Abstract

The nonlinear response of various materials extends beyond the illuminating beam. We present what is to our knowledge the first analytically tractable model for the dynamics of beams in partially nonlocal media. As far as an isolated beam is concerned, propagation is qualitatively the same, independently of the radius of nonlocality.

© 1999 Optical Society of America

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  1. A. W. Snyder and D. J. Mitchell, Science 276, 1538 (1997).
    [CrossRef]
  2. Y. R. Shen, Science 276, 1520 (1997).
    [CrossRef]
  3. 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]
  4. M. Segev, B. Crosignani, A. Yariv, and B. Fisher, Phys. Rev. Lett. 68, 923 (1993).
    [CrossRef]
  5. P. Yeh, Introduction to Photorefractive Nonlinear Optics (Wiley, New York, 1993).
  6. F. C. Khoo, Prog. Opt. 26, 107 (1988).
  7. M. Mitchell and M. Segev, Nature (London) 387, 880 (1997).
    [CrossRef]
  8. V. Tikhonenko, Opt. Lett. 23, 594 (1998).
    [CrossRef]
  9. 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]
  10. 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. OPLEDP 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. OPLEDP 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E PLEEE8 54, 6866 (1996).
    [CrossRef] [PubMed]
  11. 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.
  12. A. W. Snyder, D. J. Mitchell, and Yu. S. Kivshar, Mod. Phys. Lett. B 9, 1479 (1995).
    [CrossRef]
  13. 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+1n02ρ Δρ2ρw2−1k2ρ2=0. Equation (4) follows immediately.
  14. A first integration of Eq. (7) yields dρdz2+1k2n02ρ¯2 ln(ρ2M2)+ρ¯2ρ2=const. Taking dρ/dz=0 and ρ=ρmax or ρ=ρmin immediately yields Eq. (10).
  15. 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).
  16. M. Shih and M. Segev, Opt. Lett. 21, 1538 (1996).
    [CrossRef] [PubMed]
  17. W. Krolikowski and S. A. Holmstrom, Opt. Lett. 22, 369 (1997).
    [CrossRef]

1998 (1)

1997 (4)

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]

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

1996 (1)

1995 (1)

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

1993 (2)

1988 (1)

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

Crosignani, B.

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

Fisher, B.

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

Holmstrom, S. A.

Khoo, F. C.

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

Kivshar, Yu. S.

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

Krolikowski, W.

Mitchell, D. J.

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]

Mitchell, M.

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

Segev, M.

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]

Shen, Y. R.

Y. R. Shen, Science 276, 1520 (1997).
[CrossRef]

Sheppard, A. P.

Shih, M.

Snyder, A. W.

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]

Tikhonenko, V.

Yariv, A.

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

Mod. Phys. Lett. B (1)

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

Nature (London) (1)

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

Opt. Lett. (4)

Phys. Rev. Lett. (1)

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

Prog. Opt. (1)

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

Science (2)

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

Y. R. Shen, Science 276, 1520 (1997).
[CrossRef]

Other (7)

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]

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

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+1n02ρ Δρ2ρw2−1k2ρ2=0. Equation (4) follows immediately.

A first integration of Eq. (7) yields dρdz2+1k2n02ρ¯2 ln(ρ2M2)+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).

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. OPLEDP 19, 1714 (1994); M. Segev, A. Yariv, B. Crosignani, P. DiPorto, G. Duree, G. Salamo, and E. Sharp, Opt. Lett. OPLEDP 19, 1296 (1994); N. Fressegeas, J. Maufoy, and G. Kugel, Phys. Rev. E PLEEE8 54, 6866 (1996).
[CrossRef] [PubMed]

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.

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Figures (1)

Fig. 1
Fig. 1

Relative change ρmax/ρmin of soliton radius for a given initial radius ρ0 compared with the radius ρsol necessary for the beam to be a stationary soliton.

Equations (14)

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δn2=Δf[I(x), M(x)]general,
=Δ ln[I(x), M(x)]forf=ln,
M(x)=Mm exp(-r2/ρM2),
I(x, z)=Im(z)exp[-r2/ρ2(z)],
d2ρdz2+1k2n02ρ¯2ρρW2-ρ¯2ρ3=0,
δn2(x, z)=Δ ln{g(z)MmIm exp[-r2/ρW2(z)]}=Δ ln[g(z)MmIm]-Δr2/ρW2(z),
ρsol2ρ¯2=121+1+4ρM2ρ¯21/2,
Ω2=2k2n02ρsol41+ρM2ρsol2+ρM2.
lnρM2+ρmax2ρM2+ρmin2=ρ¯2ρmin2-ρ¯2ρmax2.
ρ2(z)=ρ02[cos2 qz+(ρsol/ρ0)4 sin2(qz)],
I(x, z)=Im(z)exp[-x2/ρx(z)-y2/ρy(z)].
δn2(x, z)=Δ ln[g(z)MmIm]-Δ[x2/(ρM2+ρx2)+y2/(ρM2+ρy2)].
d2ρdz2+1n02ρΔρ2ρw2-1k2ρ2=0.
dρdz2+1k2n02ρ¯2ln(ρ2+ρM2)+ρ¯2ρ2=const.

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