Abstract

We report on a theoretical study of activated de-correlation of pump and signal states of polarization in a fiber Raman amplifier based on 10 km of fiber with two-scale fiber spinning profile. As a result of the de-correlation, polarization dependent gain can be suppressed to 0.11 dB, PMD to 0.037 ps/km1/2 and gain can be increased to 15 dB.

©2012 Optical Society of America

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References

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  1. V. V. Kozlov, J. Nuño, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35(23), 3970–3972 (2010).
    [Crossref] [PubMed]
  2. M. Martinelli, M. Cirigliano, M. Ferrario, L. Marazzi, and P. Martelli, “Evidence of Raman-induced polarization pulling,” Opt. Express 17(2), 947–955 (2009).
    [Crossref] [PubMed]
  3. L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23(4), 254–256 (2011).
    [Crossref]
  4. S. V. Sergeyev, “Activated polarization pulling and de-correlation of signal and pump states of polarization in a fiber Raman amplifier,” Opt. Express 19(24), 24268–24279 (2011).
    [Crossref] [PubMed]
  5. S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48(1), 56–60 (2012).
    [Crossref]
  6. S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46(10), 1492–1497 (2010).
    [Crossref]
  7. N. J. Muga, M. F. S. Ferreira, and A. N. Pinto, “Broadband polarization pulling using Raman amplification,” Opt. Express 19(19), 18707–18712 (2011).
    [Crossref] [PubMed]
  8. P. Morin, S. Pitois, and J. Fatome, “Simultaneous polarization attraction and Raman amplification of a light beam in optical fibers,” J. Opt. Soc. Am. B 29(8), 2046–2052 (2012).
    [Crossref]
  9. V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23(15), 1088–1090 (2011).
    [Crossref]
  10. A. Zadok, E. Zilka, A. Eyal, L. Thévenaz, and M. Tur, “Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers,” Opt. Express 16(26), 21692–21707 (2008).
    [Crossref] [PubMed]
  11. J. E. Heebner, R. S. Bennink, R. W. Boyd, and R. A. Fisher, “Conversion of unpolarized light to polarized light with greater than 50% efficiency by photorefractive two-beam coupling,” Opt. Lett. 25(4), 257–259 (2000).
    [Crossref] [PubMed]
  12. P. Morin, J. Fatome, C. Finot, S. Pitois, R. Claveau, and G. Millot, “All-optical nonlinear processing of both polarization state and intensity profile for 40 Gbit/s regeneration applications,” Opt. Express 19(18), 17158–17166 (2011).
    [Crossref] [PubMed]
  13. M. Guasoni and S. Wabnitz, “Nonlinear polarizers based on four-wave mixing in high-birefringence optical fibers,” J. Opt. Soc. Am. B 29(6), 1511–1520 (2012).
    [Crossref]
  14. J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008).
    [Crossref] [PubMed]
  15. Q. Lin and G. P. Agrawal, “Vector theory of stimulated Raman scattering and its application to fiber-based Raman amplifiers,” J. Opt. Soc. Am. B 20(8), 1616–1631 (2003).
    [Crossref]
  16. P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
    [Crossref]
  17. A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R. Menyuk, “An analytical formula for the mean differential group delay of randomly-birefringent spun fibers,” J. Lightwave Technol. 21(7), 1635–1643 (2003).
    [Crossref]
  18. F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002).
    [Crossref] [PubMed]
  19. B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004).
    [Crossref]
  20. M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
    [Crossref] [PubMed]
  21. S. Sergeyev, S. Popov, and A. T. Friberg, “Polarization dependent gain and gain fluctuations in a fiber Raman amplifier,” J. Opt. A, Pure Appl. Opt. 9(12), 1119–1122 (2007).
    [Crossref]

2012 (3)

2011 (5)

2010 (2)

V. V. Kozlov, J. Nuño, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35(23), 3970–3972 (2010).
[Crossref] [PubMed]

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46(10), 1492–1497 (2010).
[Crossref]

2009 (1)

2008 (2)

A. Zadok, E. Zilka, A. Eyal, L. Thévenaz, and M. Tur, “Vector analysis of stimulated Brillouin scattering amplification in standard single-mode fibers,” Opt. Express 16(26), 21692–21707 (2008).
[Crossref] [PubMed]

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008).
[Crossref] [PubMed]

2007 (1)

S. Sergeyev, S. Popov, and A. T. Friberg, “Polarization dependent gain and gain fluctuations in a fiber Raman amplifier,” J. Opt. A, Pure Appl. Opt. 9(12), 1119–1122 (2007).
[Crossref]

2004 (1)

B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004).
[Crossref]

2003 (2)

2002 (1)

F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002).
[Crossref] [PubMed]

2001 (1)

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

2000 (1)

1996 (1)

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Agrawal, G. P.

Ania-Castañón, J. D.

V. V. Kozlov, J. Nuño, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35(23), 3970–3972 (2010).
[Crossref] [PubMed]

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008).
[Crossref] [PubMed]

Balle, S.

F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002).
[Crossref] [PubMed]

Barland, S.

F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002).
[Crossref] [PubMed]

Bennink, R. S.

Boyd, R. W.

Cirigliano, M.

Claveau, R.

Dykman, M. I.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

Eyal, A.

Fatome, J.

Ferrario, M.

Ferreira, M. F. S.

Finot, C.

Fisher, R. A.

Friberg, A. T.

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46(10), 1492–1497 (2010).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Polarization dependent gain and gain fluctuations in a fiber Raman amplifier,” J. Opt. A, Pure Appl. Opt. 9(12), 1119–1122 (2007).
[Crossref]

Galtarossa, A.

García-Ojalvo, J.

B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004).
[Crossref]

Giudici, M.

F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002).
[Crossref] [PubMed]

Golding, B.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

Guasoni, M.

Harper, P.

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008).
[Crossref] [PubMed]

Heebner, J. E.

Karalekas, V.

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008).
[Crossref] [PubMed]

Kozlov, V. V.

V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23(15), 1088–1090 (2011).
[Crossref]

V. V. Kozlov, J. Nuño, J. D. Ania-Castañón, and S. Wabnitz, “Theory of fiber optic Raman polarizers,” Opt. Lett. 35(23), 3970–3972 (2010).
[Crossref] [PubMed]

Lin, Q.

Lindner, B.

B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004).
[Crossref]

Luchinsky, D. G.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

Mannella, R.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

Marazzi, L.

Marino, F.

F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002).
[Crossref] [PubMed]

Marks, B. S.

Martelli, P.

Martinelli, M.

McCann, L. I.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

McClintock, P. V. E.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

Menyak, C. R.

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Menyuk, C. R.

Millot, G.

Morin, P.

Muga, N. J.

Neiman, A.

B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004).
[Crossref]

Nuño, J.

Palmieri, L.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23(4), 254–256 (2011).
[Crossref]

A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R. Menyuk, “An analytical formula for the mean differential group delay of randomly-birefringent spun fibers,” J. Lightwave Technol. 21(7), 1635–1643 (2003).
[Crossref]

Pinto, A. N.

Pitois, S.

Pizzinat, A.

Popov, S.

S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48(1), 56–60 (2012).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46(10), 1492–1497 (2010).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Polarization dependent gain and gain fluctuations in a fiber Raman amplifier,” J. Opt. A, Pure Appl. Opt. 9(12), 1119–1122 (2007).
[Crossref]

Santagiustina, M.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23(4), 254–256 (2011).
[Crossref]

Schimansky-Geier, L.

B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004).
[Crossref]

Sergeyev, S.

S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48(1), 56–60 (2012).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46(10), 1492–1497 (2010).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Polarization dependent gain and gain fluctuations in a fiber Raman amplifier,” J. Opt. A, Pure Appl. Opt. 9(12), 1119–1122 (2007).
[Crossref]

Sergeyev, S. V.

Smelyanskiy, V. N.

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

Thévenaz, L.

Tur, M.

Turitsyn, S. K.

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008).
[Crossref] [PubMed]

Ursini, L.

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23(4), 254–256 (2011).
[Crossref]

Wabnitz, S.

Wai, P. K. A.

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

Zadok, A.

Zilka, E.

Chaos (1)

M. I. Dykman, B. Golding, L. I. McCann, V. N. Smelyanskiy, D. G. Luchinsky, R. Mannella, and P. V. E. McClintock, “Activated escape of periodically driven systems,” Chaos 11(3), 587–594 (2001).
[Crossref] [PubMed]

IEEE J. Quantum Electron. (2)

S. Sergeyev and S. Popov, “Two-section fiber optic Raman polarizer,” IEEE J. Quantum Electron. 48(1), 56–60 (2012).
[Crossref]

S. Sergeyev, S. Popov, and A. T. Friberg, “Virtually isotropic transmission media with fiber Raman amplifier,” IEEE J. Quantum Electron. 46(10), 1492–1497 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (2)

V. V. Kozlov and S. Wabnitz, “Suppression of relative intensity noise in fiber-optic Raman polarizers,” IEEE Photon. Technol. Lett. 23(15), 1088–1090 (2011).
[Crossref]

L. Ursini, M. Santagiustina, and L. Palmieri, “Raman nonlinear polarization pulling in the pump depleted regime in randomly birefringent fibers,” IEEE Photon. Technol. Lett. 23(4), 254–256 (2011).
[Crossref]

J. Lightwave Technol. (2)

P. K. A. Wai and C. R. Menyak, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14(2), 148–157 (1996).
[Crossref]

A. Galtarossa, L. Palmieri, A. Pizzinat, B. S. Marks, and C. R. Menyuk, “An analytical formula for the mean differential group delay of randomly-birefringent spun fibers,” J. Lightwave Technol. 21(7), 1635–1643 (2003).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

S. Sergeyev, S. Popov, and A. T. Friberg, “Polarization dependent gain and gain fluctuations in a fiber Raman amplifier,” J. Opt. A, Pure Appl. Opt. 9(12), 1119–1122 (2007).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Express (5)

Opt. Lett. (2)

Phys. Rep. (1)

B. Lindner, J. García-Ojalvo, A. Neiman, and L. Schimansky-Geier, “Effects of noise in excitable systems,” Phys. Rep. 392(6), 321–424 (2004).
[Crossref]

Phys. Rev. Lett. (2)

F. Marino, M. Giudici, S. Barland, and S. Balle, “Experimental evidence of stochastic resonance in an excitable optical system,” Phys. Rev. Lett. 88(4), 040601 (2002).
[Crossref] [PubMed]

J. D. Ania-Castañón, V. Karalekas, P. Harper, and S. K. Turitsyn, “Simultaneous spatial and spectral transparency in ultralong fiber lasers,” Phys. Rev. Lett. 101(12), 123903 (2008).
[Crossref] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

a): Polarization dependent gain PDG, b): averaged part of the gain related to the pump-signal SOPs interaction, c): mean-square gain fluctuations σ, d): PMD parameter Dp as a function of fiber spinning frequency klf and correlation length Lc. Parameters: λp = 1460 nm, λs = 1550 nm, g = 2.3 dBW−1km−1, Pin = 5 W, Lb = 8.3 m, A0 = 3 (in units of rad), khf = 6π/Lbp, klf = [0…300/L], Lc = [5m…205m], a = −2, L = 10 km, αs = 0.2 dB/km.

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Fig. 2
Fig. 2

a), b): Averaged part of the gain related to the pump-signal SOPs interaction, polarization dependent gain PDG, mean-square gain fluctuations σ; c, d): PMD parameter Dp as a function of fiber spinning frequency klf and correlation length Lc. Parameters are the same as for Fig. 1. Gain (solid line), PDG (dotted line), σ (dashed line): a) Lc = 30 m (thin line), Lc = 65 m (thick line); b) klf = 3 km−1 (thin line), klf = 5 km−1 (thick line). PMD: c) Lc = 20 m (solid line), Lc = 30 m (dashed line); Lc = 55 m (dotted line); d) klf = 3 km−1 (solid line), klf = 5 km−1 (dashed line), klf = 15 km−1 (dotted line).

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Fig. 3
Fig. 3

Evolution of angle Φ between pump and signal SOPs and averaged part of gain related to the pump-signal SOPs interaction (Lc = 55m and klf = 3 km−1), initial signal SOP (linearly polarized) coincide with the pump SOP (solid line) and is orthogonal to the pump SOP (dotted line).

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Equations (9)

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PDG≡10log( 〈 s 0,max ( L ) 〉 / 〈 s 0,min ( L ) 〉 ),  G=10log( 〈 s 0 ( L ) 〉 / 〈 s 0 ( 0 ) 〉 ),   σ max(min) =  〈 s 0,max(min) 2 ( L ) 〉− 〈 s 0,max(min) ( L ) 〉 2 / 〈 s 0,max(min) ( L ) 〉 ,
ds dz = g 2 P 0 ( z ) s 0 p ^ +( W s + W s (NL) )×s ,  d  p ^ dz =( W p + W p (NL) )× p ^ ,
dθ dz =β(z),  〈 β(z) 〉=0,  〈 β(z)β(z') 〉= σ 2 δ(z−z'),
R 3 ( γ )=[ cos γ −sin γ 0 sin γ cos γ 0 0 0 1 ] .
dΩ dz = ∂ W s ∂ω + W s ×Ω ,  SIRF= 〈 | Ω( L ) | sp 2 〉 / 〈 | Ω( L ) | un 2 〉 ,  D ps = λ s 2 L c L bs  c SIRF
α am (z')= A 0 k hf L( cos( k lf Lz')+a )cos( k hf Lz').
d〈 s 0 〉 d z ′ = ε 1 exp( − ε 2 z ′ )〈 x 〉 ,  d〈 x 〉 d z ′ = ε 1 exp( − ε 2 z ′ )〈 s 0 〉− ε 3 〈 y 〉, d〈 y 〉 d z ′ = ε 3 [ 〈 x 〉−〈 p ˜ ^ 1 s ˜ 1 〉 ]−2α(z')〈 u 〉− 〈 y 〉L 2 L c ,  d〈 u 〉 d z ′ =2α(z')〈 y 〉− 〈 u 〉L 2 L c  , d〈 s 0 2 〉 d z ′ =2 ε 1 exp( − ε 2 z ′ )〈 s 0 x 〉 ,  d〈 s 0 x 〉 d z ′ = ε 1 exp( − ε 2 z ′ )( s 0 2 + x 2 )− ε 3 〈 y s 0 〉, d〈 s 0 y 〉 d z ′ = ε 1 exp( − ε 2 z ′ )〈 xy 〉+ ε 3 [ 〈 s 0 x 〉−〈 y 2 〉−〈 s 0 〉〈 p ˜ ^ 1   s ˜ 1 〉 ]−2α(z')〈 s 0 u 〉− 〈 s 0 y 〉L 2 L c , d〈 x 2 〉 d z ′ =2 ε 1 exp( − ε 2 z ′ )〈 s 0 x 〉−2 ε 3 〈 xy 〉, d〈 xy 〉 d z ′ = ε 1 exp( − ε 2 z ′ )〈 s 0 y 〉+ ε 3 [ 〈 x 2 〉−〈 x 〉〈 p ˜ ^ 1   s ˜ 1 〉 ]−2α(z')〈 xu 〉− 〈 xy 〉L 2 L c , d〈 u s 0 〉 d z ′ = ε 1 exp( − ε 2 z ′ )〈 xu 〉+2α(z')〈 y s 0 〉− 〈 u s 0 〉L 2 L c , d〈 xu 〉 d z ′ = ε 1 exp( − ε 2 z ′ )〈 s 0 u 〉− ε 3 〈 yu 〉+2α(z')〈 xy 〉− 〈 xu 〉L 2 L c , d〈 yu 〉 d z ′ = ε 3 ( 〈 xu 〉−〈 u 〉〈 p ˜ ^ 1   s ˜ 1 〉 )+2α(z')( 〈 y 2 〉−〈 u 2 〉 )+ L〈 yu 〉 2 L c , d〈 u 2 〉 d z ′ =2α(z')〈 yu 〉+ L L c ( 〈 y 2 〉−〈 u 2 〉 ),  d〈 y 2 〉 d z ′ =2 ε 3 [ 〈 yx 〉−〈 y 〉〈 p ˜ ^ 1   s ˜ 1 〉 ]−2α(z')〈 yu 〉− L L c ( 〈 y 2 〉−〈 u 2 〉 ).
d SIR F 2 d z ′ =〈 Ω ^ 1 〉, d〈 Ω ^ 1 〉 d z ′ =−〈 Ω ^ 1 〉L/ L c +2α(z)〈 Ω ^ 2 〉+L/ L c  , d〈 Ω ^ 2 〉 d z ′ =−2α(z)〈 Ω ^ 1 〉−〈 Ω ^ 2 〉L/ L c − ε 5 〈 Ω ^ 3 〉, d〈 Ω ^ 3 〉 d z ′ = ε 5 〈 Ω ^ 2 〉.
D ps = λ s 2 L c L bs  c SIRF .

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