Abstract

We show that the set of two coupled differential equations governing a four-wave mixing (FWM) component and its associated idler in a one-pump fiber optical parametric amplifier are the same as for standard parametric amplification, but with additional source terms arising from FWM between already-calculated waves. We show how to solve this new set of equations in general: this involves multiplication by an exponential matrix and its inverse, and integration. Hence it is in principle possible to write the exact expressions for the cross-talk fields, in terms of a large number of exponentials. The results show that the power of the first- and second-order cross-talk terms scales, respectively, like the first and second power of the signal power. In the case of two input signals, the powers of the cross-talk terms are independent of the phases of the signals, hence phase control cannot be used for reducing cross talk. In a particular case, cross talk scales like the inverse of pump power, which confirms a general trend noted in numerical simulations, and experiments.

© 2008 Optical Society of America

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References

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  1. K. Krastev and J. Rothman, “Crosstalk in fiber parametric amplifier,” in Proceedings of the 27th European Conference on Optical Communication (ECOC '01) (2001), Vol. 1, pp. 378-379.
  2. F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Crosstalk mitigation in fiber optic parametric amplifier,” in Proceedings of the 15th Laser and Electro-Optics Society Annual Meeting, LEOS 2002 (2002), pp. 383-384.
    [CrossRef]
  3. J. L. Blows, “Cross talk in a fibre parametric wavelength converter,” in Proceedings of the Optical Fiber Communication Conference (OFC '03), OSA Technical Digest (Optical Society of America, 2003), Vol. 2, pp. 565-566.
  4. T. Torounidis, H. Sunnerud, P. O. Hedekvist, and P. A. Andrekson, “Amplification of WDM signals in fiber-based optical parametric amplifiers,” IEEE Photon. Technol. Lett. 15, 1061-1063 (2003).
    [CrossRef]
  5. F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16, 434-436 (2004).
    [CrossRef]
  6. J. L. Blows and P.-F. Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B 21, 989-995 (2004).
    [CrossRef]
  7. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007), pp. 387-401.
  8. M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge Univ. Press, 2007). Chap. 13.
    [CrossRef]
  9. J. M. Chavez Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun. 259, 94-103 (2006).
    [CrossRef]
  10. J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun. 236, 115-122 (2004).
    [CrossRef]
  11. http://www.wolfram.com.

2006 (1)

J. M. Chavez Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun. 259, 94-103 (2006).
[CrossRef]

2004 (3)

J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun. 236, 115-122 (2004).
[CrossRef]

J. L. Blows and P.-F. Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B 21, 989-995 (2004).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16, 434-436 (2004).
[CrossRef]

2003 (1)

T. Torounidis, H. Sunnerud, P. O. Hedekvist, and P. A. Andrekson, “Amplification of WDM signals in fiber-based optical parametric amplifiers,” IEEE Photon. Technol. Lett. 15, 1061-1063 (2003).
[CrossRef]

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007), pp. 387-401.

Andrekson, P. A.

T. Torounidis, H. Sunnerud, P. O. Hedekvist, and P. A. Andrekson, “Amplification of WDM signals in fiber-based optical parametric amplifiers,” IEEE Photon. Technol. Lett. 15, 1061-1063 (2003).
[CrossRef]

Blows, J. L.

J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun. 236, 115-122 (2004).
[CrossRef]

J. L. Blows and P.-F. Hu, “Cross-talk-induced limitations of two-pump optical fiber parametric amplifiers,” J. Opt. Soc. Am. B 21, 989-995 (2004).
[CrossRef]

J. L. Blows, “Cross talk in a fibre parametric wavelength converter,” in Proceedings of the Optical Fiber Communication Conference (OFC '03), OSA Technical Digest (Optical Society of America, 2003), Vol. 2, pp. 565-566.

Callegari, F. A.

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16, 434-436 (2004).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Crosstalk mitigation in fiber optic parametric amplifier,” in Proceedings of the 15th Laser and Electro-Optics Society Annual Meeting, LEOS 2002 (2002), pp. 383-384.
[CrossRef]

Chavez Boggio, J. M.

J. M. Chavez Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun. 259, 94-103 (2006).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16, 434-436 (2004).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Crosstalk mitigation in fiber optic parametric amplifier,” in Proceedings of the 15th Laser and Electro-Optics Society Annual Meeting, LEOS 2002 (2002), pp. 383-384.
[CrossRef]

Fragnito, H. L.

J. M. Chavez Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun. 259, 94-103 (2006).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16, 434-436 (2004).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Crosstalk mitigation in fiber optic parametric amplifier,” in Proceedings of the 15th Laser and Electro-Optics Society Annual Meeting, LEOS 2002 (2002), pp. 383-384.
[CrossRef]

Hedekvist, P. O.

T. Torounidis, H. Sunnerud, P. O. Hedekvist, and P. A. Andrekson, “Amplification of WDM signals in fiber-based optical parametric amplifiers,” IEEE Photon. Technol. Lett. 15, 1061-1063 (2003).
[CrossRef]

Hu, P.-F.

Krastev, K.

K. Krastev and J. Rothman, “Crosstalk in fiber parametric amplifier,” in Proceedings of the 27th European Conference on Optical Communication (ECOC '01) (2001), Vol. 1, pp. 378-379.

Marconi, J. D.

J. M. Chavez Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun. 259, 94-103 (2006).
[CrossRef]

Marhic, M. E.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge Univ. Press, 2007). Chap. 13.
[CrossRef]

Rothman, J.

K. Krastev and J. Rothman, “Crosstalk in fiber parametric amplifier,” in Proceedings of the 27th European Conference on Optical Communication (ECOC '01) (2001), Vol. 1, pp. 378-379.

Sunnerud, H.

T. Torounidis, H. Sunnerud, P. O. Hedekvist, and P. A. Andrekson, “Amplification of WDM signals in fiber-based optical parametric amplifiers,” IEEE Photon. Technol. Lett. 15, 1061-1063 (2003).
[CrossRef]

Torounidis, T.

T. Torounidis, H. Sunnerud, P. O. Hedekvist, and P. A. Andrekson, “Amplification of WDM signals in fiber-based optical parametric amplifiers,” IEEE Photon. Technol. Lett. 15, 1061-1063 (2003).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

T. Torounidis, H. Sunnerud, P. O. Hedekvist, and P. A. Andrekson, “Amplification of WDM signals in fiber-based optical parametric amplifiers,” IEEE Photon. Technol. Lett. 15, 1061-1063 (2003).
[CrossRef]

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Spurious four-wave mixing in two-pump fiber-optic parametric amplifiers,” IEEE Photon. Technol. Lett. 16, 434-436 (2004).
[CrossRef]

J. Opt. Soc. Am. B (1)

Opt. Commun. (2)

J. M. Chavez Boggio, J. D. Marconi, and H. L. Fragnito, “Crosstalk in double-pumped fiber optic parametric amplifiers for wavelength division multiplexing systems,” Opt. Commun. 259, 94-103 (2006).
[CrossRef]

J. L. Blows, “Design strategy for controlling four-wave mixing-induced crosstalk between channels in a fibre optical parametric amplifier,” Opt. Commun. 236, 115-122 (2004).
[CrossRef]

Other (6)

http://www.wolfram.com.

K. Krastev and J. Rothman, “Crosstalk in fiber parametric amplifier,” in Proceedings of the 27th European Conference on Optical Communication (ECOC '01) (2001), Vol. 1, pp. 378-379.

F. A. Callegari, J. M. Chavez Boggio, and H. L. Fragnito, “Crosstalk mitigation in fiber optic parametric amplifier,” in Proceedings of the 15th Laser and Electro-Optics Society Annual Meeting, LEOS 2002 (2002), pp. 383-384.
[CrossRef]

J. L. Blows, “Cross talk in a fibre parametric wavelength converter,” in Proceedings of the Optical Fiber Communication Conference (OFC '03), OSA Technical Digest (Optical Society of America, 2003), Vol. 2, pp. 565-566.

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007), pp. 387-401.

M. E. Marhic, Fiber Optical Parametric Amplifiers, Oscillators and Related Devices (Cambridge Univ. Press, 2007). Chap. 13.
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Frequencies under consideration.

Equations (54)

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i d A 9 d Z = 2 P 1 A 9 + 2 A 1 A 3 A 5 * exp ( i Δ β 1 , 3 , 5 , 9 z ) + 2 A 1 A 6 A 4 * exp ( i Δ β 1 , 6 , 4 , 9 z ) + A 1 2 A 10 * exp ( i Δ β 1 , 1 , 9 , 10 z ) ,
i d A 10 d Z = 2 P 1 A 10 + 2 A 1 A 5 A 3 * exp ( i Δ β 1 , 5 , 3 , 10 z ) + 2 A 1 A 4 A 6 * exp ( i Δ β 1 , 4 , 6 , 10 z ) + A 1 2 A 9 * exp ( i Δ β 1 , 1 , 9 , 10 z ) ,
A k = B k exp ( 2 i P 1 Z ) , k = 9 , 10 .
i d B 9 d Z = 2 A 1 A 3 A 5 * exp [ i ( Δ β 1 , 3 , 5 , 9 2 γ P 1 ) z ] + 2 A 1 A 6 A 4 * exp [ i ( Δ β 1 , 6 , 4 , 9 2 γ P 1 ) z ] + A 1 2 B 10 * exp [ i ( Δ β 1 , 1 , 9 , 10 4 γ P 1 ) z ] ,
i d B 10 d Z = 2 A 1 A 5 A 3 * exp [ i ( Δ β 1 , 5 , 3 , 10 2 γ P 1 ) z ] + 2 A 1 A 4 A 6 * exp [ i ( Δ β 1 , 4 , 6 , 10 2 γ P 1 ) z ] + A 1 2 B 9 * exp [ i ( Δ β 1 , 1 , 9 , 10 4 γ P 1 ) z ] .
i d B 9 d Z = 2 A 1 0 A 3 A 5 * exp [ i ( Δ β 1 , 3 , 5 , 9 γ P 1 ) z ] + 2 A 1 0 A 6 A 4 * exp [ i ( Δ β 1 , 6 , 4 , 9 γ P 1 ) z ] + P 1 B 10 * exp [ i ( Δ β 1 , 1 , 9 , 10 2 γ P 1 ) z ] ,
i d B 10 d Z = 2 A 1 0 A 5 A 3 * exp [ i ( Δ β 1 , 5 , 3 , 10 γ P 1 ) z ] + 2 A 1 0 A 4 A 6 * exp [ i ( Δ β 1 , 4 , 6 , 10 γ P 1 ) z ] + P 1 B 9 * exp [ i ( Δ β 1 , , 1 , 9 , 10 2 γ P 1 ) z ] .
d d z [ C 9 C 10 * ] = N 9 [ C 9 C 10 * ] + [ S 9 S 10 * ] ,
N 9 = i [ κ 9 2 γ P 1 γ P 1 κ 9 2 ] ,
S 9 = 2 i γ A 1 0 [ A 3 A 5 * exp ( i Δ β 1 , 3 , 5 , 9 z ) + A 6 A 4 * exp ( i Δ β 1 , 6 , 4 , 9 z ) ] exp ( i Δ β 1 , 1 , 9 , 10 z 2 ) ,
S 10 = 2 i γ A 1 0 [ A 5 A 3 * exp ( i Δ β 1 , 5 , 3 , 10 z ) + A 4 A 6 * exp ( i Δ β 1 , 4 , 6 , 10 z ) , ] exp ( i Δ β 1 , 1 , 9 , 10 z 2 ) .
[ C 9 C 10 * ] = M 9 ζ = 0 z M 9 1 [ S 9 S 10 * ] d ζ ,
M 9 = exp ( N 9 z ) = [ cosh ( g 9 z ) + i κ 9 2 g 9 sinh ( g 9 z ) i γ P 1 g 9 sinh ( g 9 z ) i γ P 1 g 9 sinh ( g 9 z ) cosh ( g 9 z ) i κ 9 2 g 9 sinh ( g 9 z ) ] ,
σ = [ 1 1 1 1 ] .
C = ( I + i γ P 1 z σ ) ζ = 0 z ( I i γ P 1 ζ σ ) S d ζ = ( I + i γ P 1 z σ ) [ ζ = 0 z S d ζ i γ P 1 σ ζ = 0 z ζ S d ζ ] = ( I + i γ P 1 z σ ) ζ = 0 z S d ζ i γ P 1 σ ζ = 0 z ζ S d ζ = ( I + i γ P 1 z σ ) ζ = 0 z S d ζ i γ P 1 σ ( z ζ = 0 z S d ζ ζ = 0 z d ζ η = 0 ζ S d η ) = ζ = 0 z S d ζ + i γ P 1 σ ζ = 0 z d ζ η = 0 ζ S d η .
S 9 = K [ exp ( a z ) + exp ( b z ) ] , ( S 10 ) * = K [ exp ( c z ) + exp ( d z ) ] ,
K = i γ A 1 0 A 3 0 ( A 5 0 ) * 2 ,
a = g 3 + g 5 + i Δ β 1 , 3 , 5 , 9 i Δ β 1 , 1 , 9 , 10 2 i Δ β 3 , 4 , 5 , 6 2 ,
b = g 3 + g 5 + i Δ β 1 , 6 , 4 , 9 i Δ β 1 , 1 , 9 , 10 2 + i Δ β 3 , 4 , 5 , 6 2 ,
c = g 3 + g 5 i Δ β 1 , 5 , 3 , 10 + i Δ β 1 , 1 , 9 , 10 2 i Δ β 3 , 4 , 5 , 6 2 ,
d = g 3 + g 5 i Δ β 1 , 4 , 6 , 10 + i Δ β 1 , 1 , 9 , 10 2 + i Δ β 3 , 4 , 5 , 6 2 .
C 9 = K { e a z 1 a + e b z 1 b + i γ P 1 [ e a z a z 1 a 2 + e b z b z 1 b 2 e c z c z 1 c 2 e d z d z 1 d 2 ] } ,
C 10 * = K { e c z 1 a + e d z 1 b + i γ P 1 [ e a z a z 1 a 2 + e b z b z 1 b 2 e c z c z 1 c 2 e d z d z 1 d 2 ] } .
C 9 2 K e a z a .
X T p s = P 9 P 3 , out = C 9 2 G P 3 0 4 K 2 [ exp ( 2 γ P 1 z ) ( 2 γ P 1 ) ] 2 [ exp ( 2 γ P 1 z ) 4 ] P 3 0 4 P 3 , out P 1 ,
i d A 11 d Z = 2 P 1 A 11 + ( A 5 ) 2 A 1 * exp ( i Δ β 5 , 5 , 1 , 11 z ) + 2 A 1 A 5 A 6 * exp ( i Δ β 1 , 5 , 6 , 11 z ) + A 1 2 A 12 * exp ( i Δ β 1 , 1 , 11 , 12 z ) ,
i d A 12 d Z = 2 P 1 A 12 + ( A 6 ) 2 A 1 * exp ( i Δ β 6 , 6 , 1 , 12 z ) + 2 A 1 A 6 A 5 * exp ( i Δ β 1 , 6 , 5 , 12 z ) + A 1 2 A 11 * exp ( i Δ β 1 , 1 , 11 , 12 z ) .
d d z [ C 11 C 12 * ] = N 11 [ C 11 C 12 * ] + [ S 11 S 12 * ] ,
S 11 = i γ A 1 0 { ( A 5 ) 2 exp [ i ( Δ β 5 , 5 , 1 , 11 2 γ P 1 ) z ] + 2 A 5 A 6 * exp ( i Δ β 1 , 5 , 6 , 11 z ) } exp ( i Δ β 1 , 1 , 11 , 12 z 2 ) ,
S 12 = i γ A 1 0 { ( A 6 ) 2 exp [ i ( Δ β 6 , 6 , 1 , 12 2 γ P 1 ) z ] + 2 A 6 A 5 * exp ( i Δ β 1 , 6 , 5 , 12 z ) } exp ( i Δ β 1 , 1 , 11 , 12 z 2 ) .
i d A 13 d Z = 2 P 1 A 13 + ( A 3 ) 2 A 1 * exp ( i Δ β 3 , 3 , 1 , 13 z ) + 2 A 1 A 3 A 4 * exp ( i Δ β 1 , 3 , 4 , 13 z ) + A 1 2 A 14 * exp ( i Δ β 1 , 1 , 13 , 14 z ) ,
i d A 14 d Z = 2 P 1 A 14 + ( A 4 ) 2 A 1 * exp ( i Δ β 4 , 4 , 1 , 14 z ) + 2 A 1 A 4 A 3 * exp ( i Δ β 1 , 4 , 3 , 14 z ) + A 1 2 A 13 * exp ( i Δ β 1 , 1 , 13 , 14 z ) .
d d z [ C 13 C 14 * ] = N 13 [ C 13 C 14 * ] + [ S 13 S 14 * ] ,
S 13 = i γ A 1 0 { ( A 3 ) 2 exp [ i ( Δ β 3 , 3 , 1 , 13 2 γ P 1 ) z ] + 2 A 3 A 4 * exp ( i Δ β 1 , 3 , 4 , 13 z ) } exp ( i Δ β 1 , 1 , 13 , 14 z 2 ) ,
S 14 = i γ A 1 0 { ( A 4 ) 2 exp [ i ( Δ β 4 , 4 , 1 , 14 2 γ P 1 ) z ] + 2 A 4 A 3 * exp ( i Δ β 1 , 4 , 3 , 14 z ) } exp ( i Δ β 1 , 1 , 13 , 14 z 2 ) .
i d A 15 d Z = 2 P 1 A 15 + 2 A 3 A 5 A 1 * exp ( i Δ β 3 , 5 , 1 , 15 z ) + 2 A 1 A 3 A 6 * exp ( i Δ β 1 , 3 , 6 , 15 z ) + 2 A 1 A 5 A 4 * exp ( i Δ β 1 , 5 , 4 , 15 z ) + A 1 2 A 16 * exp ( i Δ β 1 , 1 , 15 , 16 z ) ,
i d A 16 d Z = 2 P 1 A 16 + 2 A 4 A 6 A 1 * exp ( i Δ β 4 , 6 , 1 , 16 z ) + 2 A 1 A 4 A 5 * exp ( i Δ β 1 , 4 , 5 , 16 z ) + 2 A 1 A 6 A 3 * exp ( i Δ β 1 , 6 , 3 , 16 z ) + A 1 2 A 15 * exp ( i Δ β 1 , 1 , 15 , 16 z ) .
d d z [ C 15 C 16 * ] = N 15 [ C 15 C 16 * ] + [ S 15 S 16 * ] ,
S 15 = 2 i γ A 1 0 { A 3 A 5 exp [ i ( Δ β 3 , 5 , 1 , 15 2 γ P 1 ) z ] + A 3 A 6 * exp ( i Δ β 1 , 3 , 6 , 15 z ) + A 5 A 4 * exp ( i Δ β 1 , 5 , 4 , 15 z ) } exp ( i Δ β 1 , 1 , 15 , 16 z 2 ) ,
S 16 = 2 i γ A 1 0 { A 4 A 6 exp [ i ( Δ β 4 , 6 , 1 , 16 2 γ P 1 ) z ] + A 4 A 5 * exp ( i Δ β 1 , 4 , 5 , 16 z ) + A 6 A 3 * exp ( i Δ β 1 , 6 , 3 , 16 z ) } exp ( i Δ β 1 , 1 , 15 , 16 z 2 ) .
( X T ) p s x 4 x 2 = x 2 signal input power.
i d A 7 d Z = 2 P 1 A 7 + ( A 3 ) 2 A 5 * exp ( i Δ β 3 , 3 , 5 , 7 z ) + 2 A 3 A 6 A 4 * exp ( i Δ β 3 , 6 , 4 , 7 z ) + 2 A 1 A 3 A 10 * exp ( i Δ β 1 , 3 , 10 , 7 z ) + 2 A 3 A 9 A 1 * exp ( i Δ β 3 , 9 , 1 , 7 z ) + 2 A 1 A 13 A 5 * exp ( i Δ β 1 , 13 , 5 , 7 z ) + 2 A 1 A 6 A 14 * exp ( i Δ β 1 , 6 , 14 , 7 z ) + A 1 2 A 8 * exp ( i Δ β 1 , 1 , 7 , 8 z ) ,
i d A 8 d Z = 2 P 1 A 8 + ( A 4 ) 2 A 6 * exp ( i Δ β 4 , 4 , 6 , 8 z ) + 2 A 4 A 5 A 3 * exp ( i Δ β 4 , 5 , 3 , 8 z ) + 2 A 1 A 4 A 9 * exp ( i Δ β 1 , 4 , 9 , 8 z ) + 2 A 4 A 10 A 1 * exp ( i Δ β 4 , 10 , 1 , 8 z ) + 2 A 1 A 14 A 6 * exp ( i Δ β 1 , 14 , 6 , 8 z ) + 2 A 1 A 5 A 13 * exp ( i Δ β 1 , 5 , 13 , 8 z ) + A 1 2 A 8 * exp ( i Δ β 1 , 1 , 7 , 8 z ) .
d d z [ C 7 C 8 * ] = N 7 [ C 7 C 8 * ] + [ S 7 S 8 * ] ,
S 7 exp ( i Δ β 1 , 1 , 7 , 8 z 2 ) ( i γ ) = ( A 3 ) 2 A 5 * exp [ i ( Δ β 3 , 3 , 5 , 7 γ P 1 ) z ] + 2 A 3 A 6 A 4 * exp ( i Δ β 3 , 6 , 4 , 7 z ) + 2 A 1 0 { A 3 A 10 * exp ( i Δ β 1 , 3 , 10 , 7 z ) + 2 A 3 A 9 exp [ i ( Δ β 3 , 9 , 1 , 7 2 γ P 1 ) z ] + 2 A 13 A 5 * exp ( i Δ β 1 , 13 , 5 , 7 z ) + 2 A 6 A 14 * exp ( i Δ β 1 , 6 , 14 , 7 z ) } ,
S 8 exp ( i Δ β 1 , 1 , 7 , 8 z 2 ) ( i γ ) = ( A 4 ) 2 A 6 * exp [ i ( Δ β 4 , 4 , 6 , 8 γ P 1 ) z ] + 2 A 4 A 5 A 3 * exp ( i Δ β 4 , 5 , 3 , 8 z ) + 2 A 1 0 { A 4 A 9 * exp ( i Δ β 1 , 4 , 9 , 8 z ) + 2 A 4 A 10 exp [ i ( Δ β 4 , 10 , 1 , 8 2 γ P 1 ) z ] + 2 A 14 A 6 * exp ( i Δ β 1 , 14 , 6 , 8 z ) + 2 A 5 A 13 * exp ( i Δ β 1 , 5 , 13 , 8 z ) } .
( X T ) s s x 6 x 2 = x 4 square of signal input power.
d C d z = N C + S ,
M = exp ( N z ) .
M 1 = exp ( N z ) .
d d z ( M 1 C ) = N M 1 C + M 1 d C d z .
d C d z N C = M d d z ( M 1 C ) .
M d d z ( M 1 C ) = S .
C = M ζ = 0 z M 1 S d ζ + M C 0 ,

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