M. Mitchell and M. Segev, Nature (London) 387, 880 (1997).

[CrossRef]

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).

[CrossRef]

Y. R. Shen, Science 276, 1520 (1997).

[CrossRef]

A. W. Snyder and D. J. Mitchell, [Opt. Lett. 22, 16 (1997)] describe the mighty morphing spatial solitons and bullets that are characteristic of a highly saturating medium. The ln I law had been introduced previously, unbeknown to us, in a different context. See, for example, I. Bialynicki-Birula and J. Mycielski, Phys. Scr. 20, 539 (1979). But no one seems to have recognized its relevance to optical spatial solitons, possibly because of its singularity at I=0. However, Gausian beams in a ln I medium induce parabolic index waveguides, which are a standard model in linear optics. Coming at the problem, as we do, from the linear perspective shows from the outset that ln I is, like the parabolic index fiber, a good approximation.

[CrossRef]
[PubMed]

W. Krolikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).

[CrossRef]

A. W. Snyder, D. J. Mitchell, and Yu. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 440 (1993); D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E 54, 6866 (1996).

[CrossRef]
[PubMed]

F. C. Khoo, Prog. Opt. 26, 107 (1988).

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, Phys. Rev. Lett. 68, 923 (1993).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 440 (1993); D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E 54, 6866 (1996).

[CrossRef]
[PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 440 (1993); D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E 54, 6866 (1996).

[CrossRef]
[PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, Phys. Rev. Lett. 68, 923 (1993).

[CrossRef]

F. C. Khoo, Prog. Opt. 26, 107 (1988).

A. W. Snyder, D. J. Mitchell, and Yu. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).

[CrossRef]

A. W. Snyder and D. J. Mitchell, [Opt. Lett. 22, 16 (1997)] describe the mighty morphing spatial solitons and bullets that are characteristic of a highly saturating medium. The ln I law had been introduced previously, unbeknown to us, in a different context. See, for example, I. Bialynicki-Birula and J. Mycielski, Phys. Scr. 20, 539 (1979). But no one seems to have recognized its relevance to optical spatial solitons, possibly because of its singularity at I=0. However, Gausian beams in a ln I medium induce parabolic index waveguides, which are a standard model in linear optics. Coming at the problem, as we do, from the linear perspective shows from the outset that ln I is, like the parabolic index fiber, a good approximation.

[CrossRef]
[PubMed]

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).

[CrossRef]

A. W. Snyder, D. J. Mitchell, and Yu. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).

[CrossRef]

M. Mitchell and M. Segev, Nature (London) 387, 880 (1997).

[CrossRef]

M. Mitchell and M. Segev, Nature (London) 387, 880 (1997).

[CrossRef]

M. Shih and M. Segev, Opt. Lett. 21, 1538 (1996).

[CrossRef]
[PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, Phys. Rev. Lett. 68, 923 (1993).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 440 (1993); D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E 54, 6866 (1996).

[CrossRef]
[PubMed]

A. W. Snyder and D. J. Mitchell, [Opt. Lett. 22, 16 (1997)] describe the mighty morphing spatial solitons and bullets that are characteristic of a highly saturating medium. The ln I law had been introduced previously, unbeknown to us, in a different context. See, for example, I. Bialynicki-Birula and J. Mycielski, Phys. Scr. 20, 539 (1979). But no one seems to have recognized its relevance to optical spatial solitons, possibly because of its singularity at I=0. However, Gausian beams in a ln I medium induce parabolic index waveguides, which are a standard model in linear optics. Coming at the problem, as we do, from the linear perspective shows from the outset that ln I is, like the parabolic index fiber, a good approximation.

[CrossRef]
[PubMed]

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).

[CrossRef]

A. W. Snyder, D. J. Mitchell, and Yu. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).

[CrossRef]

A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 482 (1993). A simple qualitative theory is advanced here to predict when annihilation, fusion, or birth occurs.

[CrossRef]
[PubMed]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, Phys. Rev. Lett. 68, 923 (1993).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 440 (1993); D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E 54, 6866 (1996).

[CrossRef]
[PubMed]

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

A. W. Snyder, D. J. Mitchell, and Yu. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).

[CrossRef]

M. Mitchell and M. Segev, Nature (London) 387, 880 (1997).

[CrossRef]

V. Tikhonenko, Opt. Lett. 23, 594 (1998).

[CrossRef]

A. W. Snyder and A. P. Sheppard, Opt. Lett. 18, 482 (1993). A simple qualitative theory is advanced here to predict when annihilation, fusion, or birth occurs.

[CrossRef]
[PubMed]

A. W. Snyder and D. J. Mitchell, [Opt. Lett. 22, 16 (1997)] describe the mighty morphing spatial solitons and bullets that are characteristic of a highly saturating medium. The ln I law had been introduced previously, unbeknown to us, in a different context. See, for example, I. Bialynicki-Birula and J. Mycielski, Phys. Scr. 20, 539 (1979). But no one seems to have recognized its relevance to optical spatial solitons, possibly because of its singularity at I=0. However, Gausian beams in a ln I medium induce parabolic index waveguides, which are a standard model in linear optics. Coming at the problem, as we do, from the linear perspective shows from the outset that ln I is, like the parabolic index fiber, a good approximation.

[CrossRef]
[PubMed]

M. Shih and M. Segev, Opt. Lett. 21, 1538 (1996).

[CrossRef]
[PubMed]

W. Krolikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fisher, Phys. Rev. Lett. 68, 923 (1993).

[CrossRef]

M. Segev, B. Crosignani, A. Yariv, and B. Fischer, Phys. Rev. Lett. 68, 923 (1992); B. Crosignani, M. Segev, D. Engin, P. DiPorto, A. Yariv, and G. Salamo, J. Opt. Soc. Am. B 10, 440 (1993); D. N. Christodoulides and M. I. Carvalho, Opt. Lett. 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E 54, 6866 (1996).

[CrossRef]
[PubMed]

F. C. Khoo, Prog. Opt. 26, 107 (1988).

A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).

[CrossRef]

Y. R. Shen, Science 276, 1520 (1997).

[CrossRef]

P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).

Section 8 of Ref. 12 shows that Maxwell’s equations for transverse field E are equivalent to the Schrödinger equation, provided only that the medium is homogeneous. This is so because the maximum nonlinear induced refractive-index change is small.

We substitute ψ=B exp(ax2) or ψ=B exp(ar2) into the Schrödinger equation. Taking B=b-1/2 exp(iϕ) in two dimensions or B=b-1 exp(iϕ) in three dimensions, where ϕ=(k/2n0)∫[n2(0, z)-n02]dz, and using Eq. (5), we find that a=ikn0(db/dz)/(2b) and 2ikn0(da/dz)+4a2-(k2Δ/ρw2)=0. Now the radius is given by 1/ρ2=-2 Re(a). Thus, after a little algebra, we obtain d2ρdz2+1n02ρΔρ2ρw2-1k2ρ2=0. Equation (4) follows immediately.

A first integration of Eq. (7) yields dρdz2+1k2n02ρ¯2ln(ρ2+ρM2)+ρ¯2ρ2=const. Taking dρ/dz=0 and ρ=ρmax or ρ=ρmin immediately yields Eq. (10).

Following Ref. 13, we substitute ψ=B exp(axx2)exp(ayy2) into Schrödinger equation. Taking B=b-1/2 exp(iϕ), where ϕ=(k/2n0)∫[n2(0, z)-n02]dz, we find that ikn0(db/dz)/(2b)=ax+ay, and ax and ay both satisfy 2ikn0(da/dz)+4a2-(k2Δ/ρw2)=0, with ρ replaced by ρx and ρy, respectively. Thus in Ref. 13 both ρx and ρy satisfy Eq. (4).