Abstract

Restoration in both shape and spectrum of a (train of) 6.4-psec optical pulses has been observed at the soliton period in a single-mode fiber. The source was an F2+ color-center laser at 1.55 μm, and the fiber was 1.3 km long, which was one soliton period for this pulse width and wavelength. As predicted by the nonlinear Schrödinger equation, pulse restoration occurs despite initial spectral broadening from self-phase modulation and temporal compression as a result of negative group-velocity dispersion acting on the chirped pulse.

© 1983 Optical Society of America

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References

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  1. A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  2. J. Satsuma, N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
    [CrossRef]
  3. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  4. A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145 (1981).
    [CrossRef]
  5. W. J. Tomlinson, J. P. Gordon, P. W. Smith, A. E. Kaplan, “Reflection of a Gaussian beam at a nonlinear interface,” Appl. Opt. 21, 2041 (1982). To verify the accuracy of the numerical solutions, we made use of the fact that for integer values of A the exact solutions are symmetric about z/z0 = ½. The grid spacing in the t/t0 coordinate was reduced until the numerical solution for the next-higher integer A value was symmetric to within about one part in 103. For A = 3 this required a grid spacing of Δ(t/t0) = 0.04.
    [CrossRef] [PubMed]
  6. For typical core sizes P1 ≃ z0−1 W, where z0 is measured in kilometers. In Ref. 3 the power of the fundamental soliton P1 was designated by P0. N2 = (4πn2/nc) × 107, where n2 is the self-focusing coefficient in electrostatic units.
  7. R. H. Stolen, “Nonlinear properties of optical fibers,” in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), Chap. 5, p. 125; B. Bendow, P. D. Gianino, N. Tzoar, M. Jain, “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980); B. Crosignani, P. DiPorto, C. H. Papas, “Coupled-mode theory approach to nonlinear pulse propagation in optical fibers,” Opt. Lett. 6, 61 (1981); R. H. Stolen, J. E. Bjorkholm, “Parameter amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062 (1982).
    [CrossRef] [PubMed]
  8. J. Botineau, R. H. Stolen, “Effect of polarization on spectral broadening in optical fibers,” J. Opt. Soc. Am. 72, 1592 (1982).
    [CrossRef]
  9. B. Bendow, P. D. Gianino, “Theory of nonlinear pulse propagation in inhomogeneous waveguides,” Opt. Lett. 4, 164 (1979).
    [CrossRef] [PubMed]
  10. R. H. Stolen, J. Botineau, A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512 (1982).
    [CrossRef] [PubMed]

1982 (3)

1981 (1)

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

1980 (1)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

1979 (1)

1974 (1)

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

1973 (1)

A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Ashkin, A.

Bendow, B.

Botineau, J.

Gianino, P. D.

Gordon, J. P.

Hasegawa, A.

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Kaplan, A. E.

Kodama, Y.

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Satsuma, J.

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

Smith, P. W.

Stolen, R. H.

J. Botineau, R. H. Stolen, “Effect of polarization on spectral broadening in optical fibers,” J. Opt. Soc. Am. 72, 1592 (1982).
[CrossRef]

R. H. Stolen, J. Botineau, A. Ashkin, “Intensity discrimination of optical pulses with birefringent fibers,” Opt. Lett. 7, 512 (1982).
[CrossRef] [PubMed]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

R. H. Stolen, “Nonlinear properties of optical fibers,” in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), Chap. 5, p. 125; B. Bendow, P. D. Gianino, N. Tzoar, M. Jain, “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980); B. Crosignani, P. DiPorto, C. H. Papas, “Coupled-mode theory approach to nonlinear pulse propagation in optical fibers,” Opt. Lett. 6, 61 (1981); R. H. Stolen, J. E. Bjorkholm, “Parameter amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef] [PubMed]

Tappert, F.

A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Tomlinson, W. J.

Yajima, N.

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

Appl. Opt. (1)

Appl. Phys. Lett. (1)

A. Hasegawa, F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (2)

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Proc. IEEE (1)

A. Hasegawa, Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

Prog. Theor. Phys. Suppl. (1)

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

Other (2)

For typical core sizes P1 ≃ z0−1 W, where z0 is measured in kilometers. In Ref. 3 the power of the fundamental soliton P1 was designated by P0. N2 = (4πn2/nc) × 107, where n2 is the self-focusing coefficient in electrostatic units.

R. H. Stolen, “Nonlinear properties of optical fibers,” in Optical Fiber Telecommunications, S. E. Miller, A. G. Chynoweth, eds. (Academic, New York, 1979), Chap. 5, p. 125; B. Bendow, P. D. Gianino, N. Tzoar, M. Jain, “Theory of nonlinear pulse propagation in optical waveguides,” J. Opt. Soc. Am. 70, 539 (1980); B. Crosignani, P. DiPorto, C. H. Papas, “Coupled-mode theory approach to nonlinear pulse propagation in optical fibers,” Opt. Lett. 6, 61 (1981); R. H. Stolen, J. E. Bjorkholm, “Parameter amplification and frequency conversion in optical fibers,” IEEE J. Quantum Electron. QE-18, 1062 (1982).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

Perspective plot of the temporal pulse shape at various points along a fiber for the A = 3 soliton. The intensity variable is defined as |V(z, t)/A|2. The parameters z0 and t0 and the input pulse shape are defined by Eqs. (2)(4).

Fig. 2
Fig. 2

Calculated pulse shapes and frequency spectra at fiber lengths z0/2 and z0 for various values of A. In addition to the amplitude parameter A, the relative pulse powers are listed in terms of the fundamental soliton power P1 The two sets of curves show the evolution of the pulse shape and spectrum as the input power is increased. The input (z = 0) pulse shape and spectrum are not shown but are identical with those at z = z0 for A = 1, 2, 3. The frequency variable δω ≡ (ωω0)t0, where ω0 is the center frequency of the pulse. The intensity variable is |V/A|2.

Fig. 3
Fig. 3

Experimental pulse shapes (autocorrelation) and spectra. All curves are normalized to the same peak intensity. The first line shows the shape and the spectrum for the input pulse, and the lower lines show the shapes and the spectra of the output pulses for different input powers.

Equations (5)

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i d V d ξ = 1 2 d 2 V d s 2 + V 2 V ,
ξ = π 2 z z 0 ,             z 0 = 0.322 π 2 c 2 τ 2 D ( λ ) λ 0 ,
s = t t 0 = | t - z v g | / t 0 ,             t 0 = 0.568 τ ,
V ( z = 0 , t ) = A sech ( t / t 0 ) .
P 1 = λ 0 A eff 4 N 2 z 0 ,

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