Abstract

We show that solitarylike stimulated Raman excitations are possible in massive wavelength-division-multiplexed fiber-optic systems. These self-similar states can propagate across channels at a constant rate that depends on the total power launched in the system and the wave’s spectral width. The evolution of these excitations under the action of noise and during collisions is investigated in detail.

© 1999 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
    [CrossRef]
  2. A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.
  3. A. R. Chraplyvy, J. Lightwave Technol. 8, 1548 (1990).
    [CrossRef]
  4. R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
    [CrossRef]
  5. A. R. Chraplyvy and R. W. Tkach, IEEE Photon. Technol. Lett. 5, 666 (1993).
    [CrossRef]
  6. G. P. Agrawal, Nonlinear Fiber Optics2nd ed. (Academic, San Diego, Calif., 1995).
  7. S. Tariq and J. C. Palais, J. Lightwave Technol. 11, 1914 (1993)M. S. Kao and J. Wu, J. Lightwave Technol. 7, 1290 (1989)X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, Opt. Commun. 107, 358 (1994).
    [CrossRef]
  8. A. R. Chraplyvy, Electron. Lett. 20, 58 (1984).
    [CrossRef]
  9. D. N. Christodoulides and R. B. Jander, IEEE Photon. Technol. Lett. 8, 1722 (1996).
    [CrossRef]
  10. D. Cotter and A. M. Hill, Electron. Lett. 20, 185 (1984)A. R. Chraplyvy and P. S. Henry, Electron. Lett. 19, 641 (1983).
    [CrossRef]
  11. D. N. Christodoulides and R. I. Joseph, IEEE J. Quantum Electron. 25, 273 (1989).
    [CrossRef]
  12. If walk-off effects were to be included, then the operator ?z in Eq.??(3) should be replaced with ?z+fx?t, where fx is a polynomial of x.?A linear fx implies only first-order dispersion, etc.
  13. M. Zirngibl, Electron. Lett. 34, 789 (1998).
    [CrossRef]

1998 (2)

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

M. Zirngibl, Electron. Lett. 34, 789 (1998).
[CrossRef]

1996 (1)

D. N. Christodoulides and R. B. Jander, IEEE Photon. Technol. Lett. 8, 1722 (1996).
[CrossRef]

1993 (2)

A. R. Chraplyvy and R. W. Tkach, IEEE Photon. Technol. Lett. 5, 666 (1993).
[CrossRef]

S. Tariq and J. C. Palais, J. Lightwave Technol. 11, 1914 (1993)M. S. Kao and J. Wu, J. Lightwave Technol. 7, 1290 (1989)X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, Opt. Commun. 107, 358 (1994).
[CrossRef]

1990 (2)

A. R. Chraplyvy, J. Lightwave Technol. 8, 1548 (1990).
[CrossRef]

R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
[CrossRef]

1989 (1)

D. N. Christodoulides and R. I. Joseph, IEEE J. Quantum Electron. 25, 273 (1989).
[CrossRef]

1984 (2)

D. Cotter and A. M. Hill, Electron. Lett. 20, 185 (1984)A. R. Chraplyvy and P. S. Henry, Electron. Lett. 19, 641 (1983).
[CrossRef]

A. R. Chraplyvy, Electron. Lett. 20, 58 (1984).
[CrossRef]

Abramov, A. A.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Agrawal, G. P.

G. P. Agrawal, Nonlinear Fiber Optics2nd ed. (Academic, San Diego, Calif., 1995).

Braun, R-P.

R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
[CrossRef]

Chen, C. D.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Chraplyvy, A. R.

A. R. Chraplyvy and R. W. Tkach, IEEE Photon. Technol. Lett. 5, 666 (1993).
[CrossRef]

A. R. Chraplyvy, J. Lightwave Technol. 8, 1548 (1990).
[CrossRef]

A. R. Chraplyvy, Electron. Lett. 20, 58 (1984).
[CrossRef]

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Christodoulides, D. N.

D. N. Christodoulides and R. B. Jander, IEEE Photon. Technol. Lett. 8, 1722 (1996).
[CrossRef]

D. N. Christodoulides and R. I. Joseph, IEEE J. Quantum Electron. 25, 273 (1989).
[CrossRef]

Cotter, D.

D. Cotter and A. M. Hill, Electron. Lett. 20, 185 (1984)A. R. Chraplyvy and P. S. Henry, Electron. Lett. 19, 641 (1983).
[CrossRef]

Espindola, R. P.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Ferrand, D. A.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Friesem, A. A.

R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
[CrossRef]

Hill, A. M.

D. Cotter and A. M. Hill, Electron. Lett. 20, 185 (1984)A. R. Chraplyvy and P. S. Henry, Electron. Lett. 19, 641 (1983).
[CrossRef]

Jander, R. B.

D. N. Christodoulides and R. B. Jander, IEEE Photon. Technol. Lett. 8, 1722 (1996).
[CrossRef]

Joseph, R. I.

D. N. Christodoulides and R. I. Joseph, IEEE J. Quantum Electron. 25, 273 (1989).
[CrossRef]

Judkins, J. B.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Kim, I.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Li, Y. P.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Lichtman, E.

R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
[CrossRef]

Mizuhara, O.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Monnard, R.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Nguyen, T. V.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Ogawa, K.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Palais, J. C.

S. Tariq and J. C. Palais, J. Lightwave Technol. 11, 1914 (1993)M. S. Kao and J. Wu, J. Lightwave Technol. 7, 1290 (1989)X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, Opt. Commun. 107, 358 (1994).
[CrossRef]

Pedrazzani, J. R.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Srivastava, A. K.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Strasser, T. A.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Sulhoff, J. W.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Sun, Y.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Tariq, S.

S. Tariq and J. C. Palais, J. Lightwave Technol. 11, 1914 (1993)M. S. Kao and J. Wu, J. Lightwave Technol. 7, 1290 (1989)X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, Opt. Commun. 107, 358 (1994).
[CrossRef]

Tench, R. E.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Tkach, R. W.

A. R. Chraplyvy and R. W. Tkach, IEEE Photon. Technol. Lett. 5, 666 (1993).
[CrossRef]

Tzeng, L. D.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Vengsarkar, A. M.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Waarts, R. G.

R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
[CrossRef]

Wolf, C.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Wysocki, P. F.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Yaffe, H. H.

R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
[CrossRef]

Yeates, P. D.

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

Zhou, J.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Zirngibl, M.

M. Zirngibl, Electron. Lett. 34, 789 (1998).
[CrossRef]

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Zyskind, J. L.

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

Electron. Lett. (4)

C. D. Chen, I. Kim, O. Mizuhara, T. V. Nguyen, K. Ogawa, R. E. Tench, L. D. Tzeng, and P. D. Yeates, Electron. Lett. 34, 2370 (1998).
[CrossRef]

A. R. Chraplyvy, Electron. Lett. 20, 58 (1984).
[CrossRef]

D. Cotter and A. M. Hill, Electron. Lett. 20, 185 (1984)A. R. Chraplyvy and P. S. Henry, Electron. Lett. 19, 641 (1983).
[CrossRef]

M. Zirngibl, Electron. Lett. 34, 789 (1998).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. N. Christodoulides and R. I. Joseph, IEEE J. Quantum Electron. 25, 273 (1989).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

D. N. Christodoulides and R. B. Jander, IEEE Photon. Technol. Lett. 8, 1722 (1996).
[CrossRef]

A. R. Chraplyvy and R. W. Tkach, IEEE Photon. Technol. Lett. 5, 666 (1993).
[CrossRef]

J. Lightwave Technol. (2)

A. R. Chraplyvy, J. Lightwave Technol. 8, 1548 (1990).
[CrossRef]

S. Tariq and J. C. Palais, J. Lightwave Technol. 11, 1914 (1993)M. S. Kao and J. Wu, J. Lightwave Technol. 7, 1290 (1989)X. Zhang, B. F. Jorgensen, F. Ebskamp, and R. J. Pedersen, Opt. Commun. 107, 358 (1994).
[CrossRef]

Proc. IEEE (1)

R. G. Waarts, A. A. Friesem, E. Lichtman, H. H. Yaffe, and R-P. Braun, Proc. IEEE 78, 1344 (1990).
[CrossRef]

Other (3)

A. K. Srivastava, Y. Sun, J. W. Sulhoff, C. Wolf, M. Zirngibl, R. Monnard, A. R. Chraplyvy, A. A. Abramov, R. P. Espindola, T. A. Strasser, J. R. Pedrazzani, A. M. Vengsarkar, J. L. Zyskind, J. Zhou, D. A. Ferrand, P. F. Wysocki, J. B. Judkins, and Y. P. Li, in Optical Fiber Communication Conference, Vol.??2 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), postdeadline paper PD10.

G. P. Agrawal, Nonlinear Fiber Optics2nd ed. (Academic, San Diego, Calif., 1995).

If walk-off effects were to be included, then the operator ?z in Eq.??(3) should be replaced with ?z+fx?t, where fx is a polynomial of x.?A linear fx implies only first-order dispersion, etc.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

(a) Power-distribution envelope of a SRS solitary excitation when xo=10 and JT=1 W at (dotted curve) z=0, (dashed curve) z=49 km, and (solid curve) z=98 km. (b) Same as in (a) but with fiber losses 0.2 dB/km included.

Fig. 2
Fig. 2

Propagation of a SRS solitary excitation when xo=10, JT=1 W, and the noise level per channel is 10 nW.

Fig. 3
Fig. 3

(a) Collision of two Gaussian self-similar states when the narrow beam leads the broad one. (b) Collision of two identical states.

Equations (6)

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

dPnzdz+αPnz+gΔf2APnzm=1Nm-nPmz=0,
Pnz=PnoJTexp-αzexpnGJTZem=1NPmoexpmGJTZe,
Pz,xz+αPz,x+GPz,x-+η-xPz,ηdη=0.
Pz,x=PoxJTexp-αzexpGJTZexPoxexpGJTZexdx,
Pz,x=Poexp-αzexp-x-vZe2xo2.
v=πGPoxo3/2.

Metrics