Abstract

We show that pulse spectral broadening in normally dispersive nonlinear fiber amplifiers may be enhanced by introducing a suitable dispersion tapering. We obtain an analytical dispersion profile that permits one to reduce pulse propagation in a varying dispersion fiber to the case of an equivalent fiber with constant parameters.

© 2008 Optical Society of America

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References

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  1. D. Anderson, M. Desaix, M. Karlsson, M. Lisak, and M. L. Quiroga-Teixeiro, “Wave-breaking-free pulses in nonlinear-optical fibers,” J. Opt. Soc. Am. B 10, 1185-1190 (1993).
    [CrossRef]
  2. J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity and scaling phenomena in nonlinear ultrafast optics,” Nat. Phys. 3, 597-603 (2007).
    [CrossRef]
  3. M. E. Fernmann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
    [CrossRef]
  4. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 1753-1755 (2000).
    [CrossRef]
  5. V. I. Kruglov, A. C. Peacock, J. D. Harvey, and J. M. Dudley, “Self-similar propagation of parabolic pulses in normal dispersion fiber amplifiers,” J. Opt. Soc. Am. B 19, 461-469 (2002).
    [CrossRef]
  6. C. Finot, G. Millot, and J. M. Dudley, “Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers,” Opt. Lett. 29, 2533-2535 (2004).
    [CrossRef] [PubMed]
  7. C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, “Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers,” IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
    [CrossRef]
  8. C. Billet, C. M. Dudley, N. Joly, and J. C. Knight, “Intermediate asymptotic evolution and photonic bandgap fiber compression of optical similaritons around 1550 nm,” Opt. Express 13, 3236-3241 (2005).
    [CrossRef] [PubMed]
  9. O. Ilday, J. R. Buckley, W. J. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 231902 (2004).
    [CrossRef]
  10. C. Finot, F. Parmigiani, P. Petropoulos, and D. J. Richardson, “Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime,” Opt. Express 14, 3161-3170 (2006).
    [CrossRef] [PubMed]
  11. S. Wabnitz, “Analytical dynamics of parabolic pulses in nonlinear optical fiber amplifiers,” IEEE Photon. Technol. Lett. 19, 507-509 (2007).
    [CrossRef]
  12. T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group velocity dispersion,” Opt. Lett. 29, 498-500 (2004).
    [CrossRef] [PubMed]
  13. B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comb-like profiled dispersion decreasing fibre,” Electron. Lett. 42, 965-966 (2006).
    [CrossRef]
  14. A. Plocky, A. A. Sysoliatin, A. I. Latkin, V. F. Khopin, P. Harper, J. Harrison, and S. K. Turitsyn, “Experiments on the generation of parabolic pulses in waveguides with length-varying normal chromatic dispersion,” JETP Lett. 85, 319-322 (2007).
    [CrossRef]
  15. A. Latkin, S. Turitsyn, and A. Sysoliatin, “Theory of parabolic pulse generation in tapered fiber,” Opt. Lett. 32, 331-333 (2007).
    [CrossRef] [PubMed]
  16. C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express 15, 15824-15835 (2007).
    [CrossRef] [PubMed]
  17. Y. Kodama, “The Whitham equations for optical communications: mathematical theory of NRZ,” SIAM J. Appl. Math. 59, 2162-2192 (1999).
    [CrossRef]

2007 (5)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity and scaling phenomena in nonlinear ultrafast optics,” Nat. Phys. 3, 597-603 (2007).
[CrossRef]

S. Wabnitz, “Analytical dynamics of parabolic pulses in nonlinear optical fiber amplifiers,” IEEE Photon. Technol. Lett. 19, 507-509 (2007).
[CrossRef]

A. Plocky, A. A. Sysoliatin, A. I. Latkin, V. F. Khopin, P. Harper, J. Harrison, and S. K. Turitsyn, “Experiments on the generation of parabolic pulses in waveguides with length-varying normal chromatic dispersion,” JETP Lett. 85, 319-322 (2007).
[CrossRef]

A. Latkin, S. Turitsyn, and A. Sysoliatin, “Theory of parabolic pulse generation in tapered fiber,” Opt. Lett. 32, 331-333 (2007).
[CrossRef] [PubMed]

C. Finot, B. Barviau, G. Millot, A. Guryanov, A. Sysoliatin, and S. Wabnitz, “Parabolic pulse generation with active or passive dispersion decreasing optical fibers,” Opt. Express 15, 15824-15835 (2007).
[CrossRef] [PubMed]

2006 (2)

C. Finot, F. Parmigiani, P. Petropoulos, and D. J. Richardson, “Parabolic pulse evolution in normally dispersive fiber amplifiers preceding the similariton formation regime,” Opt. Express 14, 3161-3170 (2006).
[CrossRef] [PubMed]

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comb-like profiled dispersion decreasing fibre,” Electron. Lett. 42, 965-966 (2006).
[CrossRef]

2005 (1)

2004 (4)

O. Ilday, J. R. Buckley, W. J. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 231902 (2004).
[CrossRef]

C. Finot, G. Millot, and J. M. Dudley, “Asymptotic characteristics of parabolic similariton pulses in optical fiber amplifiers,” Opt. Lett. 29, 2533-2535 (2004).
[CrossRef] [PubMed]

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, “Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers,” IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

T. Hirooka and M. Nakazawa, “Parabolic pulse generation by use of a dispersion-decreasing fiber with normal group velocity dispersion,” Opt. Lett. 29, 498-500 (2004).
[CrossRef] [PubMed]

2002 (1)

2000 (2)

M. E. Fernmann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef]

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Self-similar propagation of high power parabolic pulses in optical fiber amplifiers,” Opt. Lett. 25, 1753-1755 (2000).
[CrossRef]

1999 (1)

Y. Kodama, “The Whitham equations for optical communications: mathematical theory of NRZ,” SIAM J. Appl. Math. 59, 2162-2192 (1999).
[CrossRef]

1993 (1)

Electron. Lett. (1)

B. Kibler, C. Billet, P. A. Lacourt, R. Ferriere, L. Larger, and J. M. Dudley, “Parabolic pulse generation in comb-like profiled dispersion decreasing fibre,” Electron. Lett. 42, 965-966 (2006).
[CrossRef]

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

C. Finot, G. Millot, S. Pitois, C. Billet, and J. M. Dudley, “Numerical and experimental study of parabolic pulses generated via Raman amplification in standard optical fibers,” IEEE J. Sel. Top. Quantum Electron. 10, 1211-1218 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

S. Wabnitz, “Analytical dynamics of parabolic pulses in nonlinear optical fiber amplifiers,” IEEE Photon. Technol. Lett. 19, 507-509 (2007).
[CrossRef]

J. Opt. Soc. Am. B (2)

JETP Lett. (1)

A. Plocky, A. A. Sysoliatin, A. I. Latkin, V. F. Khopin, P. Harper, J. Harrison, and S. K. Turitsyn, “Experiments on the generation of parabolic pulses in waveguides with length-varying normal chromatic dispersion,” JETP Lett. 85, 319-322 (2007).
[CrossRef]

Nat. Phys. (1)

J. M. Dudley, C. Finot, G. Millot, and D. J. Richardson, “Self-similarity and scaling phenomena in nonlinear ultrafast optics,” Nat. Phys. 3, 597-603 (2007).
[CrossRef]

Opt. Express (3)

Opt. Lett. (4)

Phys. Rev. Lett. (2)

M. E. Fernmann, V. I. Kruglov, B. C. Thomsen, J. M. Dudley, and J. D. Harvey, “Self-similar propagation and amplification of parabolic pulses in optical fibers,” Phys. Rev. Lett. 84, 6010-6013 (2000).
[CrossRef]

O. Ilday, J. R. Buckley, W. J. Clark, and F. W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 231902 (2004).
[CrossRef]

SIAM J. Appl. Math. (1)

Y. Kodama, “The Whitham equations for optical communications: mathematical theory of NRZ,” SIAM J. Appl. Math. 59, 2162-2192 (1999).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Fiber length dependence of the dispersion profile with (a) δ = 0.5 and different values of δ 0 or (b) with δ 0 = 0.05 , and δ = 0.125 , 0.25, 1.

Fig. 2
Fig. 2

Length of DDF (km) as a function of the physical gain δ 0 , for a virtual gain (a) δ = 0.5 and CDF length of 150 m (the horizontal arrow indicates a passive DDF); (b) δ = 0.25 , and CDF length of 350 m . Vertical arrows at δ 0 = 0.05 point to a DDF length of 1 km in both cases.

Fig. 3
Fig. 3

Minimum distance for entering the self-similar regime for the equivalent CDF with δ = 0.25 , versus the input parabolic pulse rms time width, with input pulse energy U 0 = 94 pJ .

Fig. 4
Fig. 4

Dependence of the ratio R of gain coefficient-fiber length product δ 0 L of the DDF and of the equivalent CDF, versus the DDF gain δ 0 , for δ = 0.25 : the arrow indicates the δ 0 = 0.05 case.

Fig. 5
Fig. 5

(a) Dispersion profile of the DDF for the nonlinear distance L NL = 16 m ( 31 m ) so that δ 0 = 0.025 ( δ 0 = 0.05 ) ; the virtual δ = 0.25 in both cases: (b) DDF length versus δ 0 for L NL = 16 m , with δ = 0.25 and the CDF length of 175 m . The arrow indicates the 1 km DDF length for a physical gain of 14 dB km .

Fig. 6
Fig. 6

Comparison of the evolution with distance of the rms time width of a parabolic (Gaussian) pulse in a CDF [dashed (dotted) curve] and in a DDF [solid (dotted-dashed) curve] with a physical gain δ 0 = 0.1 in each case, and a virtual gain coefficient δ = 0.5 for the DDF.

Fig. 7
Fig. 7

Same as in Fig. 6, for the rms spectral width.

Fig. 8
Fig. 8

Comparison of the output intensity profile versus time (top) and frequency (bottom), for a parabolic (Gaussian) pulse after a 1 km long CDF [gray dashed thick (thin) curves] or DDF [black solid thick (thin) curves] with the physical and virtual gains as in Fig. 6.

Fig. 9
Fig. 9

Evolution of the rms (a) time and (b) frequency width of a parabolic pulse in a CDF as in Figs. 6, 7, but with the linear gain δ = 0.5 .

Fig. 10
Fig. 10

Dependence of optimal parabolic pulse parameters versus its input energy, for a fixed DDF length of 1 km , the physical gain coefficient δ 0 = 0.05 , and different virtual gains δ: (a) optimal input FWHM time width; (b): associated output FWHM spectral width.

Fig. 11
Fig. 11

Same as in Fig. 10, for a DDF gain coefficient δ 0 = 0.025 (with L NL = 16 m ), and δ = 0.25 .

Fig. 12
Fig. 12

Input (gray dashed curves) and output (solid curves) pulse power profile versus time and frequency for a 1 km long DDF with (a) β 3 = 0 with δ 0 = 0.1 , δ = 0.5 ; (b) for a 755 m long DDF with β 3 = 0.025 ps 3 km .

Fig. 13
Fig. 13

Same as Fig. 11b, for (a) a 779 m long DDF with β 3 = 0.025 ps 3 km ; (b) a 974 m long DDF with β 3 = 0.015 ps 3 km .

Equations (23)

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

i q Z β 2 D ( Z ) 2 q T 2 + q 2 q = i δ 0 q ,
ξ 0 Z D ( X ) d X ,
u = q D ( Z ) .
u ξ = Z ( q ( Z , T ) D ( Z ) ) d Z d ξ = q Z 1 D 3 2 1 2 q D 1 D 2 d D d Z ,
i u ξ β 2 2 u T 2 + u 2 u = i δ u ,
Γ = 1 D ( Z ) ( Γ 0 1 D ( Z ) d D ( Z ) d Z ) .
0 L DDF D ( Z ) d Z = L .
d D d Z = Γ 0 D Γ D 2 , D ( Z = 0 ) = 1 .
D ( Z ) = Γ 0 Γ { 1 + Γ Γ 0 Γ 0 + Γ ( e Γ 0 Z 1 ) } ,
L = 0 L DDF D ( Z ) d Z = 1 Γ [ log { Γ ( e Γ 0 L DDF 1 ) + Γ 0 } log Γ 0 ] ,
L DDF = 1 Γ 0 [ log { Γ 0 ( e Γ L 1 ) + Γ } log Γ ] .
u ( T , ξ ) = ρ ( T , ξ ) exp [ i ϕ ] = ρ ( T , ξ ) exp [ i T ω ( T , ξ ) d T ] .
ρ ( T , ξ ) = ρ 0 ( ξ ) ( 1 T 2 T 0 2 ( ξ ) ) ,
ω ( T , ξ ) = C 0 ( ξ ) T ,
ρ 0 S = U 0 2 δ 2 3 exp [ 4 δ ξ 3 ] ( 2 β 3 ) ρ 0 S S exp [ 4 δ ξ 3 ] ,
T 0 S = 3 U 0 β δ 2 3 exp [ 2 δ ξ 3 ] 2 T 0 S S exp [ 2 δ ξ 3 ] ,
C 0 S = 2 δ 3 β C 0 S S ,
i Q z + β 2 ( z ) 2 2 Q T 2 i β 3 6 3 Q T 3 + γ Q 2 Q = i δ 0 Q ,
β 3 β 2 3 > 1 4 L NL ρ 0 ,
D ¯ = Γ 0 Γ .
d ( Z ) = 1 D ( Z ) D ¯ ,
d = Γ 0 d + Γ ,
d ( Z ) = C e Γ 0 Z Γ Γ 0 .

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