Abstract

We present a statistical description of the propagation of short pulses in long optical fibers, taking into account the Kerr and nonlocal nonlinearities on an equal footing. We use the Wigner approach on the modified nonlinear Schrödinger equation to obtain a wave kinetic equation and a nonlinear dispersion relation. The latter shows that the optical pulse decoherence reduces the growth rate of the modulational instability and thereby contributes to the nonlinear stability of the pulses in long optical fibers. It is also found that the interaction between spectral broadening and nonlocality tends to extend the instability region.

© 2005 Optical Society of America

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  1. A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  2. A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 171 (1973).
    [CrossRef]
  3. Yu. Kivshar and G. P. Agrawal, Optical Solitons, From Fibers to Photonic Crystals (Academic, 2003).
  4. A. Hasegawa, Phys. Plasmas 8, 1763 (2001).
    [CrossRef]
  5. C. C. Jeng, M. F. Shih, K. Motzek, and Y. S. Kivshar, Phys. Rev. Lett. 92, 043904 (2004).
    [CrossRef]
  6. T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
    [CrossRef]
  7. E. Wigner, Phys. Rev. 40, 749 (1932).
    [CrossRef]
  8. J. T. Mendonça and N. L. Tsintsadze, Phys. Rev. E 62, 4276 (2000).
    [CrossRef]
  9. R. Fedele and D. Anderson, J. Opt. B: Quantum Semiclassical Opt. 2, 207 (2000).
    [CrossRef]
  10. B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
    [CrossRef]
  11. L. Helczynski, M. Lisak, and D. Anderson, Phys. Rev. E 67, 026602 (2003).
    [CrossRef]
  12. D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
    [CrossRef]
  13. D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 70, 026603 (2004).
    [CrossRef]
  14. B. Hall, M. Lisak, D. Anderson, and V. E. Semenov, Phys. Rev. A 321, 255 (2004).
  15. Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
    [CrossRef]
  16. A. Hasegawa and Y. Kodama, Solitons and Optical Communications (Oxford U. Press, 1995).
  17. P. K. Shukla and J. J. Rasmussen, Opt. Lett. 11, 171 (1986).
    [CrossRef]
  18. J. T. Mendonça, Theory of Photon Acceleration (IOP Publishing, 2001).
    [CrossRef]
  19. I. M. Besieris and F. D. Tappert, J. Math. Phys. 17, 734 (1976).
    [CrossRef]
  20. R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).
  21. D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 (2002).
    [CrossRef]
  22. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

2005

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

2004

C. C. Jeng, M. F. Shih, K. Motzek, and Y. S. Kivshar, Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 70, 026603 (2004).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, and V. E. Semenov, Phys. Rev. A 321, 255 (2004).

2003

L. Helczynski, M. Lisak, and D. Anderson, Phys. Rev. E 67, 026602 (2003).
[CrossRef]

2002

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 (2002).
[CrossRef]

2001

A. Hasegawa, Phys. Plasmas 8, 1763 (2001).
[CrossRef]

2000

J. T. Mendonça and N. L. Tsintsadze, Phys. Rev. E 62, 4276 (2000).
[CrossRef]

R. Fedele and D. Anderson, J. Opt. B: Quantum Semiclassical Opt. 2, 207 (2000).
[CrossRef]

1987

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

1986

1976

I. M. Besieris and F. D. Tappert, J. Math. Phys. 17, 734 (1976).
[CrossRef]

1973

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

1932

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Agrawal, G. P.

Yu. Kivshar and G. P. Agrawal, Optical Solitons, From Fibers to Photonic Crystals (Academic, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

Anderson, D.

B. Hall, M. Lisak, D. Anderson, and V. E. Semenov, Phys. Rev. A 321, 255 (2004).

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 70, 026603 (2004).
[CrossRef]

L. Helczynski, M. Lisak, and D. Anderson, Phys. Rev. E 67, 026602 (2003).
[CrossRef]

D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 (2002).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

R. Fedele and D. Anderson, J. Opt. B: Quantum Semiclassical Opt. 2, 207 (2000).
[CrossRef]

Besieris, I. M.

I. M. Besieris and F. D. Tappert, J. Math. Phys. 17, 734 (1976).
[CrossRef]

Fedele, R.

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

R. Fedele and D. Anderson, J. Opt. B: Quantum Semiclassical Opt. 2, 207 (2000).
[CrossRef]

Hall, B.

B. Hall, M. Lisak, D. Anderson, and V. E. Semenov, Phys. Rev. A 321, 255 (2004).

D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 (2002).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Phys. Plasmas 8, 1763 (2001).
[CrossRef]

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons and Optical Communications (Oxford U. Press, 1995).

Helczynski, L.

L. Helczynski, M. Lisak, and D. Anderson, Phys. Rev. E 67, 026602 (2003).
[CrossRef]

Helczynski-Wolf, L.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 70, 026603 (2004).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

Jeng, C. C.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. S. Kivshar, Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef]

Kivshar, Y. S.

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

C. C. Jeng, M. F. Shih, K. Motzek, and Y. S. Kivshar, Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef]

Kivshar, Yu.

Yu. Kivshar and G. P. Agrawal, Optical Solitons, From Fibers to Photonic Crystals (Academic, 2003).

Kodama, Y.

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

A. Hasegawa and Y. Kodama, Solitons and Optical Communications (Oxford U. Press, 1995).

Ku, T. S.

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

Lisak, M.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 70, 026603 (2004).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, and V. E. Semenov, Phys. Rev. A 321, 255 (2004).

L. Helczynski, M. Lisak, and D. Anderson, Phys. Rev. E 67, 026602 (2003).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 (2002).
[CrossRef]

Loudon, R.

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

Marklund, M.

D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 (2002).
[CrossRef]

Mendonça, J. T.

J. T. Mendonça and N. L. Tsintsadze, Phys. Rev. E 62, 4276 (2000).
[CrossRef]

J. T. Mendonça, Theory of Photon Acceleration (IOP Publishing, 2001).
[CrossRef]

Motzek, K.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. S. Kivshar, Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef]

Rasmussen, J. J.

Semenov, V.

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 70, 026603 (2004).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

Semenov, V. E.

B. Hall, M. Lisak, D. Anderson, and V. E. Semenov, Phys. Rev. A 321, 255 (2004).

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

Shih, M. F.

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

C. C. Jeng, M. F. Shih, K. Motzek, and Y. S. Kivshar, Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef]

Shukla, P. K.

Sukhorukov, A. A.

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

Tappert, F. D.

I. M. Besieris and F. D. Tappert, J. Math. Phys. 17, 734 (1976).
[CrossRef]

Tappert, T.

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Tsintsadze, N. L.

J. T. Mendonça and N. L. Tsintsadze, Phys. Rev. E 62, 4276 (2000).
[CrossRef]

Wigner, E.

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Appl. Phys. Lett.

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

A. Hasegawa and T. Tappert, Appl. Phys. Lett. 23, 171 (1973).
[CrossRef]

IEEE J. Quantum Electron.

Y. Kodama and A. Hasegawa, IEEE J. Quantum Electron. 23, 510 (1987).
[CrossRef]

J. Math. Phys.

I. M. Besieris and F. D. Tappert, J. Math. Phys. 17, 734 (1976).
[CrossRef]

J. Opt. B: Quantum Semiclassical Opt.

R. Fedele and D. Anderson, J. Opt. B: Quantum Semiclassical Opt. 2, 207 (2000).
[CrossRef]

Opt. Lett.

Phys. Plasmas

A. Hasegawa, Phys. Plasmas 8, 1763 (2001).
[CrossRef]

Phys. Rev.

E. Wigner, Phys. Rev. 40, 749 (1932).
[CrossRef]

Phys. Rev. A

B. Hall, M. Lisak, D. Anderson, and V. E. Semenov, Phys. Rev. A 321, 255 (2004).

Phys. Rev. E

J. T. Mendonça and N. L. Tsintsadze, Phys. Rev. E 62, 4276 (2000).
[CrossRef]

D. Anderson, B. Hall, M. Lisak, and M. Marklund, Phys. Rev. E 65, 046417 (2002).
[CrossRef]

B. Hall, M. Lisak, D. Anderson, R. Fedele, and V. E. Semenov, Phys. Rev. E 65, 035602(R) (2002).
[CrossRef]

L. Helczynski, M. Lisak, and D. Anderson, Phys. Rev. E 67, 026602 (2003).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 69, 025601 (2004).
[CrossRef]

D. Anderson, L. Helczynski-Wolf, M. Lisak, and V. Semenov, Phys. Rev. E 70, 026603 (2004).
[CrossRef]

Phys. Rev. Lett.

C. C. Jeng, M. F. Shih, K. Motzek, and Y. S. Kivshar, Phys. Rev. Lett. 92, 043904 (2004).
[CrossRef]

T. S. Ku, M. F. Shih, A. A. Sukhorukov, and Y. S. Kivshar, Phys. Rev. Lett. 94, 063904 (2005).
[CrossRef]

Other

Yu. Kivshar and G. P. Agrawal, Optical Solitons, From Fibers to Photonic Crystals (Academic, 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

R. Loudon, The Quantum Theory of Light (Oxford U. Press, 2000).

A. Hasegawa and Y. Kodama, Solitons and Optical Communications (Oxford U. Press, 1995).

J. T. Mendonça, Theory of Photon Acceleration (IOP Publishing, 2001).
[CrossRef]

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Figures (1)

Fig. 1
Fig. 1

Effects of spectral broadening and nonlocality. Normalizing variables K and Ω, as well as parameters Ω T and γ, such that α = β = 1 , we have plotted the imaginary part κ = Im ( K ) as a function of Ω, when the frequency shift Ω 0 is zero. From the peaks of the curves downward, we have used I 0 = 0.5 , and the solid curve represents Ω T = γ = 0 and shows the regular modulational instability growth rate. The dashed curve gives κ for Ω T = 0.1 and γ = 0 , while the dashed–dotted curve uses Ω T = 0 and γ = 1 , and the fourth (dashed–dotted) curve has Ω T = 0.1 and γ = 1 . The last two curves (dashed and dotted, respectively), where Ω T = 0.1 , γ = 1.9 , and Ω T = 0 , γ = 1.9 , respectively, clearly show the character of the combined effect of broadening and nonlocality, namely, a widening instability region.

Equations (10)

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

i ( z + Γ ) E + α t 2 E + β I E + i γ t ( I E ) = 0 ,
F ( z , t , k , ω ) = 1 ( 2 π ) 2 d ζ d τ exp [ i ( k ζ ω τ ) ] C ( z + , z , t + , t ) ,
I ( z , t ) = ( 2 π ) 2 d k d ω F ( z , t , k , ω ) .
2 ω α t F z F + 2 β I sin ( 1 2 t ω ) F + γ { t [ I cos ( 1 2 t ω ) F ] + 2 ω I sin ( 1 2 t ω ) F } = 2 Γ F ,
1 = 1 2 α Ω d ω [ β + γ ( ω + Ω 2 ) ] F 0 ( ω Ω 2 ) [ β + γ ( ω Ω 2 ) ] F 0 ( ω + Ω 2 ) ω + ( K γ Ω I 0 ) 2 α Ω ,
K = 2 ( γ I 0 α Ω 0 ) Ω ± [ γ 2 I 0 2 Ω 2 + α 2 Ω 4 2 α I 0 ( β + γ Ω 0 ) Ω 2 ] 1 2 .
F 0 ( ω ) = I 0 π Ω T ( ω Ω 0 ) 2 + Ω T 2 ,
1 = I 0 Ω 2 γ [ K γ I 0 Ω + α Ω ( Ω 0 i Ω T ) ] 2 α β Ω ( K γ I 0 Ω + Ω 0 i Ω T ) 2 α 2 Ω 4 ,
K = 2 [ γ I 0 α ( Ω 0 i Ω T ) ] Ω ± { γ 2 I 0 2 Ω 2 + α 2 Ω 4 2 α I 0 [ β + γ ( Ω 0 i Ω T ) ] Ω 2 } 1 2 .
K Ω 2 γ I 0 2 α Ω 0 ± f 1 2 + 2 i α Ω T ± i α γ I 0 Ω T f 1 2

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