Abstract

Using a scale transformation, I relate the solution of a beam propagating in a parabolic-index Kerr material to its solution in a homogeneous Kerr material. This technique is then applied to prove that a two-dimensional cw beam that propagates as a spatial soliton in a homogeneous Kerr medium would collapse if instead it were propagating in a parabolic-index material and that its mode of collapse is different from that of a low-intensity beam propagating in the same parabolic medium.

© 1992 Optical Society of America

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References

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  1. V E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  2. J. S. Aitchinson, A. M. Weiner, Y. Silberberg, M. K. Oliver, J. L. Jackel, D. E. Leaird, E. M. Vogel, P. W. E. Smith, Opt. Lett. 15, 471 (1990).
    [Crossref]
  3. F. Reynaud, A. Barthelemy, Europhys. Lett. 12, 5 (1990).
    [Crossref]
  4. See, for example, A. Yariv, Quantum Electronics (Wiley, New York, 1976).
  5. F. Cornolti, M. Lucchesi, B. Zambon, Opt. Commun. 75, 129 (1990).
    [Crossref]
  6. Y. Silberberg, Opt. Lett. 15, 1282 (1990).
    [Crossref] [PubMed]
  7. J. T. Manassah, Opt. Lett. 16, 563 (1991).
    [Crossref] [PubMed]
  8. J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
    [Crossref]

1991 (1)

1990 (4)

1972 (1)

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

1965 (1)

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Aitchinson, J. S.

Barthelemy, A.

F. Reynaud, A. Barthelemy, Europhys. Lett. 12, 5 (1990).
[Crossref]

Cornolti, F.

F. Cornolti, M. Lucchesi, B. Zambon, Opt. Commun. 75, 129 (1990).
[Crossref]

Gordon, J. P.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Jackel, J. L.

Leaird, D. E.

Leite, R. C.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Lucchesi, M.

F. Cornolti, M. Lucchesi, B. Zambon, Opt. Commun. 75, 129 (1990).
[Crossref]

Manassah, J. T.

Moore, R. S.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Oliver, M. K.

Porto, S. P. S.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Reynaud, F.

F. Reynaud, A. Barthelemy, Europhys. Lett. 12, 5 (1990).
[Crossref]

Shabat, A. B.

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

Silberberg, Y.

Smith, P. W. E.

Vogel, E. M.

Weiner, A. M.

Whinnery, J. R.

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Yariv, A.

See, for example, A. Yariv, Quantum Electronics (Wiley, New York, 1976).

Zakharov, V E.

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

Zambon, B.

F. Cornolti, M. Lucchesi, B. Zambon, Opt. Commun. 75, 129 (1990).
[Crossref]

Europhys. Lett. (1)

F. Reynaud, A. Barthelemy, Europhys. Lett. 12, 5 (1990).
[Crossref]

J. Appl. Phys. (1)

J. P. Gordon, R. C. Leite, R. S. Moore, S. P. S. Porto, J. R. Whinnery, J. Appl. Phys. 36, 3 (1965).
[Crossref]

Opt. Commun. (1)

F. Cornolti, M. Lucchesi, B. Zambon, Opt. Commun. 75, 129 (1990).
[Crossref]

Opt. Lett. (3)

Sov. Phys. JETP (1)

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

Other (1)

See, for example, A. Yariv, Quantum Electronics (Wiley, New York, 1976).

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Figures (3)

Fig. 1
Fig. 1

Beam x width plotted as a function of the normalized distance of propagation = z/ka02 for an initial beam ellipticity of (a) 1/2 and (b) 0.1. Curve (i), low-intensity homogeneous medium; curve (ii), low-intensity parabolic medium, ka02g = 0.5; curve (iii), waveguiding intensity in a homogeneous medium; curve (iv), same intensity as in curve (iii) but for a parabolic medium, ka02g = 0.5.

Fig. 2
Fig. 2

Beam y width plotted as a function of the normalized distance of propagation = z/ka02 for an initial beam ellipticity of (a) 1/2 and (b) 0.1. Curve (i), low-intensity homogeneous medium; curve (ii), low-intensity parabolic medium, ka02g = 0.5; curve (iii), waveguiding intensity in a homogeneous medium; curve (iv), same intensity as in curve (iii) but for a parabolic medium, ka02g = 0.5.

Fig. 3
Fig. 3

Ratio of the beam x width to its y width plotted as function of the normalized distance of propagation = z/ka02 for an initial beam ellipticity of (a) 1/2 and (b) 0.1. Curve (i), low-intensity, homogeneous medium; curve (ii), low-intensity parabolic medium, ka02g = 0.5; curve (iii), waveguiding intensity in a homogeneous medium; curve (iv), same intensity as in curve (iii) but for a parabolic medium, ka02g = 0.5.

Equations (12)

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n = n 0 + n 2 | E | 2 g 2 r 2 2 ,
i E z = 1 2 k T 2 E + k 2 g 2 r 2 E n 2 k n 0 | E | 2 E .
E = E 0 ψ , z ¯ = p z / k a 0 2 , x ¯ = p 1 / 2 x / a 0 , y ¯ = p 1 / 2 y / a 0 ,
i ψ z ¯ = 1 2 ( 2 ψ x ¯ 2 + 2 ψ y ¯ 2 ) + ω 2 2 ( x ¯ 2 + y ¯ 2 ) ψ | ψ | 2 ψ ,
d 2 ζ d z ¯ 2 + ω 2 ζ = 0 .
Z ( z ¯ ) = 0 z ¯ d z ζ 2 ( z ) = 1 ω tan ( ω z ¯ ) ,
ψ ( x ¯ , y ¯ , z ¯ ) = 1 ζ ( z ¯ ) φ [ x ¯ ζ ( z ¯ ) , y ¯ ζ ( z ¯ ) , Z ( z ¯ ) ] × exp [ i d ζ ( z ¯ ) d z ¯ 2 ζ ( z ¯ ) ( x ¯ 2 + y ¯ 2 ) ] ,
i Z φ = 1 2 ( 2 φ x ¯ 2 + 2 φ y ¯ 2 ) | φ | 2 φ .
φ ( x ¯ , y ¯ , Z ) = lim λ η sech ( η x ¯ ) sech ( y ¯ λ ) exp ( i 2 η 2 Z ) ,
ψ ( x ¯ , y ¯ , z ¯ ) = lim λ η cos ( ω z ¯ ) sech [ η x ¯ cos ( ω z ¯ ) ] × sech [ y ¯ λ cos ( ω z ¯ ) ] × exp { i 2 [ η 2 ω ω ( x ¯ 2 + y ¯ 2 ) ] tan ( ω z ¯ ) } .
n = n 0 ( 1 + δ r H 2 r 2 ) ,
δ = 0.12 P b π n 0 κ ( d n d T ) 8 D t r H 2 + 8 D t ,

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