Abstract

We discover analytically an extensive family of optical similaritons, propagating inside graded-index nonlinear waveguide amplifiers. We show that there exists a one-to-one correspondence between these novel similaritons and standard solitons of the homogeneous nonlinear Schrödinger equation. We demonstrate that for certain inhomogeneity and gain profiles, the newly discovered similaritons turn into solitons over sufficiently long propagation distances.

© 2007 Optical Society of America

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References

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  1. G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).
  2. Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).
  3. H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976).
    [CrossRef]
  4. R. Balakrishnan, Phys. Rev. A 32, 1144 (1985).
    [CrossRef] [PubMed]
  5. V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007).
    [CrossRef] [PubMed]
  6. V. N. Serkin and A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002).
    [CrossRef]
  7. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000).
    [CrossRef]
  8. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, J. Opt. Soc. Am. B 10, 1185 (1993).
    [CrossRef]
  9. M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
    [CrossRef] [PubMed]
  10. C. Finot and G. Millot, Opt. Express 13, 7653 (2005).
    [CrossRef] [PubMed]
  11. J. D. Moores, Opt. Lett. 21, 555 (1996).
    [CrossRef] [PubMed]
  12. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
    [CrossRef] [PubMed]
  13. V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
    [CrossRef]
  14. V. I. Kruglov and J. D. Harvey, J. Opt. Soc. Am. B 23, 2541 (2006).
    [CrossRef]
  15. S. A. Ponomarenko and G. P. Agrawal, Opt. Express 15, 2963 (2007).
    [CrossRef] [PubMed]
  16. S. A. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006).
    [CrossRef] [PubMed]
  17. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  18. Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
    [CrossRef]
  19. G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).
  20. P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, J. Opt. Soc. Am. B 11, 1254 (1994).
    [CrossRef]

2007 (2)

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, Opt. Express 15, 2963 (2007).
[CrossRef] [PubMed]

2006 (2)

S. A. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

V. I. Kruglov and J. D. Harvey, J. Opt. Soc. Am. B 23, 2541 (2006).
[CrossRef]

2005 (2)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

C. Finot and G. Millot, Opt. Express 13, 7653 (2005).
[CrossRef] [PubMed]

2003 (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

2002 (1)

V. N. Serkin and A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002).
[CrossRef]

2000 (2)

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

1998 (1)

Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

1996 (1)

1994 (1)

1993 (1)

1985 (1)

R. Balakrishnan, Phys. Rev. A 32, 1144 (1985).
[CrossRef] [PubMed]

1976 (1)

H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Agrawal, G. P.

S. A. Ponomarenko and G. P. Agrawal, Opt. Express 15, 2963 (2007).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Anderson, D.

Balakrishnan, R.

R. Balakrishnan, Phys. Rev. A 32, 1144 (1985).
[CrossRef] [PubMed]

Belyaeva, T. L.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

Bosshard, C.

Chen, H.-H.

H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976).
[CrossRef]

Desaix, M.

Dudley, J. D.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Dudley, J. M.

Fermann, M. E.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Finot, C.

Harvey, J. D.

V. I. Kruglov and J. D. Harvey, J. Opt. Soc. Am. B 23, 2541 (2006).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000).
[CrossRef]

Hasegawa, A.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

V. N. Serkin and A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002).
[CrossRef]

Karlsson, M.

Kivshar, Y. S.

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

Kivshar, Yu. S.

Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Kruglov, V. I.

V. I. Kruglov and J. D. Harvey, J. Opt. Soc. Am. B 23, 2541 (2006).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000).
[CrossRef]

Lamb, G. L.

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

Lisak, M.

Liu, C.-S.

H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976).
[CrossRef]

Luther-Davies, B.

Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Mamyshev, P. V.

Millot, G.

Moores, J. D.

Peacock, A. C.

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000).
[CrossRef]

Ponomarenko, S. A.

S. A. Ponomarenko and G. P. Agrawal, Opt. Express 15, 2963 (2007).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

Quiroga-Teixeiro, M. L.

Serkin, V. N.

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

V. N. Serkin and A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Stegeman, G. I.

Thomsen, B. C.

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

IEEE J. Sel. Top. Quantum Electron. (1)

V. N. Serkin and A. Hasegawa, IEEE J. Sel. Top. Quantum Electron. 8, 418 (2002).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (2)

Phys. Rep. (1)

Yu. S. Kivshar and B. Luther-Davies, Phys. Rep. 298, 81 (1998).
[CrossRef]

Phys. Rev. A (1)

R. Balakrishnan, Phys. Rev. A 32, 1144 (1985).
[CrossRef] [PubMed]

Phys. Rev. E (1)

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. E 71, 056619 (2005).
[CrossRef]

Phys. Rev. Lett. (5)

H.-H. Chen and C.-S. Liu, Phys. Rev. Lett. 37, 693 (1976).
[CrossRef]

V. I. Kruglov, A. C. Peacock, and J. D. Harvey, Phys. Rev. Lett. 90, 113902 (2003).
[CrossRef] [PubMed]

S. A. Ponomarenko and G. P. Agrawal, Phys. Rev. Lett. 97, 013901 (2006).
[CrossRef] [PubMed]

V. N. Serkin, A. Hasegawa, and T. L. Belyaeva, Phys. Rev. Lett. 98, 074102 (2007).
[CrossRef] [PubMed]

M. E. Fermann, V. I. Kruglov, B. C. Thomsen, J. D. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Other (3)

G. P. Agrawal, Nonlinear Fiber Optics, 4th ed. (Academic, 2007).

Y. S. Kivshar and G. P. Agrawal, Optical Solitons: from Fibers to Photonic Crystals (Academic, 2003).

G. L. Lamb, Elements of Soliton Theory (Wiley, 1980).

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Figures (4)

Fig. 1
Fig. 1

Gain, width, and tapering profiles, plotted as functions of Z, for two specific choices of F ( Z ) discussed in the text.

Fig. 2
Fig. 2

Evolution of the fundamental (a) bright and (b) dark similaritons over two diffraction lengths under conditions of Fig. 1a. Other parameters are specified in the text.

Fig. 3
Fig. 3

Evolution of (a) bright and (b) dark similaritons over 3 diffraction lengths for the width and gain profiles of Fig. 1b. Notice that solitons already form within 2 diffraction lengths.

Fig. 4
Fig. 4

Collision of two bright similaritons, which turn into solitons for large Z.

Equations (16)

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n ( x , z ) = n 0 + n 1 F ( z ) x 2 + n 2 I ( x , z ) ,
i U Z + 1 2 2 U X 2 + F ( Z ) X 2 2 U i 2 G ( Z ) U ± U 2 U = 0 ,
U ( X , Z ) = A ( Z ) Ψ [ X X c ( Z ) W ( Z ) , ζ ( Z ) ] e i Φ ( X , Z ) ,
Φ ( X , Z ) = C ( Z ) X 2 2 + B ( Z ) X + D ( Z ) .
i Ψ ζ + 1 2 2 Ψ χ 2 ± Ψ 2 Ψ = 0 .
ζ ( Z ) = ζ 0 + 0 Z d S W 2 ( S ) ,
χ ( X , Z ) = [ X X c ( Z ) ] W ( Z ) .
X c ( Z ) = ( X 0 + B 0 Z ) W ( Z ) ,
U ( X , Z ) = 1 W ( Z ) Ψ [ X ( X 0 + B 0 Z ) W W , ζ ( Z ) ] × exp [ i X 2 2 W d W d Z + i B 0 X W i B 0 2 2 0 Z d S W 2 ( S ) ] ,
d 2 W d Z 2 F ( Z ) W = 0 .
G ( Z ) = d [ ln W ( Z ) ] d Z .
I B ( χ , ζ ) = ( a 2 W 2 ) sech 2 [ a ( χ v ζ ) ] ,
I D ( χ , ζ ) = ( u 0 2 W 2 ) [ cos 2 ( ϕ ) tanh 2 ( Θ ) + sin 2 ( ϕ ) ] ,
Θ ( χ , ζ ) = u 0 cos ( ϕ ) [ χ u 0 ζ sin ( ϕ ) ] .
F ( Z ) = 1 2 sech 2 ( Z ) , W ( Z ) = sech ( Z ) .
W Z = tanh ( Z + Z * ) tanh ( Z * ) .

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