Abstract

We present an analytical solution for propagating pulses in normal dispersion fiber amplifiers, including the effect of third-order dispersion. The solution of the generalized nonlinear Schrödinger equation is based on asymptotic methods, first-order perturbation theory, and a renormalization procedure and leads to determination of the critical length corresponding to pulse breakup. We have also found and confirmed numerically the condition on the parameters that govern the propagation, as is necessary to ensure a highly accurate analytical description of the pulses and critical lengths in fiber amplifiers with third-order dispersion.

© 2010 Optical Society of America

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  1. M. E. Fermann, V. I. Kruglov, B. C. Thomson, J. M. Dudley, and J. D. Harvey, Phys. Rev. Lett. 84, 6010 (2000).
    [CrossRef] [PubMed]
  2. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, Opt. Lett. 25, 1753 (2000).
    [CrossRef]
  3. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, J. Opt. Soc. Am. B 19, 461 (2002).
    [CrossRef]
  4. S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
    [CrossRef]
  5. V. I. Kruglov and J. D. Harvey, J. Opt. Soc. Am. B 23, 2541 (2006).
    [CrossRef]
  6. J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nature Phys. 3, 597 (2007).
    [CrossRef]
  7. P. D. Drummond and K. V. Kheruntsyan, Phys. Rev. A 63, 013605 (2000).
    [CrossRef]
  8. V. I. Kruglov, M. K. Olsen, and M. J. Collett, Phys. Rev. A 72, 033604 (2005).
    [CrossRef]
  9. L. Shah, Z. Liu, I. Hartl, G. Imeshev, G. C. Cho, and M. E. Fermann, Opt. Express 13, 4717 (2005).
    [CrossRef] [PubMed]
  10. S. Zhou, L. Kuznetsova, A. Chong, and F. W. Wise, Opt. Express 13, 4869 (2005).
    [CrossRef] [PubMed]
  11. Y. Logvin, V. P. Kalosha, and H. Anis, Opt. Express 15, 985 (2007).
    [CrossRef] [PubMed]
  12. A. Ruchl, O. Prochnow, M. Schultz, D. Wandt, and D. Kracht, Opt. Lett. 32, 2590 (2007).
    [CrossRef]
  13. A. I. Latkin, S. K. Turitsyn, and A. A. Sysoliatin, Opt. Lett. 32, 331 (2007).
    [CrossRef] [PubMed]
  14. S. Wabnitz and C. Finot, J. Opt. Soc. Am. B 25, 614 (2008).
    [CrossRef]
  15. B. G. Bale and S. Boscolo, J. Opt. 12, 015202 (2010).
    [CrossRef]
  16. S. Zhang, C. Jin, Y. Meng, X. Wang, and H. Li, J. Opt. Soc. Am. B 27, 1272 (2010).
    [CrossRef]
  17. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2001).

2010 (2)

2008 (1)

2007 (4)

2006 (1)

2005 (3)

2002 (2)

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, J. Opt. Soc. Am. B 19, 461 (2002).
[CrossRef]

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

2001 (1)

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

2000 (3)

M. E. Fermann, V. I. Kruglov, B. C. Thomson, J. M. 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]

P. D. Drummond and K. V. Kheruntsyan, Phys. Rev. A 63, 013605 (2000).
[CrossRef]

Agrawal, G. P.

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

Anis, H.

Bale, B. G.

B. G. Bale and S. Boscolo, J. Opt. 12, 015202 (2010).
[CrossRef]

Boscolo, S.

B. G. Bale and S. Boscolo, J. Opt. 12, 015202 (2010).
[CrossRef]

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Cho, G. C.

Chong, A.

Collett, M. J.

V. I. Kruglov, M. K. Olsen, and M. J. Collett, Phys. Rev. A 72, 033604 (2005).
[CrossRef]

Drummond, P. D.

P. D. Drummond and K. V. Kheruntsyan, Phys. Rev. A 63, 013605 (2000).
[CrossRef]

Dudley, J. M.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nature Phys. 3, 597 (2007).
[CrossRef]

V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, J. Opt. Soc. Am. B 19, 461 (2002).
[CrossRef]

M. E. Fermann, V. I. Kruglov, B. C. Thomson, J. M. 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]

Fermann, M. E.

L. Shah, Z. Liu, I. Hartl, G. Imeshev, G. C. Cho, and M. E. Fermann, Opt. Express 13, 4717 (2005).
[CrossRef] [PubMed]

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

Finot, C.

S. Wabnitz and C. Finot, J. Opt. Soc. Am. B 25, 614 (2008).
[CrossRef]

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nature Phys. 3, 597 (2007).
[CrossRef]

Hartl, I.

Harvey, J. D.

Imeshev, G.

Jin, C.

Kalosha, V. P.

Kheruntsyan, K. V.

P. D. Drummond and K. V. Kheruntsyan, Phys. Rev. A 63, 013605 (2000).
[CrossRef]

Kracht, D.

Kruglov, V. I.

Kuznetsova, L.

Latkin, A. I.

Li, H.

Liu, Z.

Logvin, Y.

Meng, Y.

Millot, G.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nature Phys. 3, 597 (2007).
[CrossRef]

Nijhof, J. H. B.

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Novokshenov, V. Y.

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Olsen, M. K.

V. I. Kruglov, M. K. Olsen, and M. J. Collett, Phys. Rev. A 72, 033604 (2005).
[CrossRef]

Peacock, A. C.

Prochnow, O.

Richardson, D. J.

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nature Phys. 3, 597 (2007).
[CrossRef]

Ruchl, A.

Schultz, M.

Shah, L.

Sysoliatin, A. A.

Thomson, B. C.

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

Turitsyn, S. K.

A. I. Latkin, S. K. Turitsyn, and A. A. Sysoliatin, Opt. Lett. 32, 331 (2007).
[CrossRef] [PubMed]

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Wabnitz, S.

Wandt, D.

Wang, X.

Wise, F. W.

Zhang, S.

Zhou, S.

J. Opt. (1)

B. G. Bale and S. Boscolo, J. Opt. 12, 015202 (2010).
[CrossRef]

J. Opt. Soc. Am. B (4)

Nature Phys. (1)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, Nature Phys. 3, 597 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lett. (3)

Phys. Rev. A (2)

P. D. Drummond and K. V. Kheruntsyan, Phys. Rev. A 63, 013605 (2000).
[CrossRef]

V. I. Kruglov, M. K. Olsen, and M. J. Collett, Phys. Rev. A 72, 033604 (2005).
[CrossRef]

Phys. Rev. Lett. (1)

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

Theor. Math. Phys. (1)

S. Boscolo, S. K. Turitsyn, V. Y. Novokshenov, and J. H. B. Nijhof, Theor. Math. Phys. 133, 1647 (2002).
[CrossRef]

Other (1)

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

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Figures (2)

Fig. 1
Fig. 1

Numerical and analytical (for σ = 2.01 and σ = 1.5 ) dependence of the dimensionless critical length ξ c = g z c on the dimensionless input-pulse energy E 0 .

Fig. 2
Fig. 2

Normalized pulse power and chirp of analytical and numerical solutions for propagating distances (a), (c) z = 0.7 z c and (b), (d) z = 0.97 z c , with E 0 = 1 and κ = 0.029 . Here ξ c = 11.03 , and normalization power parameter P max = 7.359 for Fig. 2a and P max = 19.86 for Fig. 2b.

Equations (16)

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i ψ z = β 2 2 ψ τ τ + i β 3 6 ψ τ τ τ γ | ψ | 2 ψ + i g 2 ψ ,
U ( z , τ ) = 1 2 ( 2 g 2 E 0 2 γ β 2 ) 1 / 6 exp ( 1 3 g z ) Q ( z , τ ) I ( z , τ ) ,
Q ( z , τ ) = 1 ε ( z ) ( τ τ q ( z ) ) ( τ τ q ( z ) ) 2 + 8 3 ε ( z ) ( τ τ q ( z ) ) 3 .
τ q ( z ) = 4 τ p ( z ) 3 f ( z ) , τ p ( z ) = 3 ( γ β 2 E ( z ) 2 g 2 ) 1 / 3 ,
ε ( z ) = β 3 g 6 β 2 2 ( γ β 2 E ( z ) 2 g 2 ) 1 / 3 ,
I ( z , τ ) = { 1 if τ 1 ( z ) τ τ 2 ( z ) 0 otherwise ,
f ( z ) = ( T 2 ( z ) T 1 ( z ) ) ε ( z ) 2 ( T 2 2 ( z ) T 1 2 ( z ) ) 1 3 ( T 2 3 ( z ) T 1 3 ( z ) ) + 2 ε ( z ) 3 ( T 2 4 ( z ) T 1 4 ( z ) ) ,
Φ ( z , τ ) = ϕ 0 + 3 8 ( 2 γ 2 E ( z ) 2 β 2 g ) 1 / 3 β 3 g 72 β 2 2 ( 2 γ 2 E ( z ) 2 β 2 g ) 1 / 3 τ g 6 β 2 τ 2 + 7 β 3 g 2 486 β 2 3 τ 3 .
Ω ( z , τ ) = β 3 g e 2 g z / 3 72 β 2 2 ( 2 γ 2 E 0 2 β 2 g ) 1 / 3 + g 3 β 2 τ 7 β 3 g 2 162 β 2 3 τ 2 .
32 ε c 4 429 ε c 2 + 12 = 0.
z c = 3 g ln | ε c | | ε 0 | , ε 0 = β 3 6 β 2 2 ( 1 2 γ β 2 g E 0 ) 1 / 3 ,
z c = 1 g ln ( σ β 2 5 γ g E 0 | β 3 | 3 ) , σ = 432 | ε c | 3 = 2.01 .
u ( ξ , ζ ) = γ g ψ ( z , τ ) , ξ = g z , ζ = τ τ g ,
i u ξ = 1 2 u ζ ζ + i κ 6 u ζ ζ ζ | u | 2 u + i 2 u ,
ξ c g z c = ln ( σ E 0 | κ | 3 ) ,
δ = E 0 | κ | 3 = γ g E 0 | β 3 | 3 β 2 5 .

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