Abstract

We present analytical results, based on a variational approach, on the spatiotemporal evolution of a pulse in a nonlinear waveguide with a periodic refractive-index profile. Stationary pulses, along with their stability properties, are predicted and then compared with numerical simulations of the governing equation.

© 1993 Optical Society of America

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References

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  1. See the feature issue on nonlinear guided-wave phenomena, J. Opt. Soc. Am. B 5, 264–574 (1988).
  2. Y. Silberberg, Opt. Lett. 15, 1282 (1990).
    [Crossref] [PubMed]
  3. V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).
  4. J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
    [Crossref]
  5. S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).
  6. A. Y. Wong, P. Y. Cheung, Phys. Rev. Lett. 22, 1222 (1984).
    [Crossref]
  7. M. Desaix, D. Anderson, M. Lisak, J. Opt. Soc. Am. B 8, 2082 (1991).
    [Crossref]
  8. B. Crosignani, P. Di Porto, S. Piazzolla, Pure Appl. Opt. 1, 7 (1992); B. Crosignani, P. Di Porto, Opt. Commun. 89, 453 (1992).
    [Crossref]
  9. R. A. Sammut, C. Pask, Opt. Lett. 16, 70 (1991).
    [Crossref] [PubMed]
  10. C. M. Bender, S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).
  11. A. G. Litvak, A. M. Sergeev, N. A. Shakova, Sov. Tech. Phys. Lett. 5, 33 (1979).

1992 (1)

B. Crosignani, P. Di Porto, S. Piazzolla, Pure Appl. Opt. 1, 7 (1992); B. Crosignani, P. Di Porto, Opt. Commun. 89, 453 (1992).
[Crossref]

1991 (2)

1990 (1)

1988 (1)

1986 (1)

J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[Crossref]

1984 (1)

A. Y. Wong, P. Y. Cheung, Phys. Rev. Lett. 22, 1222 (1984).
[Crossref]

1983 (1)

S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).

1979 (1)

A. G. Litvak, A. M. Sergeev, N. A. Shakova, Sov. Tech. Phys. Lett. 5, 33 (1979).

1972 (1)

V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).

Anderson, D.

Anisimov, S. I.

S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).

Bender, C. M.

C. M. Bender, S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Berezovskii, M. A.

S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).

Cheung, P. Y.

A. Y. Wong, P. Y. Cheung, Phys. Rev. Lett. 22, 1222 (1984).
[Crossref]

Crosignani, B.

B. Crosignani, P. Di Porto, S. Piazzolla, Pure Appl. Opt. 1, 7 (1992); B. Crosignani, P. Di Porto, Opt. Commun. 89, 453 (1992).
[Crossref]

Desaix, M.

Di Porto, P.

B. Crosignani, P. Di Porto, S. Piazzolla, Pure Appl. Opt. 1, 7 (1992); B. Crosignani, P. Di Porto, Opt. Commun. 89, 453 (1992).
[Crossref]

Lisak, M.

Litvak, A. G.

A. G. Litvak, A. M. Sergeev, N. A. Shakova, Sov. Tech. Phys. Lett. 5, 33 (1979).

Orzag, S. A.

C. M. Bender, S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

Pask, C.

Petrov, I. V.

S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).

Piazzolla, S.

B. Crosignani, P. Di Porto, S. Piazzolla, Pure Appl. Opt. 1, 7 (1992); B. Crosignani, P. Di Porto, Opt. Commun. 89, 453 (1992).
[Crossref]

Rasmussen, J.

J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[Crossref]

Rubenchik, A. M.

S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).

Rypdal, K.

J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[Crossref]

Sammut, R. A.

Sergeev, A. M.

A. G. Litvak, A. M. Sergeev, N. A. Shakova, Sov. Tech. Phys. Lett. 5, 33 (1979).

Shakova, N. A.

A. G. Litvak, A. M. Sergeev, N. A. Shakova, Sov. Tech. Phys. Lett. 5, 33 (1979).

Silberberg, Y.

Wong, A. Y.

A. Y. Wong, P. Y. Cheung, Phys. Rev. Lett. 22, 1222 (1984).
[Crossref]

Zakharov, V. E.

S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).

V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).

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

Opt. Lett. (2)

Phys. Rev. Lett. (1)

A. Y. Wong, P. Y. Cheung, Phys. Rev. Lett. 22, 1222 (1984).
[Crossref]

Phys. Scr. (1)

J. Rasmussen, K. Rypdal, Phys. Scr. 33, 481 (1986).
[Crossref]

Pure Appl. Opt. (1)

B. Crosignani, P. Di Porto, S. Piazzolla, Pure Appl. Opt. 1, 7 (1992); B. Crosignani, P. Di Porto, Opt. Commun. 89, 453 (1992).
[Crossref]

Sov. Phys. JETP (2)

S. I. Anisimov, M. A. Berezovskii, V. E. Zakharov, I. V. Petrov, A. M. Rubenchik, Sov. Phys. JETP 57, 1192 (1983).

V. E. Zakharov, Sov. Phys. JETP 35, 908 (1972).

Sov. Tech. Phys. Lett. (1)

A. G. Litvak, A. M. Sergeev, N. A. Shakova, Sov. Tech. Phys. Lett. 5, 33 (1979).

Other (1)

C. M. Bender, S. A. Orzag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978).

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Figures (1)

Fig. 1
Fig. 1

Evolution of α (solid curve) and β (dotted–dashed curve) as computed from Eq. (1) for parameter values = 1, w = 1, and q = 1 and initial power of (a) E0 = 1 and (b) E0 = 3.

Equations (14)

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i z u + x x u + q t t u + 2 | u | 2 u + 2 n c n 0 ( x , ω ) u = 0 ,
z = Z 2 β k 0 , x = X k 0 , t = k 0 δ ( T Z k 0 k ω 0 β ) , δ = χ k ω 0 2 + k k ω ω 0 ,
L = i 2 ( u z u * u * z u ) + | u x | 2 + q | u t | 2 | u | 4 cos ( w x ) | u | 2 ,
u ( x , t , z ) = A ( z ) exp ( x x 0 ) 2 2 α 2 exp ( t 2 2 β 2 ) × exp [ i a ( x x 0 ) 2 ] exp ( i b t 2 ) ,
L = + + L d x d t = π 2 [ i α β ( A A z * A * A z ) + α 3 β | A | 2 ( 4 a 2 + 1 α 4 ) ± α β 3 | A | 2 ( 4 b 2 + 1 β 4 ) α β | A | 4 + 2 | A | 2 α β cos ( w x 0 ) exp ( α 2 w 2 4 ) + a z | A | 2 α 3 β + b z | A | 2 α β 3 ] .
d 2 α d z 2 = 4 1 α 3 2 E 0 α 2 β 2 cos ( w x 0 ) α w 2 exp ( α 2 w 2 4 ) ,
d 2 β dz 2 = 4 1 β 3 2 E 0 α β 2 ,
a = 1 4 α d α d z ,
b = ± 1 4 β d β d z .
α = β [ 8 ( E 0 2 ) 1 / 4 ( z c z ) 1 / 2 ,
V ( α , β ) = 2 α 2 2 E 0 α β + 2 β 2 + 4 cos ( w x 0 ) exp ( α 2 w 2 4 ) .
E 0 2 4 = 2 w 2 α 4 exp ( a 2 w 2 4 ) cos ( w x 0 ) ,
β = 2 α E 0 .
J = E 0 4 α 8 ( 4 E 0 2 ) [ 1 ( α w 2 2 ) 2 ] .

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