Abstract

It is shown that partially incoherent spatial solitons are possible in saturable nonlinear media. This is demonstrated by means of an exact Gaussian solution in the case when the saturable nonlinearity is of the logarithmic type. The conditions necessary to establish these soliton states as well as their associated characteristics are discussed in detail. Pertinent examples are provided to elucidate their behavior further.

© 1997 Optical Society of America

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References

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  1. M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
    [CrossRef] [PubMed]
  2. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
    [CrossRef] [PubMed]
  3. D. N. Christodoulides and M. I. Carvalho, J. Opt. Soc. Am. B 12, 1628 (1995).
    [CrossRef]
  4. M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
    [CrossRef] [PubMed]
  5. G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
    [CrossRef] [PubMed]
  6. M. D. Castillo, P. A. Aguilar, J. J. Mondragon, and S. Stepanov, Appl. Phys. Lett. 64, 408 (1994).
    [CrossRef]
  7. J. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap.  5.
  8. D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
    [CrossRef]
  9. A. W. Snyder and J. D. Mitchell, Opt. Lett. 22, 16 (1997).
    [CrossRef] [PubMed]
  10. In Ref.  8 the process was described in terms of the so-called “coherent components,” which are in fact a discrete version of the coherent density employed here.
  11. L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), Chap.  4, p. 74.

1997 (2)

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

A. W. Snyder and J. D. Mitchell, Opt. Lett. 22, 16 (1997).
[CrossRef] [PubMed]

1996 (1)

M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

1995 (2)

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

D. N. Christodoulides and M. I. Carvalho, J. Opt. Soc. Am. B 12, 1628 (1995).
[CrossRef]

1994 (2)

M. D. Castillo, P. A. Aguilar, J. J. Mondragon, and S. Stepanov, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

1993 (1)

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Aguilar, P. A.

M. D. Castillo, P. A. Aguilar, J. J. Mondragon, and S. Stepanov, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Carvalho, M. I.

Castillo, M. D.

M. D. Castillo, P. A. Aguilar, J. J. Mondragon, and S. Stepanov, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Chen, Z.

M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Christodoulides, D. N.

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

D. N. Christodoulides and M. I. Carvalho, J. Opt. Soc. Am. B 12, 1628 (1995).
[CrossRef]

Coskun, T. H.

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

Crosignani, B.

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

DiPorto, P.

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Duree, G. C.

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Goodman, J.

J. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap.  5.

Mitchell, J. D.

Mitchell, M.

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

Mondragon, J. J.

M. D. Castillo, P. A. Aguilar, J. J. Mondragon, and S. Stepanov, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Neurgaonkar, R.

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Salamo, G.

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Schiff, L. I.

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), Chap.  4, p. 74.

Segev, M.

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Sharp, E.

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Shih, M. F.

M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

Shultz, J. L.

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Snyder, A. W.

Stepanov, S.

M. D. Castillo, P. A. Aguilar, J. J. Mondragon, and S. Stepanov, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Valley, G. C.

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Yariv, A.

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

Appl. Phys. Lett. (1)

M. D. Castillo, P. A. Aguilar, J. J. Mondragon, and S. Stepanov, Appl. Phys. Lett. 64, 408 (1994).
[CrossRef]

Electron. Lett. (1)

M. F. Shih, M. Segev, G. C. Valley, G. Salamo, B. Crosignani, and P. DiPorto, Electron. Lett. 31, 826 (1995); M. F. Shih, P. Leach, M. Segev, M. Garret, G. Salamo, and G. C. Valley, Opt. Lett. 21, 324 (1996).
[CrossRef] [PubMed]

J. Opt. Soc. Am. B (1)

Opt. Lett. (1)

Phys. Rev. Lett. (4)

G. C. Duree, J. L. Shultz, G. Salamo, M. Segev, A. Yariv, B. Crosignani, P. DiPorto, E. Sharp, and R. Neurgaonkar, Phys. Rev. Lett. 71, 533 (1993).
[CrossRef] [PubMed]

D. N. Christodoulides, T. H. Coskun, M. Mitchell, and M. Segev, Phys. Rev. Lett. 78, 646 (1997).
[CrossRef]

M. Mitchell, Z. Chen, M. F. Shih, and M. Segev, Phys. Rev. Lett. 77, 490 (1996).
[CrossRef] [PubMed]

M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, Phys. Rev. Lett. 73, 3211 (1994).
[CrossRef] [PubMed]

Other (3)

In Ref.  8 the process was described in terms of the so-called “coherent components,” which are in fact a discrete version of the coherent density employed here.

L. I. Schiff, Quantum Mechanics, 3rd ed. (McGraw-Hill, New York, 1968), Chap.  4, p. 74.

J. Goodman, Statistical Optics, 1st ed. (Wiley, New York, 1985), Chap.  5.

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Figures (2)

Fig. 1
Fig. 1

(a) Stationary propagation of a partially incoherent spatial soliton when x0=18 µm, θ0=4.5 mrad, and n2=10-4; (b) evolution of the normalized coherent density function f associated with this incoherent soliton.

Fig. 2
Fig. 2

Propagation of a partially incoherent Gaussian beam in a logarithmically saturable nonlinear medium when x0=18 µm and (a) θ0=6 mrad, (b) θ0=4 mrad.

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

n2I=n02+n2 lnI/It,
ifz+θfx+12k2fx2+k0n22n0lnINx, zf=0,
INx, z=-fx, z, θ2dθ
fz=0, x, θ=r1/2GN1/2θϕ0x.
INx, z=0=rexp-x2/x02.
iFξ+αFs+122Fs2-βs2F=0,
F=Aξexp-η22w2ξexpigξ+ηpξ+η2qξ,
vξ=α2β-1/2 sin2β1/2ξ,
w2ξ=2β+14β+2β-14β cos8β1/2ξ.
IN=rQ1/2exp-s2Q,
Q=w2ξ+k2x02θ022βsin22β1/2ξ.
x0=1k0n2-n02θ021/2.

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