1 ) | 2 | Δ Ω N ¯ ( k 2 ) | 2 ,
J N N ¯ = 2 Δ k π 3 / 2 ( Ω N + Ω N ¯ ) v N P ( v N v N ¯ Ω N Ω N ¯ Ω N P 4 ) 1 ( Δ ω ( k N , k N ¯ ) 2 v N P k N P ) 2 + ( Ω N + Ω N ¯ ) 2 , whereas for Ω N P T 1 | β ϕ N N ¯ ( k 1 , k 2 ) / β N N ¯ | 2 = 2 π ( v N P 2 ( Δ ω ( k 1 , k 2 ) / v N P 2 k N P ) 2 + 4 Ω N P 2 ) 1 J N N ¯ Δ k 2 v N P 2 | Δ Ω N ( k 1 ) | 2 | Δ Ω N ¯ ( k 2 ) | 2 , with J N N ¯ = π 3 ( Ω N + Ω N ¯ + 2 Ω N P ) ( Δ k 2 v N v N ¯ v N P 2 Ω N Ω N ¯ Ω N P ) 1 ( Δ ω ( k N , k N ¯ ) 2 v N P k N P ) 2 + ( Ω N + Ω N ¯ + 2 Ω N P ) 2 .
P N ¯ = ( γ P N P L ) 2 | F N | 2 | F N ¯ | 2 | F N P | 4 P N ,
2 | β N N ¯ | 2 = ( γ P N P L ) 2 16 ω N ω N ¯ v 3 Λ ω N P 2 Ω ( ( ω N + ω N ¯ 2 ω N P ) 2 + 4 Ω 2 ) .
P N ¯ = ( γ P N P L ) 2 16 ω N 2 ω N ¯ v 4 ω N P 2 Ω ( ( ω N + ω N ¯ 2 ω N P ) 2 + 4 Ω 2 ) .
P N ¯ = ( γ P N P L ) 2 | F 0 | 6 ω 0 v / ( 2 L ) ,

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