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

Full Article  |  PDF Article

References

  • View by:
  • |

  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. Technol. 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, 23�??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�??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]

Electron. Lett.

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]

European Conf. on Optical Comm. 2003

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.

IEEE J. Quantum Electron.

C. R. Menyuk, �??Nonlinear pulse propagation in birefringent optical fibers,�?? IEEE J. Quantum Electron. 23, 174�??176 (1987).
[CrossRef]

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

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. Technol. Lett.

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]

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. Technol. Lett. 14, 1406�??1408 (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]

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]

J. Appl. Phys.

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.

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

OFC 2003

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, 23�??28 March 2003, paper ThK3.

Opt. Express.

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.

Optical Fiber Telecommunications IVB

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�??861.

OSA Annual Meeting 2003

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.

Phys. Rev. E

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]

Pro. IRE

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]

Proc. IRE

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

Proc. Nat. Acad. Sci.

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

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

Other

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

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


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 ) .

Metrics