Abstract

The sync-pumped fiber Raman soliton ring laser is analyzed. The dependence of pulse width on the gain and gain saturation and of the carrier frequency on the length of the ring is determined. The frequency shift caused by the noninstantaneous response of the nonlinear medium is determined, and it is found that bandwidth limiting is necessary to contain it. Limits on achievable pulse width are given.

© 1987 Optical Society of America

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References

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  1. L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13 (1984).
    [CrossRef] [PubMed]
  2. H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172(1985).
    [CrossRef]
  3. R. H. Stolen, Chin-lon Lin, and R. K. Jain, “A time-dispersion-tuned fiber Raman oscillator,” Appl. Phys. Lett. 30, 340 (1977).
    [CrossRef]
  4. Chin-lon Lin, R. H. Stolen, and L. G. Cohen, “A tunable 1.1 μ m fiber Raman oscillator,” Appl. Phys. Lett. 31, 97 (1977).
    [CrossRef]
  5. L. F. Mollenauer, “Soliton laser,” U.S. Patent4,635,263 (January6, 1987).
  6. M. Nakazawa, “Synchronously pumped fiber Raman gyroscope,” Opt. Lett. 10, 193 (1985).
    [CrossRef] [PubMed]
  7. M. N. Islam and L. F. Mollenauer, “Fiber Raman amplification soliton laser,” presented at the XIV International Quantum Electronics Conference, June 9–13, 1986, San Francisco, California.
  8. V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 36, 61 (1971).
  9. F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self frequency shift,” Opt. Lett. 11, 659 (1986).
    [CrossRef] [PubMed]
  10. J. P. Gordon, “Theory of the soliton self frequency shift,” Opt. Lett. 11, 662 (1986).
    [CrossRef] [PubMed]
  11. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).
  12. J. P. Gordon, “Interaction forces among solitons in optical fibers,” Opt. Lett. 8, 596–598 (1983).
    [CrossRef] [PubMed]
  13. M. Delfou, M. Forton, and G. Payre, “Finite difference solutions of a nonlinear Schroedinger equation,” J. Comp. Phys. 44, 277, 1981.
    [CrossRef]
  14. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
    [CrossRef]
  15. C. Lin, “Optical frequency conversion in fiber Raman lasers,” in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, New York, to be published).
  16. If the bandwidth limiting is provided by the frequency-dependent loss, then operator (6.7) should not depend on the gain saturation. But this is a detail that does not affect the results in an essential way.
  17. W. J. Tomlinson, H. A. Haus, and R. H. Stolen, “Curious features of nonlinear pulse propagation in single-mode optical fibers,” J. Opt. Soc. Am. A 2(13), p33 (1985).

1986 (2)

1985 (3)

M. Nakazawa, “Synchronously pumped fiber Raman gyroscope,” Opt. Lett. 10, 193 (1985).
[CrossRef] [PubMed]

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172(1985).
[CrossRef]

W. J. Tomlinson, H. A. Haus, and R. H. Stolen, “Curious features of nonlinear pulse propagation in single-mode optical fibers,” J. Opt. Soc. Am. A 2(13), p33 (1985).

1984 (1)

1983 (1)

1981 (1)

M. Delfou, M. Forton, and G. Payre, “Finite difference solutions of a nonlinear Schroedinger equation,” J. Comp. Phys. 44, 277, 1981.
[CrossRef]

1977 (2)

R. H. Stolen, Chin-lon Lin, and R. K. Jain, “A time-dispersion-tuned fiber Raman oscillator,” Appl. Phys. Lett. 30, 340 (1977).
[CrossRef]

Chin-lon Lin, R. H. Stolen, and L. G. Cohen, “A tunable 1.1 μ m fiber Raman oscillator,” Appl. Phys. Lett. 31, 97 (1977).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

1971 (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 36, 61 (1971).

Cohen, L. G.

Chin-lon Lin, R. H. Stolen, and L. G. Cohen, “A tunable 1.1 μ m fiber Raman oscillator,” Appl. Phys. Lett. 31, 97 (1977).
[CrossRef]

Delfou, M.

M. Delfou, M. Forton, and G. Payre, “Finite difference solutions of a nonlinear Schroedinger equation,” J. Comp. Phys. 44, 277, 1981.
[CrossRef]

Forton, M.

M. Delfou, M. Forton, and G. Payre, “Finite difference solutions of a nonlinear Schroedinger equation,” J. Comp. Phys. 44, 277, 1981.
[CrossRef]

Gordon, J. P.

Haus, H. A.

W. J. Tomlinson, H. A. Haus, and R. H. Stolen, “Curious features of nonlinear pulse propagation in single-mode optical fibers,” J. Opt. Soc. Am. A 2(13), p33 (1985).

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172(1985).
[CrossRef]

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Islam, M. N.

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172(1985).
[CrossRef]

M. N. Islam and L. F. Mollenauer, “Fiber Raman amplification soliton laser,” presented at the XIV International Quantum Electronics Conference, June 9–13, 1986, San Francisco, California.

Jain, R. K.

R. H. Stolen, Chin-lon Lin, and R. K. Jain, “A time-dispersion-tuned fiber Raman oscillator,” Appl. Phys. Lett. 30, 340 (1977).
[CrossRef]

Lin, C.

C. Lin, “Optical frequency conversion in fiber Raman lasers,” in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, New York, to be published).

Lin, Chin-lon

R. H. Stolen, Chin-lon Lin, and R. K. Jain, “A time-dispersion-tuned fiber Raman oscillator,” Appl. Phys. Lett. 30, 340 (1977).
[CrossRef]

Chin-lon Lin, R. H. Stolen, and L. G. Cohen, “A tunable 1.1 μ m fiber Raman oscillator,” Appl. Phys. Lett. 31, 97 (1977).
[CrossRef]

Mitschke, F. M.

Mollenauer, L. F.

F. M. Mitschke and L. F. Mollenauer, “Discovery of the soliton self frequency shift,” Opt. Lett. 11, 659 (1986).
[CrossRef] [PubMed]

L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13 (1984).
[CrossRef] [PubMed]

M. N. Islam and L. F. Mollenauer, “Fiber Raman amplification soliton laser,” presented at the XIV International Quantum Electronics Conference, June 9–13, 1986, San Francisco, California.

L. F. Mollenauer, “Soliton laser,” U.S. Patent4,635,263 (January6, 1987).

Nakazawa, M.

Payre, G.

M. Delfou, M. Forton, and G. Payre, “Finite difference solutions of a nonlinear Schroedinger equation,” J. Comp. Phys. 44, 277, 1981.
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 36, 61 (1971).

Stolen, R. H.

W. J. Tomlinson, H. A. Haus, and R. H. Stolen, “Curious features of nonlinear pulse propagation in single-mode optical fibers,” J. Opt. Soc. Am. A 2(13), p33 (1985).

L. F. Mollenauer and R. H. Stolen, “The soliton laser,” Opt. Lett. 9, 13 (1984).
[CrossRef] [PubMed]

Chin-lon Lin, R. H. Stolen, and L. G. Cohen, “A tunable 1.1 μ m fiber Raman oscillator,” Appl. Phys. Lett. 31, 97 (1977).
[CrossRef]

R. H. Stolen, Chin-lon Lin, and R. K. Jain, “A time-dispersion-tuned fiber Raman oscillator,” Appl. Phys. Lett. 30, 340 (1977).
[CrossRef]

Tomlinson, W. J.

W. J. Tomlinson, H. A. Haus, and R. H. Stolen, “Curious features of nonlinear pulse propagation in single-mode optical fibers,” J. Opt. Soc. Am. A 2(13), p33 (1985).

Yajima, N.

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 36, 61 (1971).

Appl. Phys. Lett. (2)

R. H. Stolen, Chin-lon Lin, and R. K. Jain, “A time-dispersion-tuned fiber Raman oscillator,” Appl. Phys. Lett. 30, 340 (1977).
[CrossRef]

Chin-lon Lin, R. H. Stolen, and L. G. Cohen, “A tunable 1.1 μ m fiber Raman oscillator,” Appl. Phys. Lett. 31, 97 (1977).
[CrossRef]

IEEE J. Quantum Electron. (1)

H. A. Haus and M. N. Islam, “Theory of the soliton laser,” IEEE J. Quantum Electron. QE-21, 1172(1985).
[CrossRef]

J. Comp. Phys. (1)

M. Delfou, M. Forton, and G. Payre, “Finite difference solutions of a nonlinear Schroedinger equation,” J. Comp. Phys. 44, 277, 1981.
[CrossRef]

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

W. J. Tomlinson, H. A. Haus, and R. H. Stolen, “Curious features of nonlinear pulse propagation in single-mode optical fibers,” J. Opt. Soc. Am. A 2(13), p33 (1985).

Opt. Lett. (5)

Suppl. Prog. Theor. Phys. (1)

J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Suppl. Prog. Theor. Phys. 55, 284–306 (1974).
[CrossRef]

Zh. Eksp. Teor. Fiz. (1)

V. E. Zakharov and A. B. Shabat, “Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media,” Zh. Eksp. Teor. Fiz. 36, 61 (1971).

Other (5)

C. Lin, “Optical frequency conversion in fiber Raman lasers,” in Tunable Lasers, L. F. Mollenauer and J. C. White, eds. (Springer-Verlag, New York, to be published).

If the bandwidth limiting is provided by the frequency-dependent loss, then operator (6.7) should not depend on the gain saturation. But this is a detail that does not affect the results in an essential way.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984).

M. N. Islam and L. F. Mollenauer, “Fiber Raman amplification soliton laser,” presented at the XIV International Quantum Electronics Conference, June 9–13, 1986, San Francisco, California.

L. F. Mollenauer, “Soliton laser,” U.S. Patent4,635,263 (January6, 1987).

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Figures (4)

Fig. 1
Fig. 1

Configuration of synch-pumped FRASL. To make a high-Q cavity for the Stokes pulse, a dichroic laser mirror is used; the pump pulse exits after one circulation.

Fig. 2
Fig. 2

The FRASL with coupler.

Fig. 3
Fig. 3

Gain history of a segment of the Stokes soliton. zi is a displacement from the pump center.

Fig. 4
Fig. 4

Plot of the closure condition (9.1).

Tables (1)

Tables Icon

Table 1 Parameter Values for Fig. 4

Equations (77)

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β = β ( ω 0 ) + d β d ω ( ω - ω 0 ) + 1 2 d 2 β d ω 2 ( ω - ω 0 ) 2 ,
i A z - 1 2 d 2 β d ω 2 2 A τ 2 + δ β A = 0 ,
δ β = ω 0 2 μ 0 0 β ( ω 0 ) d a n 0 n 2 U 4 d a U 2 A 2 ,
cross section d a U 2 = 1 ,
U 2 = 1 α eff
κ = ω 0 2 μ 0 0 n 0 n 2 β ( ω 0 ) α eff = ( 2 π / λ ) 2 1 β ( ω 0 ) n 0 n 2 α eff .
κ P 0 1 z 0 .
( - i ) u q = 1 2 2 u s 2 + u 2 u ,
s τ / τ 0 ,
τ 0 = | d 2 β d ω 2 | z 0
q z / z 0 .
u = 2 η exp ( i 2 η 2 q ) cosh 2 η s .
W = P 0 - u 2 d τ = d 2 β / d ω 2 κ z 0 - u 2 d s = 4 η d 2 β / d ω 2 κ z 0 .
s 1 = 2 ln ( 1 + 2 ) = 1.76.
z 0 = τ 1 2 ( 1.76 ) 2 d 2 β / d ω 2 ,
τ D = d d ω ( 1 / v g ) Δ ω l = d 2 β d ω 2 Δ ( 2 π c λ ) l = - 2 π c λ 2 ( d 2 β d ω 2 ) l Δ λ .
D = | τ D l Δ λ | = 2 π c λ 2 | d 2 β d ω 2 | .
z 0 = 2 π c τ 1 2 ( 1.76 ) 2 D λ 2 = 0.322 ( 2 π c λ 2 ) τ 1 2 / D .
W min 2 d 2 β / d ω 2 κ l .
i d v 1 d s + u v 2 = ζ v 1 ,
i d v 2 d s + u * v 1 = - ζ v 2 .
λ j c j exp ( i ζ i s )
ψ 1 j + k = 1 N λ j λ k * ζ j - ζ k * ψ 2 k * = 0 ,
- k = 1 N λ k λ j * ζ j * - ζ k ψ 1 k + ψ 2 j * = λ j * ,
u ( s ) = - 2 k = 1 N λ k * ψ 2 k * .
λ j = c j exp { - η j ( s + 2 ξ j q ) + i [ ξ j s + ( ξ j 2 - η j 2 ) q ] } ,
c 2 η exp ( 2 η s 0 ) e i ϕ ,
u ( s , q ) = - 2 η sech 2 η [ ( s - s 0 ) + 2 ξ q ] exp { - 2 i [ ξ ( s - s 0 ) + ( ξ 2 - η 2 ) q ] - 2 i ξ s 0 - i ϕ } .
i d d s Δ v 1 + Δ u v 2 + u Δ v 2 = Δ ζ v 1 + ζ Δ v 1 ,
i d d s Δ v 2 + Δ u * v 1 + u * Δ v 1 = - Δ ζ v 2 - ζ Δ v 2 .
Δ ζ = d s ( Δ u v 2 2 - Δ u * v 1 2 ) 2 d s v 1 v 2 .
u ( s , q = 0 ) = 2 η sech 2 η s exp ( - 2 i ξ s ) ,
v 1 = 1 2 i sech 2 η s exp [ - ( η + i ξ ) s ] ,
v 2 = i 2 sech 2 η s exp [ + ( η + i ξ ) s ] .
Δ ζ = i 4 d ( 2 η s ) [ Δ u exp ( 2 η s ) exp ( 2 i ξ s ) + Δ u * exp ( - 2 η s ) exp ( - 2 i ξ s ) ] sech 2 2 η s , = i 2 d ( 2 η s ) [ Δ u exp ( 2 i ξ s + Δ u * ) exp ( - 2 i ξ s ) ] sech 2 η s + 1 2 d ( 2 η s ) [ i Δ u exp ( 2 i ξ s ) - i Δ u * exp ( - 2 i ξ s ) ] tanh 2 η s sech 2 η s .
t s = l g ( 1 v g S - 1 v g P ) ,
v g S = l / T p .
Δ v g v g P = 1 - l v g P T p
| Δ v g v g P 2 | = Δ ω s | d 2 β d ω 2 | ,
Δ ω s = 1 | d 2 β d ω 2 | v g P | 1 - l v g P T p | .
α g ( t , z ) = α g exp [ - ( t - z / v g P τ p ) 2 ] ,
0 l g α g ( t i , z ) d z = α g 0 l g d z exp [ - ( t i + z v g S - z v g P ) 2 / τ p 2 ] α g l g { 1 - 1 3 l g ( 1 v g S - 1 v g P ) τ p 2 × [ ( t i + l g v g S - l g v g P ) 3 - t i 3 ] } ,
Δ t s = l g ( 1 v g S - 1 v g P ) = l g Δ v g v g 2 ,
v g S v g P v g
Δ v g = v g P - v g S .
Δ t 0 + l g 2 ( 1 v g S - 1 v g P ) = Δ t 0 - Δ t s 2 .
t i = τ 0 s + Δ t 0 - ½ Δ t s .
gain = α g l g ( 1 - Δ t s 2 12 τ p 2 - Δ t 0 2 τ p 2 ) - α g l g τ 0 2 τ p 2 ( 2 Δ t 0 τ 0 s + s 3 ) .
α g l g [ 1 - ( Δ ω ω g ) 2 ]
α g l g ( 1 + 1 ω g 2 τ 0 2 2 s 2 ) .
α g l g ( 1 - Δ t s 2 12 τ p 2 - Δ t 0 2 τ p ) - α g l g τ 0 2 τ p 2 ( 2 Δ t 0 τ 0 s + s 2 ) + α g l g 1 ω g 2 τ 0 2 2 s 2 .
I p z + 1 v g P t I p = - γ I s I p ,
I s z + 1 v g S t I s = γ I p I s .
α g = α g 0 1 + I s I sat ,
I sat = 1 γ l g
α g 0 = γ 2 I p 0 ,
Δ n = n 2 A 2 u 2 ,
Δ n τ = - Δ n τ n + n 2 A 2 u 2 τ n .
Δ n n 2 A 2 u 2 - τ n τ ( n 2 A 2 u 2 ) .
( + i ) u q + 1 2 2 u s 2 + u 2 u = τ n τ u 2 u ,
Δ u = - i l z 0 τ n ( 2 η ) 2 τ ( sech 2 τ τ s ) u = 2 i l z 0 τ n τ 0 ( 2 η ) 3 tanh ( 2 η s ) sech 2 ( 2 η s ) u .
Δ u gain = α g l g ( 1 - Δ t s 2 12 τ p 2 - Δ t 0 2 τ p 2 ) u - α g l g τ 0 2 τ p 2 ( 2 Δ t 0 τ 0 s + s 2 ) u + α g l g 1 ω g 2 τ 0 2 2 s 2 u .
Δ u loss = - α l l u .
Δ u nonl = 2 i l z 0 τ n τ 0 ( 2 η ) 3 tanh ( τ / τ s ) sech 2 ( τ / τ s ) u .
- sech 2 x d x = 2
- x 2 sech 2 x d x = π 2 6 .
Δ ζ = η i [ 4 α g l g ( 1 - Δ t s 2 12 τ p 2 - Δ t 0 2 τ p 2 ) - π 2 3 α g l g τ 0 2 τ p 2 ( 1 2 η ) 2 - 4 3 α g l g ω g 2 τ 0 2 ( 2 η ) 2 - 4 α g l g 1 ω g 2 τ 0 2 ( 2 ξ ) 2 - 4 α l l ] - η [ 4 3 α g l g ω g 2 τ 0 2 2 ( 2 η ) ( 2 ξ ) + 16 15 ( 2 η ) 3 l z 0 τ n τ 0 ] .
2 ξ = - 2 5 ( 2 η ) 2 l z 0 τ n τ 0 ω g 2 τ 0 2 α g l g .
4 α g 0 l g 1 + η 2 η s 2 [ 1 - Δ t s 2 12 τ p 2 - Δ t 0 2 τ p 2 - ( 1 2 η ) 2 π 2 12 τ 0 2 τ p 2 - ( 2 η ) 2 1 3 ω g 2 τ 0 2 - ( 2 η ) 4 ω g 2 τ 0 2 ( 2 5 l z 0 τ n τ 0 1 α g l g ) 2 ] = 4 α l l ,
η s 2 κ α eff γ l g τ 0 2 | d 2 β d ω 2 | .
4 α g 0 l g 1 + η 2 η s 2 [ 1 + Δ t 2 2 12 τ p 2 + Δ t 0 2 τ p 2 + ( 1 2 η ) 2 π 2 12 τ 0 2 τ p 2 + ( 2 η ) 2 1 3 ω g 2 τ 0 2 + ( 2 η ) 4 ω g 2 τ 0 2 ( 2 5 l z 0 τ n τ 0 1 α g l g ) 2 ] - 1 = 4 α l l .
n t = - n τ n + n 2 I τ n ,
n t = - n τ n + n 2 τ n { E s 2 + E s 2 + E p E * s exp [ j ( ω p - ω s ) t ] + c . c . } .
δ n = n 2 E * p E s j ( ω p - ω s ) τ n + 1 .
E s z = j ω c ( n + δ n ) ( E p + E s ) at frequency ω s = j ω c n E s + ω c n 2 E p 2 E s ( ω p - ω s ) 2 τ n 2 + 1 ( ω p - ω s ) τ n .
ω c n 2 ( ω p - ω s ) τ n E p 2 1 + ( ω p - ω s ) 2 τ n 2 .
τ n = 5.9 fsec .

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