Abstract

We investigate the Brillouin dynamic grating generation and detection process in polarization-maintaining fibers for the case of continuous wave operation both theoretically and experimentally. The four interacting light waves couple together through the material density variation due to stimulated Brillouin scattering. The four coupled equations describing this process are derived and solved analytically for two cases: moving fiber Bragg grating approximation and undepleted pump and probe waves approximation. We show that the conventional grating model oversimplifies the Brillouin dynamic grating generation and detection process, since it neglects the coupling between all the four waves, while the four-wave mixing model clearly demonstrates this coupling process and it is verified experimentally by measuring the reflection of the Brillouin dynamic grating. The trends of the theoretical calculation and experimental results agree well with each other confirming that the Brillouin dynamic grating generation and detection process is indeed a four-wave mixing process.

© 2011 OSA

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  1. R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964).
    [CrossRef]
  2. Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
    [CrossRef] [PubMed]
  3. L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
    [CrossRef]
  4. G. P. Agrawal, “Nonlinear Fiber Optics,” 4th edition, Academic Press2007.
  5. X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
    [CrossRef]
  6. K. Y. Song, W. Zou, Z. He, and K. Hotate, “All-optical dynamic grating generation based on Brillouin scattering in polarization-maintaining fiber,” Opt. Lett. 33, 926–928 (2008).
    [CrossRef] [PubMed]
  7. Y. Dong, X. Bao, and L. Chen, “Distributed temperature sensing based on birefringence effect on transient Brillouin grating in a polarization-maintaining photonic crystal fiber,” Opt. Lett. 34, 2590–2592 (2009).
    [CrossRef] [PubMed]
  8. K. Y. Song and H. J. Yoon, “High-resolution Brillouin optical time domain analysis based on Brillouin dynamic grating,” Opt. Lett. 35, 52–54 (2010).
    [CrossRef] [PubMed]
  9. K. Y. Song, S. Chin, N. Primerov, and L. Thévenaz, “Time-domain distributed fiber sensor with 1 cm spatial resolution based on Brillouin dynamic grating,” J. Lightwave Technol. 28, 2062–2067 (2010).
    [CrossRef]
  10. S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011).
    [CrossRef]
  11. W. Zou, Z. He, and K. Hotate, “Complete discrimination of strain and temperature using Brillouin frequency shift and birefringence in a polarization-maintaining fiber,” Opt. Express 17, 1248–1255 (2009).
    [CrossRef] [PubMed]
  12. W. Zou, Z. He, and K. Hotate, “Demonstration of Brillouin distributed discrimination of strain and temperature using a polarization-maintaining optical fiber,” IEEE Photon. Technol. Lett. 22, 526–528 (2010).
    [CrossRef]
  13. Y. Dong, L. Chen, and X. Bao, “High-spatial-resolution time-domain simultaneous strain and temperature sensor using Brillouin scattering and birefringence in a polarization-maintaining fiber,” IEEE Photon. Technol. Lett. 22, 1364–1366 (2010).
    [CrossRef]
  14. W. Zou, Z. He, and K. Hotate, “One-laser-based generation/detection of Brillouin dynamic grating and its application to distributed discrimination of strain and temperature,” Opt. Express 19, 2363–2370 (2011).
    [CrossRef] [PubMed]
  15. K. Y. Song, K. Lee, and S. B. Lee, “Tunable optical delays based on Brillouin dynamic grating in optical fibers,” Opt. Express 17, 10344–10349 (2009).
    [CrossRef] [PubMed]
  16. Y. Dong, L. Chen, and X. Bao, “Truly distributed birefringence measurement of polarization-maintaining fibers based on transient Brillouin grating,” Opt. Lett. 35, 193–195 (2010).
    [CrossRef] [PubMed]
  17. R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).
  18. T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
    [CrossRef]
  19. Y. Dong, L. Chen, and X. Bao, “Characterization of the Brillouin grating spectra in a polarization-maintaining fiber,” Opt. Express 18, 18960–18967 (2010).
    [CrossRef] [PubMed]
  20. K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. 35, 2958–2960 (2010).
    [CrossRef] [PubMed]
  21. K. Y. Song, W. Zou, Z. He, and K. Hotate, “Optical time-domain measurement of Brillouin dynamic grating spectrum in a polarization-maintaining fiber,” Opt. Lett. 34, 1381–1383 (2009).
    [CrossRef] [PubMed]
  22. A. Kobyakov, S. Kumar, D. Q. Chowdhury, A. B. Ruffin, M. Sauer, S. R. Bichham, and R. Mishra, “Design concept for optical fibers with enhaced SBS threshold,” Opt. Express 13, 5338–5346 (2005).
    [CrossRef] [PubMed]
  23. A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2, 1–59 (2010).
    [CrossRef]

2011 (3)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
[CrossRef]

S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011).
[CrossRef]

W. Zou, Z. He, and K. Hotate, “One-laser-based generation/detection of Brillouin dynamic grating and its application to distributed discrimination of strain and temperature,” Opt. Express 19, 2363–2370 (2011).
[CrossRef] [PubMed]

2010 (8)

W. Zou, Z. He, and K. Hotate, “Demonstration of Brillouin distributed discrimination of strain and temperature using a polarization-maintaining optical fiber,” IEEE Photon. Technol. Lett. 22, 526–528 (2010).
[CrossRef]

Y. Dong, L. Chen, and X. Bao, “High-spatial-resolution time-domain simultaneous strain and temperature sensor using Brillouin scattering and birefringence in a polarization-maintaining fiber,” IEEE Photon. Technol. Lett. 22, 1364–1366 (2010).
[CrossRef]

A. Kobyakov, M. Sauer, and D. Chowdhury, “Stimulated Brillouin scattering in optical fibers,” Adv. Opt. Photon. 2, 1–59 (2010).
[CrossRef]

K. Y. Song and H. J. Yoon, “High-resolution Brillouin optical time domain analysis based on Brillouin dynamic grating,” Opt. Lett. 35, 52–54 (2010).
[CrossRef] [PubMed]

Y. Dong, L. Chen, and X. Bao, “Truly distributed birefringence measurement of polarization-maintaining fibers based on transient Brillouin grating,” Opt. Lett. 35, 193–195 (2010).
[CrossRef] [PubMed]

K. Y. Song, S. Chin, N. Primerov, and L. Thévenaz, “Time-domain distributed fiber sensor with 1 cm spatial resolution based on Brillouin dynamic grating,” J. Lightwave Technol. 28, 2062–2067 (2010).
[CrossRef]

Y. Dong, L. Chen, and X. Bao, “Characterization of the Brillouin grating spectra in a polarization-maintaining fiber,” Opt. Express 18, 18960–18967 (2010).
[CrossRef] [PubMed]

K. Y. Song and H. J. Yoon, “Observation of narrowband intrinsic spectra of Brillouin dynamic gratings,” Opt. Lett. 35, 2958–2960 (2010).
[CrossRef] [PubMed]

2009 (4)

2008 (2)

2007 (1)

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef] [PubMed]

2005 (1)

1997 (1)

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

1964 (1)

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, “Nonlinear Fiber Optics,” 4th edition, Academic Press2007.

Bao, X.

Bichham, S. R.

Boyd, R. W.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef] [PubMed]

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

Chen, L.

Chiao, R. Y.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964).
[CrossRef]

Chin, S.

S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011).
[CrossRef]

K. Y. Song, S. Chin, N. Primerov, and L. Thévenaz, “Time-domain distributed fiber sensor with 1 cm spatial resolution based on Brillouin dynamic grating,” J. Lightwave Technol. 28, 2062–2067 (2010).
[CrossRef]

Chowdhury, D.

Chowdhury, D. Q.

Dong, Y.

Erdogan, T.

T. Erdogan, “Fiber grating spectra,” J. Lightwave Technol. 15, 1277–1294 (1997).
[CrossRef]

Gauthier, D. J.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef] [PubMed]

He, Z.

Hotate, K.

Kobyakov, A.

Kumar, S.

Lee, K.

Lee, S. B.

Mishra, R.

Primerov, N.

S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011).
[CrossRef]

K. Y. Song, S. Chin, N. Primerov, and L. Thévenaz, “Time-domain distributed fiber sensor with 1 cm spatial resolution based on Brillouin dynamic grating,” J. Lightwave Technol. 28, 2062–2067 (2010).
[CrossRef]

Ruffin, A. B.

Sauer, M.

Song, K. Y.

Stoicheff, B. P.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964).
[CrossRef]

Thévenaz, L.

S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011).
[CrossRef]

K. Y. Song, S. Chin, N. Primerov, and L. Thévenaz, “Time-domain distributed fiber sensor with 1 cm spatial resolution based on Brillouin dynamic grating,” J. Lightwave Technol. 28, 2062–2067 (2010).
[CrossRef]

L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
[CrossRef]

Townes, C. H.

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964).
[CrossRef]

Yoon, H. J.

Zhu, Z.

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef] [PubMed]

Zou, W.

Adv. Opt. Photon. (1)

IEEE Photon. Technol. Lett. (2)

W. Zou, Z. He, and K. Hotate, “Demonstration of Brillouin distributed discrimination of strain and temperature using a polarization-maintaining optical fiber,” IEEE Photon. Technol. Lett. 22, 526–528 (2010).
[CrossRef]

Y. Dong, L. Chen, and X. Bao, “High-spatial-resolution time-domain simultaneous strain and temperature sensor using Brillouin scattering and birefringence in a polarization-maintaining fiber,” IEEE Photon. Technol. Lett. 22, 1364–1366 (2010).
[CrossRef]

IEEE Sens. J. (1)

S. Chin, N. Primerov, and L. Thévenaz, “Sub-centimetre spatial resolution in distributed fibre sensing, based on dynamic Brillouin grating in optical fibers,” to appear in IEEE Sens. J. (2011).
[CrossRef]

J. Lightwave Technol. (2)

Nat. Photonics (1)

L. Thévenaz, “Slow and fast light in optical fibres,” Nat. Photonics 2, 474–481 (2008).
[CrossRef]

Opt. Express (5)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

R. Y. Chiao, C. H. Townes, and B. P. Stoicheff, “Stimulated Brillouin scattering and coherent generation of intense hypersonic waves,” Phys. Rev. Lett. 12, 592–595 (1964).
[CrossRef]

Science (1)

Z. Zhu, D. J. Gauthier, and R. W. Boyd, “Stored light in an optical fiber via stimulated Brillouin scattering,” Science 318, 1748–1750 (2007).
[CrossRef] [PubMed]

Sensors (1)

X. Bao and L. Chen, “Recent progress in Brillouin scattering based fiber sensors,” Sensors 11, 4152–4187 (2011).
[CrossRef]

Other (2)

R. W. Boyd, Nonlinear Optics, 3rd ed. (Academic Press, 2008).

G. P. Agrawal, “Nonlinear Fiber Optics,” 4th edition, Academic Press2007.

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Figures (7)

Fig. 1
Fig. 1

Diagram of the theoretical model. BDG generation and detection scheme: an FWM process.

Fig. 2
Fig. 2

Properties of the tanh2(x) and tanh 2 ( x ) functions.

Fig. 3
Fig. 3

Theoretically calculated BDG reflectivity with respect to (a) pump 1 power |A1|2 for different pump 2 and probe powers (b) pump 2 power |A2|2 for different probe powers and fixed pump 1 power (c) probe power |A3|2 for different pump 1 powers and fixed pump 2 power.

Fig. 4
Fig. 4

Experiment Setup. PC: polarization controller; PD: photodetector; EDFA: erbium-doped fiber amplifier; ISO: isolator; PBS/PBC: polarization beam splitter/combiner; PM: power meter; TF: tunable filer; PMF: polarization-maintaining fiber.

Fig. 5
Fig. 5

Typical BDG spectra for (a) 13-m Panda and (b) 10-m Bow-Tie fiber.

Fig. 6
Fig. 6

BDG spectra for different pump 1 power, with the pump 2 power of 10 mW and probe power of 100 mW.

Fig. 7
Fig. 7

Experimentally obtained BDG reflectivity with respect to (a) pump 1 power |A1|2 for different pump 2 and probe powers (b) pump 2 power |A2|2 for different probe powers and fixed pump 1 power (c) probe power |A3|2 for different pump 1 powers and fixed pump 2 power.

Equations (51)

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2 ρ ˜ t 2 Γ A 2 ρ ˜ t v A 2 2 ρ ˜ = 1 2 ɛ 0 γ e 2 E ˜ tot 2 ,
E ˜ x = 1 2 x ^ [ F ( x , y ) A ˜ 1 ( z , t ) e i ( k 1 z ω 1 t ) + F ( x , y ) A ˜ 2 ( z , t ) e i ( k 2 z ω 2 t ) ] + c . c . ,
E ˜ y = 1 2 y ^ [ F ( x , y ) A ˜ 3 ( z , t ) e i ( k 3 z ω 3 t ) + F ( x , y ) A ˜ 4 ( z , t ) e i ( k 4 z ω 4 t ) ] + c . c . ,
A 1 z = κ ( A 1 | A 2 | 2 + A 2 A 3 A 4 * e i Δ k z ) ,
A 2 z = κ ( | A 1 | 2 A 2 + A 1 A 3 * A 4 e i Δ kz ) ,
A 3 z = κ ( A 3 | A 4 | 2 + A 1 A 2 * A 4 e i Δ k z ) ,
A 4 z = κ ( | A 3 | 2 A 4 + A 1 * A 2 A 3 e i Δ k z ) ,
κ = 8 π 3 γ e 2 c ρ 0 λ 3 Ω B Γ B A eff ao ,
A eff ao = [ F 2 ( x , y ) F 2 ( x , y ) F A ( x , y ) ] 2 F A 2 ( x , y ) ,
A 3 z = κ A 1 A 2 * A 4 e i Δ kz ,
A 4 z = κ A 1 * A 2 A 3 e i Δ kz ,
A 4 ( z ) = 2 κ exp ( i Δ kz / 2 ) sinh [ g ( L z ) / 2 ] A 1 * A 2 A 3 ( 0 ) g cosh ( gL / 2 ) i Δ k sinh ( gL / 2 ) ,
R FBG = | A 4 ( 0 ) A 3 ( 0 ) | 2 = tanh 2 ( κ | A 1 | | A 2 | L ) .
A 2 z = κ ( | A 1 | 2 A 2 + A 1 A 3 * A 4 e i Δ k z ) ,
A 4 z = κ ( | A 3 | 2 A 4 + A 1 * A 2 A 3 e i Δ kz ) ,
A 4 ( z ) = 2 κ A 1 * A 2 ( L ) A 3 sinh [ Φ 2 + 4 Ψ ( L z ) / 2 ] Φ 2 + 4 Ψ e Φ ( L z ) / 2 + i Δ kL ,
R FWM = | A 4 ( 0 ) A 3 | 2 = | A 1 | 2 | A 2 ( L ) | 2 [ e κ ( | A 1 | 2 + | A 3 | 2 ) L 1 | A 1 | 2 + | A 3 | 2 ] 2 .
2 E ˜ tot 2 1 2 F 2 [ A ˜ 1 A ˜ 2 * q 1 2 e i ( q 1 z Ω t ) + A ˜ 3 A ˜ 4 * q 2 2 e i ( q 2 z Ω t ) ] + c . c . ,
ρ ˜ = ρ ˜ 1 + ρ ˜ 2 = 1 2 F A ( x , y ) [ Q 1 ( z , t ) e i ( q 1 z Ω t ) + Q 2 ( z , t ) e i ( q 2 z Ω t ) ] + c . c .
2 F A ( x , y ) + ( Ω Bm 2 v A 2 q m 2 ) F A ( x , y ) = 0 , ( m = 1 , 2 )
1 2 { i Ω Γ B 1 F A Q 1 + v A 2 Q 1 [ 2 F A + ( Ω 2 v A 2 q 1 2 ) F A ] } = 1 4 ɛ 0 γ e F 2 q 1 2 A ˜ 1 A ˜ 2 * ,
1 2 { i Ω Γ B 2 F A Q 2 + v A 2 Q 2 [ 2 F A + ( Ω 2 v A 2 q 2 2 ) F A ] } = 1 4 ɛ 0 γ e F 2 q 2 2 A ˜ 3 A ˜ 4 * ,
Q 1 ( z , t ) = ɛ 0 γ e q 1 2 A ˜ 1 ( z , t ) A ˜ 2 * ( z , t ) 2 ( Ω B 1 2 Ω 2 i Ω Γ B 1 ) F 2 ( x , y ) F A ( x , y ) F A 2 ( x , y ) ,
Q 2 ( z , t ) = ɛ 0 γ e q 2 2 A ˜ 3 ( z , t ) A ˜ 4 * ( z , t ) 2 ( Ω B 2 2 Ω 2 i Ω Γ B 2 ) F 2 ( x , y ) F A ( x , y ) F A 2 ( x , y ) ,
q 1 = ω 1 n eff , x ( ω 1 ) c + ( ω 1 Ω ) n eff , x ( ω 1 ) c ,
q 2 = ω 3 n eff , y ( ω 3 ) c + ( ω 3 Ω ) n eff , y ( ω 3 ) c ,
2 F A ( x , y ) + ( Ω 2 v A 2 q m 2 ) F A ( x , y ) = 0 , ( m = 1 , 2 )
q 1 n eff , x ( ω 1 ) c ( 2 ω 1 Ω ) 2 ω 1 n eff , x ( ω 1 ) c 2 ω 1 n x c ,
q 2 n eff , y ( ω 3 ) c ( 2 ω 3 Ω ) 2 ω 3 n eff , y ( ω 3 ) c 2 ω 3 n y c ,
ρ ˜ ( x , y , z , t ) = 1 4 F A F 2 F A F A 2 ɛ 0 γ e q 2 ( A ˜ 1 A ˜ 2 * + A ˜ 3 A ˜ 4 * ) Ω B 2 Ω 2 i Ω Γ B e i ( qz Ω t ) + c . c ..
2 E ˜ j n 2 c 2 2 E ˜ j t 2 = 1 ɛ 0 c 2 2 P ˜ j t 2 ,
P ˜ tot = ɛ 0 γ e ρ 0 ρ ˜ E ˜ tot = ɛ 0 ɛ NL E ˜ tot ,
ɛ NL = F A U { A ˜ 1 A ˜ 2 * e i [ ( k 1 + k 2 ) z Ω t ] + A ˜ 3 A ˜ 4 * e i [ ( k 3 + k 4 ) z Ω t ] } + c . c .
U = ɛ 0 γ e 2 q 2 4 ρ 0 ( Ω B 2 Ω 2 i Ω Γ B ) F 2 F A F A 2 .
P ˜ 1 = 1 2 ɛ 0 U F F A ( A ˜ 1 | A ˜ 2 | 2 + A ˜ 2 A ˜ 3 A ˜ 4 * e i Δ k z ) e i ( k 1 z ω 1 t ) + c . c . ,
P ˜ 2 = 1 2 ɛ 0 U * F F A ( | A ˜ 1 | 2 A ˜ 2 + A ˜ 1 A ˜ 3 * A ˜ 4 e i Δ k z ) e i ( k 2 z ω 2 t ) + c . c . ,
P ˜ 3 = 1 2 ɛ 0 U F F A ( A ˜ 3 | A ˜ 4 | 2 + A ˜ 1 A ˜ 2 * A ˜ 4 e i Δ kz ) e i ( k 3 z ω 3 t ) + c . c . ,
P ˜ 4 = 1 2 ɛ 0 U * F F A ( | A ˜ 3 | 2 A ˜ 4 + A ˜ 1 * A ˜ 2 A ˜ 3 e i Δ kz ) e i ( k 4 z ω 4 t ) + c . c . ,
2 F ( x , y ) + ( ω j 2 n 2 c 2 k j 2 ) F ( x , y ) = 0 ,
A 1 z = i η 1 ( A 1 | A 2 | 2 + A 2 A 3 A 4 * e i Δ kz ) ,
A 2 z = i η 1 * ( | A 1 | 2 A 2 + A 1 A 3 * A 4 e i Δ kz ) ,
A 3 z = i η 2 ( A 3 | A 4 | 2 + A 1 A 2 * A 4 e i Δ kz ) ,
A 4 z = i η 2 * ( | A 3 | 2 A 4 + A 1 * A 2 A 3 e i Δ kz ) ,
η 1 = γ e 2 ω 1 3 ρ 0 c 4 ( Ω B 2 Ω 2 i Ω Γ B ) A eff ao ,
η 2 = γ e 2 ω 3 3 ρ 0 c 4 ( Ω B 2 Ω 2 i Ω Γ B ) A eff ao ,
2 A 4 z 2 + Φ A 4 z Ψ A 4 = 0 ,
A 4 ( z ) = C e p + z + D e p z ,
A 4 | z = L = 0 ,
A 4 z | z = L = κ A 1 * A 2 ( L ) A 3 e i Δ kL .
C = κ A 1 * A 2 ( L ) A 3 e i Δ kL e p + L p p + ,
D = κ A 1 * A 2 ( L ) A 3 e i Δ kL e p L p p + .

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