Abstract

Optical frequency conversion by four-wave mixing (Bragg scattering) in a fiber is considered. If the frequencies and polarizations of the waves are chosen judiciously, Bragg scattering enables the translation of individual and entangled states, without the noise pollution associated with parametric amplification (modulation instability or phase conjugation), and with reduced noise pollution associated with stimulated Raman scattering.

©2005 Optical Society of America

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References

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  1. J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
    [Crossref]
  2. S. Radic and C. J. McKinstrie , “ Two-pump fiber parametric amplifiers ,” Opt. Fiber Technol.   9 , 7 – 23 ( 2003 ).
    [Crossref]
  3. W. H. Louisell , Radiation and Noise in Quantum Electronics ( McGraw-Hill , 1964 ).
  4. R. Loudon , The Quantum Theory of Light, 3rd Ed . ( Oxford University Press , 2000 ).
  5. C. J. McKinstrie , S. Radic , and M. G. Raymer , “ Quantum noise properties of parametric amplifiers driven by two pump waves ,” Opt. Express   12 , 5037 – 5066 ( 2004 ).
    [Crossref] [PubMed]
  6. C. J. McKinstrie , M. Yu , M. G. Raymer , and S. Radic , “ Quantum noise properties of parametric processes ,” Opt. Express   13 , 4986 – 5012 ( 2005 ).
    [Crossref] [PubMed]
  7. M. G. Raymer , “ Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering ,” J. Mod. Opt.   51 , 1739 – 1759 ( 2004 ).
  8. H. Kogelnik , R. M. Jopson , and L. E. Nelson , “ Polarization-mode dispersion ,” in Optical Fiber Telecommunications IVB , edited by I. Kaminow and T. Li ( Academic Press , 2002 ), pp. 725 – 861 .
  9. C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves ,” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
    [Crossref]
  10. C. J. McKinstrie , S. Radic , and C. Xie , “ Parametric instabilities driven by orthogonal pump waves in birefringent fibers ,” Opt. Express   11 , 2619 – 2633 ( 2003 ).
    [Crossref] [PubMed]
  11. C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence ,” Opt. Express   12 , 2033 – 2055 ( 2004 ).
    [Crossref] [PubMed]
  12. M. Yu , C. J. McKinstrie , and G. P. Agrawal , “ Modulational instabilities in dispersion-flattened fibers ,” Phys. Rev. E   52 , 1072 – 1080 ( 1995 ).
    [Crossref]
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    [Crossref] [PubMed]
  15. T. Tanemura , J. Suzuki , K. Katoh , and K. Kikuchi , “ Polarization-insensitive all-optical wavelength conversion using cross-phase modulation in twisted fiber and optical filtering ,” IEEE Photon. Technol. Lett.   17 , 1052 – 1054 ( 2005 ).
    [Crossref]
  16. R. H. Stolen and E. P. Ippen , “ Raman gain in glass optical waveguides ,” Appl. Phys. Lett.   22 , 276 – 278 ( 1973 ).
    [Crossref]
  17. R. H. Stolen , “ Polarization effects in fiber Raman and Brillouin lasers ,” IEEE J. Quantum Electron.   15 , 1157 – 1160 ( 1979 ).
    [Crossref]
  18. M. G. Raymer and I. A. Walmsley , “ Quantum coherence properties of stimulated Raman scattering ,” in Progress in Optics, Vol. 28 , edited by E. Wolf ( North-Holland , 1990 ), pp. 181 – 270 .
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  22. P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength ,” Phys. Rev.   137 , A801 – A818 ( 1965 ).
    [Crossref]
  23. P. K. A. Wai , C. R. Menyuk , and H. H. Chen , “ Stability of solitons in randomly varying birefringent fibers ,” Opt. Lett.   16 , 1231 – 1233 ( 1991 ).
    [Crossref] [PubMed]
  24. S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons ,” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
    [Crossref]
  25. C. R. Menyuk , “ Nonlinear pulse propagation in birefringent optical fibers ,” IEEE J. Quantum Electron.   23 , 174 – 176 ( 1987 ).
    [Crossref]
  26. A. J. Barlow , J. J. Ramskov-Hansen , and D. N. Payne , “ Birefringence and polarization-mode dispersion in spun single-mode fibers ,” Appl. Opt.   20 , 2962 – 2968 ( 1981 ).
    [Crossref] [PubMed]

2005 (4)

2004 (5)

2003 (3)

2002 (2)

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[Crossref]

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves ,” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[Crossref]

1995 (1)

M. Yu , C. J. McKinstrie , and G. P. Agrawal , “ Modulational instabilities in dispersion-flattened fibers ,” Phys. Rev. E   52 , 1072 – 1080 ( 1995 ).
[Crossref]

1992 (1)

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons ,” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[Crossref]

1991 (1)

1987 (1)

C. R. Menyuk , “ Nonlinear pulse propagation in birefringent optical fibers ,” IEEE J. Quantum Electron.   23 , 174 – 176 ( 1987 ).
[Crossref]

1981 (1)

1979 (1)

R. H. Stolen , “ Polarization effects in fiber Raman and Brillouin lasers ,” IEEE J. Quantum Electron.   15 , 1157 – 1160 ( 1979 ).
[Crossref]

1973 (1)

R. H. Stolen and E. P. Ippen , “ Raman gain in glass optical waveguides ,” Appl. Phys. Lett.   22 , 276 – 278 ( 1973 ).
[Crossref]

1965 (1)

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength ,” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[Crossref]

Agrawal, G. P.

M. Yu , C. J. McKinstrie , and G. P. Agrawal , “ Modulational instabilities in dispersion-flattened fibers ,” Phys. Rev. E   52 , 1072 – 1080 ( 1995 ).
[Crossref]

Andrekson, P. A.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[Crossref]

Barlow, A. J.

Bergano, N. S.

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons ,” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[Crossref]

Chen, A. Y. H.

Chen, H. H.

Chen, J.

Chraplyvy, A. R.

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves ,” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[Crossref]

Coen, S.

Devgan, P.

Evangelides, S. G.

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons ,” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[Crossref]

Gordon, J. P.

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons ,” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[Crossref]

Hansryd, J.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[Crossref]

Harvey, J. D.

Hedekvist, P. O.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[Crossref]

Ippen, E. P.

R. H. Stolen and E. P. Ippen , “ Raman gain in glass optical waveguides ,” Appl. Phys. Lett.   22 , 276 – 278 ( 1973 ).
[Crossref]

Jopson, R. M.

C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence ,” Opt. Express   12 , 2033 – 2055 ( 2004 ).
[Crossref] [PubMed]

H. Kogelnik , R. M. Jopson , and L. E. Nelson , “ Polarization-mode dispersion ,” in Optical Fiber Telecommunications IVB , edited by I. Kaminow and T. Li ( Academic Press , 2002 ), pp. 725 – 861 .

Kanaev, A. V.

Katoh, K.

T. Tanemura , J. Suzuki , K. Katoh , and K. Kikuchi , “ Polarization-insensitive all-optical wavelength conversion using cross-phase modulation in twisted fiber and optical filtering ,” IEEE Photon. Technol. Lett.   17 , 1052 – 1054 ( 2005 ).
[Crossref]

Kikuchi, K.

T. Tanemura , J. Suzuki , K. Katoh , and K. Kikuchi , “ Polarization-insensitive all-optical wavelength conversion using cross-phase modulation in twisted fiber and optical filtering ,” IEEE Photon. Technol. Lett.   17 , 1052 – 1054 ( 2005 ).
[Crossref]

Knight, J. C.

Kogelnik, H.

C. J. McKinstrie , H. Kogelnik , R. M. Jopson , S. Radic , and A. V. Kanaev , “ Four-wave mixing in fibers with random birefringence ,” Opt. Express   12 , 2033 – 2055 ( 2004 ).
[Crossref] [PubMed]

H. Kogelnik , R. M. Jopson , and L. E. Nelson , “ Polarization-mode dispersion ,” in Optical Fiber Telecommunications IVB , edited by I. Kaminow and T. Li ( Academic Press , 2002 ), pp. 725 – 861 .

Kumar, P.

Lasri, J.

Lee, K. F.

Leonhardt, R.

Li, J.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[Crossref]

Li, X.

Loudon, R.

R. Loudon , The Quantum Theory of Light, 3rd Ed . ( Oxford University Press , 2000 ).

Louisell, W. H.

W. H. Louisell , Radiation and Noise in Quantum Electronics ( McGraw-Hill , 1964 ).

Maker, P. D.

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength ,” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[Crossref]

McKinstrie, C. J.

Menyuk, C. R.

P. K. A. Wai , C. R. Menyuk , and H. H. Chen , “ Stability of solitons in randomly varying birefringent fibers ,” Opt. Lett.   16 , 1231 – 1233 ( 1991 ).
[Crossref] [PubMed]

C. R. Menyuk , “ Nonlinear pulse propagation in birefringent optical fibers ,” IEEE J. Quantum Electron.   23 , 174 – 176 ( 1987 ).
[Crossref]

Mollenauer, L. F.

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons ,” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[Crossref]

Murdoch, S. G.

Nelson, L. E.

H. Kogelnik , R. M. Jopson , and L. E. Nelson , “ Polarization-mode dispersion ,” in Optical Fiber Telecommunications IVB , edited by I. Kaminow and T. Li ( Academic Press , 2002 ), pp. 725 – 861 .

Payne, D. N.

Radic, S.

Ramskov-Hansen, J. J.

Raymer, M. G.

C. J. McKinstrie , M. Yu , M. G. Raymer , and S. Radic , “ Quantum noise properties of parametric processes ,” Opt. Express   13 , 4986 – 5012 ( 2005 ).
[Crossref] [PubMed]

C. J. McKinstrie , S. Radic , and M. G. Raymer , “ Quantum noise properties of parametric amplifiers driven by two pump waves ,” Opt. Express   12 , 5037 – 5066 ( 2004 ).
[Crossref] [PubMed]

M. G. Raymer , “ Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering ,” J. Mod. Opt.   51 , 1739 – 1759 ( 2004 ).

M. G. Raymer and I. A. Walmsley , “ Quantum coherence properties of stimulated Raman scattering ,” in Progress in Optics, Vol. 28 , edited by E. Wolf ( North-Holland , 1990 ), pp. 181 – 270 .
[Crossref]

Russell, P. S. J.

Stolen, R. H.

R. H. Stolen , “ Polarization effects in fiber Raman and Brillouin lasers ,” IEEE J. Quantum Electron.   15 , 1157 – 1160 ( 1979 ).
[Crossref]

R. H. Stolen and E. P. Ippen , “ Raman gain in glass optical waveguides ,” Appl. Phys. Lett.   22 , 276 – 278 ( 1973 ).
[Crossref]

Suzuki, J.

T. Tanemura , J. Suzuki , K. Katoh , and K. Kikuchi , “ Polarization-insensitive all-optical wavelength conversion using cross-phase modulation in twisted fiber and optical filtering ,” IEEE Photon. Technol. Lett.   17 , 1052 – 1054 ( 2005 ).
[Crossref]

Tanemura, T.

T. Tanemura , J. Suzuki , K. Katoh , and K. Kikuchi , “ Polarization-insensitive all-optical wavelength conversion using cross-phase modulation in twisted fiber and optical filtering ,” IEEE Photon. Technol. Lett.   17 , 1052 – 1054 ( 2005 ).
[Crossref]

Tang, R.

Terhune, R. W.

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength ,” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[Crossref]

Voss, P. L.

Wadsworth, W. J.

Wai, P. K. A.

Walmsley, I. A.

M. G. Raymer and I. A. Walmsley , “ Quantum coherence properties of stimulated Raman scattering ,” in Progress in Optics, Vol. 28 , edited by E. Wolf ( North-Holland , 1990 ), pp. 181 – 270 .
[Crossref]

Westlund, M.

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[Crossref]

Wong, G. K. L.

Xie, C.

Yu, M.

C. J. McKinstrie , M. Yu , M. G. Raymer , and S. Radic , “ Quantum noise properties of parametric processes ,” Opt. Express   13 , 4986 – 5012 ( 2005 ).
[Crossref] [PubMed]

M. Yu , C. J. McKinstrie , and G. P. Agrawal , “ Modulational instabilities in dispersion-flattened fibers ,” Phys. Rev. E   52 , 1072 – 1080 ( 1995 ).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

R. H. Stolen and E. P. Ippen , “ Raman gain in glass optical waveguides ,” Appl. Phys. Lett.   22 , 276 – 278 ( 1973 ).
[Crossref]

IEEE J. Quantum Electron. (2)

R. H. Stolen , “ Polarization effects in fiber Raman and Brillouin lasers ,” IEEE J. Quantum Electron.   15 , 1157 – 1160 ( 1979 ).
[Crossref]

C. R. Menyuk , “ Nonlinear pulse propagation in birefringent optical fibers ,” IEEE J. Quantum Electron.   23 , 174 – 176 ( 1987 ).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (2)

C. J. McKinstrie , S. Radic , and A. R. Chraplyvy , “ Parametric amplifiers driven by two pump waves ,” IEEE J. Sel. Top. Quantum Electron.   8 , 538 – 547 and 956 ( 2002 ).
[Crossref]

J. Hansryd , P. A. Andrekson , M. Westlund , J. Li , and P. O. Hedekvist , “ Fiber-based optical parametric amplifiers and their applications ,” IEEE J. Sel. Top. Quantum Electron.   8 , 506 – 520 ( 2002 ).
[Crossref]

IEEE Photon. Technol. Lett. (1)

T. Tanemura , J. Suzuki , K. Katoh , and K. Kikuchi , “ Polarization-insensitive all-optical wavelength conversion using cross-phase modulation in twisted fiber and optical filtering ,” IEEE Photon. Technol. Lett.   17 , 1052 – 1054 ( 2005 ).
[Crossref]

J. Lightwave Technol. (1)

S. G. Evangelides , L. F. Mollenauer , J. P. Gordon , and N. S. Bergano , “ Polarization muliplexing with solitons ,” J. Lightwave Technol.   10 , 28 – 35 ( 1992 ).
[Crossref]

J. Mod. Opt. (1)

M. G. Raymer , “ Quantum state entanglement and readout of collective atomic-ensemble modes and optical wave packets by stimulated Raman scattering ,” J. Mod. Opt.   51 , 1739 – 1759 ( 2004 ).

Opt. Express (5)

Opt. Fiber Technol. (1)

S. Radic and C. J. McKinstrie , “ Two-pump fiber parametric amplifiers ,” Opt. Fiber Technol.   9 , 7 – 23 ( 2003 ).
[Crossref]

Opt. Lett. (5)

Phys. Rev. (1)

P. D. Maker and R. W. Terhune , “ Study of optical effects due to an induced polarization third order in the electric field strength ,” Phys. Rev.   137 , A801 – A818 ( 1965 ).
[Crossref]

Phys. Rev. E (1)

M. Yu , C. J. McKinstrie , and G. P. Agrawal , “ Modulational instabilities in dispersion-flattened fibers ,” Phys. Rev. E   52 , 1072 – 1080 ( 1995 ).
[Crossref]

Other (4)

M. G. Raymer and I. A. Walmsley , “ Quantum coherence properties of stimulated Raman scattering ,” in Progress in Optics, Vol. 28 , edited by E. Wolf ( North-Holland , 1990 ), pp. 181 – 270 .
[Crossref]

W. H. Louisell , Radiation and Noise in Quantum Electronics ( McGraw-Hill , 1964 ).

R. Loudon , The Quantum Theory of Light, 3rd Ed . ( Oxford University Press , 2000 ).

H. Kogelnik , R. M. Jopson , and L. E. Nelson , “ Polarization-mode dispersion ,” in Optical Fiber Telecommunications IVB , edited by I. Kaminow and T. Li ( Academic Press , 2002 ), pp. 725 – 861 .

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Figures (7)
Fig. 1.
Fig. 1. Illustration of the constituent two-mode processes in a four-mode parametric interaction driven by two pump waves.

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Fig. 2.
Fig. 2. Eigenpolarizations of BS driven by parallel pumps.

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Fig. 3.
Fig. 3. Eigenpolarizations of BS driven by perpendicular pumps. The dashed lines denote sidebands that propagate independently.

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Fig. 4.
Fig. 4. Signal transmittance (solid curve) and idler transmittance (dashed curve) plotted as functions of the pump frequency ω 1. The vertical line denotes the signal frequency ωs . (a) ωs = -5.0 and ω 2 = 5.0 Tr/s. (b) ωs = -5.05 and ω 2 = 4.95.

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Fig. 5.
Fig. 5. Signal transmittance (solid curve) and idler transmittance (dashed curve) plotted as functions of the pump frequency ω 1. The vertical line denotes the signal frequency ωs . (a) ωs = -5.53 and ω 2 = 4.47 Tr/s. (b) ωs = -40.31 and ω 2 = 39.69.

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Fig. 6.
Fig. 6. Signal transmittance (solid curve) and idler transmittance (dashed curve) plotted as functions of the signal frequency ωs . (a) ω 1 = -20 and ω 2 = 5 Tr/s. (b) ω 1 = -10 and ω 2 = 10.

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Fig. 7.
Fig. 7. Signal transmittance (solid curve) and idler transmittance (dashed curve) plotted as functions of the signal frequency ωs . The pump frequencies ω 1 = -10 and ω 2 = 10 Tr/s. (a) β 3 = 0.1 ps3/Km, γ= 20/Km-W and P = 1.0 W. (b) β 3 = 0.01, γ = 10 and P = 0.3.

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Equations (29)

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

H = δ ( a s † a s − a i † a i ) + κ a s † a i + κ * a s a i † ,
d z a j = i [ a j , H ] ,
d z a s = i δ a s + i κ a i ,
d z a i = i κ * a s − i δ a i .
a s ( z ) = μ ¯ ( z ) a s ( 0 ) + ν ¯ ( z ) a i ( 0 ) ,
a i ( z ) = − ν ¯ * ( z ) a s ( 0 ) + μ ¯ * ( z ) a i ( 0 ) ,
μ ¯ ( z ) = cos ⁡ ( k z ) + i δ sin ⁡ ( k z ) k ,
ν ¯ ( z ) = i κ sin ⁡ ( k z ) k
∣ 1,0 〉 in = μ ¯ ∣ 1,0 〉 out − ν ¯ * ∣ 0,1 〉 out .
∣ 1,0 ; 1,0 〉 in = μ ¯ μ ' ¯ ∣ 1,0 ; 1,0 〉 out − μ ¯ ( ν ' ¯ ) * ∣ 1,0 ; 0,1 〉 out
− ν ¯ * μ ' ¯ ∣ 0,1 ; 1,0 〉 out + ν ¯ * ( ν ' ¯ ) * ∣ 1,0 ; 0,1 〉 out .
δ = β 2 ( ω b 2 − ω c 2 ) / 2 + β 4 ( ω b 4 − ω c 4 ) / 24 ,
= [ ( ω b 2 − ω c 2 ) / 2 ] [ β 2 + β 4 ( ω b 2 + ω c 2 ) / 12 ] ,
d δ d ω 1 = ω c ( β 2 + β 4 ω c 2 6 ) ,
d 2 δ d ω 1 2 = − ( β 2 + β 4 ω c 2 2 ) .
d δ / d ω s = β 3 ( ω b 2 − ω c 2 ) / 4 − β 3 ω a ( ω b + ω c ) / 2 − β 4 ( ω b 3 + ω c 3 ) / 12 ,
d 2 δ / d ω s 2 = − β 3 ( ω b + ω c ) / 2 + β 4 ( ω b 2 − ω c 2 ) / 8 .
E ( t , z ) = A ( t , z ) exp ⁡ [ i ( k 0 z − ω 0 t ) ] ,
P = γ ¯ [ 2 ( A · A * ) A + ( A · A ) A * ] / 3 ,
− i ∂ z A ∥ = β ( i ∂ t ) A ∥ + γ ( ∣ A ∥ ∣ 2 + ∣ A ⊥ ∣ 2 ) A ∥ ,
− i ∂ z A ⊥ = β ( i ∂ t ) A ⊥ + γ ( ∣ A ∥ ∣ 2 + ∣ A ⊥ ∣ 2 ) A ⊥ ,
P x = γ ¯ ( ∣ X ∣ 2 + 2 ∣ Y ∣ 2 / 3 ) X + γ ¯ Y 2 X * / 3 ,
P y = γ ¯ ( 2 ∣ X ∣ 2 / 3 + ∣ Y ∣ 2 ) Y + γ ¯ X 2 Y * / 3 .
− i ∂ z X = β x ( i ∂ t ) X + γ ( ∣ X ∣ 2 + ε ∣ Y ∣ 2 ) X ,
− i ∂ z Y = β y ( i ∂ t ) Y + γ ( ε ∣ X ∣ 2 + ∣ Y ∣ 2 ) Y ,
P r = ( 2 γ ¯ / 3 ) ( ∣ R ∣ 2 + 2 ∣ L ∣ 2 ) R ,
P l = ( 2 γ ¯ / 3 ) ( 2 ∣ R ∣ 2 + ∣ L ∣ 2 ) L .
− i ∂ z R = β ( i ∂ t ) R + γ ( ∣ R ∣ 2 + ε ∣ L ∣ 2 ) R ,
− i ∂ z L = β ( i ∂ t ) L + γ ( ε ∣ R ∣ 2 + ∣ L ∣ 2 ) L ,

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