## Soliton dynamics and self-induced transparency in nonlinear nanosuspensions

Optics Express, Vol. 15, Issue 16, pp. 10207-10218 (2007)

http://dx.doi.org/10.1364/OE.15.010207

Acrobat PDF (930 KB)

### 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

## 1. Introduction

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]

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]

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]

8. S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. **58**, 699–739 (1986). [CrossRef]

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]

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]

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]

12. R. Gordon, J.T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A. **75**055801:1–4 (2007). [CrossRef]

12. R. Gordon, J.T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A. **75**055801:1–4 (2007). [CrossRef]

**6**, 284–286 (1981). [CrossRef] [PubMed]

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]

## 2. Theoretical analysis

*ρ*represents the particle concentration and

*J*⃗ is the particle current density. In these systems, the mechanisms contributing to the particle current density are described by the Nernst-Planck equation [14]:

*∂/∂t*=0. In addition, under equilibrium the current density is zero,

*J*⃗=0, i.e. drift is balanced by diffusion. In the case where the particle size is small compared to the wavelength (Rayleigh regime), the average optical gradient force on this nanoparticle can be obtained within the dipole approximation [6

6. J.P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A **8**, 14–21 (1973). [CrossRef]

8. S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. **58**, 699–739 (1986). [CrossRef]

*α*represents the particle polarizability and the quantity

*I*=

*E⃗·E⃗** is associated with the light intensity through the peak spatial field amplitude

*E*⃗. In the dipole approximation, the polarizability

*α*of a spherical particle having a refractive index

*n*is given by [15]:

_{p}*V*=4

_{p}*πa*is the volume of the particle,

^{3}/3*ε*is the free space permittivity,

_{0}*n*is the refractive index of the background medium and the dimensionless parameter

_{b}*m*=

*n*represents the ratio of the particle’s refractive index

_{p}/n_{b}*n*to

_{p}*n*. It is important to note that

_{b}*α*can be positive or negative depending on whether the refractive index of the particle is higher (

*m*>1) or lower (

*m*<1) than that of the background.

*I=0*),

*F*⃗=

*v*⃗=

*0*, the particle density obeys Laplace’s equation ∇

^{2}

*ρ*=0 under steady state conditions. Moreover, at the container boundaries, the normal component of the diffusion current ∇

*ρ*is zero. In this case, this Newman boundary value problem dictates that the particle density is everywhere constant. This uniform distribution is also the one that leads to maximum entropy (in the absence of external constraints).

*J*⃗=0) leads to (

*αµ*/4)

*ρ∇I*-

*D∇ρ*=0. This last partial differential equation can be directly integrated and gives

*µ/D*=1/

*k*we finally obtain:

_{B}T*k*is the thermal energy and

_{B}T*ρ*stands for the unperturbed uniform particle density (in the absence of light-when the container is large). Given the fact that

_{0}*f(I)*=

*f*(0)exp(

*αI/4k*). Moreover it is important to emphasize that this exponential law is only applicable in the case of relatively low concentrations (or filling factors) since the diffusion equation itself ignores particle-particle interactions. As we will see, in most typical cases Rayleigh scattering losses naturally provide an upper bound on particle concentration.

_{B}T*n*is higher than that of the background

_{p}*n*(

_{b}*α*>0). The converse it true in the other regime (

*α*<0), i.e. the particles will escape from the high intensity regions when their refractive index is lower than that of the surrounding medium. As a result the refractive index is locally perturbed due to this intensity dependent change in the particle concentration. To calculate this local index change we use the Maxwell-Garnett formula given by [16

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]

*n*is the effective refractive index of the medium and

_{eff}*f*is the volume filling factor given by the ratio of the volume of the particles to the total volume. If we expand the right hand side of Eq. (7) and by assuming a relatively small index contrast (i.e. |

*m*-1| being small) we get:

*n*=(1-

_{eff}*f*)

*n*+

_{b}*fn*. This result could have been intuitively anticipated based on fractional composition arguments. The change in the refractive index is then given by Δ

_{p}*n*=

*n*-

_{eff}*n*=(

_{b}*n*)

_{p}-n_{b}*f*where the particle volume filling factor is defined as:

*f*=(Δ

*N*/Δ

_{p}*V*)

*V*. This together with Eq. (6) provides the optical nonlinearity of such nanoparticle suspensions [11,12

_{p}=ρV_{p}12. R. Gordon, J.T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A. **75**055801:1–4 (2007). [CrossRef]

*λ*, the scattering cross section can be determined in the Rayleigh regime [18], that is:

_{0}*a*is the particle radius.

^{2}

*E+k*=0, and by assuming a slowly varying field envelope

^{2}_{0}n^{2}_{eff}E*φ(x, y, z)*, that is,

*E(x, y, z)*=

*φ(x, y, z)*exp

*(ik*, we find that:

_{0}n_{b}z)*γ*in the last term represents the loss coefficient and

*k*=

_{0}*2π*/

*λ*. If we now keep only the loss term in the last equation, we find that |

_{0}*φ*|

^{2}=|

*φ*|

_{0}^{2}exp(-2

*γz*). Given that |

*φ*|

^{2}=|

*φ*|2 exp (-

_{0}*α*) where the loss coefficient is given by

_{l}z*α*we finally obtain 2

_{l}=σ_{ρ}*γ*=

*σρ=σρ*exp(

_{0}*αI/4k*). This final expression for the loss coefficient is important since it demonstrates that the scattering losses are actually nonlinear, i.e. they depend on the beam intensity. As will be shown later, these nonlinear Rayleigh losses will play a crucial role in the beam propagation dynamics. From these latter results, Eq. (11) takes the form:

_{B}T*α*is positive or negative. If we first consider the case of positive polarizability and by introducing the following normalizations,

*ξ=z/2k*, Eq.(12) takes the form:

_{0}n_{b}w^{2}, X=x/w, Y=y/w, w^{-2}=2k^{2}_{0}n_{b}|n_{p}-n_{b}|V_{p}ρ0 φ=(4k_{B}T/|α|)^{1/2}U*U*is the normalized field amplitude, w is a characteristic beam width, and the normalized loss

*δ*is given by

*δ=σ*/(2

*k*). In the same manner, if the polarizability is negative, we obtain

_{0}|n_{p}-n_{b}|V_{p}*u*instead of

*U*. If now introduce the transformation

*u=Ue-iξ*in equation (14) we obtain:

*α*of each particle is positive and thus the particles are attracted toward the high intensity region, i.e. to the center of the beam, thus elevating the effective refractive index of the system (Fig. 1(a)). This will of course increase the nonlinear scattering losses as well. On the other hand, particles having a lower refractive index than that of the background and hence a negative polarizability will be repelled away from the center of the beam, again raising the refractive index at the center (Fig. 1(b)). In this latter case however, the nonlinear losses decrease at the beam center (due to the reduction in the particle concentration), thus increasing the transparency of the system. As it will be shown, this difference in the character of the exponential optical nonlinearity will have a profound effect on the beam dynamics of spatial solitons.

## 3. Soliton dynamics and stability properties

*δ*=0).

### 3.1 1D soliton solutions

*U(X, ξ)=g(X)exp(iκξ)*where

*κ*represents the soliton eigenvalue. Substituting this latter expression into Eqs. (13) and (15) gives:

*g*|

*X*→∞=0 and

*C*

_{1}=

*C*

_{2}=1. By rearranging Eqs.(17) we directly obtain:

*X*=0. Using the boundary conditions at the beam center, namely that

*g(0)*=

*g*and

_{0}*g*(0)=0 we can now numerically integrate Eqs. (18).

_{X}*κ*=3. As one can see, the beam width monotonically decreases as the soliton peak intensity increases. The stability properties of this class of solutions can be systematically examined using the power-eigenvalue (

*P-κ*) diagram where

*P*=∫|

*U*|

^{2}

*dX*. Following Vakhitov and Kolokolov [19], this solution is stable whenever the slope of the curve is positive (for

*κ*<2.49) and is unstable for higher eigenvalues where the slope is negative as shown in Fig. 2(b).

20. L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. **303**, 259–370 (1998). [CrossRef]

*κ*=2.49) these 1D solutions tend to catastrophically collapse into a singularity. This behavior can be qualitatively explained based on the Taylor series expansion

*U*|

^{4}) necessary for 1D systems to exhibit collapse [20

20. L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. **303**, 259–370 (1998). [CrossRef]

*µm*. The nano-suspensions involve polystyrene nano-particles (refractive index

*n*=1.56) of radius 50

_{p}*nm*suspended in water (

*n*=1.33) at a concentration of 7×10

_{b}^{11}cm

^{-3}(or

*f*=3.5×10

^{-4}). Under linear conditions, a 10 micron beam (FWHM in width) expands considerably because of diffraction (3 times) and loses 13% of its power because of scattering losses. Conversely, in the nonlinear regime, this same beam can propagate up to 4 diffraction lengths (2

*mm*) without any appreciable distortion-limited only by the nonlinear losses (20%), as clearly shown in Fig. 3.

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]

*κ*=0.95. The power-eigenvalue

*P-κ*diagram in this regime is shown in Fig. 4(b). This graph indicates that in the saturable case (of negative polarizabilities) the 1D soliton solutions are always stable since

*dP/dκ*>0.

*nm*air nano-bubbles (

*n*=1) floating in water (

_{p}*n*=1.33). Again the wavelength is taken to be 0.532

_{b}*µm*. The nano-bubble concentration is assumed to be 2×10

^{12}

*cm*

^{-3}or

*f*=10

^{-3}. Under linear or low power conditions, the beam diffracts considerably (more than 10 times) and loses almost all its energy (97%) as clearly indicated in Fig. 5(a).

*self-induced transparency*and self-trapping effects. In other words, at high powers the beam can effectively reduce the “haze” while at the same time can establish its own waveguide structure In this case the overall losses drop from 97% to 20%. A direct simulation of this beam (based on Eq. (15)) shows that the soliton (10 micron FWHM) can propagate up to 12 diffraction lengths (Fig. 5(b)), i.e. approximately 4 times more than in the first case. This behavior is understood by recalling that the nonlinear losses are exponentially growing in the first case whereas are exponentially decaying in the second one.

### 3.2 2D soliton solutions

*u(r,ξ)=g(r)exp(iκξ)*and we keep in mind the boundary conditions

*g*=0 at

*r*→∞ and

*dg/dr*=0 at

*r*=0.

*P-κ*) diagram associated with these solutions is shown in Fig. 6(a). The monotonically decreasing behavior of this latter curve clearly indicates that the 2D soliton solutions in exponentially nonlinear nanosuspensions are always unstable and tend to catastrophically collapse. Again as in the 1D case, collapse will be prevented because of nonlinear Rayleigh scattering and/or saturation effects in the particle concentration. In addition, the intensity profiles of this class of waves exhibit a cusp-like shape as a result of the exponential nonlinearity, as shown in Fig. 6(b). As an example we study the propagation of a 10 micron width (FWHM) 2D beam in water containing polystyrene nanospheres. All the physical parameters are the same as those used in the corresponding 1D system except for the volume filling factor which taken here to be

*f*=10

^{-4}. At low power levels the beam expands because of diffraction (2 times) and loses 2% of its power as a result of Rayleigh scattering, as shown in Fig. 6(c). On the other hand, at 5W (at soliton power), this same beam can propagate up to 1

*mm*(3.5 diffraction lengths) without any appreciable expansion and in spite of the nonlinear Rayleigh losses (5%) as accounted in Eq.(13), as demonstrated in Fig. 6(d). We note that in our simulations this beam would have otherwise undergone a collapse had not been for Rayleigh scattering. This collapse behavior is illustrated in Fig. 6(e) in the absence of nonlinear losses and by neglecting saturation effects in the particle density.

23. P.K. Kaw, K. Nishikawa, Y. Yoshida, and A. Hasagawa, “Two-Dimetional and Three-Dimentional Envelope Solitons,” Phys. Rev. Lett. **35**88–91 (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]

*P-κ*) stability diagram associated with these solutions is shown in Fig. 7(a) and indicates that these self-trapped states are always stable since

*dP/dκ*>0.

*f*=10

^{-3}. Fig. 7(b) shows the intensity profile of a 10

*µm*beam (FWHM) that is possible in this system at a power level of 2.8 Watts. At very low intensities, after 3.5

*mm*of propagation, the beam linearly diffracts (7 times) and loses 80% of its power because of substantial Rayleigh scattering, Fig. 7(c). On the other hand, when the beam input power is 6 W, the beam self-traps and at the same time increases the transparency of the system by optically expelling the nano-particles from its center. In this latter case, the beam expands only by 10% and loses a small fraction of energy (20%) after 3.5

*mm*of propagation as shown in Fig. 7(d). As in the 1D case, this self-induced transparency effect is again a result of the specific nature of the optical gradient force.

## 4. Engineering nonlinearities in nano-suspension systems

*j*runs over all different kinds of nano-particles. In the case of diluted suspensions particle-particle interactions can be neglected. As a result each current component

*J*⃗j vanishes independently and the statistical distribution for each type of nano-particles is represented by a Boltzmann distribution, i.e.

## 5. Conclusion

## References and links

1. | G. G. Hammes, |

2. | P.W. Smith, A. Ashkin, and W.J. Tomlinson, “Four-wave mixing in an artificial Kerr medium,” Opt. Lett. |

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. |

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+ |

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. |

6. | J.P. Gordon, “Radiation Forces and Momenta in Dielectric Media,” Phys. Rev. A |

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. |

8. | S. Stenholm, “The semiclassical theory of laser cooling,” Rev. Mod. Phys. |

9. | C. Conti, G. Ruocco, and S. Trillo, “Optical Spatial Solitons in Soft Matter,” Phys. Rev. Lett. |

10. | C. Conti, N. Ghofraniha, G. Ruocco, and S. Trillo, “Laser Beam Filamentation in Fractal Aggregates,” Phys. Rev. Lett. |

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. |

13. | D. Rogovin and S.O. Sari, “Phase conjugation in liquid suspenstions of microspheres in the diffusive limit,” Phys. Rev. A |

14. | B.J. berne and R. Pecora, |

15. | J.D. Jackson, |

16. | J.M.C. Garnett, “Colors in Metal Glasses and in Metallic Films,” Philos. Trans. R. Soc. London |

17. | J.M.C. Garnett, “Colors in Metal Glasses, in Metallic Films and in Metallic solutions,” Philos. Trans. R. Soc. London |

18. | H.C. van de Hulst, |

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. 16 1020 (1973) [Radiophys. Quantum Electron. |

20. | L. Berge, “Wave collapse in physics: principles and applications to light and plasma waves,” Phys. Rep. |

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. |

22. | D.N. Christodoulides and M.I. Carvalho, “Bright, dark and gray spatial soltion states in photorefractive media,” J. Opt. Soc. Am B |

23. | P.K. Kaw, K. Nishikawa, Y. Yoshida, and A. Hasagawa, “Two-Dimetional and Three-Dimentional Envelope Solitons,” Phys. Rev. Lett. |

24. | J.Z. Wilcox and T.J. Wilcox, “Stability of Localized Plasma in Two and Three Dimentions,” Phys. Rev. Lett. |

**OCIS Codes**

(190.3970) Nonlinear optics : Microparticle nonlinear optics

(190.5940) Nonlinear optics : Self-action effects

(290.5870) Scattering : Scattering, Rayleigh

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: June 5, 2007

Revised Manuscript: June 28, 2007

Manuscript Accepted: June 28, 2007

Published: July 27, 2007

**Citation**

R. El-Ganainy, D. N. Christodoulides, C. Rotschild, and M. Segev, "Soliton dynamics and self-induced transparency in nonlinear nanosuspensions," Opt. Express **15**, 10207-10218 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-16-10207

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### References

- G. G. Hammes, Thermodynamics and Kinetics for the Biological Sciences, (John Wiley and Sons 2000).
- 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]
- 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]
- 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]
- 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]
- J.P. Gordon, "Radiation Forces and Momenta in Dielectric Media," Phys. Rev. A 8, 14-21 (1973). [CrossRef]
- 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]
- S. Stenholm, "The semiclassical theory of laser cooling," Rev. Mod. Phys. 58, 699-739 (1986). [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]
- 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.
- R. Gordon, J.T. Blakely and D. Sinton, "Particle-optical self-trapping," Phys. Rev. A. 75055801:1-4 (2007). [CrossRef]
- 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]
- 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).
- 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]
- H.C. van de Hulst, Light Scattering by Small Particles, (Dover Publication, Inc. New York 1981).
- 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)].
- L. Berge, "Wave collapse in physics: principles and applications to light and plasma waves," Phys. Rep. 303, 259-370 (1998). [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]
- 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]
- 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]

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