Abstract

Pulse propagation in high-gain optical fiber amplifiers with normal group-velocity dispersion has been studied by self-similarity analysis of the nonlinear Schrödinger equation with gain. For an amplifier with a constant distributed gain, an exact asymptotic solution has been found that corresponds to a linearly chirped parabolic pulse that propagates self-similarly in the amplifier, subject to simple scaling rules. The evolution of an arbitrary input pulse to an asymptotic solution is associated with the development of low-amplitude wings on the parabolic pulse whose functional form has also been found by means of self-similarity analysis. These theoretical results have been confirmed with numerical simulations. A series of guidelines for the practical design of fiber amplifiers to operate in the asymptotic parabolic pulse regime has also been developed.

© 2002 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics (Cambridge U. Press, Cambridge, 1996).
  2. A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
    [CrossRef]
  3. S. An and J. E. Sipe, “Universality in the dynamics of phase grating formation in optical fiber,” Opt. Lett. 16, 1478–1480 (1991).
    [CrossRef] [PubMed]
  4. C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
    [CrossRef] [PubMed]
  5. T. M. Monro, P. D. Millar, L. Poladian, and C. M. de Sterke, “Self-similar evolution of self-written waveguides,” Opt. Lett. 23, 268–270 (1998).
    [CrossRef]
  6. M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
    [CrossRef]
  7. S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
    [CrossRef] [PubMed]
  8. 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]
  9. 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]
  10. 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]
  11. K. Tamura and M. Nakazawa, “Pulse compression by nonlinear pulse evolution with reduced optical wave breaking in erbium-doped fiber amplifiers,” Opt. Lett. 21, 68–70 (1996).
    [CrossRef] [PubMed]
  12. A. Galvanauskas and M. E. Fermann, “13-W average power ultrafast fiber laser,” in Conference on Lasers and Electro-Optics, Vol. 39 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), paper CPD3–1.
  13. P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer-Verlag, New York, 1993).
  14. F. W. J. Olver, Asymptotics and Special Functions (Academic, Orlando, Fla. 1974).
  15. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).
  16. V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Stability of asymptotic parabolic pulse solutions to the nonlinear Schrödinger equation with gain,” to be submitted to Opt. Commun.

2000 (4)

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

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, 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]

1998 (1)

1996 (1)

1993 (1)

1992 (1)

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

1991 (2)

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

S. An and J. E. Sipe, “Universality in the dynamics of phase grating formation in optical fiber,” Opt. Lett. 16, 1478–1480 (1991).
[CrossRef] [PubMed]

Afanas’ev, A. A.

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

An, S.

Anderson, D.

Bergman, K.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

de Sterke, C. M.

Desaix, M.

Dudley, J. M.

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, 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]

Fermann, M. E.

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]

Harvey, J. D.

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, 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]

Jakyte, R.

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Karlsson, M.

Kruglov, V. I.

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]

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]

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Krylov, D.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Levi, D.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Lisak, M.

Menyuk, C. R.

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Millar, P. D.

Monro, T. M.

Nakazawa, M.

Peacock, A. C.

Poladian, L.

Quiroga-Teixeiro, M. L.

Samson, B. A.

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Sears, S.

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Segev, M.

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Sipe, J. E.

Soljacic, M.

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

Tamura, K.

Thomsen, B. C.

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]

Volkov, V. M.

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

Winternitz, P.

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

A. A. Afanas’ev, V. I. Kruglov, B. A. Samson, R. Jakyte, and V. M. Volkov, “Self-action of counterpropagating axially symmetric light beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[CrossRef]

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

Opt. Lett. (4)

Phys. Rev. E (1)

M. Soljacic, M. Segev, and C. R. Menyuk, “Self-similarity and fractals in soliton-supporting systems,” Phys. Rev. E 61, R1048–R1051 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

S. Sears, M. Soljacic, M. Segev, D. Krylov, and K. Bergman, “Cantor set fractals from solitons,” Phys. Rev. Lett. 84, 1902–1905 (2000).
[CrossRef] [PubMed]

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]

C. R. Menyuk, D. Levi, and P. Winternitz, “Self-similarity in transient stimulated Raman scattering,” Phys. Rev. Lett. 69, 3048–3051 (1992).
[CrossRef] [PubMed]

Other (6)

G. I. Barenblatt, Scaling, Self-similarity, and Intermediate Asymptotics (Cambridge U. Press, Cambridge, 1996).

A. Galvanauskas and M. E. Fermann, “13-W average power ultrafast fiber laser,” in Conference on Lasers and Electro-Optics, Vol. 39 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2000), paper CPD3–1.

P. J. Olver, Applications of Lie Groups to Differential Equations, 2nd ed. (Springer-Verlag, New York, 1993).

F. W. J. Olver, Asymptotics and Special Functions (Academic, Orlando, Fla. 1974).

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions (Dover, New York, 1964).

V. I. Kruglov, A. C. Peacock, J. M. Dudley, and J. D. Harvey, “Stability of asymptotic parabolic pulse solutions to the nonlinear Schrödinger equation with gain,” to be submitted to Opt. Commun.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Evolution of a Gaussian input pulse toward a parabolic output pulse in a 6-m normal-GVD fiber amplifier. (a) Three-dimensional representation. (b) Intensity in 1-m increments on a logarithmic scale. (c) Normalized intensity in 1-m increments on a linear scale. Amplifier and pulse parameters are given in the text.

Fig. 2
Fig. 2

(a) Intensity (left axis) and chirp (right axis) for the parabolic output pulse corresponding to Fig. 1. Simulation results (curves) are compared with asymptotic theoretical predictions (open circles). (b) Corresponding spectra for simulation results (curves) and asymptotic theoretical predictions (open circles).

Fig. 3
Fig. 3

Evolution of pulse amplitude (top) and pulse width (bottom) as functions of propagation distance in the 6-m normal-GVD fiber amplifier. The theoretical predictions for the asymptotic parabolic pulse evolution (solid curves) are compared with the results obtained for simulations with Gaussian pulses of different pulse durations yet identical energy of 12 pJ. Amplifier parameters are given in the text.

Fig. 4
Fig. 4

Pulse characteristics for input pulses with (a) Gaussian (leftmost column), (b) hyperbolic secant (middle column), and (c) super-Gaussian (rightmost column) profiles. The top figure in each column shows the input pulse; the middle figure shows the output pulse from simulations (curves) compared with asymptotic theoretical predictions (open circles), and the bottom figure shows the corresponding spectra for simulation results (curves) and asymptotic theoretical predictions (open circles). Amplifier and pulse parameters are given in the text.

Fig. 5
Fig. 5

(a) Input pulse duration Tp(0) as a function of amplifier gain; we use Eq. (28) to obtain optimal evolution into the parabolic pulse regime. (b) Input pulse energy Uin as a function of amplifier gain, we use Eq. (29) to obtain optimal evolution into the parabolic pulse regime. Other parameters are given in the text.

Fig. 6
Fig. 6

Characteristic propagation distance zc(100) corresponding to an amplifier length sufficient for parabolic pulse characteristics to be observed, plotted as a function of amplifier gain. Other parameters are given in the text.

Fig. 7
Fig. 7

Simulation results (curves) compared with theoretical asymptotic and intermediate asymptotic predictions (open circles) over the central parabolic region and the wings for (a) a Gaussian and (b) a hyperbolic secant input pulse. Amplifier and pulse parameters are given in the text.

Equations (49)

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

iΨz=β222ΨT2-γ|Ψ|2Ψ+ig2Ψ.
U(z)=Uin exp(gz),
Az=β2ATΦT+β22A2ΦT2+g2A,
β22ΦT2-ΦzA=β222AT2-γA3.
A(z, T)=f(z)F(z, T)=f(z)F(ϑ),
Φ(z, T)=φ(z)+C(z)T2,
ϑ=f2(z)exp(-gz)T.
U(z)=f2(z)- F2(ϑ)dT=exp(gz)- F2(ϑ)dϑ,
U(0)=f2(0)- F2(f2(0)T)dT=- F2(ϑ)dϑ.
dfdz=β2Cf+g2f,
2β2C2-dCdz1f6exp(2gz)ϑ2-1f2dφdz=β22f2Fd2Fdϑ2exp(-2gz)-γF2.
2β2C2-dCdz1f6exp(2gz)=aγ,
1f2dφdz=γ.
F(ϑ)=1-aϑ2,|ϑ|1/a,
ddz1fdfdz-21fdfdz-g22+β2γaf6 exp(-2gz)=0,
f(z)=A0 expg3z,
ϑ=A02 exp-g3zT,
A(z, T)=A0 expg3z1-T2Tp2(z)1/2
Tp(z)=6(γβ2/2)1/2A0gexpg3z.
U(z)=exp(gz)-1/a1/a(1-aϑ2)dϑ=8γβ2/2A03gexp(gz),
A0=12gUinγβ2/21/3.
C(z)=-g6β2,
Φ(z, T)=φ0+3γA022gexp23gz-g6β2T2
Ωc(T)=-ΦT=g3β2T,|T|Tp(z).
Ψ˜(z, ω)=12π- Ψ(z, T)exp(iωT)dT,
|Ψ˜(z, ω)|2=3|β2|A02gexp23gz1-ω2ωp2(z)
ωp(z)=2γβ2A0 expg3z,
Tp(0)=6γβ2/2A0g,
Uin=2Tp3(0)g227γβ2.
G=|γF2|β22Fd2Fdϑ2f2 exp(-2gz)
zc(N)=32glnNg6|γ|A02,
Aw(z, T)=fw(z)Fw(z, T)=fw(z)Fw(ϑw),
Φw(z, T)=φw(z)+Cw(z)T2,
ϑw=fw2(z)exp(-gz)T,
dfwdz=β2Cwfw+g2fw,
2β2Cw2-dCwdzexp(4gz)ϑw2fw8-exp(2gz)fw4dφwdz=β22Fwd2Fwdϑw2,
2β2Cw2-dCwdzfw-8 exp(4gz)=-β22b,
fw-4 exp(2gz)dφwdz=-β2b0,
d2Fwdϑw2=(b0-bϑw2)Fw,
Fw(ϑw)=F0 exp(-λ|ϑw|),
ddz1fwdfwdz-21fwdfwdz-g22=0.
fw(z)=Bwzexpg2z,
ϑw=Bw2Tz,
Aw(z, T)=Bwzexpg2zexp-Λ|T|z
Cw(z)=-12β2z.
Φw(z, T)=φ0+β2Λ22z-T22β2z,|T|>Tp(z),
Ωw,c(z, T)=-ΦwT=Tβ2z,|T|>Tp(z).
Λ103Δτinβ2,Bw250β2-2γ3/2Uin2Δτin2,
Λ74Δτinβ2,Bw4β2-1γ1/4g1/4Uin3/4Δτin5/4.

Metrics