Abstract

We consider the self-similar amplification of two optical pulses of different wavelengths in order to investigate the effects of a collision between two similaritons. We theoretically demonstrate that similaritons are stable against collisions in a Raman amplifier: similaritons evolve separately in the amplifier without modification of the scaling of their temporal width and chirp and by conserving their velocities, only interact during their overlap and regain their parabolic form after collision. We show both theoretically and experimentally that the collision of two similaritons induces a sinusoidal modulation inside the overlap region, whose frequency decreases during the interaction. Theoretical and experimental studies of the pulse spectrum evidence that similaritons interact with each other through cross phase modulation.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]

Electron. Lett. (1)

Y. Ozeki, Y. Takushima, K. Aiso, K. Taira, and K. Kikuchi, "Generation of 10 GHz similariton pulse trains from 1,2 km-long erbium-doped fibre amplifier for application to multi-wavelength pulse sources," Electron. Lett. 40, 1103-1104 (2004).
[CrossRef]

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

C. Finot, S. Pitois, G. Millot, 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]

J. Opt. Soc. Am. B (1)

J. Opt. Soc. Amer. B (1)

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. Amer. B 19, 461-469 (2002).
[CrossRef]

Opt. Comm. (1)

C. Finot, "Influence of the pumping configuration on the generation of optical similaritons in optical fibers," Opt. Comm. 249, 553-561 (2005).
[CrossRef]

Opt. Commun. (2)

A.C. Peacock, R.J. Kruhlak, J.D. Harvey, and J.M. Dudley, "Solitary pulse propagation in high gain optical fiber amplifiers with normal group velocity dispersion," Opt. Commun. 206, 171-177 (2002).
[CrossRef]

A.C. Peacock, N.G.R. Broderick, and T.M. Monro, "Numerical study of parabolic pulse generation in microstructured fibre Raman amplifiers," Opt. Commun. 218, 167-172 (2003).
[CrossRef]

Opt. Express (8)

V.I. Kruglov, D. Méchin, and J.D. Harvey, "Self-similar solutions of the generalized Schrödinger equation with distributed coefficients," Opt. Express 12, 6198-6207 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-25-6198">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-25-6198</a>.
[CrossRef] [PubMed]

C. Finot, G. Millot, C. Billet, and J.M. Dudley, "Experimental generation of parabolic pulses via Raman amplification in optical fiber," Opt. Express 11, 1547-1552 (2003), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1547">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-13-1547</a>.
[CrossRef] [PubMed]

C. Billet, J.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), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-9-3236">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-9-3236</a>.
[CrossRef] [PubMed]

J. H. V. Price, W. Belardi, T. M. Monro, A. Malinowski, A. Piper, and D. J. Richardson, "Soliton transmission and supercontinuum generation in holey fiber, using a diode pumped Ytterbium fiber source,�?? Opt. Express 10, 382-387 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-382">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-8-382</a>
[PubMed]

J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H. -J. Fuchs, E. -B Bley, H. Zellmer, and A. Tünnermann, �??High Power femtosecond Yb-doped fiber amplifier,�?? Opt. Express 10, 628-638 (2002), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-14-628">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-14-628</a>
[PubMed]

C. Finot and G. Millot, "Interaction between optical parabolic pulses in a Raman fiber amplifier," Opt. Express 13, 5825-5830 (2005), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-15-5825">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-15-5825</a>.
[CrossRef] [PubMed]

C. Finot and G. Millot, "Synthesis of optical pulses by use of similaritons," Opt. Express 12, 5104-5109 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5104">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-21-5104</a>.
[CrossRef] [PubMed]

J.W. Nicholson, A. Yablon, P.S. Westbrook, K.S. Feder, and M.F. Yan, "High power, single mode, all-fiber source of femtosecond pulses at 1550 nm and its use in supercontinuum generation," Opt. Express 12, 3025-3034 (2004), <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-3025">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-13-3025</a>.
[CrossRef] [PubMed]

Opt. Lett. (7)

Phys. Rev. E (1)

S. Chen and L. Yi, "Chirped self-similar solutions of a generalized nonlinear Schrödinger equatin model," Phys. Rev. E 71, 016606 (2005).
[CrossRef]

Phys. Rev. Lett. (3)

M. E. Fermann, 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] [PubMed]

V.I. Kruglov, A.C. Peacock, and J.D. Harvey, "Exact self-similar solutions of the generalized nonlinear Schrödinger equation with distributed coefficients," Phys. Rev. Lett. 90, 113902 (2004).
[CrossRef]

F. �?. Ilday, J. R. Buckley, W. G. Clark, and F. W. Wise, "Self-similar evolution of parabolic pulses in a laser," Phys. Rev. Lett. 92, 213902 (2004).
[CrossRef] [PubMed]

Theor. Math. Phys. (1)

S. Boscolo, S.K. Turitsyn, V.Y. Novokshenov, and J.H.B. Nijhof, "Self-similar parabolic optical solitary waves," Theor. Math. Phys. 133, 1647-1656 (2002).
[CrossRef]

Other (1)

G.P. Agrawal, Nonlinear Fiber Optics, Third Edition, 2001, San Francisco, CA : Academic Press.

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Figures (11)

Fig. 1
Fig. 1

Evolution of ΔTS as a function of the propagation distance for three different values of the frequency separation : Ω = 0.75 THz (dashed red line) Ω = 1 THz (dotted blue line) and Ω = 1 25 THz (solid line) The analytical evolution of the temporal width Tp of the similaritons is shown by circles [4].

Fig. 2.
Fig. 2.

Contour plot of the normalized intensity and spectral profiles of two similaritons of different wavelengths colliding in a Raman amplifier. Numerical calculations of the intensity profiles (a,c) and power spectra (b,d). The input Gaussian pulses are frequency separated by Ω = 1.25 THz (a,b) or Ω = 0.75 THz (c,d).

Fig. 3.
Fig. 3.

Evolution of the intensity profile (a-f) and chirp (g,h) of the pair of pulses at different distances of propagation: (a) Initial pulses. (b,g) Pulses just before the overlap (z = 2110 m). (c) Beginning of the collision (2360 m). (d) Complete overlap of the two similaritons (z = zc = 3000 m). (e) End of the collision (z = 4290 m). (f,h) Similaritons after the collision (z = 5184 m). Blue circles correspond to numerical calculations of the intensity profiles (b,d,f) and chirp (g,h) of a single similariton in the absence of its neighbor.

Fig. 4.
Fig. 4.

Evolution of the modulation frequency fs in the overlap region as a function of the time separation ΔTL , for two pulses frequency shifted by Ω = 1.25 THz : numerical simulations (solid line), linear superposition (dotted red line), experimental results (circles) and linear fit of the experimental data (dashed blue line).

Fig. 5.
Fig. 5.

(a) Evolution of the normalized spectral profile as a function of ΔTL . Variation of ΔS vs ΔTL obtained from numerical integrations of Eq. (2(b)) or Eqs. 6 (c).

Fig. 6.
Fig. 6.

Comparison of the intensity profiles (a) and spectra (b) of the similaritons before the collision (ΔTL = - 90 ps, solid blue line) and after the collision (ΔTL = 100 ps, circles). The results come from numerical integration of the NLSE with a constant gain (Eq. (2)).

Fig. 7.
Fig. 7.

Experimental setup.

Fig. 8.
Fig. 8.

Experimental spectra of the initial pulses (a), of the similaritons after amplification in the EDFA and propagation in the NZ-DSF (b) and of the pulses after spectral slicing (c). (d) Autocorrelation of the pulse generated at 1550 nm.

Fig. 9.
Fig. 9.

(a) Autocorrelation of the central part of the similariton overlap for ΔTL = 0 (complete superposition of the two similaritons): experimental results (circles), numerical simulations (solid red line), sinusoidal fit of the modulation (dotted blue line). (b) Spectrum of the pulses at the Raman amplifier output in the absence of collision (ΔTL = - 90 ps): experimental results (circles), numerical simulations based, either on the NLSE with a constant gain (Eq. (2), dotted red line), or on a generalized NLSE ([12], solid blue line).

Fig. 10.
Fig. 10.

Evolution of ΔS as a function of ΔTL . (a) Results based on numerical integration of the generalized NLSE with a frequency-dependent Raman gain [12]. (b) Experimental results.

Fig. 11.
Fig. 11.

Autocorrelation (a) and spectra (b) of the similariton at 1550 nm before the collision (ΔTL = - 90 ps, circles) and after the collision (ΔTL = 100 ps, solid blue line).

Equations (9)

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Δ T s ( z ) = Δ T ( z ) 2 T p ( z )
= Δ T 0 β 2 Ω z 2 T p ( z )
i ψ z = β 2 2 2 ψ T 2 γ ψ 2 ψ + i g 2 ψ .
Δ T L = Δ T ( z = L )
ψ ( t ) 2 = 2 A p 2 { 1 1 T p 2 ( t 2 + Δ T 2 4 )
+ cos ( 2 π f s Δ T ) 1 ( t + Δ T / 2 T p ) 2 1 ( t + Δ T / 2 T p ) 2 }
f s = 1 2 π ( Ω + C p Δ T ) .
{ i ψ z = i g 2 ψ + i δ 2 ψ t + β 2 2 2 ψ t 2 γ ψ ( ψ 2 + 2 ψ + 2 ) i ψ + z = i g 2 ψ + i δ 2 ψ + t + β 2 2 2 ψ + t 2 γ ψ + ( ψ + 2 + 2 ψ 2 )
ψ ( z , t ) = ψ + ( z , t ) exp ( Ωt / 2 ) + ψ ( z , t ) exp ( Ωt / 2 ) .

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