Abstract

One can transform an optical pulse containing higher-order soliton modes and/or radiation modes into a compressed almost ideal single-soliton pulse by passing it through an initial adaptive fiber of suitably chosen length and dispersion. Analytical approximations, in good agreement with numerical simulations, are found for the optimal values of the length and the dispersion of the adaptive fiber. The technique is applied to the case of nonadiabatically amplified soliton pulses, for which the transformation is shown to result in efficiently compressed soliton pulses.

© 1995 Optical Society of America

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References

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  1. H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910–913 (1981).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, and W. J. Tomlinson, "Extreme picosecond pulse narrowing by means of soliton effect in single-mode optical fibers," Opt. Lett. 8, 289–291 (1983).
    [CrossRef] [PubMed]
  3. S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, "Soliton pulse compression in dispersion-decreasing fiber," Opt. Lett. 18, 476–478 (1993).
    [CrossRef] [PubMed]
  4. S. V. Chernikov, J. R. Taylor, and R. Kashyap, "Comblike dispersion profiled fiber for soliton pulse train generation," Opt. Lett. 539–541 (1994).
    [CrossRef]
  5. D. Anderson, M. Lisak, B. Malomed, and M. Quiroga- Teixeiro, "Tunneling of an optical soliton through a fiber junction," J. Opt. Soc. Am. B 11, 2380–2384 (1994).
    [CrossRef]
  6. T. Aakjer, K. Bertilson, M. L. Quiroga-Teixeiro, P. O. Hedekvist, and P. A. Andrekson, "Investigation of soliton compression by propagation through fiber junctions," Opt. Fiber Technol. (to be published).
  7. A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142–144 (1973).
    [CrossRef]
  8. V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62–72 (1972) [Zh. Eksp. Teor. Fiz. 61, 118–XXX (1971)].
  9. Y. S. Kivshar, "On the soliton generation in optical fibers," J. Phys. A 22, 337–340 (1989).
    [CrossRef]
  10. G. Boffetta and A. R. Osborne, "Computation of the direct scattering transform for the nonlinear Schroedinger equation," J. Comput. Phys. 102, 252–264 (1992).
    [CrossRef]
  11. J. P. Gordon, "Dispersive perturbations of solitons of the nonlinear Schrödinger equation," J. Opt. Soc. Am. B 9, 91–97 (1992).
    [CrossRef]
  12. J. Satsuma and N. Yajima, "Initial value problems of onedimensional self-modulation of nonlinear waves in dispersive media," Progr. Theor. Phys. Suppl. 55, 284–300 (1974).
    [CrossRef]
  13. W. J. Tomlinson, R. H. Stolen, and C. V. Shank, "Compression of optical pulses chirped by self-phase modulation in fibers," J. Opt. Soc. Am. B 1, 139–151 (1984).
    [CrossRef]
  14. Y. Kodama and A. Hasegawa, "Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect," Opt. Lett. 17, 31–33 (1992).
    [CrossRef] [PubMed]
  15. Y. Kodama and S. Wabnitz, "Reduction of the soliton interaction forces by bandwidth limited amplification," Electron. Lett. 27, 1931–1933 (1991).
    [CrossRef]
  16. V. V. Afanasjev, "Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification," Opt. Lett. 18, 790–792 (1993).
    [CrossRef] [PubMed]
  17. V. N. Serkin, V. V. Afanasjev, and V. A. Vysloukh, "Amplification and compression of femtosecond optical solitons in active fibers," Sov. Lightwave Commun. 2, 35–58 (1992).

1994 (2)

S. V. Chernikov, J. R. Taylor, and R. Kashyap, "Comblike dispersion profiled fiber for soliton pulse train generation," Opt. Lett. 539–541 (1994).
[CrossRef]

D. Anderson, M. Lisak, B. Malomed, and M. Quiroga- Teixeiro, "Tunneling of an optical soliton through a fiber junction," J. Opt. Soc. Am. B 11, 2380–2384 (1994).
[CrossRef]

1993 (2)

1992 (4)

Y. Kodama and A. Hasegawa, "Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect," Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

J. P. Gordon, "Dispersive perturbations of solitons of the nonlinear Schrödinger equation," J. Opt. Soc. Am. B 9, 91–97 (1992).
[CrossRef]

G. Boffetta and A. R. Osborne, "Computation of the direct scattering transform for the nonlinear Schroedinger equation," J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

V. N. Serkin, V. V. Afanasjev, and V. A. Vysloukh, "Amplification and compression of femtosecond optical solitons in active fibers," Sov. Lightwave Commun. 2, 35–58 (1992).

1991 (1)

Y. Kodama and S. Wabnitz, "Reduction of the soliton interaction forces by bandwidth limited amplification," Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

1989 (1)

Y. S. Kivshar, "On the soliton generation in optical fibers," J. Phys. A 22, 337–340 (1989).
[CrossRef]

1984 (1)

1983 (1)

1981 (1)

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910–913 (1981).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, "Initial value problems of onedimensional self-modulation of nonlinear waves in dispersive media," Progr. Theor. Phys. Suppl. 55, 284–300 (1974).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62–72 (1972) [Zh. Eksp. Teor. Fiz. 61, 118–XXX (1971)].

Aakjer, T.

T. Aakjer, K. Bertilson, M. L. Quiroga-Teixeiro, P. O. Hedekvist, and P. A. Andrekson, "Investigation of soliton compression by propagation through fiber junctions," Opt. Fiber Technol. (to be published).

Afanasjev, V. V.

V. V. Afanasjev, "Interpretation of the effect of reduction of soliton interaction by bandwidth-limited amplification," Opt. Lett. 18, 790–792 (1993).
[CrossRef] [PubMed]

V. N. Serkin, V. V. Afanasjev, and V. A. Vysloukh, "Amplification and compression of femtosecond optical solitons in active fibers," Sov. Lightwave Commun. 2, 35–58 (1992).

Anderson, D.

Andrekson, P. A.

T. Aakjer, K. Bertilson, M. L. Quiroga-Teixeiro, P. O. Hedekvist, and P. A. Andrekson, "Investigation of soliton compression by propagation through fiber junctions," Opt. Fiber Technol. (to be published).

Balant, A. C.

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910–913 (1981).
[CrossRef]

Bertilson, K.

T. Aakjer, K. Bertilson, M. L. Quiroga-Teixeiro, P. O. Hedekvist, and P. A. Andrekson, "Investigation of soliton compression by propagation through fiber junctions," Opt. Fiber Technol. (to be published).

Boffetta, G.

G. Boffetta and A. R. Osborne, "Computation of the direct scattering transform for the nonlinear Schroedinger equation," J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

Chernikov, S. V.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, "Comblike dispersion profiled fiber for soliton pulse train generation," Opt. Lett. 539–541 (1994).
[CrossRef]

S. V. Chernikov, E. M. Dianov, D. J. Richardson, and D. N. Payne, "Soliton pulse compression in dispersion-decreasing fiber," Opt. Lett. 18, 476–478 (1993).
[CrossRef] [PubMed]

Dianov, E. M.

Gordon, J. P.

Grischkowsky, D.

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910–913 (1981).
[CrossRef]

Hasegawa, A.

Y. Kodama and A. Hasegawa, "Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect," Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Hedekvist, P. O.

T. Aakjer, K. Bertilson, M. L. Quiroga-Teixeiro, P. O. Hedekvist, and P. A. Andrekson, "Investigation of soliton compression by propagation through fiber junctions," Opt. Fiber Technol. (to be published).

Kashyap, R.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, "Comblike dispersion profiled fiber for soliton pulse train generation," Opt. Lett. 539–541 (1994).
[CrossRef]

Kivshar, Y. S.

Y. S. Kivshar, "On the soliton generation in optical fibers," J. Phys. A 22, 337–340 (1989).
[CrossRef]

Kodama, Y.

Y. Kodama and A. Hasegawa, "Generation of asymptotically stable optical solitons and suppression of the Gordon–Haus effect," Opt. Lett. 17, 31–33 (1992).
[CrossRef] [PubMed]

Y. Kodama and S. Wabnitz, "Reduction of the soliton interaction forces by bandwidth limited amplification," Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

Lisak, M.

Malomed, B.

Mollenauer, L. F.

Nakatsuka, H.

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910–913 (1981).
[CrossRef]

Osborne, A. R.

G. Boffetta and A. R. Osborne, "Computation of the direct scattering transform for the nonlinear Schroedinger equation," J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

Payne, D. N.

Quiroga- Teixeiro, M.

Quiroga-Teixeiro, M. L.

T. Aakjer, K. Bertilson, M. L. Quiroga-Teixeiro, P. O. Hedekvist, and P. A. Andrekson, "Investigation of soliton compression by propagation through fiber junctions," Opt. Fiber Technol. (to be published).

Richardson, D. J.

Satsuma, J.

J. Satsuma and N. Yajima, "Initial value problems of onedimensional self-modulation of nonlinear waves in dispersive media," Progr. Theor. Phys. Suppl. 55, 284–300 (1974).
[CrossRef]

Serkin, V. N.

V. N. Serkin, V. V. Afanasjev, and V. A. Vysloukh, "Amplification and compression of femtosecond optical solitons in active fibers," Sov. Lightwave Commun. 2, 35–58 (1992).

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62–72 (1972) [Zh. Eksp. Teor. Fiz. 61, 118–XXX (1971)].

Shank, C. V.

Stolen, R. H.

Tappert, F.

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Taylor, J. R.

S. V. Chernikov, J. R. Taylor, and R. Kashyap, "Comblike dispersion profiled fiber for soliton pulse train generation," Opt. Lett. 539–541 (1994).
[CrossRef]

Tomlinson, W. J.

Vysloukh, V. A.

V. N. Serkin, V. V. Afanasjev, and V. A. Vysloukh, "Amplification and compression of femtosecond optical solitons in active fibers," Sov. Lightwave Commun. 2, 35–58 (1992).

Wabnitz, S.

Y. Kodama and S. Wabnitz, "Reduction of the soliton interaction forces by bandwidth limited amplification," Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

Yajima, N.

J. Satsuma and N. Yajima, "Initial value problems of onedimensional self-modulation of nonlinear waves in dispersive media," Progr. Theor. Phys. Suppl. 55, 284–300 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62–72 (1972) [Zh. Eksp. Teor. Fiz. 61, 118–XXX (1971)].

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, "Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion," Appl. Phys. Lett. 23, 142–144 (1973).
[CrossRef]

Electron. Lett. (1)

Y. Kodama and S. Wabnitz, "Reduction of the soliton interaction forces by bandwidth limited amplification," Electron. Lett. 27, 1931–1933 (1991).
[CrossRef]

J. Comput. Phys. (1)

G. Boffetta and A. R. Osborne, "Computation of the direct scattering transform for the nonlinear Schroedinger equation," J. Comput. Phys. 102, 252–264 (1992).
[CrossRef]

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

J. Phys. A (1)

Y. S. Kivshar, "On the soliton generation in optical fibers," J. Phys. A 22, 337–340 (1989).
[CrossRef]

Opt. Lett. (5)

Phys. Rev. Lett. (1)

H. Nakatsuka, D. Grischkowsky, and A. C. Balant, "Nonlinear picosecond-pulse propagation through optical fibers with positive group velocity dispersion," Phys. Rev. Lett. 47, 910–913 (1981).
[CrossRef]

Progr. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, "Initial value problems of onedimensional self-modulation of nonlinear waves in dispersive media," Progr. Theor. Phys. Suppl. 55, 284–300 (1974).
[CrossRef]

Sov. Lightwave Commun. (1)

V. N. Serkin, V. V. Afanasjev, and V. A. Vysloukh, "Amplification and compression of femtosecond optical solitons in active fibers," Sov. Lightwave Commun. 2, 35–58 (1992).

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, "Exact theory of two-dimensional self-focusing and one-dimensional selfmodulation of waves in nonlinear media," Sov. Phys. JETP 34, 62–72 (1972) [Zh. Eksp. Teor. Fiz. 61, 118–XXX (1971)].

Other (1)

T. Aakjer, K. Bertilson, M. L. Quiroga-Teixeiro, P. O. Hedekvist, and P. A. Andrekson, "Investigation of soliton compression by propagation through fiber junctions," Opt. Fiber Technol. (to be published).

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Figures (10)

Fig. 1
Fig. 1

Propagation dynamics of a soliton passing though an intermediate fiber section with α = 0.7 located between z = 5 and z = 7. The circles denote the energy of the soliton (2a), the continuous curve represents the maximal amplitude, and the dashed curve represents the energy in the radiation component.

Fig. 2
Fig. 2

Diffusion of an N = 2 soliton spectrum into smaller components to increase the length of the intermediate fiber and α = 16. In this case the second fiber spreads the pulse, converting the N = 2 soliton pulse into several smaller solitons and radiation. The circles represent the energy of the soliton components (2ai), and the dotted-dashed curve represent the energy of the radiation component.

Fig. 3
Fig. 3

Degradation of the pulse spectrum to increase the length of the intermediate fiber in the case α = 5. The circles represent the energy of the soliton components, and the dotted–dashed curve represents the energy of the radiation component.

Fig. 4
Fig. 4

Examples of nonlinear mode conversion showing, for the case α = 16/9 = 4/9A2, the variation with intermediate fiber length of the energy of the soliton components for the cases of initial dynamical phase ΔΦ = 0 (open circles) and ΔΦ = π (filled circles). Note the occurrence of an optimal conversion length (z ≈ 0.41) in the case of ΔΦ = 0 and the transformation of the N = 2 solitons into two twin solitons with equal amplitudes and with oppositely directed, but equal, velocities (υ ≈ ±0.62) in the case ΔΦ = π.

Fig. 5
Fig. 5

(a) Transformation of the initial pulse 2 sech(t) into approximately a single soliton pulse. The parameters of the intermediate section, which begins at z = 8z0 ≈ 6.28 (ΔΦ = 0), are Lopt = 0.43 and α = 2. The output soliton then contains 92.6% of the input energy. The open circles repre sent the energies of the solitons, and the continuous curve shows the maximum amplitude of the pulse. (b) Growth of the continuous component of the spectrum. Before the pulse enters the intermediate fiber section no radiation is present. In the intermediate fiber section the radiation grows (e.g., for z = 6.46, dotted-dashed curve) and reaches its final value when the dispersion is restored to its original value (z = 6.71, continuous curve). (c) Pulse transformation illustrated by a contour plot. Note the two quasi-linear wave packets corresponding to the two spectral peaks observed in (b).

Fig. 6
Fig. 6

Configuration of equivalent potentials in the case of optimal conversion. The particle at position b = 1/α = 1 jumps from the potential corresponding to the main fiber (continuous curve) to the potential of the intermediate fiber (dotted-dashed curve), where it performs half a period of oscillation and then jumps back to the original potential curve.

Fig. 7
Fig. 7

Comparison between the predicted and the numerically obtained optimal dispersion as a function of the initial amplitude of the pulse. The continuous curve represents the analytically obtained optimal dispersion, and the filled circles give the result from direct numerical simulations. The dashed curves represent the limiting values of the dispersion in the intermediate fiber corresponding to the different conversion regimes; see text.

Fig. 8
Fig. 8

Comparison of the predicted fiber length for optimal conversion (normalized to the oscillation period of the N = 2 soliton) as a function of the dispersion of the intermediate fiber for several initial pulse amplitudes A. The filled circles represent the numerically obtained values, and the curves show the analytical predictions: A = 2 (dotted-dashed curves), A = 1.8 (dashed curve), and A = 1.6 (continuous curve).

Fig. 9
Fig. 9

Evolution of (top) the real and (bottom) the imaginary parts of the eigenvalues ξk with the length of the intermediate fiber section for α = 4/9. At this value of α an N = 3 soliton is generated in the intermediate fiber for incident pulses of the form A sech(t) with A = 2. Note the generation of twin solitons with equal amplitudes and opposite velocities in the intervals (0.22 – 0.4) and (1.38 –1.55).

Fig. 10
Fig. 10

Pulse evolution during a nonadiabatic amplification. An initial pulse 0.6 sech(0.6t) is rapidly amplified with G0 = 3 until z = 0.3. A subsequent fiber with α = 2 and length Lopt = 0.624 nonlinearly transforms the pulse, eliminating most of the inherent pulse dynamics caused by the nonadiabatic amplification. The open circles and the dotted-dashed curve represent the discrete eigenvalues and the maximum amplitude of the pulse, respectively, without transformation. The filled circles, the continuous curve, and the dashed curves represent the discrete eigenvalues, the maximum pulse amplitude, and the energy of the radiation component, respectively, in the presence of nonlinear transformation. Note the strongly suppressed amplitude dynamics in this case.

Tables (2)

Tables Icon

Table 1 Optimized Conversion Efficiency η for Different α and ΔΦ = 0

Tables Icon

Table 2 Influence of the Internal Phase ΔΦ on the Optimal Conversion Efficiency for α = 2

Equations (11)

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i u z + α u t t + κ | u | 2 u = 0 ,
+ | u ( 0 , t ) | d t > π / 2 ,
u s ( z , t ) = a exp [ i υ t i ( υ 2 a 2 ) z ] cosh [ a ( t 2 υ z ] ,
Ψ 1 t = i ξ Ψ 1 + u ( 0 , t ) Ψ 2 , Ψ 2 t = + i ξ Ψ 2 + u * ( 0 , t ) Ψ 1 ,
ξ = υ / 2 + i a / 2 .
N 1 / 2 < A / α < N + 1 / 2 .
U ( b ) = 2 α π 2 ( α b 2 E b )
L opt = π 2 8 1 A 2 ( 1 + α 2 A 2 α ) 2 α 2 A 2 α ,
α opt = 2 A 2 1 + A 2 ,
i u z + u t t + 2 | u | 2 u = i G 0 u .
A 2 n a sech ( A 2 n a t ) .

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