Abstract

In a soliton transmission system using lumped amplifiers, pseudo phase matching allows four-wave mixing fields from soliton–soliton collisions to grow uncontrollably and inflict severe penalties. Through numerical simulation, we show that this growth can be eliminated, or at least greatly reduced, through the use of fiber whose dispersion is tapered, either continuously or in steps, in conformity with the fiber loss curve.

© 1996 Optical Society of America

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References

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  1. L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
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  5. For a discussion of soliton units, seeMollenauerL. F.GordonJ. P.IslamM. N., IEEE J. Quantum Electron. QE-22, 157 (1986), App.
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  7. V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
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1996

1995

1992

1991

A. Mecozzi, J. D. Moores, H. A. Haus, Y. Lai, Opt. Lett. 16, 1841 (1991).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
[CrossRef]

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

1987

1986

For a discussion of soliton units, seeMollenauerL. F.GordonJ. P.IslamM. N., IEEE J. Quantum Electron. QE-22, 157 (1986), App.
[CrossRef]

Bogatyrev, V. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Boyd, R. W.

Bubnov, M. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Chernikov, S. V.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Devyatykh, G. G.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Dianov, E. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Evangelides, S. G.

L. F. Mollenauer, J. P. Gordon, S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
[CrossRef]

Evans, A. F.

Gordon, J. P.

L. F. Mollenauer, J. P. Gordon, S. G. Evangelides, Opt. Lett. 17, 1575 (1992).
[CrossRef] [PubMed]

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
[CrossRef]

Gur’yanov, A. N.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Haus, H. A.

Hawegawa, A.

Kodama, Y.

Kurkov, A. S.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Lai, Y.

Mamyshev, P. V.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Mecozzi, A.

Miroshnichenko, S. I.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Mollenauer, L. F.

Moores, J. D.

Prokorov, A. M.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Runyantsev, S. D.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Semenov, S. L.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Semonov, V. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Stentz, A. J.

Sysoliatin, A. A.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

Tajima, K.

IEEE J. Quantum Electron.

For a discussion of soliton units, seeMollenauerL. F.GordonJ. P.IslamM. N., IEEE J. Quantum Electron. QE-22, 157 (1986), App.
[CrossRef]

J. Lightwave Technol.

V. A. Bogatyrev, M. M. Bubnov, E. M. Dianov, A. S. Kurkov, P. V. Mamyshev, A. M. Prokorov, S. D. Runyantsev, V. A. Semonov, S. L. Semenov, A. A. Sysoliatin, S. V. Chernikov, A. N. Gur’yanov, G. G. Devyatykh, S. I. Miroshnichenko, J. Lightwave Technol. 9, 561 (1991).
[CrossRef]

L. F. Mollenauer, S. G. Evangelides, J. P. Gordon, J. Lightwave Technol. 9, 362 (1991).
[CrossRef]

Opt. Lett.

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Figures (5)

Fig. 1
Fig. 1

Pulses that have traversed a 10-Mm transmission line with Lamp = 33 km and constant D = 0.5 ps/(nm km) and that have undergone collisions with an adjacent channel 0.6 nm away, containing all 1’s. In this numerical simulation a small seed of noise was added, but (to avoid conventional amplitude and Gordon–Haus jitter) the seed was added only in the FWM sidebands. Also, there were no guiding filters. Note the large resultant amplitude and timing jitter.

Fig. 2
Fig. 2

Growth of FWM energy during a single soliton– soliton collision for three different conditions: (small; smooth curve) lossless fiber with constant dispersion, (small; smaller jagged curve) real fiber with lumped amplifiers spaced 33.3 km apart and exponentially tapered dispersion, (large; larger jagged curve) real fiber with lumped amplifiers spaced 33.3 km apart and constant dispersion. The FWM energy is for a single sideband and is normalized to the soliton pulse energy. Note that for the first two cases the FWM energy disappears completely following the collision, whereas for the third case, for which there is effective pseudo phase matching, the FWM energy builds to a large residual value.

Fig. 3
Fig. 3

Ideal exponential taper of D and the best three-step approximation to it for a fiber span with Lamp = 33.3 km and a loss rate of 0.21 dB/km. Inset: Locus of the FWM field vector in the complex plane, as Δk ranges from 0 to 2π, for the three-step approximation. Note that the figure approximates the perfect closed circle obtained with the exponential taper.

Fig. 4
Fig. 4

Residual FWM energy following a single collision of 20-ps solitons in channels spaced 0.6 nm apart in a chain of fiber spans with = 0.5 ps/(nm km) as a function of the amplifier spacing, for constant D and for the optimal two-, three-, and four-step approximations to the ideal exponential taper. The FWM energy is for a single sideband and is normalized to the soliton pulse energy. No noise seed was used in these simulations.

Fig. 5
Fig. 5

Same as Fig. 4, except that here the channel spacing is twice as great, i.e., it is 1.2 nm.

Equations (3)

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Δ k = ( k 2 + k s - k 1 - k 1 or k 1 + k α - k 2 - k 2 ) = 2 k ω 2 Δ ω 2 = - λ 2 D 2 π c Δ ω 2 ,
E i ( z ) E j 2 E k * i Δ k [ exp ( i Δ k z ) - 1 ] ,
L amp = N L res 2 π N / Δ k             ( N = 1 , 2 , 3 , ) .

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