Abstract

Exact formulas are obtained for the amplitudes of light waves involved in four-wave-mixing cascades near the zero-dispersion frequency of a fiber. The cascade that is initiated by two strong pump waves is phase insensitive, whereas the cascade that is initiated by two strong pump waves and a weak signal wave is phase sensitive. In both cascades, the number of waves that have significant power increases with distance.

© 2006 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, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Electron. E88C, 859–869 (2005).
    [CrossRef]
  3. 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]
  4. C. J. McKinstrie and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973–4979 (2004).
    [CrossRef] [PubMed]
  5. 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]
  6. C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
    [CrossRef]
  7. I. P. Kaminow and T. L. Koch, Editors, Optical Fiber Telecommunications IIIA and IIIB (Academic Press, 1997).
  8. S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D: Appl. Phys. 35, R43–R59 (2002).
    [CrossRef]
  9. I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1994), pp. 987 and 994.
  10. G. Cappellini and S. Trillo, “Third-order three-wave mixing in single-mode fibers: Exact solutions and spatial instability effects,” J. Opt. Soc. Am. B 8, 824–838 (1991).
    [CrossRef]
  11. 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).
    [CrossRef]
  12. B. I. Cohen, A. N. Kaufman, and K. M. Watson, “Beat heating of a plasma,” Phys. Rev. Lett. 29, 581–584 (1972).
    [CrossRef]
  13. S. J. Karttunen and R. R. E. Salomaa, “Electromagnetic field cascading in the beat-wave generation of plasma waves,” Phys. Rev. Lett. 56, 604–607 (1986).
    [CrossRef] [PubMed]
  14. K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
    [CrossRef]
  15. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
    [CrossRef]
  16. R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
    [CrossRef]

2006 (1)

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

2005 (3)

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Electron. E88C, 859–869 (2005).
[CrossRef]

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]

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

2004 (1)

2002 (3)

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]

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D: Appl. Phys. 35, R43–R59 (2002).
[CrossRef]

1995 (1)

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

1994 (1)

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

1993 (1)

1991 (1)

1986 (1)

S. J. Karttunen and R. R. E. Salomaa, “Electromagnetic field cascading in the beat-wave generation of plasma waves,” Phys. Rev. Lett. 56, 604–607 (1986).
[CrossRef] [PubMed]

1972 (1)

B. I. Cohen, A. N. Kaufman, and K. M. Watson, “Beat heating of a plasma,” Phys. Rev. Lett. 29, 581–584 (1972).
[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]

Cao, X. D.

Cappellini, G.

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]

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

Cohen, B. I.

B. I. Cohen, A. N. Kaufman, and K. M. Watson, “Beat heating of a plasma,” Phys. Rev. Lett. 29, 581–584 (1972).
[CrossRef]

Cundiff, S. T.

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D: Appl. Phys. 35, R43–R59 (2002).
[CrossRef]

Derosier, R. M.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

Devgan, P.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Forghieri, F.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

Gnauck, A. H.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

Gradsteyn, I. S.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1994), pp. 987 and 994.

Grigoryan, V. S.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[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]

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]

Inoue, K.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

Karttunen, S. J.

S. J. Karttunen and R. R. E. Salomaa, “Electromagnetic field cascading in the beat-wave generation of plasma waves,” Phys. Rev. Lett. 56, 604–607 (1986).
[CrossRef] [PubMed]

Kaufman, A. N.

B. I. Cohen, A. N. Kaufman, and K. M. Watson, “Beat heating of a plasma,” Phys. Rev. Lett. 29, 581–584 (1972).
[CrossRef]

Kumar, P.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

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, J. S.

McKinstrie, C. J.

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Electron. E88C, 859–869 (2005).
[CrossRef]

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 and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973–4979 (2004).
[CrossRef] [PubMed]

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]

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).
[CrossRef]

Nakanishi, K.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

Oda, K.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

Radic, S.

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Electron. E88C, 859–869 (2005).
[CrossRef]

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 and S. Radic, “Phase-sensitive amplification in a fiber,” Opt. Express 12, 4973–4979 (2004).
[CrossRef] [PubMed]

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]

Raymer, M. G.

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

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]

Ryzhik, I. M.

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1994), pp. 987 and 994.

Salomaa, R. R. E.

S. J. Karttunen and R. R. E. Salomaa, “Electromagnetic field cascading in the beat-wave generation of plasma waves,” Phys. Rev. Lett. 56, 604–607 (1986).
[CrossRef] [PubMed]

Tang, R.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Tkach, R. W.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

Toba, H.

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

Trillo, S.

Vasilyev, M. V.

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

Voss, P. L.

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

Watson, K. M.

B. I. Cohen, A. N. Kaufman, and K. M. Watson, “Beat heating of a plasma,” Phys. Rev. Lett. 29, 581–584 (1972).
[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]

Yu, M.

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]

IEEE J. Sel. Top. Quantum. Electron. (1)

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]

IEEE Photon. Technol. Lett. (1)

R. Tang, P. Devgan, P. L. Voss, V. S. Grigoryan, and P. Kumar, “In-line frequency-nondegenerate phase-sensitive fiber-optical parametric amplifier,” IEEE Photon. Technol. Lett. 17, 1845–1847 (2005).
[CrossRef]

IEICE Trans. Electron. (1)

S. Radic and C. J. McKinstrie, “Optical amplification and signal processing in highly-nonlinear optical fiber,” IEICE Trans. Electron. E88C, 859–869 (2005).
[CrossRef]

J. Lightwave Technol. (2)

K. Inoue, K. Nakanishi, K. Oda, and H. Toba, “Crosstalk and power penalty due to fiber four-wave mixing in multichannel transmissions,” J. Lightwave Technol. 12, 1423–1439 (1994).
[CrossRef]

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, and R. M. Derosier, “Four-photon mixing and high-speed WDM systems,” J. Lightwave Technol. 13, 841–849 (1995).
[CrossRef]

J. Opt. Soc. Am. B (2)

J. Phys. D: Appl. Phys. (1)

S. T. Cundiff, “Phase stabilization of ultrashort optical pulses,” J. Phys. D: Appl. Phys. 35, R43–R59 (2002).
[CrossRef]

Opt. Commun. (1)

C. J. McKinstrie, M. G. Raymer, S. Radic, and M. V. Vasilyev, “Quantum mechanics of phase-sensitive amplification in a fiber,” Opt. Commun. 257, 146–163 (2006).
[CrossRef]

Opt. Express (2)

Phys. Rev. Lett. (2)

B. I. Cohen, A. N. Kaufman, and K. M. Watson, “Beat heating of a plasma,” Phys. Rev. Lett. 29, 581–584 (1972).
[CrossRef]

S. J. Karttunen and R. R. E. Salomaa, “Electromagnetic field cascading in the beat-wave generation of plasma waves,” Phys. Rev. Lett. 56, 604–607 (1986).
[CrossRef] [PubMed]

Other (2)

I. P. Kaminow and T. L. Koch, Editors, Optical Fiber Telecommunications IIIA and IIIB (Academic Press, 1997).

I. S. Gradsteyn and I. M. Ryzhik, Tables of Integrals, Series and Products (Academic Press, 1994), pp. 987 and 994.

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Figures (6)

Fig. 1.
Fig. 1.

Mode powers plotted as functions of mode number for (a) x=1 (left) and (b) x=3 (right).

Fig. 2.
Fig. 2.

(a) Mode powers plotted as functions of the distance parameter x. The solid, dot-dashed and dashed curves represent modes -1, -3 and -5, respectively. (b) Power asymmetry plotted as a function of distance.

Fig. 3.
Fig. 3.

Mode powers plotted as functions of mode number for (a) x=1 and (b) x=3. Red bars denote pumps, whereas blue bars denote signals. The input-signal phase ϕ0=0.

Fig. 4.
Fig. 4.

(a) Mode powers plotted as functions of the distance parameter x. The solid, dot-dashed and dashed curves represent modes 0, -2 and -4, respectively. (b) Total pump and signal powers plotted as functions of distance. The solid and dashed curves represent the pumps and signals, respectively. The input-signal phase ϕ0=0.

Fig. 5.
Fig. 5.

Mode powers plotted as functions of mode number for (a) x=1 and (b) x=3. Red bars denote pumps, whereas blue bars denote signals. The input-signal phase ϕ 0=π/2.

Fig. 6.
Fig. 6.

(a) Mode powers plotted as functions of the distance parameter x. The solid, dot-dashed and dashed curves represent modes 0, -2 and -4, respectively. (b) Total pump and signal powers plotted as functions of distance. The solid and dashed curves represent the pumps and signals, respectively. The input-signal phase ϕ0=π/2.

Equations (38)

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

i z A = β ( i τ ) A + γ A 2 A ,
A ( τ , z ) = A ( τ , 0 ) exp [ i γ A ( τ , 0 ) 2 z ] .
A ( τ , 0 ) = ρ + exp ( i ϕ ) + ρ exp ( i ϕ ) ,
exp ( ix cos θ ) = m i m J m ( x ) exp ( im θ ) ,
A ( τ , z ) = n A n ( x ) exp ( in ϕ ) ,
A n ( x ) = i ( n 1 ) 2 ρ + J ( n 1 ) 2 ( x ) + i ( n + 1 ) 2 ρ J ( n + 1 ) 2 ( x ) ,
n = J n 2 ( x ) = 1 .
n A n ( x ) 2 = ρ + 2 + ρ 2 ,
n = 1 [ A n ( x ) 2 A n ( x ) 2 ] = ( ρ + 2 ρ 2 ) J 0 2 ( x ) .
n = 1 A n ( x ) A n ( x ) constant .
A ( τ , 0 ) = ρ exp ( i ϕ ) + ρ 0 exp ( i ϕ 0 ) + ρ exp ( i ϕ ) ,
A n ( x ) = m i n m J m ( x ) [ ρ 0 exp ( i ϕ 0 ) J n 2 m ( ε 0 x ) 2 i ρ J n 2 m ( ε 0 x ) ] ,
A n ( x ) i n 2 ρ 0 { J n 2 ( x ) [ exp ( i ϕ 0 ) + 2 ix cos ϕ 0 ] + J n 2 ( x ) 2 x cos ϕ 0 } ,
d z B 3 i 1 c B 3 + i 1 d + i 2 n B 3 * ,
B 3 ( z ) i 1 d z + 1 d ( 2 n 1 c ) z 2 2 .
d z B 1 i 1 d B 3 B 1 B 1 * + i 2 n B 3 B 3 B 1 * ,
B 1 ( z ) 1 1 d 2 z 2 2 .
A 1 ( z ) exp ( iz ) 1 z 2 2 ,
A 3 ( z ) exp ( iz ) iz + z 2 2 ,
A 0 ( z ) exp ( iz ) ( c 0 + i s 0 ) + ( s 0 + 3 i c 0 ) z 3 ( c 0 + i s 0 ) z 2 2 ,
A 2 ( z ) exp ( iz ) ( 3 i c 0 s 0 ) z + ( i s 0 c 0 ) z 2 ,
d z B 0 = i 1 c B 0 + i 2 m B 0 * ,
B 0 ( z ) = ( c 0 + i s 0 ) + [ ( 2 m 1 c ) s 0 + i ( 2 m + 1 c ) c 0 ] z
+ ( 2 m 2 1 c 2 ) ( c 0 + i s 0 ) z 2 2 .
d z B 2 i 1 c B 2 + i 1 m B 0 * + i 2 b B 0 + i 2 p B 2 * ,
B 2 [ i ( 1 m + 2 b ) c 0 + ( 1 m 2 b ) s 0 ] z .
d z B 2 i 2 b ( i 1 d z ) B 0 + i 2 p ( i 1 d z ) B 0 * + i 2 b ( i 1 d z ) B 0 ,
B 2 1 d 2 p ( i s 0 c 0 ) z 2 2 .
d z B 0 i 1 m B 2 * + i 2 b B 2 + i 1 m B 2 * + i 2 b B 2 ,
B 0 ( 2 b 2 1 m 2 ) ( c 0 + i s 0 ) z 2 .
d z B 0 = i δ B 0 + i 2 γ B 1 B 1 B 0 * ,
d z B 0 = i δ B 0 + i 2 ρ 2 B 0 * ,
B 0 ( z ) = μ ( z ) B 0 ( 0 ) + ν ( z ) B 0 * ( 0 ) ,
μ ( z ) = cosh ( κ z ) + i δ sinh ( κ z ) κ ,
ν ( z ) = i 2 ρ 2 sinh ( κ z ) κ ,
B 0 ( z ) 2 = ρ 0 2 [ μ 2 + ν i 2 + 2 μ r ν i sin ( 2 ϕ 0 ) + 2 μ i ν i cos ( 2 ϕ 0 ) ] ,
2 ϕ m = tan 1 ( μ r μ i ) .
B 0 ( z ) ( c 0 + i s 0 ) + ( s 0 + 3 i c 0 ) ( ρ 2 z ) + 3 ( c 0 + i s 0 ) ( ρ 2 z ) 2 2 ,

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