Abstract

The conditions necessary for the amplitude or the temporal width of an N = 1 soliton to increase, remain constant, or decrease with distance on an axially nonuniform optical fiber are derived. Numerical results for a step-index fiber with an axially tapered core are presented. The effects of such a fiber on higher-order solitons are also presented.

© 1988 Optical Society of America

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References

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers. I. Anomalous dispersion,” Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  2. A. Hasegawa and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145 (1981).
    [CrossRef]
  3. K. J. Blow and N. J. Doran, “High bit rate communication systems using non-linear effects,” Opt. Commun. 42, 403 (1982).
    [CrossRef]
  4. N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
    [CrossRef]
  5. A. Hasegawa, “Amplification and reshaping of optical solitons in a glass fiber—IV: use of the stimulated Raman process,” Opt. Lett. 8, 650 (1983).
    [CrossRef] [PubMed]
  6. A. Hasegawa, “Numerical study of optical soliton transmission amplified periodically by the simulated Raman process,” Appl. Opt. 23, 3302 (1984).
    [CrossRef]
  7. L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157 (1986).
    [CrossRef]
  8. K. Tajima, “Compensation of soliton broadening in nonlinear optical fibers with loss,” Opt. Lett. 12, 54 (1987).
    [CrossRef] [PubMed]
  9. J. Satsuma and N. Yajima, “Initial value problems of one-dimensional self-modulation of nonlinear waves in dispersive media,” Prog. Theor. Phys. Suppl. 55, 284 (1974).
    [CrossRef]
  10. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, “Experimental observation of picosecond pulse narrowing and solitons in optical fibers,” Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  11. H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 282.
  12. G. P. Agrawal and M. J. Potasek, “Nonlinear pulse distortion in single-mode optical fibers at the zero-dispersion wavelength,” Phys. Rev. A 33, 1765 (1986).
    [CrossRef] [PubMed]
  13. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252 (1971).
    [CrossRef] [PubMed]
  14. N. Asano, “Wave propagations in non-uniform media,” Progr. Theor. Phys. Suppl. 55, 52 (1974).
    [CrossRef]
  15. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 2.
  16. R. Grimshaw, “Slowly varying solitary waves. II. Nonlinear Schrödinger equation,” Proc. R. Soc. London Ser. A 368, 377 (1979).
    [CrossRef]
  17. D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653 (1980).
    [CrossRef] [PubMed]
  18. H. Murata and H. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
    [CrossRef]

1987 (1)

1986 (2)

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

G. P. Agrawal and M. J. Potasek, “Nonlinear pulse distortion in single-mode optical fibers at the zero-dispersion wavelength,” Phys. Rev. A 33, 1765 (1986).
[CrossRef] [PubMed]

1984 (1)

1983 (2)

1982 (1)

K. J. Blow and N. J. Doran, “High bit rate communication systems using non-linear effects,” Opt. Commun. 42, 403 (1982).
[CrossRef]

1981 (2)

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

H. Murata and H. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

1980 (2)

D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653 (1980).
[CrossRef] [PubMed]

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

1979 (1)

R. Grimshaw, “Slowly varying solitary waves. II. Nonlinear Schrödinger equation,” Proc. R. Soc. London Ser. A 368, 377 (1979).
[CrossRef]

1974 (2)

N. Asano, “Wave propagations in non-uniform media,” Progr. Theor. Phys. Suppl. 55, 52 (1974).
[CrossRef]

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

1971 (1)

Agrawal, G. P.

G. P. Agrawal and M. J. Potasek, “Nonlinear pulse distortion in single-mode optical fibers at the zero-dispersion wavelength,” Phys. Rev. A 33, 1765 (1986).
[CrossRef] [PubMed]

Asano, N.

N. Asano, “Wave propagations in non-uniform media,” Progr. Theor. Phys. Suppl. 55, 52 (1974).
[CrossRef]

Blow, K. J.

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
[CrossRef]

K. J. Blow and N. J. Doran, “High bit rate communication systems using non-linear effects,” Opt. Commun. 42, 403 (1982).
[CrossRef]

Doran, N. J.

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
[CrossRef]

K. J. Blow and N. J. Doran, “High bit rate communication systems using non-linear effects,” Opt. Commun. 42, 403 (1982).
[CrossRef]

Gloge, D.

Gordon, J. P.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

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

Grimshaw, R.

R. Grimshaw, “Slowly varying solitary waves. II. Nonlinear Schrödinger equation,” Proc. R. Soc. London Ser. A 368, 377 (1979).
[CrossRef]

Hasegawa, A.

A. Hasegawa, “Numerical study of optical soliton transmission amplified periodically by the simulated Raman process,” Appl. Opt. 23, 3302 (1984).
[CrossRef]

A. Hasegawa, “Amplification and reshaping of optical solitons in a glass fiber—IV: use of the stimulated Raman process,” Opt. Lett. 8, 650 (1983).
[CrossRef] [PubMed]

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

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

Haus, H. A.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 282.

Inagaki, H.

H. Murata and H. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

Islam, M. N.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

Kodama, Y.

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

Marcuse, D.

D. Marcuse, “Pulse distortion in single-mode fibers,” Appl. Opt. 19, 1653 (1980).
[CrossRef] [PubMed]

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 2.

Mollenauer, L. F.

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

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

Murata, H.

H. Murata and H. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

Potasek, M. J.

G. P. Agrawal and M. J. Potasek, “Nonlinear pulse distortion in single-mode optical fibers at the zero-dispersion wavelength,” Phys. Rev. A 33, 1765 (1986).
[CrossRef] [PubMed]

Satsuma, J.

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

Stolen, R. H.

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

Tajima, K.

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 (1973).
[CrossRef]

Yajima, N.

J. Satsuma and 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. (3)

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 (1973).
[CrossRef]

IEEE J. Quantum Electron. (3)

N. J. Doran and K. J. Blow, “Solitons in optical communications,” IEEE J. Quantum Electron. QE-19, 1883 (1983).
[CrossRef]

L. F. Mollenauer, J. P. Gordon, and M. N. Islam, “Soliton propagation in long fibers with periodically compensated loss,” IEEE J. Quantum Electron. QE-22, 157 (1986).
[CrossRef]

H. Murata and H. Inagaki, IEEE J. Quantum Electron. QE-17, 835 (1981).
[CrossRef]

Opt. Commun. (1)

K. J. Blow and N. J. Doran, “High bit rate communication systems using non-linear effects,” Opt. Commun. 42, 403 (1982).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (1)

G. P. Agrawal and M. J. Potasek, “Nonlinear pulse distortion in single-mode optical fibers at the zero-dispersion wavelength,” Phys. Rev. A 33, 1765 (1986).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and 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 and Y. Kodama, “Signal transmission by optical solitons in monomode fiber,” Proc. IEEE 69, 1145 (1981).
[CrossRef]

Proc. R. Soc. London Ser. A (1)

R. Grimshaw, “Slowly varying solitary waves. II. Nonlinear Schrödinger equation,” Proc. R. Soc. London Ser. A 368, 377 (1979).
[CrossRef]

Prog. Theor. Phys. Suppl. (1)

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

Progr. Theor. Phys. Suppl. (1)

N. Asano, “Wave propagations in non-uniform media,” Progr. Theor. Phys. Suppl. 55, 52 (1974).
[CrossRef]

Other (2)

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, New York, 1974), Chap. 2.

H. A. Haus, Waves and Fields in Optoelectronics (Prentice-Hall, Englewood Cliffs, N.J., 1984), p. 282.

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Figures (1)

Fig. 1
Fig. 1

Core radius and GVD parameter versus distance along a step-index fiber. The labels on the curves refer to an N = 1 soliton.

Equations (25)

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i ( q z + γ q + k 1 q t + f z q 2 f ) - k 2 2 2 q t 2 + ω 0 n 2 G c q 2 q = 0 ,
E ( x , y , z , t ) = q ( z , t ) U ( x , y , z ) exp [ i ( k 0 d z - ω 0 t ) ] .
G ( z ) = U 4 d x d y / U 2 d x d y ,
V = ( ω 0 a / c ) ( n 01 2 - n 02 2 ) 1 / 2 ,
f ( z ) = n 0 ( z ) U ( x , y , z ) 2 d x d y ,
z ( q 2 f ) + f k 1 t q 2 + i f k 2 2 t ( q * q t - q q * t ) = 0.
d d z [ f ( z ) - q ( z , t ) 2 d t ] = 0.
W = n 0 ( z ) U ( x , y , z ) 2 d x d y - q ( z , t ) 2 d t .
f = n 0 π [ ( V / γ ) J 1 ( κ a ) ] 2 ,
ζ = k 2 d z ,
τ = t - k 1 d z ,
u = f 1 / 2 q exp ( γ d z ) ,
i u ζ + 1 2 2 u τ 2 + σ u 2 u = 0 ,
σ ( ζ ) = ω 0 n 2 G c f k 2 exp [ - 2 ( γ / k 2 ) d ζ ] .
u ( 0 ) = K σ 1 / 2 sech ( K σ τ ) exp [ i ( K 2 / 2 ) σ 2 d ζ ] ,
q ( 0 ) = A sech [ β ( t - k 1 d z ) ] exp ( i ϕ ) ,
A ( z ) = K σ 1 / 2 f 1 / 2 exp ( - γ d z ) = K f ( ω 0 n 2 G c k 2 ) 1 / 2 exp ( - 2 γ d z ) ,
β ( z ) = K σ = ( K ω 0 n 2 G c f k 2 ) exp ( - 2 γ d z ) ,
ϕ ( z ) = ( β 2 k 2 / 2 ) d z .
1 f ( n 2 G k 2 ) 1 / 2 exp ( - 2 0 z γ d z ) = C 1 exp ( p 1 z )
n 2 G f k 2 exp ( - 2 0 z γ d z ) = C 2 exp ( p 2 z )
k 2 a 2 exp ( 2 γ z ) = const . ,
| 1 k 2 d d z ( β - 2 ) | 1.
β - 2 = ( f 0 / β 0 f ) 2 exp [ - 4 ( p 1 z + 0 z γ d z ) ] ,
q ( 0 ) = 4 A [ cosh ( 3 β τ ) + 3 cosh ( β τ ) exp ( i 8 ϕ ) ] exp ( i ϕ ) cosh ( 4 β τ ) + 4 cosh ( 2 β τ ) + 3 cos ( 8 ϕ ) ,

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