Abstract

Propagation of short pulses in birefringent optical fibers is considered in the framework of two coupled nonlinear Schrödinger equations. When the amplitudes of the pulses belonging to different polarizations are equal, we propose a simple analytical explanation of the amplitude threshold for the capture of two partial pulses into a coupled two-component pulse. Our approach is based on a soliton phenomenology. The analytical dependence of the amplitude threshold on linear birefringence is in good agreement with numerical results of Menyuk [ J. Opt. Soc. Am. B 5, 392 ( 1988)]. The influence of small dissipative losses on the effect is also discussed.

© 1990 Optical Society of America

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  1. L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1045 (1980).
    [CrossRef]
  2. A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
    [CrossRef]
  3. C. R. Menyuk, IEEE J. Quantum Electron. QE-23, 142 (1987).
  4. K. J. Blow, N. J. Doran, and D. Wood, Opt. Lett. 12, 202 (1987).
    [CrossRef] [PubMed]
  5. C. R. Menyuk, Opt. Lett. 12, 614 (1987).
    [CrossRef] [PubMed]
  6. D. N. Christodulides and R. I. Joseph, Opt. Lett. 13, 53 (1988).
    [CrossRef]
  7. C. R. Menyuk, J. Opt. Soc. Am. B 5, 392 (1988).
    [CrossRef]
  8. A. D. Boardman and G. S. Cooper, J. Opt. Soc. Am. B 5, 403 (1988).
    [CrossRef]
  9. M. V. Tratnik and J. E. Sipe, Phys. Rev. A 38, 2011 (1988).
    [CrossRef] [PubMed]
  10. S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
    [CrossRef] [PubMed]
  11. Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
    [CrossRef]
  12. V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  13. J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
    [CrossRef]
  14. A. Bondeson, D. Anderson, and M. Lisak, Phys. Scr. 20, 479 (1979).
    [CrossRef]
  15. D. Anderson and M. Lisak, Phys. Rev. A 32, 2270 (1985); Opt. Lett. 11, 174 (1986).
    [CrossRef] [PubMed]
  16. A. Hasegawa, Opt. Lett. 5, 416 (1980).
    [CrossRef] [PubMed]
  17. L. F. Mollenauer and K. Smith, Opt. Lett. 13, 675 (1988).
    [CrossRef]

1989 (1)

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

1988 (6)

1987 (3)

1985 (1)

D. Anderson and M. Lisak, Phys. Rev. A 32, 2270 (1985); Opt. Lett. 11, 174 (1986).
[CrossRef] [PubMed]

1980 (2)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1045 (1980).
[CrossRef]

A. Hasegawa, Opt. Lett. 5, 416 (1980).
[CrossRef] [PubMed]

1979 (1)

A. Bondeson, D. Anderson, and M. Lisak, Phys. Scr. 20, 479 (1979).
[CrossRef]

1974 (1)

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

1973 (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Anderson, D.

D. Anderson and M. Lisak, Phys. Rev. A 32, 2270 (1985); Opt. Lett. 11, 174 (1986).
[CrossRef] [PubMed]

A. Bondeson, D. Anderson, and M. Lisak, Phys. Scr. 20, 479 (1979).
[CrossRef]

Blow, K. J.

Boardman, A. D.

Bondeson, A.

A. Bondeson, D. Anderson, and M. Lisak, Phys. Scr. 20, 479 (1979).
[CrossRef]

Christodulides, D. N.

Cooper, G. S.

Doran, N. J.

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1045 (1980).
[CrossRef]

Hasegawa, A.

A. Hasegawa, Opt. Lett. 5, 416 (1980).
[CrossRef] [PubMed]

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Joseph, R. I.

Kivshar, Yu. S.

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Lisak, M.

D. Anderson and M. Lisak, Phys. Rev. A 32, 2270 (1985); Opt. Lett. 11, 174 (1986).
[CrossRef] [PubMed]

A. Bondeson, D. Anderson, and M. Lisak, Phys. Scr. 20, 479 (1979).
[CrossRef]

Malomed, B. A.

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Menyuk, C. R.

Mollenauer, L. F.

L. F. Mollenauer and K. Smith, Opt. Lett. 13, 675 (1988).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1045 (1980).
[CrossRef]

Satsuma, J.

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Shabat, A. B.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Sipe, J. E.

M. V. Tratnik and J. E. Sipe, Phys. Rev. A 38, 2011 (1988).
[CrossRef] [PubMed]

Smith, K.

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1045 (1980).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

Tratnik, M. V.

M. V. Tratnik and J. E. Sipe, Phys. Rev. A 38, 2011 (1988).
[CrossRef] [PubMed]

Wabnitz, S.

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

Wood, D.

Yajima, N.

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Appl. Phys. Lett. (1)

A. Hasegawa and F. Tappert, Appl. Phys. Lett. 23, 142 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

C. R. Menyuk, IEEE J. Quantum Electron. QE-23, 142 (1987).

J. Opt. Soc. Am. B (2)

Opt. Lett. (5)

Phys. Rev. A (3)

D. Anderson and M. Lisak, Phys. Rev. A 32, 2270 (1985); Opt. Lett. 11, 174 (1986).
[CrossRef] [PubMed]

M. V. Tratnik and J. E. Sipe, Phys. Rev. A 38, 2011 (1988).
[CrossRef] [PubMed]

S. Wabnitz, Phys. Rev. A 38, 2018 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

L. F. Mollenauer, R. H. Stolen, and J. P. Gordon, Phys. Rev. Lett. 45, 1045 (1980).
[CrossRef]

Phys. Scr. (1)

A. Bondeson, D. Anderson, and M. Lisak, Phys. Scr. 20, 479 (1979).
[CrossRef]

Progr. Theor. Phys. Suppl. (1)

J. Satsuma and N. Yajima, Progr. Theor. Phys. Suppl. 55, 284 (1974).
[CrossRef]

Rev. Mod. Phys. (1)

Yu. S. Kivshar and B. A. Malomed, Rev. Mod. Phys. 61, 763 (1989).
[CrossRef]

Sov. Phys. JETP (1)

V. E. Zakharov and A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

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Figures (3)

Fig. 1
Fig. 1

Symmetric pulse configuration in birefringent fibers. Solid curve, u; dashed curve, υ (arbitrary units).

Fig. 2
Fig. 2

Effective potential energy Uint(Δ) for the soliton interaction due to birefringence. Ekin is kinetic energy. The case EkinUmax corresponds to a bound state of the partial solitons.

Fig. 3
Fig. 3

Analytical (solid curve) and numerical (filled circles) dependences of the threshold amplitude on the birefringence parameter δ. The numerical points were obtained by Menyuk.5,7 The dashed curve shows the threshold estimation [relation (37)] based on the paper by Hasegawa.16

Tables (1)

Tables Icon

Table 1 Threshold Values of A Obtained Numerically by Menyuka and Analytically by Using Formula (34) at = 2/3

Equations (47)

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i ( u z + δ u t ) + 1 2 2 u t 2 + ( | u | 2 + | υ | 2 ) u = i γ u ,
i ( υ z δ υ t ) + 1 2 2 υ t 2 + ( | υ | 2 + | u | 2 ) υ = i γ υ ,
= 2 / 3 .
u ( 0 , t ) = A cos α sech t , υ ( 0 , t ) = A sin α sech t ,
u ( 0 , t ) = υ ( 0 , t ) = A 2 sech t ,
i V z + 1 2 2 V t 2 + | V | 2 V = 0 ,
V ( 0 , t ) = B sech t
V ( z , t ) = 2 η exp ( 2 i η 2 z ) sech ( 2 η t )
η = B 1 2 .
u = υ = 1 1 + 2 η exp ( 2 i η 2 z ) sech ( 2 η t ) ,
η = A 1 + 2 1 2 .
δ d t d z L = 0 ,
L = L u + L υ + L u υ ,
L u = l 2 ( u * u z u u * z ) + i 2 δ ( u * u t u u * t ) 1 2 | u t | 2 + 1 2 | u | 4 ,
L υ = L u ( u υ ) ,
L u υ = | u | 2 | υ | 2 .
[ u υ ] = 2 η j 1 + exp [ 2 i C j ( t ζ j ) + i D j ] cosh [ 2 η j ( t ζ j ) ] ,
L a = d d z [ L ( a z ) ] ,
L = L d t .
d η j d z = 0 ,
8 d d z ( η j C j ) + ζ j L u υ = 0 ,
η j d ζ j d z + ( 1 ) j η j δ 2 η j C j = 0 ,
d D j d z = 2 C j 2 + 2 η j 2 ( 1 1 + ) + 1 4 ( 1 + ) η j L u υ ,
L u υ = 8 η 1 η 2 2 ( 1 + ) d z 1 cosh 2 z 1 cosh 2 z 2 ,
η 1 = η 2 = η = const ., C 1 = C 2 = C , ζ 1 , 2 = ± Δ / 4 η ,
d Δ d z = 8 η C + 4 η δ ,
d C d z = 1 8 η Δ U int ( Δ ) ,
U int ( Δ ) = 128 η 4 ( 1 + ) [ 1 3 cosh Δ sinh 3 Δ ( Δ tanh Δ ) ]
U int ( Δ ) = 256 η 4 15 ( 1 + ) Δ 2 ,
ω 0 2 = 512 15 η 4 ( 1 + ) .
U max = 128 η 4 3 ( 1 + ) ,
u ( z , t ) = ( π / 2 z ) 1 / 2 ( 1 i ) A 2 × exp [ i ( t δ z ) 2 ] sech [ π 2 z ( t δ z ) ] ,
u ( z , t ) = ( π / 2 z ) 1 / 2 ( 1 i ) A 2 × exp [ i ( t + δ z ) 2 ] sech [ π 2 z ( t + δ z ) ] .
1 2 ( d Δ d z ) 2 + U int ( Δ ) = const .
1 2 ( 4 η δ ) 2 U max ,
A thr = 1 2 ( 1 + ) + 1 2 3 2 δ ,
A thr 0.55 + 0.75 δ .
( 2 δ ) 2 2 ( | u | 2 + | υ | 2 ) ,
A thr = 2 3 / 5 δ 1.55 δ ,
η = η ( 0 ) exp ( 2 γ z ) .
z * ( 2 γ ) 1 ln [ 4 η ( 0 ) δ 3 ( 1 + ) ] ,
L u υ ( Δ ) = 8 η 3 ( 1 + ) I ( Δ ) ,
I ( Δ ) = d x cosh 2 x cosh 2 ( x Δ ) = d x cosh 2 ( x Δ 2 ) cosh 2 ( x + Δ 2 ) = d x ( cosh 2 x cosh 2 Δ 2 sinh 2 x sinh 2 Δ 2 ) 2 .
w = tanh x , a 2 = coth 2 ( Δ / 2 ) ,
I ( Δ ) = 2 sinh 2 Δ 2 0 1 ( 1 w 2 ) d w ( a 2 w 2 ) 2 .
I ( Δ ) 4 3 8 15 Δ 2 .
U int ( Δ ) = 4 η [ L u υ ( 0 ) L u υ ( Δ ) ] .

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