Abstract

We investigate nonlinear propagation in the presence of the optical Kerr effect by relying on a rigorous generalization of the standard parabolic equation that includes nonparaxial and vectorial terms. We show that, in the 1+1D case, both soliton and propagation-invariant pattern solutions exist (while the standard hyperbolic-secant function is not a solution).

© 2004 Optical Society of America

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References

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  1. A. Ciattoni, P. Di Porto, B. Crosignani, and A. Yariv, J. Opt. Soc. Am. B 17, 809 (2000).
    [CrossRef]
  2. A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2389 (1994).
    [CrossRef]
  3. A. Ciattoni, C. Conti, E. DelRe, P. Di Porto, B. Crosignani, and A. Yariv, Opt. Lett. 27, 734 (2002).
    [CrossRef]
  4. R. de la Fuente, R. Varela, and H. Michinel, Opt. Commun. 173, 403 (2000).
    [CrossRef]
  5. J. A. Stratton, Electromagnetic Fields (McGraw-Hill, New York, 1941).
  6. S. Chi and Q. Guo, Opt. Lett. 20, 1598 (1995).
    [CrossRef] [PubMed]
  7. B. Crosignani, P. Di Porto, and A. Yariv, Opt. Lett. 22, 778 (1997).
    [CrossRef] [PubMed]
  8. S. Blair, Chaos 10, 570 (2000).
    [CrossRef]
  9. G. Fibich and B. Ilan, Physica D 157, 112 (2001).
    [CrossRef]
  10. K. Marinov, D. I. Pushkarov, and A. Shivarova, in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds. (Kluwer, Dordrecht, The Netherlands, 2001), pp. 293–316.
    [CrossRef]
  11. P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
    [CrossRef]
  12. T. A. Laine and A. T. Friberg, J. Opt. Soc. Am. B 17, 751 (2000).
    [CrossRef]

2002 (1)

2001 (2)

G. Fibich and B. Ilan, Physica D 157, 112 (2001).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
[CrossRef]

2000 (4)

1997 (1)

1995 (1)

1994 (1)

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2389 (1994).
[CrossRef]

Blair, S.

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

Chamorro-Posada, P.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
[CrossRef]

Chi, S.

Ciattoni, A.

Conti, C.

Crosignani, B.

de la Fuente, R.

R. de la Fuente, R. Varela, and H. Michinel, Opt. Commun. 173, 403 (2000).
[CrossRef]

DelRe, E.

Di Porto, P.

Fibich, G.

G. Fibich and B. Ilan, Physica D 157, 112 (2001).
[CrossRef]

Friberg, A. T.

Guo, Q.

Ilan, B.

G. Fibich and B. Ilan, Physica D 157, 112 (2001).
[CrossRef]

Laine, T. A.

Marinov, K.

K. Marinov, D. I. Pushkarov, and A. Shivarova, in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds. (Kluwer, Dordrecht, The Netherlands, 2001), pp. 293–316.
[CrossRef]

McDonald, G. S.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
[CrossRef]

Michinel, H.

R. de la Fuente, R. Varela, and H. Michinel, Opt. Commun. 173, 403 (2000).
[CrossRef]

New, G. H. C.

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
[CrossRef]

Pushkarov, D. I.

K. Marinov, D. I. Pushkarov, and A. Shivarova, in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds. (Kluwer, Dordrecht, The Netherlands, 2001), pp. 293–316.
[CrossRef]

Savchenko, A. Yu.

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2389 (1994).
[CrossRef]

Shivarova, A.

K. Marinov, D. I. Pushkarov, and A. Shivarova, in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds. (Kluwer, Dordrecht, The Netherlands, 2001), pp. 293–316.
[CrossRef]

Stratton, J. A.

J. A. Stratton, Electromagnetic Fields (McGraw-Hill, New York, 1941).

Varela, R.

R. de la Fuente, R. Varela, and H. Michinel, Opt. Commun. 173, 403 (2000).
[CrossRef]

Yariv, A.

Zel’dovich, B. Ya.

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2389 (1994).
[CrossRef]

Chaos (1)

S. Blair, Chaos 10, 570 (2000).
[CrossRef]

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

Opt. Commun. (2)

R. de la Fuente, R. Varela, and H. Michinel, Opt. Commun. 173, 403 (2000).
[CrossRef]

P. Chamorro-Posada, G. S. McDonald, and G. H. C. New, Opt. Commun. 192, 1 (2001).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. E (1)

A. Yu. Savchenko and B. Ya. Zel’dovich, Phys. Rev. E 50, 2389 (1994).
[CrossRef]

Physica D (1)

G. Fibich and B. Ilan, Physica D 157, 112 (2001).
[CrossRef]

Other (2)

K. Marinov, D. I. Pushkarov, and A. Shivarova, in Soliton-Driven Photonics, A. D. Boardman and A. P. Sukhorukov, eds. (Kluwer, Dordrecht, The Netherlands, 2001), pp. 293–316.
[CrossRef]

J. A. Stratton, Electromagnetic Fields (McGraw-Hill, New York, 1941).

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Figures (3)

Fig. 1
Fig. 1

fudu/dξ2 plotted as a function of uξ2 for different values of u0, the peak amplitude. The range of u is from 0 to u0 in each case, and fu is always positive.

Fig. 2
Fig. 2

Soliton envelope uξ obtained by solving du/dξ2=fu, where fu is obtained from Fig. 1. As the peak amplitude u0 becomes comparable to 1, the envelope widths approach a constant value.

Fig. 3
Fig. 3

For large values of u01 the normalized envelopes uξ/u0 have a constant width, independent of u0.

Equations (12)

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2E-·E+ω2/c2n2rE=0,
izA+12k2A+kn0δnrA=0.
iz+12k2x2Ax,z=-kn2n0A2A+13k2×A22x2A+83k2AxA2+56k2A22x2A*,
i/ζU+1/2U=-U2U-1/3U2U-5/6U2U*-8/3UU2,
-βu+1/2u=-u3-7/6u2u-8/3uu2,
dfdu+323+7u2uf=-12u-β+u23+7u2.
fu=-123+7u216/7×u0uu˜-β+u˜23+7u˜29/7du+3+7u0216/73+7u216/7fu0,
fu=-63+7u216/7u02u2-β+w3+7w9/7dw,=-63+7u216/7-βGu2-Gu02+u2Gu2-u02Gu02+Gu023+7u0223-Gu23+7u223,
β=0u02w3+7w9/7dw0u023+7w9/7dw=1623u02Gu02Gu02-G0-323.
α2=62314, β=α22163u02-1.
Ex,z=n0/n2u0 expi1-βkzsinαkx+γ,
2z2+2x2+k2E+2n/n0k2E=0,

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