Abstract

We study spatial soliton dynamics in nano-particle suspensions. Starting from the Nernst-Planck and Smoluchowski equations, we demonstrate that in these systems the underlying nonlinearities as well as the nonlinear Rayleigh losses depend exponentially on optical intensity. Two different nonlinear regimes are identified depending on the refractive index contrast of the nanoparticles involved and the interesting prospect of self-induced transparency is demonstrated. Soliton stability is systematically analyzed for both 1D and 2D configurations and their propagation dynamics in the presence of Rayleigh losses is examined. The possibility of synthesizing artificial nonlinearities using mixtures of nanosuspensions is also considered.

© 2007 Optical Society of America

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References

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  1. G. G. Hammes, Thermodynamics and Kinetics for the Biological Sciences, (John Wiley and Sons 2000).
  2. P.W. Smith, A. Ashkin and W.J. Tomlinson, "Four-wave mixing in an artificial Kerr medium," Opt. Lett. 6,284-286 (1981).
    [CrossRef] [PubMed]
  3. A. Ashkin, J.M. Dziedzic and P.W. Smith, "Continuous-wave self-focusing and self-trapping of light in artificial Kerr media," Opt. Lett. 7,276-278 (1982).
    [CrossRef] [PubMed]
  4. V.E. Yashin, S.A. Chizhov, R.L. Sabirov, T.V. Starchikova, N.V. Vysotina, N.N. Rozanov, V.E. Semenov, V.A. Smirnov and S.V. Fedorov, "Formation of Soliton-like Light Beams in an Aqueous Suspension of Polystyrene Particles," Opt Spectrosc+  98, 466-469 (2005).
    [CrossRef]
  5. P.J. Reece, E.M. Wright and K. Dholakia, " Experimnetal Observation of Modulation Instability and Optical Spatial Soliton Arrays in Soft Condensed Matter," Phys. Rev. Lett. 98, 203902:1-4 (2007).
    [CrossRef]
  6. J.P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8, 14-21 (1973).
    [CrossRef]
  7. A. Ashkin, J.M. Dziedzic, J.E. Bjorkholm and S. Chu, "Observation of single-beam gradient force optical trap for dielectric particles," Opt. Lett. 11, 288-290 (1986).
    [CrossRef] [PubMed]
  8. S. Stenholm, "The semiclassical theory of laser cooling," Rev. Mod. Phys. 58, 699-739 (1986).
    [CrossRef]
  9. C. Conti, G. Ruocco and S. Trillo, "Optical Spatial Solitons in Soft Matter," Phys. Rev. Lett. 95, 183902:1-4 (2005).
    [CrossRef]
  10. C. Conti, N. Ghofraniha, G. Ruocco and S. Trillo, "Laser Beam Filamentation in Fractal Aggregates," Phys. Rev. Lett. 97, 123903:1-4 (2006).
    [CrossRef]
  11. R. El-Ganainy, K. G. Makris, D. N. Christodoulides, C. Rothchild, and M. Segev, "Cusp solitons in exponentially nonlinear nanosuspensions", paper QMB4, CLEO/QELS 2007, May 6-11, Baltimore, Maryland.
  12. R. Gordon, J.T. Blakely and D. Sinton, "Particle-optical self-trapping," Phys. Rev. A. 75055801:1-4 (2007).
    [CrossRef]
  13. D. Rogovin and S.O. Sari, "Phase conjugation in liquid suspenstions of microspheres in the diffusive limit," Phys. Rev. A 31, 2375-2389 (1985).
    [CrossRef] [PubMed]
  14. B.J. berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology and Physics (Dover Publication, Inc. New York 2000)
  15. J.D. Jackson, Classical Electrodynamics, (John Wiley and Sons, New York 1999).
  16. J.M.C. Garnett, "Colors in Metal Glasses and in Metallic Films," Philos. Trans. R. Soc. London 203, 385-420 (1904).
    [CrossRef]
  17. J.M.C. Garnett, "Colors in Metal Glasses, in Metallic Films and in Metallic solutions," Philos. Trans. R. Soc. London 205, 237- 288 (1906).
    [CrossRef]
  18. H.C. van de Hulst, Light Scattering by Small Particles, (Dover Publication, Inc. New York 1981).
  19. N.G. Vakhitov and A.A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Izv. Vyssh. Uchebn. Zaved Radiofiz. 161020 (1973) [ Radiophys. Quantum Electron. 16, 783-789 (1973)].
  20. L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves," Phys. Rep. 303, 259-370 (1998).
    [CrossRef]
  21. M. Segev, G.C. Valley, B. Chosignani, P.D Portp and A. Yariv, "steady state spatial screening solitons in photorefractive material with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
    [CrossRef] [PubMed]
  22. D.N. Christodoulides and M.I. Carvalho, "Bright, dark and gray spatial soltion states in photorefractive media," J. Opt. Soc. Am B 12, 1628-1633 (1995).
    [CrossRef]
  23. P.K. Kaw, K. Nishikawa, Y. Yoshida and A. Hasagawa, "Two-Dimetional and Three-Dimentional Envelope Solitons," Phys. Rev. Lett. 3588-91 (1975).
    [CrossRef]
  24. J.Z. Wilcox and T.J. Wilcox, "Stability of Localized Plasma in Two and Three Dimentions," Phys. Rev. Lett. 34, 1160-1163 (1975)
    [CrossRef]

2007 (2)

P.J. Reece, E.M. Wright and K. Dholakia, " Experimnetal Observation of Modulation Instability and Optical Spatial Soliton Arrays in Soft Condensed Matter," Phys. Rev. Lett. 98, 203902:1-4 (2007).
[CrossRef]

R. Gordon, J.T. Blakely and D. Sinton, "Particle-optical self-trapping," Phys. Rev. A. 75055801:1-4 (2007).
[CrossRef]

2006 (1)

C. Conti, N. Ghofraniha, G. Ruocco and S. Trillo, "Laser Beam Filamentation in Fractal Aggregates," Phys. Rev. Lett. 97, 123903:1-4 (2006).
[CrossRef]

2005 (2)

V.E. Yashin, S.A. Chizhov, R.L. Sabirov, T.V. Starchikova, N.V. Vysotina, N.N. Rozanov, V.E. Semenov, V.A. Smirnov and S.V. Fedorov, "Formation of Soliton-like Light Beams in an Aqueous Suspension of Polystyrene Particles," Opt Spectrosc+  98, 466-469 (2005).
[CrossRef]

C. Conti, G. Ruocco and S. Trillo, "Optical Spatial Solitons in Soft Matter," Phys. Rev. Lett. 95, 183902:1-4 (2005).
[CrossRef]

1998 (1)

L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves," Phys. Rep. 303, 259-370 (1998).
[CrossRef]

1995 (1)

D.N. Christodoulides and M.I. Carvalho, "Bright, dark and gray spatial soltion states in photorefractive media," J. Opt. Soc. Am B 12, 1628-1633 (1995).
[CrossRef]

1994 (1)

M. Segev, G.C. Valley, B. Chosignani, P.D Portp and A. Yariv, "steady state spatial screening solitons in photorefractive material with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

1986 (2)

1985 (1)

D. Rogovin and S.O. Sari, "Phase conjugation in liquid suspenstions of microspheres in the diffusive limit," Phys. Rev. A 31, 2375-2389 (1985).
[CrossRef] [PubMed]

1982 (1)

1981 (1)

1975 (2)

P.K. Kaw, K. Nishikawa, Y. Yoshida and A. Hasagawa, "Two-Dimetional and Three-Dimentional Envelope Solitons," Phys. Rev. Lett. 3588-91 (1975).
[CrossRef]

J.Z. Wilcox and T.J. Wilcox, "Stability of Localized Plasma in Two and Three Dimentions," Phys. Rev. Lett. 34, 1160-1163 (1975)
[CrossRef]

1973 (2)

J.P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8, 14-21 (1973).
[CrossRef]

N.G. Vakhitov and A.A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Izv. Vyssh. Uchebn. Zaved Radiofiz. 161020 (1973) [ Radiophys. Quantum Electron. 16, 783-789 (1973)].

1906 (1)

J.M.C. Garnett, "Colors in Metal Glasses, in Metallic Films and in Metallic solutions," Philos. Trans. R. Soc. London 205, 237- 288 (1906).
[CrossRef]

1904 (1)

J.M.C. Garnett, "Colors in Metal Glasses and in Metallic Films," Philos. Trans. R. Soc. London 203, 385-420 (1904).
[CrossRef]

Izv. Vyssh. Uchebn. Zaved Radiofiz. (1)

N.G. Vakhitov and A.A. Kolokolov, "Stationary solutions of the wave equation in a medium with nonlinearity saturation," Izv. Vyssh. Uchebn. Zaved Radiofiz. 161020 (1973) [ Radiophys. Quantum Electron. 16, 783-789 (1973)].

J. Opt. Soc. Am B (1)

D.N. Christodoulides and M.I. Carvalho, "Bright, dark and gray spatial soltion states in photorefractive media," J. Opt. Soc. Am B 12, 1628-1633 (1995).
[CrossRef]

Opt Spectrosc (1)

V.E. Yashin, S.A. Chizhov, R.L. Sabirov, T.V. Starchikova, N.V. Vysotina, N.N. Rozanov, V.E. Semenov, V.A. Smirnov and S.V. Fedorov, "Formation of Soliton-like Light Beams in an Aqueous Suspension of Polystyrene Particles," Opt Spectrosc+  98, 466-469 (2005).
[CrossRef]

Opt. Lett. (3)

Philos. Trans. R. Soc. London (2)

J.M.C. Garnett, "Colors in Metal Glasses and in Metallic Films," Philos. Trans. R. Soc. London 203, 385-420 (1904).
[CrossRef]

J.M.C. Garnett, "Colors in Metal Glasses, in Metallic Films and in Metallic solutions," Philos. Trans. R. Soc. London 205, 237- 288 (1906).
[CrossRef]

Phys. Rep. (1)

L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves," Phys. Rep. 303, 259-370 (1998).
[CrossRef]

Phys. Rev. A (2)

D. Rogovin and S.O. Sari, "Phase conjugation in liquid suspenstions of microspheres in the diffusive limit," Phys. Rev. A 31, 2375-2389 (1985).
[CrossRef] [PubMed]

J.P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8, 14-21 (1973).
[CrossRef]

Phys. Rev. A. (1)

R. Gordon, J.T. Blakely and D. Sinton, "Particle-optical self-trapping," Phys. Rev. A. 75055801:1-4 (2007).
[CrossRef]

Phys. Rev. Lett. (6)

P.K. Kaw, K. Nishikawa, Y. Yoshida and A. Hasagawa, "Two-Dimetional and Three-Dimentional Envelope Solitons," Phys. Rev. Lett. 3588-91 (1975).
[CrossRef]

J.Z. Wilcox and T.J. Wilcox, "Stability of Localized Plasma in Two and Three Dimentions," Phys. Rev. Lett. 34, 1160-1163 (1975)
[CrossRef]

P.J. Reece, E.M. Wright and K. Dholakia, " Experimnetal Observation of Modulation Instability and Optical Spatial Soliton Arrays in Soft Condensed Matter," Phys. Rev. Lett. 98, 203902:1-4 (2007).
[CrossRef]

C. Conti, G. Ruocco and S. Trillo, "Optical Spatial Solitons in Soft Matter," Phys. Rev. Lett. 95, 183902:1-4 (2005).
[CrossRef]

C. Conti, N. Ghofraniha, G. Ruocco and S. Trillo, "Laser Beam Filamentation in Fractal Aggregates," Phys. Rev. Lett. 97, 123903:1-4 (2006).
[CrossRef]

M. Segev, G.C. Valley, B. Chosignani, P.D Portp and A. Yariv, "steady state spatial screening solitons in photorefractive material with external applied field," Phys. Rev. Lett. 73, 3211-3214 (1994).
[CrossRef] [PubMed]

Rev. Mod. Phys. (1)

S. Stenholm, "The semiclassical theory of laser cooling," Rev. Mod. Phys. 58, 699-739 (1986).
[CrossRef]

Other (5)

G. G. Hammes, Thermodynamics and Kinetics for the Biological Sciences, (John Wiley and Sons 2000).

H.C. van de Hulst, Light Scattering by Small Particles, (Dover Publication, Inc. New York 1981).

R. El-Ganainy, K. G. Makris, D. N. Christodoulides, C. Rothchild, and M. Segev, "Cusp solitons in exponentially nonlinear nanosuspensions", paper QMB4, CLEO/QELS 2007, May 6-11, Baltimore, Maryland.

B.J. berne and R. Pecora, Dynamic Light Scattering: With Applications to Chemistry, Biology and Physics (Dover Publication, Inc. New York 2000)

J.D. Jackson, Classical Electrodynamics, (John Wiley and Sons, New York 1999).

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Figures (7)

Fig. 1.
Fig. 1.

A high intensity beam (a) attracting nanoparticles with positive polarizabilities and (b) repelling nanoparticles with negative polarizabilities.

Fig. 2.
Fig. 2.

Normalized soliton FWHM width as a function of their normalized peak intensities in exponential nonlinear nanosuspentions. The inset represents the intensity profile of such a solution. (b) The corresponding P-κ diagram with S being the stable and U the unstable branch.

Fig. 3.
Fig. 3.

a) Linear propagation of an optical beam in water-polystyrene nanosuspension. (b) Nonlinear soliton effects in this same system.

Fig. 4.
Fig. 4.

a) Normalized soliton intensity FWHM width as a function of their peak intensity in exponentially saturable nanosuspentions. The inset depicts such a solution (b) Corresponding P-κ diagram indicating stability.

Fig. 5.
Fig. 5.

Linear propagation of a 10 µm beam in water-air nanobubble suspenstions where 97% of losses are expected. (b) Nonlinear soliton self-trapping and self-induced transparency effects.

Fig. 6.
Fig. 6.

a) P-κ stability diagram of 2D solitons in nanosuspentions with positive polarizabilities. (b) Soliton intensity profile at κ=1.7. (c) Beam diffraction at low power levels shown in scale. (d) Propagation dynamics of a 2D 10 µm soliton beam after 1 mm in the presence of Rayleigh losses. (e) Catastrophic collapse in the absence of nonlinear losses.

Fig. 7.
Fig. 7.

a) P-κ stability diagram of 2D solitons in nanosuspentions with negative polarizabilities. (b) Soliton intensity profile at κ=0.5. (c) Expantion and loss effects during linear propagation of a 2D 10 µm beam after 3.5 mm (d) Self-trapping and self-induced transperancy effects at 6 W of beam power in this same system.

Equations (26)

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ρ t + · J = 0 ,
J = ρ ν D ρ ,
ρ t + · ( ρ ν D ρ ) = 0 .
F = α 4 I .
α = 3 V p ε 0 n b 2 ( m 2 1 m 2 + 2 ) ,
ρ = ρ 0 exp ( α 4 k B T I ) ,
n eff 2 = n b 2 n p 2 + 2 n b 2 + 2 f ( n p 2 n b 2 ) n p 2 + 2 n b 2 f ( n p 2 n b 2 ) .
n eff 2 = n b 2 ( 1 + 2 f n p n b n b ) .
Δ n NL = n eff ( I ) n eff ( I = 0 ) = ( n p n b ) V p ρ 0 ( e α 4 k B T I 1 ) .
σ = 128 π 5 a 2 n b 4 3 ( a λ 0 ) 4 ( m 2 1 m 2 + 2 ) 2 ,
i φ z + 1 2 k 0 n b 2 φ + k 0 ( n p n b ) f φ + i γ φ = 0 ,
i φ z + 1 2 k 0 n b 2 φ + k 0 ( n p n b ) V p ρ 0 e α 4 k B T φ 2 φ + i 2 σ ρ 0 e α 4 k B T φ 2 φ = 0 .
i U ξ + U XX + U YY + e U 2 U + i δ e U 2 U = 0 , ( for n p > n b ) .
i u ξ + u XX + u YY e u 2 u + i δ e u 2 u = 0 , ( for n p < n b )
i U ξ + U XX + U YY + ( 1 e U 2 ) U + i δ e U 2 U = 0 , ( for n p < n b ) .
g XX κ g + e g 2 g = 0 ,
g XX κ g + ( 1 e g 2 ) g = 0
g X 2 κ g 2 + e g 2 = C 1
g X 2 + ( 1 κ ) g 2 + e g 2 = C 2
g 0 g dg 1 + κ g 2 exp ( g 2 ) = ± 0 X dX
g 0 g dg 1 + ( κ 1 ) g 2 exp ( g 2 ) = ± 0 X dX .
J = j J j = j ( ρ j ν j D j ρ j ) ,
ρ j = ρ j 0 exp ( α j 4 k B T I ) .
i U ξ + U XX + U YY + j = 1 , 2 , 3 , [ ( n pj + n b ) V j + ρ jo + ( n p 1 + n b ) V 1 + ρ 1 o + e α j + α 1 + U 2 U + ( n b n pj ) V j ρ jo ( n p 1 + n b ) V 1 + ρ 1 o + ( 1 e α j α 1 + U 2 ) U ]
+ j = 1 , 2 , 3 , i [ δ j + δ 1 + e α j + α 1 + U 2 + δ j δ 1 + e α j α 1 + U 2 U = 0 .
i U ξ + U XX + U YY + 2 sinh ( U 2 ) U = 0 .

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