Abstract

Cross-phase-modulation-induced instability and collisions lead to a minimum channel spacing in dispersion-managed wavelength-division-multiplexed nonreturn-to-zero transmissions. We obtain a simple analytical description of these instabilities and confirm the results by numerical simulations.

© 1996 Optical Society of America

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References

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  1. H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.
  2. N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.
  3. C. Kurtzke, IEEE Photon. Technol. Lett. 5, 1250 (1993).
    [CrossRef]
  4. E. Lichtman, S. G. Evangelides, Electron. Lett. 30, 346 (1994).
    [CrossRef]
  5. K. Sekine, N. Kikuchi, S. Sasaki, M. Aoki, Electron. Lett. 30, 1150 (1994).
    [CrossRef]
  6. R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
    [CrossRef]
  7. N. Kikuchi, K. Sekine, S. Sasaki, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 40–41; T. Ogata, Y. Aoki, T. Koga, I. Matsuoka, ibid., pp. 42–44; Z. Wang, S. T. Ho, ibid., pp. 44–45; M. I. Hayee, X. Y. Zou, A. E. Willner, ibid., pp. 78–79.
  8. Y. Kodama, S. Wabnitz, Opt. Lett. 20, 2291 (1995); Electron. Lett. 31, 1761 (1995).
    [CrossRef] [PubMed]
  9. G. B. Whitman, Linear and Nonlinear Waves (Wiley, New York, 1974), Chap. 5.
  10. A. L. Berkhoer, V. E. Zakharov, Zh. Eksp. Teor. Fiz. 58, 903 (1970) [Sov. Phys. JETP 31, 486 (1970)].
  11. G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
    [CrossRef] [PubMed]

1995 (2)

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
[CrossRef]

Y. Kodama, S. Wabnitz, Opt. Lett. 20, 2291 (1995); Electron. Lett. 31, 1761 (1995).
[CrossRef] [PubMed]

1994 (2)

E. Lichtman, S. G. Evangelides, Electron. Lett. 30, 346 (1994).
[CrossRef]

K. Sekine, N. Kikuchi, S. Sasaki, M. Aoki, Electron. Lett. 30, 1150 (1994).
[CrossRef]

1993 (1)

C. Kurtzke, IEEE Photon. Technol. Lett. 5, 1250 (1993).
[CrossRef]

1987 (1)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

1970 (1)

A. L. Berkhoer, V. E. Zakharov, Zh. Eksp. Teor. Fiz. 58, 903 (1970) [Sov. Phys. JETP 31, 486 (1970)].

Agrawal, G. P.

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Aoki, M.

K. Sekine, N. Kikuchi, S. Sasaki, M. Aoki, Electron. Lett. 30, 1150 (1994).
[CrossRef]

Bergano, N. S.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Berkhoer, A. L.

A. L. Berkhoer, V. E. Zakharov, Zh. Eksp. Teor. Fiz. 58, 903 (1970) [Sov. Phys. JETP 31, 486 (1970)].

Chikama, T.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Chraplyvy, A. R.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
[CrossRef]

Corbett, P. C.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Davidson, C. R.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Derosier, R. M.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
[CrossRef]

Evangelides, S. G.

E. Lichtman, S. G. Evangelides, Electron. Lett. 30, 346 (1994).
[CrossRef]

Evankow, J. D.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Ferguson, G. A.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Forghieri, F.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
[CrossRef]

Gnauck, A. H.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
[CrossRef]

Ishikawa, G.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Kai, Y.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Kerfoot, F. W.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Kikuchi, N.

K. Sekine, N. Kikuchi, S. Sasaki, M. Aoki, Electron. Lett. 30, 1150 (1994).
[CrossRef]

N. Kikuchi, K. Sekine, S. Sasaki, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 40–41; T. Ogata, Y. Aoki, T. Koga, I. Matsuoka, ibid., pp. 42–44; Z. Wang, S. T. Ho, ibid., pp. 44–45; M. I. Hayee, X. Y. Zou, A. E. Willner, ibid., pp. 78–79.

Kinoshita, S.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Kodama, Y.

Kurtzke, C.

C. Kurtzke, IEEE Photon. Technol. Lett. 5, 1250 (1993).
[CrossRef]

Levonas, M. D.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Lichtman, E.

E. Lichtman, S. G. Evangelides, Electron. Lett. 30, 346 (1994).
[CrossRef]

Mills, M. A.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Miyata, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Morreale, J. P.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Nagel, J. A.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Nishimoto, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Onaka, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Ooi, H.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Otsuka, K.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Pedrazzani, J. R.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Sasaki, S.

K. Sekine, N. Kikuchi, S. Sasaki, M. Aoki, Electron. Lett. 30, 1150 (1994).
[CrossRef]

N. Kikuchi, K. Sekine, S. Sasaki, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 40–41; T. Ogata, Y. Aoki, T. Koga, I. Matsuoka, ibid., pp. 42–44; Z. Wang, S. T. Ho, ibid., pp. 44–45; M. I. Hayee, X. Y. Zou, A. E. Willner, ibid., pp. 78–79.

Seino, M.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

Sekine, K.

K. Sekine, N. Kikuchi, S. Sasaki, M. Aoki, Electron. Lett. 30, 1150 (1994).
[CrossRef]

N. Kikuchi, K. Sekine, S. Sasaki, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 40–41; T. Ogata, Y. Aoki, T. Koga, I. Matsuoka, ibid., pp. 42–44; Z. Wang, S. T. Ho, ibid., pp. 44–45; M. I. Hayee, X. Y. Zou, A. E. Willner, ibid., pp. 78–79.

Sulhoff, J. W.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Tkach, R. W.

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
[CrossRef]

Trembay, M. D.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Vengsarkar, A. M.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Wabnitz, S.

Whitman, G. B.

G. B. Whitman, Linear and Nonlinear Waves (Wiley, New York, 1974), Chap. 5.

Wilson, D. L.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Zakharov, V. E.

A. L. Berkhoer, V. E. Zakharov, Zh. Eksp. Teor. Fiz. 58, 903 (1970) [Sov. Phys. JETP 31, 486 (1970)].

Zyskind, J. L.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

Electron. Lett. (2)

E. Lichtman, S. G. Evangelides, Electron. Lett. 30, 346 (1994).
[CrossRef]

K. Sekine, N. Kikuchi, S. Sasaki, M. Aoki, Electron. Lett. 30, 1150 (1994).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

C. Kurtzke, IEEE Photon. Technol. Lett. 5, 1250 (1993).
[CrossRef]

J. Lightwave Technol. (1)

R. W. Tkach, A. R. Chraplyvy, F. Forghieri, A. H. Gnauck, R. M. Derosier, J. Lightwave Technol. 13, 841 (1995).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

G. P. Agrawal, Phys. Rev. Lett. 59, 880 (1987).
[CrossRef] [PubMed]

Zh. Eksp. Teor. Fiz. (1)

A. L. Berkhoer, V. E. Zakharov, Zh. Eksp. Teor. Fiz. 58, 903 (1970) [Sov. Phys. JETP 31, 486 (1970)].

Other (4)

G. B. Whitman, Linear and Nonlinear Waves (Wiley, New York, 1974), Chap. 5.

H. Onaka, H. Miyata, G. Ishikawa, K. Otsuka, H. Ooi, Y. Kai, S. Kinoshita, M. Seino, H. Nishimoto, T. Chikama, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD19.

N. S. Bergano, C. R. Davidson, D. L. Wilson, F. W. Kerfoot, M. D. Trembay, M. D. Levonas, J. P. Morreale, J. D. Evankow, P. C. Corbett, M. A. Mills, G. A. Ferguson, A. M. Vengsarkar, J. R. Pedrazzani, J. A. Nagel, J. L. Zyskind, J. W. Sulhoff, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), paper PD23.

N. Kikuchi, K. Sekine, S. Sasaki, in Optical Fiber Communication, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), pp. 40–41; T. Ogata, Y. Aoki, T. Koga, I. Matsuoka, ibid., pp. 42–44; Z. Wang, S. T. Ho, ibid., pp. 44–45; M. I. Hayee, X. Y. Zou, A. E. Willner, ibid., pp. 78–79.

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Figures (4)

Fig. 1
Fig. 1

Contour plot of XPM-induced MI gain versus pump frequency separation Δ and sideband detuning Ω.

Fig. 2
Fig. 2

Input (dashed curve) and output signal (solid curve) in the top trace and output spectrum in the bottom trace with channel separation Δ = 8.

Fig. 3
Fig. 3

Same as in Fig. 2 but with Δ = 6.

Fig. 4
Fig. 4

Collisions between (a) short fast and (b) long slow NRZ pulses. Dashed curves, initial pulses; solid curves, pulses at 9000 km.

Equations (8)

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

i Q z - k 2 2 Q t 2 + ν Q 2 Q = 0 ,
Q = Q 1 + Q 2 ,
i q 1 Z - β 2 2 q 1 T 2 + ( α 1 q 1 2 + α 2 q 2 2 ) q 1 = 0 , i q 2 Z - β 2 2 q 2 T 2 + ( α 1 q 2 2 + α 2 q 1 2 ) q 2 = 0.
q i ( T , Z ) = [ ρ i ( T , Z ) ] 1 / 2 exp [ i - T u i ( T , Z ) d T ] .
Z ( ρ 1 u 1 ρ 2 u 2 ) = [ β u 1 β ρ 1 0 0 α 1 β u 1 α 2 0 0 0 β u 2 β ρ 2 α 2 0 α 1 β u 2 ] T ( ρ 1 u 1 ρ 2 u 2 ) .
[ ( β u 1 - σ ) 2 - α 1 β ρ 1 ] [ ( β u 2 - σ ) 2 - α 1 β ρ 2 ] - α 2 2 β 2 ρ 1 ρ 2 = 0 ,
Δ > Δ c = 2 [ ( α 1 + α 2 ) ρ β ] 1 / 2 ,
Δ λ > Δ λ c ( nm ) = 2 × 10 13 × { 30 λ ( μ m ) n 2 ( m 2 / W ) P av ( mW ) c ( m / s ) D [ ps / ( nm km ) ] A eff ( μ m 2 ) } 1 / 2 ,

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