Abstract

We present approximate but nevertheless highly accurate self-similar solutions to the generalized linear Schrödinger Equation appropriate to the description of pulse propagation in an optical fibre under the influence of distributed dispersion and gain or attenuation. These new similariton solutions apply for any shape of linearly chirped pulse for any dispersion and gain profiles and are indistinguishable from numerically generated solutions in the majority of practical applications.

©2004 Optical Society of America

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References

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  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 symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
    [Crossref]
  3. V.I. Kruglov, Yu.A. Logvin, and V.M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
    [Crossref]
  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. 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]
  8. J. Limpert, T. Schreiber, T. Clausnitzer, K. Zöllner, H.J. Fuchs, E.B. Kley, H. Zellmer, and A. Tünnermann, “High-power femtosecond Yb-doped fiber amplifier,” Opt. Express 10, 628–638 (2002).
    [Crossref] [PubMed]
  9. 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).
    [Crossref] [PubMed]
  10. F.Ö. Ilday, J.R. Buckley, W.G. Clark, and F.W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902/1–4 (2004).
    [Crossref]
  11. J. D. Moores, “Nonlinear compression of chirped solitary waves with and without phase modulation,” Opt. Lett. 21, 555–557 (1996).
    [Crossref] [PubMed]
  12. M. L. Quiroga-Teixeiro, D. Anderson, P. A. Andrekson, A. Berntson, and M. Lisak, “Efficient soliton compression by fast adiabatic amplification,” J. Opt. Soc. Am. B 13, 687–692 (1996).
    [Crossref]
  13. V.I. Kruglov, A.C. Peacock, and J.D. Harvey, “Exact self-similar solutions of the generalized nonlinear schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902/1–4 (2003).
    [Crossref]
  14. 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]
  15. 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]
  16. 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]
  17. D. Marcuse, “Pulse distortion in single-mode fibers. 3: Chirped pulses,” Appl. Opt. 20, 3573–3579 (1981).
    [Crossref] [PubMed]
  18. G.P. Agrawal, Nonlinear Fiber Optics, (Academic Press, Inc., San Diego, California, 1995).

2004 (1)

F.Ö. Ilday, J.R. Buckley, W.G. Clark, and F.W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902/1–4 (2004).
[Crossref]

2003 (2)

V.I. Kruglov, A.C. Peacock, and J.D. Harvey, “Exact self-similar solutions of the generalized nonlinear schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902/1–4 (2003).
[Crossref]

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).
[Crossref] [PubMed]

2002 (2)

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

1998 (1)

1996 (2)

1993 (1)

1992 (2)

V.I. Kruglov, Yu.A. Logvin, and V.M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[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]

1991 (1)

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

1981 (1)

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 symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[Crossref]

Agrawal, G.P.

G.P. Agrawal, Nonlinear Fiber Optics, (Academic Press, Inc., San Diego, California, 1995).

Anderson, D.

Andrekson, P. A.

Barenblatt, G.I.

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

Berntson, A.

Billet, C.

Buckley, J.R.

F.Ö. Ilday, J.R. Buckley, W.G. Clark, and F.W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902/1–4 (2004).
[Crossref]

Clark, W.G.

F.Ö. Ilday, J.R. Buckley, W.G. Clark, and F.W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902/1–4 (2004).
[Crossref]

Clausnitzer, T.

de Sterke, C. M.

Desaix, M.

Dudley, J. M.

Dudley, J.M.

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]

Finot, C.

Fuchs, H.J.

Harvey, J. D.

Harvey, J.D.

V.I. Kruglov, A.C. Peacock, and J.D. Harvey, “Exact self-similar solutions of the generalized nonlinear schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902/1–4 (2003).
[Crossref]

Ilday, F.Ö.

F.Ö. Ilday, J.R. Buckley, W.G. Clark, and F.W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902/1–4 (2004).
[Crossref]

Jakyte, R.

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

Karlsson, M.

Kley, E.B.

Kruglov, V. I.

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]

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 symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[Crossref]

Kruglov, V.I.

V.I. Kruglov, A.C. Peacock, and J.D. Harvey, “Exact self-similar solutions of the generalized nonlinear schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902/1–4 (2003).
[Crossref]

V.I. Kruglov, Yu.A. Logvin, and V.M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[Crossref]

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]

Limpert, J.

Lisak, M.

Logvin, Yu.A.

V.I. Kruglov, Yu.A. Logvin, and V.M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[Crossref]

Marcuse, D.

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.

Millot, G.

Monro, T. M.

Moores, J. D.

Peacock, A. C.

Peacock, A.C.

V.I. Kruglov, A.C. Peacock, and J.D. Harvey, “Exact self-similar solutions of the generalized nonlinear schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902/1–4 (2003).
[Crossref]

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 symmetrical light-beams in a transparent cubic-nonlinearity medium,” J. Mod. Opt. 38, 1189–1202 (1991).
[Crossref]

Schreiber, T.

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]

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]

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]

Tünnermann, A.

Volkov, V. M.

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

Volkov, V.M.

V.I. Kruglov, Yu.A. Logvin, and V.M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[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]

Wise, F.W.

F.Ö. Ilday, J.R. Buckley, W.G. Clark, and F.W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902/1–4 (2004).
[Crossref]

Zellmer, H.

Zöllner, K.

Appl. Opt. (1)

J. Mod. Opt. (2)

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

V.I. Kruglov, Yu.A. Logvin, and V.M. Volkov, “The theory of spiral laser-beams in nonlinear media,” J. Mod. Opt. 39, 2277–2291 (1992).
[Crossref]

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

Opt. Express (2)

Opt. Lett. (3)

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. (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]

F.Ö. Ilday, J.R. Buckley, W.G. Clark, and F.W. Wise, “Self-similar evolution of parabolic pulses in a laser,” Phys. Rev. Lett. 92, 213902/1–4 (2004).
[Crossref]

V.I. Kruglov, A.C. Peacock, and J.D. Harvey, “Exact self-similar solutions of the generalized nonlinear schrodinger equation with distributed coefficients,” Phys. Rev. Lett. 90, 113902/1–4 (2003).
[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]

Other (2)

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

G.P. Agrawal, Nonlinear Fiber Optics, (Academic Press, Inc., San Diego, California, 1995).

Supplementary Material (1)

» Media 1: AVI (2250 KB)     

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Figures (6)

Fig. 1.
Fig. 1. Case sign(c 0)≠sign(β 0). Top: ��D (z) for increasing values of c 0. Left: A 3(t) spreading. Right: Comparison between the input pulse (thin line), the analytical solution (thick line) and the numerical simulation (thickest line) after propagation.
Fig. 2.
Fig. 2. Case sign(c 0)=sign(β 0). Top: ��D (z) for decreasing values of c0. Left: A 3(t) compressing and then spreading. Right: Comparison between the input pulse (thin line), the analytical solution (thick line) and the numerical simulation (thickest line) after propagation. [Media 1]
Fig. 3.
Fig. 3. Left: Misfit factor and ��D (z) function of the propagation when sign(c 0)=sign(β 0), c 0=-0.05THz 2. Right: ��D (z) when sign(c 0)=sign(β 0), sign(c 0)≠sign(β 0) and c 0=0.
Fig. 4.
Fig. 4. Case sign(c 0)=sign(β (z)). Left: A3(t) compressing. Right: Comparison between the input pulse (thin line), the analytical solution (thick line) and the numerical simulation (thickest line) after propagation.
Fig. 5.
Fig. 5. Compression of a linearly chirped sin function (left) and of a linearly chirped sech pulse (right) when sign(c 0)=sign(β (z)).
Fig. 6.
Fig. 6. Comparison between the exact solution from [17,18] (square), the new analytical solution (circle), the numerical simulation (line) and the input (dashed line) for a propagating linearly chirped gaussian pulse when sign(c 0)=sign(β 0), g 0=0 and L=1km.

Equations (31)

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i z ψ ( z , τ ) = β ( z ) 2 2 τ 2 ψ ( z , τ ) + i g ( z ) 2 ψ ( z , τ ) ,
ψ ( z , τ ) = U ( z , τ ) exp ( i Φ ( z , τ ) ) ,
U ( z , τ ) = 1 Γ ( z ) F ( T ) exp ( 1 2 G ( z ) ) ,
T = τ τ c Γ ( z ) , G ( z ) = 0 z g ( z ) d z .
Φ ( z , τ ) = a ( z ) + c ( z ) ( τ τ c ) 2 ,
c ( z ) = c 0 1 c 0 D ( z ) , Γ ( z ) = 1 c 0 D ( z ) ,
D ( z ) = 2 0 z β ( z ) d z .
d 2 F d T 2 + 2 Γ 2 β d a d z F = 0 .
2 Γ ( z ) 2 β ( z ) d a d z = λ ,
λ = 2 β ( 0 ) da dz | z = 0 = constan t ,
a ( z ) = a 0 λ 0 z β ( z ) d z 2 ( 1 c 0 D ( z ) ) 2 = a 0 λ D ( z ) 4 ( 1 c 0 D ( z ) ) .
u = T τ 0 = τ τ c τ 0 ( 1 c 0 D ( z ) ) ,
d 2 d u 2 F ˜ ( u ; s ) + s 2 F ˜ ( u ; s ) = 0 .
Φ ( z , τ ; s ) = Φ 0 ( z , τ ) + Θ ( z ; s ) ,
Φ 0 ( z , τ ) = a 0 + c 0 ( τ τ c ) 2 1 c 0 D ( z ) , Θ ( z ; s ) = D ( z ) s 2 4 τ 0 2 ( 1 c 0 D ( z ) ) .
U ( z , τ ; s ) = S ( z ) F ˜ ( u ; s ) ,
S ( z ) = 1 1 c 0 D ( z ) exp ( 1 2 0 z g ( z ) d z ) ,
ψ ( c ) ( z , τ ) = S ( z ) cos ( us ) e i Φ ( z , τ ) , ψ ( s ) ( z , τ = S ( z ) sin ( us ) e i Φ ( z , τ ) .
ψ ( z , τ ) = 𝓡 ˜ ( s ) ψ ( c ) ( z , τ ) + i 𝓡 ˜ ( s ) ψ ( s ) ( z , τ ) = S ( z ) 𝓡 ˜ ( s ) e isu e i ( Φ 0 ( z , τ ) + Θ ( z ; s ) ) .
ψ ( z , τ ) = S ( z ) e i Φ Θ 0 ( z , τ ) 1 2 π + 𝓡 ˜ ( s ) e isu e i Θ ( z ; s ) d s ,
ψ ( 0 , τ ) = 𝓡 ( t ) exp ( i ( a 0 + c 0 ( τ τ c ) 2 ) ) ,
𝓡 ( t ) = 1 2 π + 𝓡 ˜ ( s ) e ist d s .
𝓡 ( s ) = 1 2 π + 𝓡 ( t ) e ist d t .
Θ ( z ; s ) = s 2 ρ ( z ) , ρ ( z ) = 4 τ 0 2 ( 1 c 0 D ( z ) ) D ( z ) .
I ( z , u ) = 1 2 π + 𝓡 ˜ ( s ) e isu e i Θ ( z ; s ) d s = 𝓡 ( u ) i ρ ( z ) d 2 d u 2 𝓡 ( u ) 1 2 ρ 2 ( z ) d 4 d u 4 𝓡 ( u ) + .
𝓛 D ( z ) = τ 0 2 1 2 c 0 0 z β ( z ) d z 0 z β ( z ) d z ,
ψ ( z , τ ) = S ( z ) 𝓡 ( u ) e i Φ 0 ( z , τ ) , u = τ τ c τ 0 ( 1 c 0 D ( z ) ) ,
ψ ( z , τ ) = S ( z ) e i Φ 0 ( z , τ ) ( 𝓡 ( u ) i ρ ( z ) d 2 d u 2 𝓡 ( u ) 1 2 ρ 2 ( z ) d 4 d u 4 𝓡 ( u ) + ) ,
𝓡 = + 𝓡 ( t ) 2 d t < ,
𝓡 ( t ) = n = 0 + a n A n ( t ) = n = 0 + a n H n ( t ) exp ( t 2 / 2 ) ,
a n = 1 2 n n ! π + ( t ) H n ( t ) exp ( t 2 / 2 ) d t .

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