Abstract

In a previous paper the generalized Schroedinger equation that governs wave propagation in a rapidly-spun fiber was derived. In this paper the aforementioned equation is used to study four-wave mixing (FWM). The properties of FWM associated with a rapidly-spun fiber are described, and contrasted to those associated with constantly-birefringent and randomly-birefringent fibers. FWM driven by perpendicular linearly-polarized pump waves, or counter-rotating circularly-polarized pump waves, provides polarization-independent signal amplification and phase-conjugation, whereas FWM driven by co-rotating circularly-polarized pump waves provides polarization-independent frequency conversion.

© 2006 Optical Society of America

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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, "Optical amplification and signal processing in highly-nonlinear optical fiber," IEICE Trans. Electron. E88C, 859-869 (2005).</jrn>
    [CrossRef]
  3. R.M. Jopson and R. E. Tench, "Polarisation-independent phase conjugation of lightwave signals," Electron. Lett. 29, 2216-2217 (1993).
    [CrossRef]
  4. K. Inoue, "Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies," J. Lightwave Technol. 12, 1916-1920 (1994).
    [CrossRef]
  5. T. Tanemura, K. Katoh and K. Kikuchi, "Polarization-insensitive asymmetric four-wave mixing using circularly polarized pumps in a twisted fiber," Opt. Express 13, 7497-7505 (2005).
    [CrossRef] [PubMed]
  6. C. J. McKinstrie, J. D. Harvey, S. Radic and M. G. Raymer, "Translation of quantum states by four-wave mixing in fibers," Opt. Express 13, 9131-9142 (2005).
    [CrossRef] [PubMed]
  7. C. J. McKinstrie and H. Kogelnik, "Nonlinear wave propagation in a rapidly-spun fiber," Opt. Express 14, 8072- 8087 (2006).
    [CrossRef] [PubMed]
  8. 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]
  9. C. R. Menyuk, "Nonlinear pulse propagation in birefringent optical fibers," IEEE J. Quantum Electron. 23, 174- 176 (1987).
    [CrossRef]
  10. 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]
  11. 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]
  12. R. W. Boyd, Nonlinear Optics (Academic Press, 1992).
  13. J. P. Gordon and H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. 97, 4541-4550 (2000).
    [CrossRef] [PubMed]
  14. P. D. Maker, R. W. Terhune and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
    [CrossRef]
  15. B. Daino, G. Gregori and S. Wabnitz, "New all-optical devices based on third-order nonlinearity of birefringent fibers," Opt. Lett. 11, 42-44 (1986).
    [CrossRef] [PubMed]
  16. L. F. Mollenauer, J. P. Gordon and F. Heismann, "Polarization scattering by soliton-soliton collisions," Opt. Lett. 20, 2060-2062 (1995).
    [CrossRef] [PubMed]
  17. K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, "CW three-wave mixing in single-mode optical fibers," J. Appl. Phys. 49, 5098-5106 (1978).
    [CrossRef]
  18. R. H. Stolen and J. E. Bjorkholm, "Parametric amplification and frequency conversion in optical fibers," IEEE J. Quantum Electron. 18, 1062-1072 (1982).
    [CrossRef]
  19. K. Inoue, "Polarization effect on four-wave mixing efficiency in a single-mode fiber," IEEE J. Quantum Electron. 28, 883-894 (1992).
    [CrossRef]
  20. 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]
  21. G. Millot, S. Pitois and P. T. Dinda, "Modulational instability processes in optical isotropic fibers under dualfrequency circular polarization pumping," J. Opt. Soc. Am. B 19, 454-460 (2002).
    [CrossRef]
  22. M. E. Marhic, K. K. Y. Wong and L. G. Kazovsky, "Fibre optical parametric amplifiers with linearly or circularly polarized waves," J. Opt. Soc. Am. B 20, 2425-2433 (2003).
    [CrossRef]
  23. E. Seve, P. Tchofo Dinda, G. Millot,M. Remoissenet, J.M. Bibault andM. Haelterman, "Modulational instability and critical regime in a highly birefringent fiber," Phys. Rev. A 54, 3519-3534 (1996).
    [CrossRef] [PubMed]
  24. 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]
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    [CrossRef]

2006 (1)

2005 (2)

2004 (2)

2003 (2)

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]

G. Millot, S. Pitois and P. T. Dinda, "Modulational instability processes in optical isotropic fibers under dualfrequency circular polarization pumping," J. Opt. Soc. Am. B 19, 454-460 (2002).
[CrossRef]

2000 (1)

J. P. Gordon and H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

1996 (1)

E. Seve, P. Tchofo Dinda, G. Millot,M. Remoissenet, J.M. Bibault andM. Haelterman, "Modulational instability and critical regime in a highly birefringent fiber," Phys. Rev. A 54, 3519-3534 (1996).
[CrossRef] [PubMed]

1995 (1)

1994 (1)

K. Inoue, "Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies," J. Lightwave Technol. 12, 1916-1920 (1994).
[CrossRef]

1993 (1)

R.M. Jopson and R. E. Tench, "Polarisation-independent phase conjugation of lightwave signals," Electron. Lett. 29, 2216-2217 (1993).
[CrossRef]

1992 (2)

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]

K. Inoue, "Polarization effect on four-wave mixing efficiency in a single-mode fiber," IEEE J. Quantum Electron. 28, 883-894 (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]

1986 (1)

1982 (1)

R. H. Stolen and J. E. Bjorkholm, "Parametric amplification and frequency conversion in optical fibers," IEEE J. Quantum Electron. 18, 1062-1072 (1982).
[CrossRef]

1978 (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, "CW three-wave mixing in single-mode optical fibers," J. Appl. Phys. 49, 5098-5106 (1978).
[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]

1964 (1)

P. D. Maker, R. W. Terhune and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Agrawal, G. P.

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]

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]

Bjorkholm, J. E.

R. H. Stolen and J. E. Bjorkholm, "Parametric amplification and frequency conversion in optical fibers," IEEE J. Quantum Electron. 18, 1062-1072 (1982).
[CrossRef]

Chen, H. H.

Daino, B.

Dinda, P. T.

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.

J. P. Gordon and H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

L. F. Mollenauer, J. P. Gordon and F. Heismann, "Polarization scattering by soliton-soliton collisions," Opt. Lett. 20, 2060-2062 (1995).
[CrossRef] [PubMed]

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]

Gregori, G.

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]

Heismann, F.

Hill, K. O.

K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, "CW three-wave mixing in single-mode optical fibers," J. Appl. Phys. 49, 5098-5106 (1978).
[CrossRef]

Inoue, K.

K. Inoue, "Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies," J. Lightwave Technol. 12, 1916-1920 (1994).
[CrossRef]

K. Inoue, "Polarization effect on four-wave mixing efficiency in a single-mode fiber," IEEE J. Quantum Electron. 28, 883-894 (1992).
[CrossRef]

Johnson, D. C.

K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, "CW three-wave mixing in single-mode optical fibers," J. Appl. Phys. 49, 5098-5106 (1978).
[CrossRef]

Jopson, R. M.

Jopson, R.M.

R.M. Jopson and R. E. Tench, "Polarisation-independent phase conjugation of lightwave signals," Electron. Lett. 29, 2216-2217 (1993).
[CrossRef]

Kanaev, A. V.

Katoh, K.

Kawasaki, B. S.

K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, "CW three-wave mixing in single-mode optical fibers," J. Appl. Phys. 49, 5098-5106 (1978).
[CrossRef]

Kazovsky, L. G.

Kikuchi, K.

Kogelnik, H.

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]

Lin, Q.

MacDonald, R. I.

K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, "CW three-wave mixing in single-mode optical fibers," J. Appl. Phys. 49, 5098-5106 (1978).
[CrossRef]

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]

P. D. Maker, R. W. Terhune and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Marhic, M. E.

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]

Millot, G.

G. Millot, S. Pitois and P. T. Dinda, "Modulational instability processes in optical isotropic fibers under dualfrequency circular polarization pumping," J. Opt. Soc. Am. B 19, 454-460 (2002).
[CrossRef]

E. Seve, P. Tchofo Dinda, G. Millot,M. Remoissenet, J.M. Bibault andM. Haelterman, "Modulational instability and critical regime in a highly birefringent fiber," Phys. Rev. A 54, 3519-3534 (1996).
[CrossRef] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, J. P. Gordon and F. Heismann, "Polarization scattering by soliton-soliton collisions," Opt. Lett. 20, 2060-2062 (1995).
[CrossRef] [PubMed]

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]

Pitois, S.

Radic, S.

Raymer, M. G.

Remoissenet, M.

E. Seve, P. Tchofo Dinda, G. Millot,M. Remoissenet, J.M. Bibault andM. Haelterman, "Modulational instability and critical regime in a highly birefringent fiber," Phys. Rev. A 54, 3519-3534 (1996).
[CrossRef] [PubMed]

Savage, C. M.

P. D. Maker, R. W. Terhune and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Seve, E.

E. Seve, P. Tchofo Dinda, G. Millot,M. Remoissenet, J.M. Bibault andM. Haelterman, "Modulational instability and critical regime in a highly birefringent fiber," Phys. Rev. A 54, 3519-3534 (1996).
[CrossRef] [PubMed]

Stolen, R. H.

R. H. Stolen and J. E. Bjorkholm, "Parametric amplification and frequency conversion in optical fibers," IEEE J. Quantum Electron. 18, 1062-1072 (1982).
[CrossRef]

Tanemura, T.

Tchofo Dinda, P.

E. Seve, P. Tchofo Dinda, G. Millot,M. Remoissenet, J.M. Bibault andM. Haelterman, "Modulational instability and critical regime in a highly birefringent fiber," Phys. Rev. A 54, 3519-3534 (1996).
[CrossRef] [PubMed]

Tench, R. E.

R.M. Jopson and R. E. Tench, "Polarisation-independent phase conjugation of lightwave signals," Electron. Lett. 29, 2216-2217 (1993).
[CrossRef]

Terhune, R. W.

P. D. Maker, R. W. Terhune and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

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]

Wabnitz, S.

Wai, P. K. A.

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, K. K. Y.

Xie, C.

Electron. Lett. (1)

R.M. Jopson and R. E. Tench, "Polarisation-independent phase conjugation of lightwave signals," Electron. Lett. 29, 2216-2217 (1993).
[CrossRef]

IEEE J. Quantum Electron. (3)

R. H. Stolen and J. E. Bjorkholm, "Parametric amplification and frequency conversion in optical fibers," IEEE J. Quantum Electron. 18, 1062-1072 (1982).
[CrossRef]

K. Inoue, "Polarization effect on four-wave mixing efficiency in a single-mode fiber," IEEE J. Quantum Electron. 28, 883-894 (1992).
[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. (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]

J. Appl. Phys. (1)

K. O. Hill, D. C. Johnson, B. S. Kawasaki and R. I. MacDonald, "CW three-wave mixing in single-mode optical fibers," J. Appl. Phys. 49, 5098-5106 (1978).
[CrossRef]

J. Lightwave Technol. (2)

K. Inoue, "Polarization independent wavelength conversion using fiber four-wave mixing with two orthogonal pump lights of different frequencies," J. Lightwave Technol. 12, 1916-1920 (1994).
[CrossRef]

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. Opt. Soc. Am. B (3)

Opt. Express (5)

Opt. Lett. (3)

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. A (1)

E. Seve, P. Tchofo Dinda, G. Millot,M. Remoissenet, J.M. Bibault andM. Haelterman, "Modulational instability and critical regime in a highly birefringent fiber," Phys. Rev. A 54, 3519-3534 (1996).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

P. D. Maker, R. W. Terhune and C. M. Savage, "Intensity-dependent changes in the refractive index of liquids," Phys. Rev. Lett. 12, 507-509 (1964).
[CrossRef]

Proc. Nat. Acad. Sci. (1)

J. P. Gordon and H. Kogelnik, "PMD fundamentals: Polarization mode dispersion in optical fibers," Proc. Nat. Acad. Sci. 97, 4541-4550 (2000).
[CrossRef] [PubMed]

Other (2)

S. Radic and C. J. McKinstrie, "Optical amplification and signal processing in highly-nonlinear optical fiber," IEICE Trans. Electron. E88C, 859-869 (2005).</jrn>
[CrossRef]

R. W. Boyd, Nonlinear Optics (Academic Press, 1992).

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Figures (14)

Fig. 1.
Fig. 1.

Polarization diagrams for degenerate FWM driven by two linearly-polarized inputs (1 and 2). Figures 1a (left) and 1b (right) correspond to columns 1 and 2 of Table 4, respectively.

Fig. 2.
Fig. 2.

Polarization diagrams for degenerate FWMdriven by two circularly-polarized inputs (1 and 2). The symbol ◦ signifies that no idler is produced. Figures 2 a and 2 b correspond to columns 3 and 4 of Table 4, respectively.

Fig. 3.
Fig. 3.

Eigenpolarizations of MI driven by a linearly-polarized pump (2). Figures 3 a and 3 b correspond to columns 1 and 2 of Table 5, respectively.

Fig. 4.
Fig. 4.

Eigenpolarizations ofMI driven by a circularly-polarized pump (2). The dashed lines denote sidebands that propagate independently. Figures 4 a and 4 b correspond to columns 3 and 4 of Table 5, respectively.

Fig. 5.
Fig. 5.

Polarization diagrams for nondegenerate FWM driven by three linearly-polarized inputs (1, 2 and 3). Figures 5 a (upper left), 5 b (upper right), 5c (lower left) and 5d (lower right) correspond to columns 1–4 of Table 6, respectively.

Fig. 6.
Fig. 6.

Polarization diagrams for nondegenerate FWM driven by three circularly-polarized inputs (1, 2 and 3). Figures 6a-6d correspond to columns 1–4 of Table 7, respectively.

Fig. 7.
Fig. 7.

Eigenpolarizations of PC driven by parallel linearly-polarized pumps (2 and 3). Figures 7 a and 7 b correspond to columns 1 and 2 of Table 8, respectively.

Fig. 8.
Fig. 8.

Eigenpolarizations of PC driven by perpendicular linearly-polarized pumps (2 and 3). Figures 8 a and 8 b correspond to columns 3 and 4 of Table 8, respectively.

Fig. 9.
Fig. 9.

Eigenpolarizations of PC driven by co-rotating circularly-polarized pumps (2 and 3). Figures 9 a and 9 b correspond to columns 1 and 2 of Table 9, respectively.

Fig. 10.
Fig. 10.

Eigenpolarizations of PC driven by counter-rotating circularly-polarized pumps (2 and 3). Figures 10 a and 10 b correspond to columns 3 and 4 of Table 9, respectively.

Fig. 11.
Fig. 11.

Eigenpolarizations of BS driven by parallel linearly-polarized pumps (1 and 3). Figures 11 a and 11 b correspond to columns 1 and 3 of Table 8, respectively.

Fig. 12.
Fig. 12.

Eigenpolarizations of BS driven by perpendicular linearly-polarized pumps (1 and 3). Figures 12 a and 12 b correspond to columns 2 and 4 of Table 8, respectively.

Fig. 13.
Fig. 13.

Eigenpolarizations of BS driven by co-rotating circularly-polarized pumps (1 and 3). Figures 13 a and 13 b correspond to columns 1 and 3 of Table 9, respectively.

Fig. 14.
Fig. 14.

Eigenpolarizations of BS driven by counter-rotating circularly-polarized pumps (1 and 3). Figures 14 a and 14 b correspond to columns 2 and 4 of Table 9, respectively.

Tables (9)

Tables Icon

Table 1. Table of acronyms, constants and reference frames

Tables Icon

Table 2. Self-phase modulation

Tables Icon

Table 3. Cross-phase modulation

Tables Icon

Table 4. Degenerate FWM driven by two input waves (1 and 2)

Tables Icon

Table 5. MI driven by one pump wave (2)

Tables Icon

Table 6. Nondegenerate FWM with three linearly-polarized input waves (1, 2 and 3)

Tables Icon

Table 7. Nondegenerate FWM with three circularly-polarized input waves (1, 2 and 3)

Tables Icon

Table 8. PC and BS driven by two linearly-polarized pump waves

Tables Icon

Table 9. PC and BS driven by two circularly-polarized pump waves

Equations (66)

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

i z B = β ( i t ) B + γ a ( B B ) B + γ b ( B t B ) B * ,
S 1 = B x 2 B y 2 ,
S 2 = B x B y * + B x * B y ,
S 3 = i ( B x B y * B x * B y ) .
D S = 2 γ b S × S 3 ,
B ( t , z ) = B 1 ( z ) exp ( i ω 1 t ) + B 2 ( z ) exp ( i ω 2 t )
D S 1 = ( γ a + 2 γ b ) S 1 × S 2 2 γ b S 1 × ( S 13 + 2 S 23 ) ,
B ( t , z ) = j = 1 3 B j ( z ) exp ( i ω j t )
D B 1 = i k 1 B 1 + i γ a ( B 3 B 2 ) B 2 + i γ b ( B 2 t B 2 ) B 3 * ,
D B 2 = i k 2 B 2 + i γ a [ ( B 2 B 3 ) B 1 + ( B 2 B 1 ) B 3 ] + 2 i γ b ( B 3 t B 1 ) B 2 * ,
D B 3 = i k 3 B 3 + i γ a ( B 1 B 2 ) B 2 + i γ b ( B 2 t B 2 ) B 1 * ,
D C 3 = i [ γ a ( C 1 C 2 ) C 2 + γ b ( C 2 t C 2 ) C 1 * ] exp ( 2 i δ z ) ,
C 3 ( z ) = C 3 ( 0 ) + i e 3 χ exp ( i δ z ) sin ( δ z ) δ ,
P 3 ( z ) = χ 2 sin 2 ( δ z ) δ 2 .
B 1 ( z ) = C 1 ( z ) exp [ i ( k 1 + 2 k 2 k 3 ) z 2 ] ,
B 3 ( z ) = C 3 ( z ) exp [ i ( k 3 + 2 k 2 k 1 ) z 2 ]
D C 1 = i δ C 1 + i γ a ( C 3 C 2 ) C 2 + i γ b ( C 2 t C 2 ) C 3 * ,
D C 3 = i δ C 3 i γ a ( C 2 C 1 ) C 2 i γ b ( C 2 C 2 * ) C 1 t ,
D C 1 x = i δ C 1 x + i ( γ a + γ b ) C 2 x 2 C 3 x * ,
D C 1 y = i δ C 1 y + i γ b C 2 x 2 C 3 y * ,
D C 3 x * = i δ C 3 x * i ( γ a + γ b ) ( C 2 x * ) 2 C 1 x ,
D C 3 y * = i δ C 3 y * i γ b ( C 2 x * ) 2 C 1 y .
D C 1 = i δ C 1 + i κ C 3 * ,
D C 3 * = i δ C 3 * i κ * C 1 .
C 1 ( z ) = μ ( z ) C 1 ( 0 ) + ν ( z ) C 3 * ( 0 ) ,
C 3 * ( z ) = ν * ( z ) C 1 ( 0 ) + μ * ( z ) C 3 * ( 0 ) ,
μ ( z ) = cos ( k z ) + i δ sin ( k z ) k ,
ν ( z ) = i κ sin ( k z ) k
P 1 ( z ) = P 1 ( 0 ) [ 1 + κ 2 sin 2 ( k z ) k 2 ] ,
P 3 ( z ) = P 1 ( 0 ) κ 2 sin 2 ( k z ) k 2 .
B ( t , z ) = j = 1 4 B j ( z ) exp ( i ω j t )
D B 1 = i k 1 B 1 + i γ a [ ( B 4 B 2 ) B 3 + ( B 4 B 3 ) B 2 ] + i 2 γ b ( B 2 t B 3 ) B 4 * ,
D B 2 = i k 2 B 2 + i γ a [ ( B 3 B 4 ) B 1 + ( B 3 B 1 ) B 4 ] + 2 i γ b ( B 4 t B 1 ) B 3 * ,
D B 3 = i k 3 B 3 + i γ a [ ( B 2 B 4 ) B 1 + ( B 2 B 1 ) B 4 ] + i 2 γ b ( B 4 t B 1 ) B 2 * ,
D B 4 = i k 4 B 4 + i γ a [ ( B 1 B 2 ) B 3 + ( B 1 B 3 ) B 2 ] + i 2 γ b ( B 2 t B 3 ) B 1 * ,
D C 4 = i { γ a [ ( C 1 C 2 ) C 3 + ( C 1 C 3 ) C 2 ] + 2 γ b ( C 2 t C 3 ) C 1 * } exp ( 2 i δ z ) ,
C 4 ( z ) = C 4 ( 0 ) + i e 4 χ exp ( i δ z ) sin ( δ z ) δ ,
P 4 ( z ) = χ 2 sin 2 ( δ z ) δ 2 .
B 1 ( z ) = C 1 ( z ) exp [ i ( k 1 + k 2 + k 3 k 4 ) z 2 ] ,
B 4 ( z ) = C 4 ( z ) exp [ i ( k 4 + k 2 + k 3 k 1 ) z 2 ]
D C 1 = i δ C 1 + i γ a [ ( C 4 C 2 ) C 3 + ( C 4 C 3 ) C 2 ] + i 2 γ b ( C 2 t C 3 ) C 4 * ,
D C 4 = i δ C 4 i γ a [ ( C 3 C 1 ) C 2 + ( C 2 C 1 ) C 3 ] i 2 γ b ( C 3 C 2 * ) C 1 t ,
D C 1 x = i δ C 1 x + i 2 ( γ a + γ b ) C 2 x C 3 x C 4 x * ,
D C 1 y = i δ C 1 y + i 2 γ b C 2 x C 3 x C 4 y * ,
D C 4 x * = i δ C 4 x * i 2 ( γ a + γ b ) C 2 x * C 3 x * C 1 x ,
D C 4 y * = i δ C 4 y * i 2 γ b C 2 x * C 3 x * C 1 y .
D C 1 = i δ C 1 + i κ C 4 * ,
D C 4 * = i δ C 4 * i κ * C 1 ,
P 1 ( z ) = P 1 ( 0 ) [ 1 + κ 2 sin 2 ( k z ) k 2 ] ,
P 4 ( z ) = P 1 ( 0 ) κ 2 sin 2 ( k z ) k 2 .
B 2 ( z ) = C 1 ( z ) exp [ i ( k 1 + k 2 k 3 + k 4 ) z 2 ] ,
B 4 ( z ) = C 4 ( z ) exp [ i ( k 1 + k 2 + k 3 + k 4 ) z 2 ]
D C 2 = i δ C 2 + i γ a [ ( C 3 C 4 ) C 1 + ( C 3 C 1 ) C 4 ] + i 2 γ b ( C 4 t C 1 ) C 3 * ,
D C 4 = i δ C 4 + i γ a [ ( C 1 C 2 ) C 3 + ( C 1 C 3 ) C 2 ] + i 2 γ b ( C 2 t C 3 ) C 1 * ,
D C 2 + = i δ C 2 + i 2 γ a C 1 + C 3 + * C 4 + ,
D C 2 = i δ C 2 + i ( γ a + 2 γ b ) C 1 + C 3 + * C 4 ,
D C 4 + = i δ C 4 + + i 2 γ a C 3 + C 1 + * C 2 + ,
D C 4 = i δ C 4 + i ( γ a + 2 γ b ) C 3 + C 1 + * C 2 ,
D C 2 = i δ C 2 + i κ C 4 ,
D C 4 = i δ C 4 + i κ * C 2 ,
C 2 ( z ) = μ ( z ) C 2 ( 0 ) + ν ( z ) C 4 ( 0 ) ,
C 4 ( z ) = ν * ( z ) C 2 ( 0 ) + μ * ( z ) C 4 ( 0 ) ,
μ ( z ) = cos ( k z ) i δ sin ( k z ) k ,
ν ( z ) = i κ sin ( k z ) k
P 2 ( z ) = P 2 ( 0 ) [ 1 κ 2 sin 2 ( k z ) k 2 ] ,
P 4 ( z ) = P 2 ( 0 ) κ 2 sin 2 ( k z ) k 2 .

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