Abstract

We apply a complete perturbation theory of solitons (for position, phase, frequency, and amplitude) to the cases of amplitude, intensity, or phase modulation, filtering, and soliton interaction. We show that on-line intensity modulation and filtering act in the same way in reducing noise-induced jitter. Soliton interaction can be reduced with initial phase and filtering control or with a periodic π-phase change for consecutive solitons (as in the case of amplitude modulation). Finally, we design a 30-Gbit/s transoceanic soliton transmission system, using filtering and initial phase control techniques.

© 1993 Optical Society of America

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  1. J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
    [Crossref] [PubMed]
  2. J. P. Gordon, “Interaction forces among solitons,” Opt. Lett. 8, 596–598 (1983).
    [Crossref] [PubMed]
  3. F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 355–357 (1987).
    [Crossref] [PubMed]
  4. L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 15, 1203–1205 (1990).
    [Crossref] [PubMed]
  5. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
    [Crossref] [PubMed]
  6. Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
    [Crossref] [PubMed]
  7. M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometres,” Electron. Lett. 27, 1270–1272 (1991).
    [Crossref]
  8. A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Modulation and filtering control of soliton transmission,” J. Opt. Soc. Am. B 9, 1350–1357 (1992).
    [Crossref]
  9. M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Electron. Lett. 28, 1099–1100 (1992).
    [Crossref]
  10. P. L. François and T. Georges, “Reduction of averaged soliton interaction forces by amplitude modulation,” Opt. Lett. 18, 583–585 (1993).
    [Crossref] [PubMed]
  11. M. Nakazawa and H. Kubota, “Physical interpretation of reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 28, 958–960 (1992).
    [Crossref]
  12. C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. IEEE 134, 145–151 (1987).
  13. T. Georges, “Amplifier noise jitter of two interacting solitons,” Opt. Commun. 85, 195–201 (1991).
    [Crossref]
  14. T. Georges and F. Favre, “Influence of soliton interaction on amplifier noise-induced jitter: a first-order analytical solution,” Opt. Lett. 16, 1656–1658 (1991).
    [Crossref] [PubMed]
  15. H. A. Haus and Y. Lai, “Quantum theory of soliton squeezing: a linearized approach,” J. Opt. Soc. Am. B 7, 386–392 (1990).
    [Crossref]
  16. H. A. Haus, “Quantum noise in a solitonlike repeater system,” J. Opt. Soc. Am. B 8, 1122–1126 (1991).
    [Crossref]
  17. A. Hasegawa and Y. Kodoma, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
    [Crossref] [PubMed]
  18. A. Hasegawa and Y. Kodoma, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
    [Crossref] [PubMed]
  19. K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
    [Crossref]
  20. L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Longdistance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
    [Crossref]
  21. Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
    [Crossref]
  22. Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
    [Crossref]
  23. J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
    [Crossref]
  24. L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
    [Crossref] [PubMed]
  25. L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
    [Crossref] [PubMed]

1993 (1)

1992 (5)

1991 (10)

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[Crossref] [PubMed]

H. A. Haus, “Quantum noise in a solitonlike repeater system,” J. Opt. Soc. Am. B 8, 1122–1126 (1991).
[Crossref]

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[Crossref]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[Crossref]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometres,” Electron. Lett. 27, 1270–1272 (1991).
[Crossref]

T. Georges, “Amplifier noise jitter of two interacting solitons,” Opt. Commun. 85, 195–201 (1991).
[Crossref]

T. Georges and F. Favre, “Influence of soliton interaction on amplifier noise-induced jitter: a first-order analytical solution,” Opt. Lett. 16, 1656–1658 (1991).
[Crossref] [PubMed]

A. Hasegawa and Y. Kodoma, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[Crossref] [PubMed]

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
[Crossref]

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Longdistance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[Crossref]

1990 (3)

1989 (2)

1987 (2)

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. IEEE 134, 145–151 (1987).

F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 355–357 (1987).
[Crossref] [PubMed]

1986 (1)

1983 (1)

Blow, K. J.

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
[Crossref]

Chu, P. L.

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. IEEE 134, 145–151 (1987).

Cohen, L. G.

Desem, C.

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. IEEE 134, 145–151 (1987).

Doran, N. J.

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
[Crossref]

Evangelides, S. G.

Favre, F.

François, P. L.

Georges, T.

Gordon, J. P.

Hasegawa, A.

Haus, H. A.

Kivshar, Y. S.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[Crossref]

Kodama, Y.

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[Crossref] [PubMed]

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[Crossref]

Kodoma, Y.

Kubota, H.

M. Nakazawa and H. Kubota, “Physical interpretation of reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 28, 958–960 (1992).
[Crossref]

M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Electron. Lett. 28, 1099–1100 (1992).
[Crossref]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometres,” Electron. Lett. 27, 1270–1272 (1991).
[Crossref]

Lai, Y.

Malomed, B. A.

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[Crossref]

Mecozzi, A.

Menyuk, C. R.

Mitschke, F. M.

Mollenauer, L. F.

Moores, J. D.

Nakazawa, M.

M. Nakazawa and H. Kubota, “Physical interpretation of reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 28, 958–960 (1992).
[Crossref]

M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Electron. Lett. 28, 1099–1100 (1992).
[Crossref]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometres,” Electron. Lett. 27, 1270–1272 (1991).
[Crossref]

Neubelt, M. J.

Simpson, J. R.

Smith, K.

Suzuki, K.

M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Electron. Lett. 28, 1099–1100 (1992).
[Crossref]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometres,” Electron. Lett. 27, 1270–1272 (1991).
[Crossref]

Wabnitz, S.

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[Crossref]

Yamada, E.

M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Electron. Lett. 28, 1099–1100 (1992).
[Crossref]

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometres,” Electron. Lett. 27, 1270–1272 (1991).
[Crossref]

Electron. Lett. (4)

M. Nakazawa, E. Yamada, H. Kubota, and K. Suzuki, “10 Gb/s soliton data transmission over one million kilometres,” Electron. Lett. 27, 1270–1272 (1991).
[Crossref]

M. Nakazawa, H. Kubota, E. Yamada, and K. Suzuki, “Infinite-distance soliton transmission with soliton controls in time and frequency domains,” Electron. Lett. 28, 1099–1100 (1992).
[Crossref]

M. Nakazawa and H. Kubota, “Physical interpretation of reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 28, 958–960 (1992).
[Crossref]

Y. Kodama and S. Wabnitz, “Reduction of soliton interaction forces by bandwidth limited amplification,” Electron. Lett. 27, 1931–1933 (1991).
[Crossref]

IEEE Photon. Technol. Lett. (1)

K. J. Blow and N. J. Doran, “Average soliton dynamics and the operation of soliton systems with lumped amplifiers,” IEEE Photon. Technol. Lett. 3, 369–371 (1991).
[Crossref]

J. Lightwave Technol. (2)

L. F. Mollenauer, S. G. Evangelides, and H. A. Haus, “Longdistance soliton propagation using lumped amplifiers and dispersion shifted fiber,” J. Lightwave Technol. 9, 194–197 (1991).
[Crossref]

J. P. Gordon and L. F. Mollenauer, “Effects of fiber nonlinearities and amplifier spacing on ultra long distance transmission,” J. Lightwave Technol. 9, 170–173 (1991).
[Crossref]

J. Opt. Soc. Am. B (3)

Opt. Commun. (1)

T. Georges, “Amplifier noise jitter of two interacting solitons,” Opt. Commun. 85, 195–201 (1991).
[Crossref]

Opt. Lett. (11)

T. Georges and F. Favre, “Influence of soliton interaction on amplifier noise-induced jitter: a first-order analytical solution,” Opt. Lett. 16, 1656–1658 (1991).
[Crossref] [PubMed]

A. Hasegawa and Y. Kodoma, “Guiding-center soliton in optical fibers,” Opt. Lett. 15, 1443–1445 (1990).
[Crossref] [PubMed]

P. L. François and T. Georges, “Reduction of averaged soliton interaction forces by amplitude modulation,” Opt. Lett. 18, 583–585 (1993).
[Crossref] [PubMed]

J. P. Gordon and H. A. Haus, “Random walk of coherently amplified solitons in optical fiber transmission,” Opt. Lett. 11, 665–667 (1986).
[Crossref] [PubMed]

J. P. Gordon, “Interaction forces among solitons,” Opt. Lett. 8, 596–598 (1983).
[Crossref] [PubMed]

F. M. Mitschke and L. F. Mollenauer, “Experimental observation of interaction forces between solitons in optical fibers,” Opt. Lett. 12, 355–357 (1987).
[Crossref] [PubMed]

L. F. Mollenauer, M. J. Neubelt, S. G. Evangelides, J. P. Gordon, J. R. Simpson, and L. G. Cohen, “Experimental study of soliton transmission over more than 10,000 km in dispersion-shifted fiber,” Opt. Lett. 15, 1203–1205 (1990).
[Crossref] [PubMed]

A. Mecozzi, J. D. Moores, H. A. Haus, and Y. Lai, “Soliton transmission control,” Opt. Lett. 16, 1841–1843 (1991).
[Crossref] [PubMed]

Y. Kodama and A. Hasegawa, “Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect,” Opt. Lett. 17, 31–33 (1992).
[Crossref] [PubMed]

L. F. Mollenauer, J. P. Gordon, and S. G. Evangelides, “The sliding-frequency guiding filter: an improved form of soliton jitter control,” Opt. Lett. 17, 1575–1577 (1992).
[Crossref] [PubMed]

L. F. Mollenauer, K. Smith, J. P. Gordon, and C. R. Menyuk, “Resistance of solitons to the effects of polarization dispersion in optical fibers,” Opt. Lett. 14, 1219–1221 (1989).
[Crossref] [PubMed]

Phys. Rev. Lett. (1)

A. Hasegawa and Y. Kodoma, “Guiding-center soliton,” Phys. Rev. Lett. 66, 161–164 (1991).
[Crossref] [PubMed]

Proc. IEEE (1)

C. Desem and P. L. Chu, “Reducing soliton interaction in single-mode optical fibres,” Proc. IEEE 134, 145–151 (1987).

Rev. Mod. Phys. (1)

Y. S. Kivshar and B. A. Malomed, “Dynamics of solitons in nearly integrable systems,” Rev. Mod. Phys. 61, 763–915 (1989).
[Crossref]

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Figures (10)

Fig. 1
Fig. 1

Interaction jitter (dashed curves) and p:n displacement (solid curves) for Θ = 0.75π (α0 = 3). p = 1, circles; p = 2, squares; p = 3, crosses. (a) No modulation, no filter; (b) π-phase modulation; (c) filter (kf = 0.2); (d) amplitude modulation (km = 0.2).

Fig. 2
Fig. 2

Interaction jitter (dashed curves) and p:n displacement (solid curves) versus Θ (α0 = 3). The symbols are the same as in Fig. 1. (a), (b), In phase; (c), (d), opposite phase.

Fig. 3
Fig. 3

Soliton interaction in presence of filters. The results of a numerical integration of the NLSE (symbols) are compared with a direct integration of Eqs. (4.4). Values of kf are 0 (crosses), 0.1 (squares), and 0.2 (circles).

Fig. 4
Fig. 4

Maximum Θ versus the number of on-line amplitude modulators for three input phases. α0 = 3.

Fig. 5
Fig. 5

Noise-induced jitter of a single soliton versus Z. The figure shows the influence of the filtering or the modulation.

Fig. 6
Fig. 6

Noise-induced jitter of interacting solitons normalized to that of a single soliton (Gordon–Haus limit). Three phases are plotted without filtering (solid curves). Φzm is plotted with filtering (dashed curve).

Fig. 7
Fig. 7

Contribution of each amplifier to the total jitter versus the amplifier’s rank for a transmission with 80 amplifiers. Θ = 0 (solid curve); Θ = 0.15π (dotted curves); Θ = 0.75π (dashed curve).

Fig. 8
Fig. 8

Noise-induced jitter of interacting solitons p:n (symbols), asymptotic values for large n (solid curves), and reference jitter level of soliton 1:1 (dotted curves), α0 = 3, Φ0 = 0.1π, and Θ = 0.75π.

Fig. 9
Fig. 9

Noise-induced jitter of interacting solitons normalized to the jitter of soliton 1:1 versus input phase. α0 = 3.

Fig. 10
Fig. 10

Limit for chromatic dispersion in a soliton transmission system with filtering and initial phase control. (T is calculated for Θ = 0.75π for the SN ratio limited dispersion.)

Equations (48)

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

Z max = Z c 3 / Z a 2
i u z + 1 2 u t t + | u | 2 u = P ( u ) ,
u s ( A , α , Φ , ω ; t ) = A sech [ A ( t α ) ] exp ( i Φ i ω t ) ,
U A = u s * , U ω = tanh [ A ( t α ) ] u s * , U α = ( t α ) u s * , U Φ = { 1 A ( t α ) tanh [ A ( t α ) ] } u s * ,
A z = Im + P ( u s ) U A d t , ω z = + P ( u s ) U ω d t , α z = ω + 1 A Im + P ( u s ) U α d t , Φ z = A 2 ω 2 2 1 A Re + P ( u s ) U Φ d t .
P ( u ) = i n ( z , t ) = i [ n 1 ( z , t ) + i n 2 ( z , t ) ] exp ( i Φ i ω t ) ,
n i ( z , t ) n j ( z , t ) = 1 2 n s p F ( G ) h ν δ i j δ ( t t ) δ ( z z ) , i , j = 1 , 2 , n i ( z , t ) = 0 ,
δ x 2 = 1 2 n s p F ( G ) | U x | 2 d t , x = A , ω , α , Φ
δ A 2 = A n s p F ( G ) , δ ω 2 = A 3 n s p F ( G ) , δ α 2 = π 2 12 A n s p F ( G ) , δ Φ 2 = 1 3 A ( π 2 12 + 1 ) n s p F ( G ) .
δ α ( z ) = p = 1 n [ δ ω p ( p n ) z A + δ α p ] .
δ α 2 ( z ) 1 3 z 3 z A δ ω 2 + z z A δ α 2 .
δ t 2 = K GH Z ( Z 2 + Z 1 2 ) ,
K GH = 0.1959 f ( G ) n s p h n 2 D A eff τ Γ , Z 1 = π 3 2 Z c , f ( G ) = F ( G ) ln ( G ) ,
P ( u 1 ) = 2 | u 1 | 2 u 2 u 1 2 u 2 * .
+ d t cosh 3 ( t ) cosh ( t + 2 α ) 4 exp ( 2 α ) , + tanh ( t ) d t cosh 3 ( t ) cosh ( t + 2 α ) 4 3 exp ( 2 α )
A z = 0 , a z = 4 A 3 exp ( 2 α ) sin 2 Φ , ω z = 4 A 3 exp ( 2 α ) cos 2 Φ , γ z = 0 , α z = ω , Φ z = a A ,
2 A = A 1 + A 2 , 2 a = A 1 A 2 , 2 ω = ω 1 ω 2 , 2 γ = ω 1 + ω 2 , 2 α = α 1 α , 2 Φ = Φ 1 Φ 2 .
Q z = ω + i a , Q z z = 4 exp ( 2 Q ) .
exp ( Q Q 0 ) = cosh ( i p z ) i [ ( Q z ) 0 / ρ ] sinh ( i ρ z ) , ρ 2 = 4 exp ( 2 Q 0 ) ( Q z ) 0 2 .
log T ( ω ) = 1 2 a f ( ω ω 0 ) 2 .
P ( u ) = i ( δ u β u u ) , β = 1.554 a f τ 2 Z c Z f < 0 ,
ω z = k f A 2 ω , A z = 2 A [ δ k f 4 ( A 2 + 3 ω 2 ) ] , α z = ω , Φ z = ( A 2 ω 2 ) 2 ,
k f = 4 3 β .
δ = k f 4 ( 1 + 3 ω 2 ) ,
α z z = k f α z .
P ( u ) = i u ( δ + ζ t 2 ) , ζ = 0.1609 a m τ 2 Z c Z m < 0 ,
ω z = 0 , A z = 2 A [ δ k m 4 A 2 ( 1 + 12 π 2 α 2 ) ] , α z = k m A 3 α ω , Φ z = ( A 2 ω 2 ) 2 ,
δ = k m 4 ( 1 + 12 π 2 α 2 ) .
α z = k m α .
α int 2 = n = 2 2 ( n + 1 ) p = 1 n α 2 ( p : n ) .
Q ( 2 z m ) = Q 0 4 z m 2 exp ( 2 Q 0 ) .
Q ( z ) = Q 0 2 z z m exp ( 2 Q 0 ) .
Q z = ω 1 + i a , ω 1 = ω + k m ( α α 0 ) , Q z z = 4 exp ( 2 Q ) ( k f + k m ) Re ( Q z ) k f k m ( α α 0 ) ,
A z = 2 A [ δ k m 4 A 2 ( 1 + 12 π 2 α 2 ) k f 4 ( A 2 + 3 ω 2 ) ] .
x = F ( x 0 , z z 0 ) .
δ x = k = 1 p A k δ x k ,
A k = F x ( x k , z z k ) ,
δ x 2 = k = 1 p B k δ x k 2 ,
A k = [ exp ( k m y ) exp ( k m y ) exp ( k f y ) k m k f 0 exp ( k f y ) ] , x = [ α ω ] ,
δ t 2 = K GH Z [ Z 2 h 1 ( k f z , k m z ) + Z 1 2 h 2 ( k m z ) ] , h 1 ( x 1 , x 2 ) = 3 ( x 2 x 1 ) 2 × [ 2 exp [ ( x 1 + x 2 ) ] 1 x 1 + x 2 exp ( 2 x 1 ) 1 2 x 1 exp ( 2 x 2 ) 1 2 x 2 ] , h 2 ( x 2 ) = 1 exp ( 2 x 2 ) 2 x 2 .
Z 2 = [ 1.5 k f k m ( k f + k m ) ] 1 / 3 Z c .
d α 2 = δ A 2 p = 1 n a p 2 + δ ω 2 p = 1 n b p 2 + δ α 2 p = 1 n c p 2 + δ Φ 2 p = 1 n d p 2 ,
a p + i b p = 1 ρ ( tanh ζ n tanh ζ p ) + i ( n p ) z a tanh ζ n tanh ζ p , c p + i d p = [ 1 i ( n p ) ρ z A tanh ζ n ] ( 1 tanh 2 ζ p ) + tanh ζ n tanh ζ p , ζ p = ζ 0 + i p ρ z A , tanh ζ 0 = i ( Q z ) 0 ρ .
T = τ 0.88 ln ( 2 Θ Z Z c ) .
τ min = 1.246 λ Z 1 / 6 Z a 1 / 3 D π c , τ max = 5.9 × 10 3 λ 4 π 2 c 2 h n 2 A eff Γ Z n s p f ( G ) D .
D min = [ 212 h n 2 Γ n s p f ( G ) λ 3 A eff ] 2 π 3 c 3 Z 7 / 3 Z a 2 / 3 , τ lim = 264 π c h n 2 Γ n s p f ( G ) A eff λ 2 Z 4 / 3 Z a 2 / 3 .
d t 2 η T 2 2 erfc 1 ( BER ) ,
D > 5 D λ Δ λ 1.5 λ 2 c τ D λ , D 10 ( Δ β h 1 / 2 ) 2 .

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