Abstract

Numerical experiments on nonlinear amplifying dispersive media such as single-mode doped light guides (exhibiting both gain saturation and linear losses) have revealed a new effect, namely, the formation of autosolitons, which are solitary waves with all their parameters independent of the initial amplifying pulse and governed only by the medium parameters. In contrast with a soliton in the Kerr dispersive medium (described by the nonlinear Schrödinger equation), the autosoliton is formed in a medium with a positive sign of dispersion as well as in a medium with a negative sign of dispersion. The dependences of the main parameters of the autosoliton on the medium parameters are described.

© 1991 Optical Society of America

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References

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  1. I. P. Alcock, A. I. Ferguson, D. C. Hanna, and A. C. Tropper, “Tunable continuous-wave neodymium-doped monomode fiber operating at 0.900–0.945 and 1.070–1.135 μm,” Opt. Lett. 11, 709–711 (1986); I. M. Jauncey, L. Reekie, R. J. Mears, and C. J. Rowe, “Narrow-linewidth fiber laser operating at 1.55 μm,” Opt. Lett. 12, 164–165 (1987).
    [Crossref] [PubMed]
  2. E. Desurvire, J. R. Simpson, and P. C. Becker, “High-gain erbium-doped traveling-wave fiber amplifier,” Opt. Lett. 12, 888–890 (1987).
    [Crossref] [PubMed]
  3. M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989); K. Suzuki, Y. Kimura, and M. Nakazawa, “Subpicosecond soliton amplification using Er3+-doped fibers pumped by InGaAsP laser diodes,” Opt. Lett. 14, 865–867 (1989).
    [Crossref] [PubMed]
  4. V. S. Grigoryan, “Formation of solitary pulses in an amplifying nonlinear medium with dispersion,” JETP Lett. 44, 575–579 (1986).
  5. K. P. Komarov, “On the theory of steady-state ultrashort pulses in solid-state lasers with passive mode locking,” Opt. Spektrosk. 60, 379–384 (1986).
  6. W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quant. Electron. 16, 541–550 (1984).
    [Crossref]
  7. P. A. Belanger, L. Gagnon, and C. Pare, “Solitary pulses in amplified nonlinear dispersive medium,” Opt. Lett. 14, 943–945 (1989).
    [Crossref]
  8. V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklyarov, “Evolution of light pulses in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530–534 (1988).

1989 (2)

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989); K. Suzuki, Y. Kimura, and M. Nakazawa, “Subpicosecond soliton amplification using Er3+-doped fibers pumped by InGaAsP laser diodes,” Opt. Lett. 14, 865–867 (1989).
[Crossref] [PubMed]

P. A. Belanger, L. Gagnon, and C. Pare, “Solitary pulses in amplified nonlinear dispersive medium,” Opt. Lett. 14, 943–945 (1989).
[Crossref]

1988 (1)

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklyarov, “Evolution of light pulses in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530–534 (1988).

1987 (1)

1986 (3)

I. P. Alcock, A. I. Ferguson, D. C. Hanna, and A. C. Tropper, “Tunable continuous-wave neodymium-doped monomode fiber operating at 0.900–0.945 and 1.070–1.135 μm,” Opt. Lett. 11, 709–711 (1986); I. M. Jauncey, L. Reekie, R. J. Mears, and C. J. Rowe, “Narrow-linewidth fiber laser operating at 1.55 μm,” Opt. Lett. 12, 164–165 (1987).
[Crossref] [PubMed]

V. S. Grigoryan, “Formation of solitary pulses in an amplifying nonlinear medium with dispersion,” JETP Lett. 44, 575–579 (1986).

K. P. Komarov, “On the theory of steady-state ultrashort pulses in solid-state lasers with passive mode locking,” Opt. Spektrosk. 60, 379–384 (1986).

1984 (1)

W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quant. Electron. 16, 541–550 (1984).
[Crossref]

Alcock, I. P.

Becker, P. C.

Belanger, P. A.

Desurvire, E.

Ferguson, A. I.

Gagnon, L.

Grigoryan, V. S.

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklyarov, “Evolution of light pulses in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530–534 (1988).

V. S. Grigoryan, “Formation of solitary pulses in an amplifying nonlinear medium with dispersion,” JETP Lett. 44, 575–579 (1986).

Hanna, D. C.

Kimura, Y.

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989); K. Suzuki, Y. Kimura, and M. Nakazawa, “Subpicosecond soliton amplification using Er3+-doped fibers pumped by InGaAsP laser diodes,” Opt. Lett. 14, 865–867 (1989).
[Crossref] [PubMed]

Komarov, K. P.

K. P. Komarov, “On the theory of steady-state ultrashort pulses in solid-state lasers with passive mode locking,” Opt. Spektrosk. 60, 379–384 (1986).

Maimistov, A. I.

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklyarov, “Evolution of light pulses in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530–534 (1988).

Nakazawa, M.

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989); K. Suzuki, Y. Kimura, and M. Nakazawa, “Subpicosecond soliton amplification using Er3+-doped fibers pumped by InGaAsP laser diodes,” Opt. Lett. 14, 865–867 (1989).
[Crossref] [PubMed]

Pare, C.

Rudolph, W.

W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quant. Electron. 16, 541–550 (1984).
[Crossref]

Simpson, J. R.

Sklyarov, Yu. M.

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklyarov, “Evolution of light pulses in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530–534 (1988).

Suzuki, K.

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989); K. Suzuki, Y. Kimura, and M. Nakazawa, “Subpicosecond soliton amplification using Er3+-doped fibers pumped by InGaAsP laser diodes,” Opt. Lett. 14, 865–867 (1989).
[Crossref] [PubMed]

Tropper, A. C.

Electron. Lett. (1)

M. Nakazawa, Y. Kimura, and K. Suzuki, “Soliton amplification and transmission with Er3+-doped fibre repeater pumped by GaInAsP laser diode,” Electron. Lett. 25, 199–200 (1989); K. Suzuki, Y. Kimura, and M. Nakazawa, “Subpicosecond soliton amplification using Er3+-doped fibers pumped by InGaAsP laser diodes,” Opt. Lett. 14, 865–867 (1989).
[Crossref] [PubMed]

JETP Lett. (1)

V. S. Grigoryan, “Formation of solitary pulses in an amplifying nonlinear medium with dispersion,” JETP Lett. 44, 575–579 (1986).

Opt. Lett. (3)

Opt. Quant. Electron. (1)

W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quant. Electron. 16, 541–550 (1984).
[Crossref]

Opt. Spektrosk. (1)

K. P. Komarov, “On the theory of steady-state ultrashort pulses in solid-state lasers with passive mode locking,” Opt. Spektrosk. 60, 379–384 (1986).

Sov. Phys. JETP (1)

V. S. Grigoryan, A. I. Maimistov, and Yu. M. Sklyarov, “Evolution of light pulses in a nonlinear amplifying medium,” Sov. Phys. JETP 67, 530–534 (1988).

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Figures (4)

Fig. 1
Fig. 1

Evolution of the secant pulse solitary wave [Eq. (6)] in the medium with αr = 5, αI = 0.1, β = 3, R = 0.5, I = S = 0.1, and κ = 1. In this and all the other figures the numbers beneath the maxima of the pulses denote the positions of maxima of the pulses along the t axis.

Fig. 2
Fig. 2

Evolution of two initial pulses:

Fig. 3
Fig. 3

Evolution of a noise pulse f(t) = 0.3 exp(−t2) (σa = σb = 0.1) in the medium with αr = 5, αI = 0.1, β = 3, I = −0.3, and S = −0.1.

Fig. 4
Fig. 4

Example of pulse evolutions in the positive dispersion medium with initial profiles (a) f(t) = sech(t) (σa = σb = 0.1), (b) f(t) = 0.3 exp(−t2) (σa = σb = 0.1), (c) f(t) = sech[(t + 1)0.7] (σa = σb = 0.1), and (d) f ( t ) = { 0 t < 0 t 2 exp ( - t ) t 0 ,             σ a = σ b = 0.2.

Equations (25)

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i A x + Q 2 A t 2 + u A A 2 = ρ - i γ A ,
ρ ˙ + ρ ( T - 1 + i Δ ) = - i A w ,
w ˙ = 4 Im ( ρ A * ) ,
Q = - v k / 2 t 0 ;             μ = n 2 k 2 v / 2 n 0 2 d 2 t 0 ;
ρ = ( 2 π N ω d 2 t 0 2 v / n 0 c ) σ ;             w = ( 2 π N ω d 2 t 0 2 v / n 0 c ) η ;
( ) = - ( ) g ¯ ( Δ ) d Δ .
i q z + g 2 q t 2 + S q t + κ q q 2 = - i α q - t q 2 d t + i β q + i a q ( - t q 2 d t ) 2 - i P q t - t q 2 d t + i r 1 q - t q t q * d t + i r 2 q - t q * t q d t .
q = κ r 0 / R 0 A ,             z = x R 0 ,             R 0 = Re ( g 0 ) , g = g 0 / R 0 ,             g 0 = 1 2 [ Q + i w 0 ( T - 1 + i Δ ) - 3 ] , S = S 0 / R 0 ,             S 0 = - i w 0 ( T - 1 + i Δ ) - 2 , κ = κ 0 / κ r 0 ,             κ r 0 = Re ( κ 0 ) , κ 0 = μ + 4 i w 0 T - 1 [ ( T - 2 + Δ 2 ) ( T - 1 + i Δ ) 2 ] - 1 , α = α 0 / κ r 0 ,             α 0 = - 4 w 0 T - 1 ( T - 2 + Δ 2 ) - 1 ( T - 1 + i Δ ) - 1 , β = β 0 / R 0 ,             β 0 = - w 0 ( T - 1 + i Δ ) - 1 - γ , a = a 0 R 0 ( κ r 0 ) - 2 , a 0 = - 8 w 0 T - 2 ( T - 2 + Δ 2 ) - 2 ( T - 2 + i Δ ) - 1 , P = P 0 / κ r 0 ,             P 0 = 4 w 0 T - 1 ( T - 2 + Δ 2 ) - 1 ( T - 1 + i Δ ) - 2 , r 1 = r 10 / κ r 0 ,             r 2 = r 20 / κ r 0 ,             r 10 = - 2 w 0 ( T - 1 + i Δ ) - 3 , r 20 = - 2 w 0 ( T - 1 + i Δ ) - 1 ( T - 1 - i Δ ) - 2 .
q = A exp ( i { B z + C t + G ln [ cosh ( D t + F z ) ] } ) cosh ( D t + F z ) ,
κ A 2 - B + i C S + g ( i G - 1 ) D 2 - g C 2 = - i α A 2 D - 1 + i β + i a A 4 D - 2 + P C A 2 D - 1 + i r 1 A 2 D - 1 [ i C - 0.5 ( i G - 1 ) D ] + i r 2 [ - i C + 0.5 ( i G + 1 ) D ] , ( i G - 1 ) ( i F + S D + 2 i g ( C D ) = A 2 D - 1 { - i α + 2 i a A 2 / D - i P [ i C + ( i G - 1 ) D ] - r 1 C + r 2 C } , g D 2 ( i G - 1 ) ( i G - 2 ) - κ A 2 = i a A 4 D - 2 + 0.5 i r 1 A 2 ( i G - 1 ) - 0.5 i r 2 A 2 ( i G + 1 ) .
G = - 3 ( R κ r + I κ I ) ± [ 9 ( R κ r + I κ I ) 2 + 8 ( I κ r - R κ I ) 2 ] 1 / 2 2 ( I κ r - R κ I ) , C = S ( 1 + G 2 ) + ξ ( α I + α r G ) 2 I ( 1 + G 2 ) , D = - α z ξ ± [ α r 2 ξ 2 - 4 ( R G - I + κ I ξ ) ( C S - I C 2 - β ) ] 1 / 2 2 ( R G - I + κ I ξ ) , B = D 2 ( κ r ξ - R - I G ) - α I ξ D - R C 2 , F = S G D - 2 D C ( R + I G ) + α r ξ D , A = ξ D ,
R = Re ( g ) ,             I = Im ( g ) ,             κ r = Re ( κ ) ,             κ I = Im ( κ ) , α r = Re ( α ) ,             α I = Im ( α ) ,             ξ = [ R ( 2 - G 2 ) + 3 I G ] / κ r .
ρ = [ C 1 1 cosh ( u ) + C 2 sinh ( u ) cosh 2 ( u ) + C 3 sinh 2 ( u ) cosh 3 ( u ) ] exp ( i y ) ,
u = D t + F x ,             y = B x + C t + G ln ( cos u ) .
D 2 ( C 2 - C 1 ) + D ( i C + T - 1 + i Δ ) = - i w 0 A D - 2 A 2 [ C 1 - C 2 - ( C 2 - C 2 * ) / 2 + ( C 3 - C 3 * ) / 3 ] , ( 2 C 3 + i G C 1 ) D 2 + D ( i C + T - 1 + i Δ ) C 2 = - 2 A 2 ( C 1 - C 1 * ) , D 2 ( i G - 2 ) C 2 + D ( i C + T - 1 + i Δ ) C 3 = - A 2 ( C 2 - C 2 * ) , 3 D 2 ( i G - 3 ) + 2 A 2 = 2 A 2 C 3 * , R A [ ( i G - 1 ) D 2 - C 2 ] - i A B - μ A 3 = C 1 - i γ A , 2 A C D ( i + G ) R - A F ( i G - 1 ) = - C 2 , A D 2 R ( i G - 1 ) ( i G - 2 ) + μ A 3 = C 3 .
i q z + R 2 q t 2 + κ q q 2 = - i α q - t q 2 d t + i β q .
U = 2 β α - 1 U 0 exp ( 2 β z ) U 0 exp ( 2 β z ) - U 0 + 2 β / α ,
q ( 0 , t ) = ψ ( t ) f ( t ) ,
q max 2 = β 4 / 3 / α ,
τ as = U as / q max 2 = 2 β - 1 / 3 .
γ ¯ = 5.15 + 0.46 β ,
i q z + g 2 q t 2 + S ^ q t + κ q q 2 = - i α q - t q 2 d t + i β q .
f ( t ) = { 0 , t < 0 t 2 exp ( - t ) , t 0 ,             σ a = σ b = 0 ,             a ¯ = b ¯ = 0 ,
f ( t ) = 0.3 exp ( - t 2 ) ,             σ a = σ b = 0.1.
f ( t ) = { 0 t < 0 t 2 exp ( - t ) t 0 ,             σ a = σ b = 0.2.

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