Abstract

The dynamics that result from the combined effects of spatial diffraction, nonlinearity, and a parabolic graded index for wave propagation in optical fibers are presented. An approximate analytical solution of the nonlinear Schrodinger equation in a graded-index fiber is obtained by using a variational approach. Particular emphasis is put on the variation of both the pulse width and the longitudinal phase delay with the distance of propagation along the fiber.

© 1992 Optical Society of America

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References

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  1. R. Y Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
    [CrossRef]
  2. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  3. J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 1090 (1988).
    [CrossRef] [PubMed]
  4. J. T. Manassah, P. L. Baldeck, R. R. Alfano, Opt. Lett. 13, 589 (1988).
    [CrossRef] [PubMed]
  5. B. Bendow, P. D. Gianino, N. Tzoar, J. Opt. Soc. Am. 71, 656 (1981).
    [CrossRef]
  6. Y. Silberberg, Opt. Lett. 15, 1282 (1990).
    [CrossRef] [PubMed]
  7. M. Karlsson, D. Anderson, M. Desaix, M. Lisak, Opt. Lett. 16, 1373 (1991).
    [CrossRef] [PubMed]
  8. M. Desaix, D. Anderson, M. Lisak, J. Opt. Soc. Am. B 8, 2082 (1991).
    [CrossRef]
  9. K. Okamoto, E. A. J. Marcatili, IEEE J. Lightwave Technol. 7, 1988 (1989).
    [CrossRef]
  10. R. A. Sammut, C. Pask, J. Opt. Soc. Am. B 8, 395 (1991).
    [CrossRef]
  11. Note that we use slightly different definitions of pulse width and nonlinear refractive index n2 compared with those in Ref. 4: a=a′/2, and n2 = n2′/2, where the variables with primes refer to those in Ref. 4.

1991 (3)

1990 (1)

1989 (1)

K. Okamoto, E. A. J. Marcatili, IEEE J. Lightwave Technol. 7, 1988 (1989).
[CrossRef]

1988 (2)

1981 (1)

1980 (1)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

1964 (1)

R. Y Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Alfano, R. R.

Anderson, D.

Baldeck, P. L.

Bendow, B.

Chiao, R. Y

R. Y Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Desaix, M.

Garmire, E.

R. Y Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Gianino, P. D.

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Karlsson, M.

Lisak, M.

Manassah, J. T.

Marcatili, E. A. J.

K. Okamoto, E. A. J. Marcatili, IEEE J. Lightwave Technol. 7, 1988 (1989).
[CrossRef]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Okamoto, K.

K. Okamoto, E. A. J. Marcatili, IEEE J. Lightwave Technol. 7, 1988 (1989).
[CrossRef]

Pask, C.

Sammut, R. A.

Silberberg, Y.

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Townes, C. H.

R. Y Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

Tzoar, N.

IEEE J. Lightwave Technol. (1)

K. Okamoto, E. A. J. Marcatili, IEEE J. Lightwave Technol. 7, 1988 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (4)

Phys. Rev. Lett. (2)

R. Y Chiao, E. Garmire, C. H. Townes, Phys. Rev. Lett. 13, 479 (1964).
[CrossRef]

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Other (1)

Note that we use slightly different definitions of pulse width and nonlinear refractive index n2 compared with those in Ref. 4: a=a′/2, and n2 = n2′/2, where the variables with primes refer to those in Ref. 4.

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Figures (3)

Fig. 1
Fig. 1

Normalized pulse width a(z)/a0 versus normalized longitudinal coordinate ( z g ) for different values of C ranging from −1.0 [curve (a)] to 2.0 [curve (b)] with steps of 0.5.

Fig. 2
Fig. 2

Plot of the longitudinal phase shift ϕ ( z g ) along the fiber when C = 1 − p and the parameter p ranges from 0.1 [curve (a)] to 1.5 [curve (b)] with steps of 0.2.

Fig.3
Fig.3

Plot illustrating the dependence of ϕ ( z g ) on the parameters p and C. The parameters are p = 0.5, C = 0.3 [curve (a)], p = 0.5, C = 3 [curve (b)], p = 0.8, C = 3 [curve (c)], and p = 0.8, C = 0.3 [curve (d)].

Equations (14)

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n = n 0 + n 2 | E | 2 g r 2 2 .
1 r r ( r E r ) 2 i k E z k 2 r 2 g E + 2 n 2 n 0 k 2 | E | 2 E = 0 ,
E ( r , z , u ) = | A ( z , u ) | exp { 1 2 [ r a ( z , u ) ] 2 + i r 2 b ( z , u ) + i ϕ ( z , u ) } .
L = r | E r | 2 i k r ( E E * z E * E z ) + k 2 r 3 g | E | 2 n 2 n 0 k 2 r | E | 4 .
z ( | A | 2 a 2 ) = 0 ,
b = k 2 a a z ,
k ϕ z = 1 a 2 ( 1 3 n 2 A 0 2 a 0 2 4 n 0 ) ,
2 a z 2 + g a + 1 a 3 ( n 2 A 0 2 a 0 2 2 n 0 1 k 2 ) = 0 .
p ( A 0 E c ) 2 = n 2 k 2 A 0 2 a 0 2 2 n 0 ,
a ( z ) = a 0 2 [ ( 1 C ) cos ( 2 z g ) + 1 + C ] 1 / 2 = a 0 [ cos 2 ( z g ) + C sin 2 ( z g ) ] 1 / 2 ,
ϕ ( z , u ) = ( 1 3 2 p ) 1 1 p arctan [ C tan ( z g ) ] , 0 < p < 1 ,
ϕ ( z , u ) = 1 2 k a 0 2 g tan ( z g ) , p = 1 ,
ϕ ( z , u ) = ( 1 3 2 p ) 1 p 1 × arctanh [ C tan ( z g ) ] , 1 < p , z < z foc .
ϕ reg ( z ) = ϕ ( z ) ϕ ( z ) | p = 0 ,

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