Abstract

Parametric amplification is made possible by four-wave mixing. In low-birefringence fibers the birefringence axes and strength vary randomly with distance. Light-wave propagation in such fibers is governed by the Manakov equation. In this paper the Manakov equation is used to study degenerate and nondegenerate four-wave mixing. The effects of linear and nonlinear wavenumber mismatches, and nonlinear polarization rotation, are included in the analysis. Formulas are derived for the initial quadratic growth of the idler power, and the subsequent exponential growth of the signal and idler powers (which continues until pump depletion occurs). These formulas are valid for arbitrary pump and signal polarizations.

© 2004 Optical Society of America

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  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]
  2. R. H. Stolen and J. E. Bjorkholm, “Parametric amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. 18, 1062–1072 (1982).
    [CrossRef]
  3. J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
    [CrossRef]
  4. S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
    [CrossRef]
  5. 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) and references therein.
    [CrossRef]
  6. 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) and references therein.
    [CrossRef]
  7. C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
    [CrossRef] [PubMed]
  8. C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
    [CrossRef]
  9. K. Inoue, “Polarization effect on four-wave mixing efficiency in a single-mode fiber,” IEEE J. Quantum Electron. 28, 883–894 (1992).
    [CrossRef]
  10. R. M. Jopson and R. E. Tench, “Polarisation-independent phase conjugation of lightwave signals,” Electron. Lett. 29, 2216–2217 (1993).
    [CrossRef]
  11. 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]
  12. K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
    [CrossRef]
  13. S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
    [CrossRef]
  14. T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photon. Technol. Lett. 15, 1573–1575 (2003).
    [CrossRef]
  15. S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.
  16. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  17. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. 23, 174–176 (1987).
    [CrossRef]
  18. 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]
  19. S. G. Evanglides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing with solitons,” J. Lightwave Technol.,  10, 28–35 (1992).
    [CrossRef]
  20. P. K. A. Wai and C. R. Menyuk, “Polarization mode dispersion, decorrelation and diffusion in optical fibers with randomly varying birefringence,” J. Lightwave Technol. 14, 148–157 (1996).
    [CrossRef]
  21. T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996).
    [CrossRef]
  22. H. Kogelnik, R. M. Jopson, and L. E. Nelson, “Polarization-mode dispersion,” in Optical Fiber Telecommunications IVB, edited by I. P. Kaminow and T. Li (Academic, San Diego,2002), pp. 725𠄿861.
  23. M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
    [CrossRef]
  24. J. P. Gordon and H. Kogelnik, “PMD fundamentals: Polarization mode dispersion in optical fibers,” Proc. Nat. Acad. Sci. 97, 4541–4550 (2000).
    [CrossRef] [PubMed]
  25. L. F. Mollenauer, J. P. Gordon, and F. Heismann, “Polarization scattering by soliton-soliton collisions,” Opt. Lett. 20, 2060–2062 (1995).
    [CrossRef] [PubMed]
  26. D. Wang and C. R. Menyuk, “Reduced model of the evolution of the polarization states in wavelength-division-multiplexed channels,” Opt. Lett. 23, 1677–1679 (1998).
    [CrossRef]
  27. R. W. Boyd, Nonlinear Optics (Academic, San Diego,1992), Sections 1.4, 2.3 and 6.5.
  28. J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
    [CrossRef]
  29. M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).
  30. M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
    [CrossRef]
  31. C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993) and references therein.
    [CrossRef]
  32. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
    [CrossRef]
  33. Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.
  34. K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
    [CrossRef]
  35. T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Widely tunable wavelength conversion using nondegenerate fiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communications, Rimini, Italy, 21–25 September 2003, paper We3.7.3.
  36. G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt. Soc. Am. B 9, 1047–1061 (1992).
    [CrossRef]

2003 (4)

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[CrossRef] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[CrossRef]

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photon. Technol. Lett. 15, 1573–1575 (2003).
[CrossRef]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[CrossRef]

2002 (4)

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (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) and references therein.
[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) and references therein.
[CrossRef]

2001 (1)

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[CrossRef]

2000 (2)

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
[CrossRef]

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

1998 (2)

1996 (2)

T. I. Lakoba, “Concerning the equations governing nonlinear pulse propagation in randomly birefringent fibers,” J. Opt. Soc. Am. B 13, 2006–2011 (1996).
[CrossRef]

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

1995 (1)

1994 (2)

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
[CrossRef]

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 (3)

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

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993) and references therein.
[CrossRef]

1992 (3)

G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt. Soc. Am. B 9, 1047–1061 (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]

S. G. Evanglides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing 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]

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]

1974 (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1957 (1)

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

1956 (1)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Agrawal, G. P.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.

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) and references therein.
[CrossRef]

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[CrossRef]

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
[CrossRef]

Benedetto, S.

Bergano, N. S.

S. G. Evanglides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing 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]

Boyd, R. W.

R. W. Boyd, Nonlinear Optics (Academic, San Diego,1992), Sections 1.4, 2.3 and 6.5.

Brar, K.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[CrossRef]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Brentel, J.

Cao, X. D.

Carena, A.

Centanni, J. C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[CrossRef]

Chen, H. H.

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) and references therein.
[CrossRef]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[CrossRef]

Curri, V.

Evanglides, S. G.

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

Gaudino, R.

Goh, C. S.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Widely tunable wavelength conversion using nondegenerate fiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communications, Rimini, Italy, 21–25 September 2003, paper We3.7.3.

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. Evanglides, L. F. Mollenauer, J. P. Gordon, and N. S. Bergano, “Polarization multiplexing 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) and references therein.
[CrossRef]

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[CrossRef]

Headley, C.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[CrossRef]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

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) and references therein.
[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, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
[CrossRef]

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.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Jopson, R. M.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[CrossRef]

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

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

Jorgensen, C.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Jorgensen, C. G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[CrossRef]

Karlsson, M.

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.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

Kikuchi, K.

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photon. Technol. Lett. 15, 1573–1575 (2003).
[CrossRef]

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Widely tunable wavelength conversion using nondegenerate fiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communications, Rimini, Italy, 21–25 September 2003, paper We3.7.3.

Kogelnik, H.

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

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

Lakoba, T. I.

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) and references therein.
[CrossRef]

Li, J. S.

Lin, Q.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[CrossRef]

Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.

Luther, G. G.

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]

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Manley, J. M.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Marhic, M. E.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

McKinstrie, C. J.

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[CrossRef]

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[CrossRef] [PubMed]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[CrossRef]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (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) and references therein.
[CrossRef]

C. J. McKinstrie, X. D. Cao, and J. S. Li, “Nonlinear detuning of four-wave interactions,” J. Opt. Soc. Am. B 10, 1856–1869 (1993) and references therein.
[CrossRef]

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

G. G. Luther and C. J. McKinstrie, “Transverse modulational instability of counterpropagating waves,” J. Opt. Soc. Am. B 9, 1047–1061 (1992).
[CrossRef]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Menyuk, C. R.

D. Wang and C. R. Menyuk, “Reduced model of the evolution of the polarization states in wavelength-division-multiplexed channels,” Opt. Lett. 23, 1677–1679 (1998).
[CrossRef]

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

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.

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

Nelson, L. E.

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

Poggiolini, P.

Radic, S.

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[CrossRef] [PubMed]

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[CrossRef]

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[CrossRef]

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (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) and references therein.
[CrossRef]

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

Raybon, G.

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[CrossRef]

Rowe, H. E.

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

Set, S. Y.

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Widely tunable wavelength conversion using nondegenerate fiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communications, Rimini, Italy, 21–25 September 2003, paper We3.7.3.

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.

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photon. Technol. Lett. 15, 1573–1575 (2003).
[CrossRef]

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Widely tunable wavelength conversion using nondegenerate fiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communications, Rimini, Italy, 21–25 September 2003, paper We3.7.3.

Tench, R. E.

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

Uesaka, K.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

Wai, P. K. A.

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

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]

Wang, D.

Weiss, M. T.

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

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) and references therein.
[CrossRef]

Wong, K. K. Y.

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

Xie, C.

C. J. McKinstrie, S. Radic, and C. Xie, “Phase conjugation driven by orthogonal pump waves in birefringent fibers,” J. Opt. Soc. Am. B 20, 1437–1446 (2003).
[CrossRef]

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[CrossRef] [PubMed]

Yu, M.

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

Electron. Lett. (2)

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

S. Radic, C. J. McKinstrie, R. M. Jopson, Q. Lin, and G. P. Agrawal, “Record performance of a parametric amplifier constructed with highly-nonlinear fiber,” Electron. Lett. 39, 838–839 (2003).
[CrossRef]

IEEE J. Quantum Electron. (3)

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]

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

IEEE J. Sel. Top. Quantum Electron. (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) and references therein.
[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) and references therein.
[CrossRef]

IEEE Photon. Tech-nol. Lett. (1)

S. Radic, C. J. McKinstrie, A. R. Chraplyvy, G. Raybon, J. C. Centanni, C. G. Jorgensen, K. Brar, and C. Headley, “Continuous-wave parametric-gain synthesis using nondegenerate-pump four-wave mixing,” IEEE Photon. Tech-nol. Lett. 14, 1406–1408 (2002).
[CrossRef]

IEEE Photon. Technol. Lett. (4)

J. Hansryd and P. A. Andrekson, “Broadband continuous-wave pumped fiber optical parametric amplifier with 49-dB gain and wavelength-conversion efficiency,” IEEE Photon. Technol. Lett. 13, 194–196 (2001).
[CrossRef]

K. K. Y. Wong, M. E. Marhic, K. Uesaka, and L. G. Kazovsky, “Polarization-independent two-pump fiber optical parametric amplifier,” IEEE Photon. Technol. Lett. 14, 911–913 (2002).
[CrossRef]

T. Tanemura and K. Kikuchi, “Polarization-independent broad-band wavelength conversion using two-pump fiber optical parametric amplification without idler spectral broadening,” IEEE Photon. Technol. Lett. 15, 1573–1575 (2003).
[CrossRef]

K. Inoue, “Tunable and selective wavelength conversion using fiber four-wave mixing with two pump lights,” IEEE Photon. Technol. Lett. 6, 1451–1453 (1994).
[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. (5)

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

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

M. Karlsson, J. Brentel, and P. A. Andrekson, “Long-term measurement of PMD and polarization drift in installed fibers,” J. Lightwave Technol. 18, 941–951 (2000).
[CrossRef]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[CrossRef]

J. Opt. Soc. Am. B (4)

Opt. Express. (1)

C. J. McKinstrie, S. Radic, and C. Xie, “Parametric instabilities driven by orthogonal pump waves in birefringent fibers,” Opt. Express. 11, 2619–2633 (2003) and references therein.
[CrossRef] [PubMed]

Opt. Lett. (3)

Phys. Rev. E (1)

M. Yu, C. J. McKinstrie, and G. P. Agrawal, “Instability due to cross-phase modulation in the normal dispersion regime,” Phys. Rev. E 48, 2178–2186 (1993).
[CrossRef]

Proc. IRE (2)

J. M. Manley and H. E. Rowe, “Some general properties of nonlinear elements—Part I. General energy relations,” Proc. IRE 44, 904–913 (1956).
[CrossRef]

M. T. Weiss, “Quantum derivation of energy relations analogous to those for nonlinear reactances,” Proc. IRE 45, 1012–1013 (1957).

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]

Sov. Phys. JETP (1)

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Other (5)

T. Tanemura, C. S. Goh, K. Kikuchi, and S. Y. Set, “Widely tunable wavelength conversion using nondegenerate fiber four-wave mixing driven by co-modulated pump waves,” European Conference on Optical Communications, Rimini, Italy, 21–25 September 2003, paper We3.7.3.

Q. Lin and G. P. Agrawal, “Effects of polarization-mode dispersion on fiber-based parametric amplification and wavelength conversion,” Annual Meeting of the Optical Society of America, Tucson, Arizona, 5–9 October 2003, paper TuP3.

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

R. W. Boyd, Nonlinear Optics (Academic, San Diego,1992), Sections 1.4, 2.3 and 6.5.

S. Radic, C. J. McKinstrie, R. Jopson, C. Jorgensen, K. Brar, and C. Headley, “Polarization-dependent parametric gain in amplifiers with orthogonally multiplexed pumps, Optical Fiber Communication conference, Atlanta, Georgia, 232013;28 March 2003, paper ThK3.

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Figures (7)

Fig. 1.
Fig. 1.

Polarization diagrams for degenerate FWM driven by two input waves. The symbols ∥ and ⊥ should be interpreted in the sense of Jones vectors, and the symbol ○ signifies that no idler is produced. Figures 1a and 1b correspond to rows 1 and 2 of Table 1, respectively.

Fig. 2.
Fig. 2.

Eigenpolarizations of MI. The dashed lines denote sidebands that propagate independently.

Fig. 3.
Fig. 3.

Polarization diagrams for nondegenerate FWM driven by three input waves. The symbols ∥ and ⊥ should be interpreted in the sense of Jones vectors, and the symbol ○ signifies that no idler is produced. Figures 3a–3d correspond to rows 1–4 of Table 2, respectively.

Fig. 4.
Fig. 4.

Eigenpolarizations of PC driven by parallel pumps. The dashed lines denote sidebands that propagate independently.

Fig. 5.
Fig. 5.

Eigenpolarizations of PC driven by perpendicular pumps.

Fig. 6.
Fig. 6.

Eigenpolarizations of BS driven by parallel pumps.

Fig. 7.
Fig. 7.

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

Tables (2)

Tables Icon

Table 1. Properties of degenerate FWM driven by two input waves

Tables Icon

Table 2. Properties of nondegenerate FWM driven by three input waves

Equations (144)

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

E x ( t , z ) = A x ( t , z ) exp [ i ( k 0 z ω 0 t ) ] + c . c . ,
E y ( t , z ) = A y ( t , z ) exp [ i ( k 0 z ω 0 t ) ] + c . c .
i z A x = β x ( i t ) A x + γ ( A x 2 + 2 A y 2 3 ) A x + γ A y 2 A x * 3 ,
i z A y = β y ( i t ) A y + γ ( 2 A x 2 3 + A y 2 ) A y + γ A x 2 A y * 3 ,
i z B x = β x ( i t ) B x + γ ( B x 2 + 2 B y 2 3 ) B x ,
i z B y = β y ( i t ) B y + γ ( 2 B x 2 3 B y 2 ) B y ,
i z A ξ = β ( i t ) A ξ + ( 8 γ 9 ) ( A ξ 2 + A η 2 ) A ξ ,
i z A η = β ( i t ) A η + ( 8 γ 9 ) ( A ξ 2 + A η 2 ) A η ,
a 1 · a 2 = a 1 · a 2 in exp ( δ ω 2 δ τ 2 3 ) ,
i z A = β ( i t ) A + γ ¯ A A A ,
A = exp ( i ω 1 t ) A 1 + exp ( i ω 2 t ) A 2
D A 1 = i β 1 A 1 + i γ [ ( P 1 + P 2 ) A 1 + A 2 A 1 A 2 ] ,
D A 2 = i β 2 A 2 + i γ [ A 1 A 2 A 1 + ( P 1 + P 2 ) A 2 ] ,
D A j = i H j A j ,
H j = [ β j + γ ( P j + 3 P k 2 ) ] I + γ a k · σ 2
H j = [ β j + γ ( P j + 3 P k 2 ) ] I + γ a t · σ 2 ,
D A j A k = i A j ( H k H j ) A j
= i [ β k β j + γ ( P j P k ) ] A j A k .
D A j σ A j = i A j [ σ , H j ] A j ,
D a j = a j × a t .
D a t = 0 ,
D ( a j · a k ) = 0 .
θ j ( z ) = β j z + γ 0 z [ P j ( z ) + 3 P k ( z ) ] dz 2 .
D B j = iH t B j ,
B j ( z ) = U t ( z ) B j ( 0 ) .
A j ( z ) = exp ( i H t z + i θ j ) A j ( 0 ) .
A = j = 1 3 exp ( i ω j t ) A j
D A 1 = i H 1 A 1 + A 3 A 2 A 2 ,
D A 2 = i H 2 A 2 + ( A 2 A 3 A 1 + A 2 A 1 A 3 ) .
H j = [ β j + γ ( P j + 3 Σ P k 2 ) ] I + γ a t · σ 2 ,
D A 1 A 1 = ( A 3 A 2 A 1 A 2 A 2 A 3 A 2 A 1 ) ,
D A 2 A 2 = 2 ( A 2 A 3 A 2 A 1 A 3 A 2 A 1 A 2 ) .
D ( P 1 P 3 ) = 0 ,
D ( P 1 + P 2 + P 3 ) = 0 .
D A 1 σ A 1 i A 1 [ σ , H t ] A 1 = ( A 3 A 2 A 1 σ A 2 A 2 A 3 A 2 σ A 1 ) ,
D A 2 σ A 2 i A 2 [ σ , H t ] A 2 = ( A 2 A 3 A 2 σ A 1 A 3 A 2 A 1 σ A 2
+ A 2 A 1 A 2 σ A 3 A 1 A 2 A 3 σ A 2 ) .
D a t = 0 .
D A l = iH l A l ,
D A 3 = iH 3 A 3 + A 1 A 2 A 2 ,
D B l = 0 ,
D B 3 = B 1 B 2 B 2 exp [ i ( 2 θ 2 θ 3 θ 1 ) ] .
P 3 ( z ) = Γ ( z ) P 2 A 1 A 2 2 ,
Γ ( z ) = γ 2 sin 2 ( kz ) k 2
k = [ β 1 2 β 2 + β 3 + γ ( 2 P 2 P 1 ) ] 2
P 3 ( z ) = Γ ( z ) P 1 P 2 2 ( 1 + e 1 · e 2 ) 2 ,
D A 1 = iH 1 A 1 + A 3 A 2 A 2 ,
D A 2 = iH 2 A 2 ,
D A 3 = iH 3 A 3 + A 1 A 2 A 2 .
D B 1 = i ( β 1 β 2 + γP 2 ) B 1 + B 3 B 2 B 2 ,
D B 2 = 0 .
D B 3 = i ( β 3 β 2 + γP 2 ) B 3 + B 1 B 2 B 2 .
D 1 B 1 = B 3 B 2 B 2 ,
D 3 * B 3 = B 2 B 1 B 2 .
( D 3 * D 1 I γ 2 P 2 B 2 B 2 ) B 1 = 0 .
[ D 3 * D 1 γ 2 P 2 2 0 0 D 3 * D 1 ] [ S S ] = 0 .
k ± = ( δ k 1 δ k 3 ) 2 ± [ ( δ k 1 + δ k 3 ) 2 4 γ 2 P 2 2 ] 1 2 ,
κ = [ γ 2 P 2 2 ( δβ + γ P 2 ) 2 ] 1 2 .
A = j = 1 4 exp ( i ω j t ) A j
D A 1 = iH 1 A 1 + ( A 4 A 2 A 3 + A 4 A 3 A 2 ) ,
D A 2 = iH 2 A 2 + ( A 3 A 4 A 1 + A 3 A 1 A 4 ) ,
D A 1 A 1 = ( A 4 A 2 A 1 A 3 + A 4 A 3 A 1 A 2
A 2 A 4 A 3 A 1 A 3 A 4 A 2 A 1 ) ,
D A 2 A 2 = ( A 3 A 4 A 2 A 1 + A 3 A 1 A 2 A 4
A 4 A 3 A 1 A 2 A 1 A 3 A 4 A 2 ) .
D ( P 1 P 4 ) = 0 ,
D ( P 2 P 3 ) = 0 ,
D ( P 1 + P 2 + P 3 + P 4 ) = 0 ,
D A 1 σ A 1 γ A 1 σ × a t A 1 = ( A 4 A 2 A 1 σ A 3 + A 4 A 3 A 1 σ A 2 )
A 2 A 4 A 3 σ A 1 A 3 A 4 A 2 σ A 1 ) ,
D A 2 σ A 2 γ A 2 σ × a t A 2 = ( A 3 A 4 A 2 σ A 1 + A 3 A 1 A 2 σ A 4 )
A 4 A 3 A 1 σ A 2 A 1 A 3 A 4 σ A 2 ) .
D ( A 1 σ A 1 + A 2 σ A 2 ) = γ ( A 1 σ × a t A 1 + A 1 σ × a t A 1 )
+ ( A 4 A 2 A 1 σ A 3 A 2 A 4 A 3 σ A 1
+ A 3 A 1 A 2 σ A 4 A 1 A 3 A 4 σ A 2 ) .
D a t = 0 .
D A l = iH l A l ,
D A 4 = iH 4 A 4 + ( A 1 A 2 A 3 + A 1 A 3 A 2 ) ,
D B l = 0 ,
D B 4 = ( B 1 B 2 B 3 + B 1 B 3 B 2 ) exp [ i ( θ 2 + θ 3 θ 4 θ 1 ) ] .
P 4 ( z ) = Γ ( z ) ( P 2 A 3 A 1 2 + P 3 A 1 A 2 2
+ A 2 A 1 A 1 A 3 A 3 A 2 + A 3 A 1 A 1 A 2 A 2 A 3 ) ,
k = [ β 1 β 2 β 3 + β 4 + γ ( P 2 + P 3 P 1 ) ] 2 .
P 4 ( z ) = Γ ( z ) P 1 P 2 P 3 ( 3 + 2 e 1 · e 2 + e 2 · e 3 + 2 e 3 · e 1 ) 2 ,
D A 1 = iH 1 A 1 + ( A 4 A 2 A 3 + A 4 A 3 A 2 ) ,
D A l = iH l A l ,
D A 4 = iH 4 A 4 + ( A 1 A 2 A 3 + A 1 A 3 A 2 ) ,
D B 1 = i ( β 1 β 2 + γP 2 ) B 1 + ( B 4 B 2 B 3 + B 4 B 3 B 2 ) ,
D B l = 0 ,
D B 4 = i ( β 4 β 3 + γP 3 ) B 4 + ( B 1 B 2 B 3 + B 1 B 3 B 2 ) .
D 1 B 1 = ( B 4 B 2 B 3 + B 4 B 3 B 2 ) ,
D 4 * B 4 = ( B 2 B 1 B 3 + B 3 B 1 B 2 ) .
[ D 4 * D 1 I γ 2 ( P 2 B 3 B 3 + P 3 B 2 B 2
+ B 2 B 3 B 2 B 3 + B 3 B 2 B 3 B 2 ) ] B 1 = 0 .
[ D 4 * D 1 γ 2 P 2 P 3 ( 1 + 3 B 2 ) 2 γ 2 P 2 P 3 B B * 2 γ 2 P 2 P 3 B * B D 4 * D 1 γ 2 P 2 P 3 B 2 ] [ S S ] = 0 .
k ± = ( δ k 1 δ k 4 ) 2 ± [ ( δ k 1 + δ k 4 ) 2 4 γ 2 P 2 P 3 Δ ± ] 1 2 ,
Δ ± = ( 1 ± B ) 2
κ ± = { γ 2 P 2 P 3 Δ ± [ δβ + γ ( P 2 + P 3 ) 2 ] 2 } 1 2 .
S S ± 2 = [ ( 1 B ) ( 1 + B ) ] ±1
S ± 2 = ( 1 ± B ) 2 .
Δ ± = ( 3 + e 2 · e 3 ) 2 ± [ 2 ( 1 + e 2 · e 3 ) ] 1 2 ,
( e 1 · e 2 ) ± = ± [ ( 1 + e 2 · e 3 ) 2 ] 1 2 ,
( e 4 · e 2 ) ± = ± [ ( 1 + e 2 · e 3 ) 2 ] 1 2 .
D A l = iH l A l ,
D A 2 = iH 2 A 2 + ( A 3 A 4 A 1 + A 3 A 1 A 4 ) ,
D A 4 = iH 4 A 4 + ( A 1 A 2 A 3 + A 1 A 3 A 2 ) ,
D B l = 0 ,
D B 2 = i ( β 2 β 1 + γP 1 ) B 2 + ( B 3 B 4 B 1 + B 3 B 1 B 4 ) ,
D B 4 = i ( β 4 β 3 + γP 3 ) B 4 + ( B 1 B 2 B 3 + B 1 B 3 B 2 ) .
D 2 B 2 = ( B 3 B 4 B 1 + B 3 B 1 B 4 ) ,
D 4 B 4 = ( B 1 B 2 B 3 + B 1 B 3 B 2 ) .
[ D 4 D 2 I + γ 2 ( P 3 B 1 B 1 + B 1 B 3 2
+ B 3 B 1 B 3 B 1 + B 1 B 3 B 1 B 3 ) ] B 2 = 0 .
[ D 4 D 2 + γ 2 P 1 P 3 ( 1 + 3 B 2 ) γ 2 P 1 P 3 B B * γ 2 P 1 P 3 B * B D 4 D 2 + γ 2 P 1 P 3 ( B 2 ) ] [ S S ] = 0 .
k ± = ( δ k 2 + δ k 4 ) 2 ± [ ( δ k 2 δ k 4 ) 2 4 γ 2 P 1 P 3 Δ ± ] 1 2 ,
Δ ± = [ ( 1 + 4 B 2 ) ± ( 1 + 8 B 2 ) 1 2 ] 2
S S ± 2 = ( 1 + 8 B 2 ) 1 2 ± ( 1 + 2 B 2 ) ( 1 + 8 B 2 ) 1 2 ( 1 + 2 B 2 )
S ± 2 = ( 1 + 8 B 2 ) 1 2 ( 1 + 2 B 2 ) 2 ( 1 + 8 B 2 ) 1 2 .
[ D 2 D 4 I + γ 2 ( P 1 B 3 B 3 + B 3 B 1 2
+ B 1 B 3 B 1 B 3 + B 3 B 1 B 3 B 1 ) ] B 2 = 0 .
[ D 2 D 4 + 4 γ 2 P 1 P 3 B 2 2 γ 2 P 1 P 3 B B * 2 γ 2 P 1 P 3 B * B D 2 D 4 + γ 2 P 1 P 3 ] [ I I ] = 0 .
I I ± 2 = ( 1 + 8 B 2 ) 1 2 ± ( 4 B 2 1 ) ( 1 + 8 B 2 ) 1 2 ( 4 B 2 1 )
I ± 2 = ( 1 + 8 B 2 ) 1 2 ( 4 B 2 1 ) 2 ( 1 + 8 B 2 ) 1 2 .
Δ ± = [ ( 3 + 2 e 1 · e 3 ) ± ( 5 + 4 e 1 · e 3 ) 1 2 ] 2 ,
( e 2 · e 1 ) ± = ( 2 + e 1 · e 3 ) ( 5 + 4 e 1 · e 3 ) 1 2 ·
( e 4 · e 1 ) ± = ( 1 + 2 e 1 · e 3 ) ( 5 + 4 e 1 · e 3 ) 1 2 .
D C 1 = ( C 4 C 2 C 3 + C 4 C 3 C 2 ) ,
D + C 4 = ( C 2 C 1 C 3 + C 3 C 1 C 2 ) ,
[ D 0 2 iγP B iγP B 0 D iγP B 0 2 iγP B iγP B D + 0 iγP B 0 0 D + ] [ S S I * I * ] = 0 ,
κ ± = ± ( γ 2 P 2 Δ ± δ + 2 ) 1 2 ,
Δ 2 2 ( 1 + B 2 ) Δ + B 4 = 0 .
Δ ± = ( 1 ± B ) 2 ,
( S ) ± = ± B 1 ± B ,
( I * ) ± = i ( κ ± i δ + ) γP ( 1 + B ) ,
( I * ) ± = iγP B κ ± + i δ + .
D C 2 = ( C 3 C 4 C 1 + C 3 C 1 C 4 ) ,
D + C 4 = ( C 1 C 2 C 3 + C 1 C 3 C 2 ) ,
[ D 0 2 iγP B iγP B 0 D 0 iγP B 2 iγP B 0 D + 0 iγP B iγP B 0 D + ] [ S S I I ] = 0 .
k ± = ± ( γ 2 P 2 Δ ± + δ 2 ) 1 2 ,
Δ 2 ( 1 + 4 B 2 ) Δ + 4 B 4 = 0 .
Δ ± = [ 1 + 4 B 2 ( 1 + 8 B 2 ) 1 2 ] 2 .
( S ) ± = B B Δ ± B 2 ,
( I ) ± = 2 γP B k ± + δ ,
( I ) ± = ( k ± δ ) B γP ( Δ ± B 2 ) .

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