Abstract

We experimentally probed the nonlinear optical response of aqueous nano-colloidal suspensions to provide a test of the theoretical approaches that have been proposed for the nonlinearity, namely an exponential model, an artificial Kerr medium, and a non-ideal gas model. The best agreement with experiment is found using the non-ideal gas model for the colloidal suspension which in turn can be used to infer values for the second virial coefficient of the medium and the nonlinear coefficients.

© 2009 OSA

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  1. P. W. Smith, P. J. Maloney, and A. Ashkin, “Use of a liquid suspension of dielectric spheres as an artificial Kerr medium,” Opt. Lett. 7(8), 347–349 (1982).
    [CrossRef] [PubMed]
  2. P. W. Smith, A. Ashkin, and W. J. Tomlinson, “Four-wave mixing in an artificial Kerr medium,” Opt. Lett. 6(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(6), 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(3), 511–514 (2005).
    [CrossRef]
  5. P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett. 98(20), 203902 (2007).
    [CrossRef] [PubMed]
  6. C. Conti, G. Ruocco, and S. Trillo, “Optical spatial solitons in soft matter,” Phys. Rev. Lett. 95(18), 183902 (2005).
    [CrossRef] [PubMed]
  7. R. El-Ganainy, D. N. Christodoulides, C. Rotschild, and M. Segev, “Soliton dynamics and self-induced transparency in nonlinear nanosuspensions,” Opt. Express 15(16), 10207–10218 (2007).
    [CrossRef] [PubMed]
  8. R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, C. Rotschild, and M. Segev, “Optical beam instabilities in nonlinear nanosuspensions,” Opt. Lett. 32(21), 3185–3187 (2007).
    [CrossRef] [PubMed]
  9. R. Gordon, J. T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A 75(5), 055801 (2007).
    [CrossRef]
  10. M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Spatial solitons and light-induced instabilities in colloidal media,” Opt. Express 16(2), 1371–1376 (2008).
    [CrossRef] [PubMed]
  11. M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79(2), 023814 (2009).
    [CrossRef]
  12. “Critical collapse beyond a self-focusing power occurs for the case sd = 4, where s is the order of the nonlinearity, s = 2 for a Kerr medium, and d the number of transverse dimensions. Super-critical collapse occurs for sd>4 and tends to be much more abrupt spatially than critical collapse, see, for example, N. E. Kosmatov, V. F. Shvets, and V. E. Zakharov, “Computer simulation of wave collapses in the nonlinear Schrodinger equation,” Physica D 52, 16–35 (1991).
  13. M. Sheik-bahae, A. A. Said, and E. W. Van Stryland, “High-sensitivity, single-beam n2 measurements,” Opt. Lett. 14(17), 955–957 (1989).
    [CrossRef] [PubMed]
  14. P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005).
    [CrossRef] [PubMed]
  15. R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in non-ideal gases of interacting colloidal nano-particles,” to be submitted (2009).
  16. J. Junio, E. Blanton, and H. D. Ou-Yang, “The Kerr effect produced by optical trapping of nanoparticles in aqueous suspension,” Optical Trapping and Optical Micromanipulation IV. Edited by Dholakia, Kishan; Spalding, Gabriel C. Proceedings of the SPIE, Volume 6644, pp. 664408 (2007).
  17. D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22(1), 105 (1979).
    [CrossRef]
  18. E. M. Wright, B. L. Lawrence, W. Torruellas, and G. I. Stegeman, “Stable self-trapping and ring formation in PTS,” Opt. Lett. 20, 2481–2483 (1995).
    [CrossRef] [PubMed]

2009

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79(2), 023814 (2009).
[CrossRef]

2008

2007

P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett. 98(20), 203902 (2007).
[CrossRef] [PubMed]

R. El-Ganainy, D. N. Christodoulides, C. Rotschild, and M. Segev, “Soliton dynamics and self-induced transparency in nonlinear nanosuspensions,” Opt. Express 15(16), 10207–10218 (2007).
[CrossRef] [PubMed]

R. El-Ganainy, D. N. Christodoulides, Z. H. Musslimani, C. Rotschild, and M. Segev, “Optical beam instabilities in nonlinear nanosuspensions,” Opt. Lett. 32(21), 3185–3187 (2007).
[CrossRef] [PubMed]

R. Gordon, J. T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A 75(5), 055801 (2007).
[CrossRef]

2005

C. Conti, G. Ruocco, and S. Trillo, “Optical spatial solitons in soft matter,” Phys. Rev. Lett. 95(18), 183902 (2005).
[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(3), 511–514 (2005).
[CrossRef]

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005).
[CrossRef] [PubMed]

1995

1989

1982

1981

1979

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22(1), 105 (1979).
[CrossRef]

Anderson, D.

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22(1), 105 (1979).
[CrossRef]

Ashkin, A.

Bhatia, V. K.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005).
[CrossRef] [PubMed]

Blakely, J. T.

R. Gordon, J. T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A 75(5), 055801 (2007).
[CrossRef]

Bonnedal, M.

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22(1), 105 (1979).
[CrossRef]

Chizhov, S. A.

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(3), 511–514 (2005).
[CrossRef]

Christodoulides, D. N.

Conti, C.

C. Conti, G. Ruocco, and S. Trillo, “Optical spatial solitons in soft matter,” Phys. Rev. Lett. 95(18), 183902 (2005).
[CrossRef] [PubMed]

Dholakia, K.

P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett. 98(20), 203902 (2007).
[CrossRef] [PubMed]

Dziedzic, J. M.

El-Ganainy, R.

Fedorov, S. V.

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(3), 511–514 (2005).
[CrossRef]

Gordon, R.

R. Gordon, J. T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A 75(5), 055801 (2007).
[CrossRef]

Hansen, P. M.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005).
[CrossRef] [PubMed]

Harrit, N.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005).
[CrossRef] [PubMed]

Kivshar, Y. S.

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79(2), 023814 (2009).
[CrossRef]

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Spatial solitons and light-induced instabilities in colloidal media,” Opt. Express 16(2), 1371–1376 (2008).
[CrossRef] [PubMed]

Krolikowski, W.

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79(2), 023814 (2009).
[CrossRef]

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Spatial solitons and light-induced instabilities in colloidal media,” Opt. Express 16(2), 1371–1376 (2008).
[CrossRef] [PubMed]

Lawrence, B. L.

Maloney, P. J.

Matuszewski, M.

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79(2), 023814 (2009).
[CrossRef]

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Spatial solitons and light-induced instabilities in colloidal media,” Opt. Express 16(2), 1371–1376 (2008).
[CrossRef] [PubMed]

Musslimani, Z. H.

Oddershede, L.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005).
[CrossRef] [PubMed]

Reece, P. J.

P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett. 98(20), 203902 (2007).
[CrossRef] [PubMed]

Rotschild, C.

Rozanov, N. N.

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(3), 511–514 (2005).
[CrossRef]

Ruocco, G.

C. Conti, G. Ruocco, and S. Trillo, “Optical spatial solitons in soft matter,” Phys. Rev. Lett. 95(18), 183902 (2005).
[CrossRef] [PubMed]

Sabirov, R. L.

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(3), 511–514 (2005).
[CrossRef]

Said, A. A.

Segev, M.

Semenov, V. E.

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(3), 511–514 (2005).
[CrossRef]

Sheik-bahae, M.

Sinton, D.

R. Gordon, J. T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A 75(5), 055801 (2007).
[CrossRef]

Smirnov, V. A.

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(3), 511–514 (2005).
[CrossRef]

Smith, P. W.

Starchikova, T. V.

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(3), 511–514 (2005).
[CrossRef]

Stegeman, G. I.

Tomlinson, W. J.

Torruellas, W.

Trillo, S.

C. Conti, G. Ruocco, and S. Trillo, “Optical spatial solitons in soft matter,” Phys. Rev. Lett. 95(18), 183902 (2005).
[CrossRef] [PubMed]

Van Stryland, E. W.

Vysotina, N. V.

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(3), 511–514 (2005).
[CrossRef]

Wright, E. M.

P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett. 98(20), 203902 (2007).
[CrossRef] [PubMed]

E. M. Wright, B. L. Lawrence, W. Torruellas, and G. I. Stegeman, “Stable self-trapping and ring formation in PTS,” Opt. Lett. 20, 2481–2483 (1995).
[CrossRef] [PubMed]

Yashin, V. E.

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(3), 511–514 (2005).
[CrossRef]

Nano Lett.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, “Expanding the optical trapping range of gold nanoparticles,” Nano Lett. 5(10), 1937–1942 (2005).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Opt. Spectrosc.

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(3), 511–514 (2005).
[CrossRef]

Phys. Fluids

D. Anderson and M. Bonnedal, “Variational approach to nonlinear self-focusing of Gaussian laser beams,” Phys. Fluids 22(1), 105 (1979).
[CrossRef]

Phys. Rev. A

R. Gordon, J. T. Blakely, and D. Sinton, “Particle-optical self-trapping,” Phys. Rev. A 75(5), 055801 (2007).
[CrossRef]

M. Matuszewski, W. Krolikowski, and Y. S. Kivshar, “Soliton interactions and transformations in colloidal media,” Phys. Rev. A 79(2), 023814 (2009).
[CrossRef]

Phys. Rev. Lett.

P. J. Reece, E. M. Wright, and K. Dholakia, “Experimental observation of modulation instability and optical spatial soliton arrays in soft condensed matter,” Phys. Rev. Lett. 98(20), 203902 (2007).
[CrossRef] [PubMed]

C. Conti, G. Ruocco, and S. Trillo, “Optical spatial solitons in soft matter,” Phys. Rev. Lett. 95(18), 183902 (2005).
[CrossRef] [PubMed]

Other

“Critical collapse beyond a self-focusing power occurs for the case sd = 4, where s is the order of the nonlinearity, s = 2 for a Kerr medium, and d the number of transverse dimensions. Super-critical collapse occurs for sd>4 and tends to be much more abrupt spatially than critical collapse, see, for example, N. E. Kosmatov, V. F. Shvets, and V. E. Zakharov, “Computer simulation of wave collapses in the nonlinear Schrodinger equation,” Physica D 52, 16–35 (1991).

R. El-Ganainy, D. N. Christodoulides, E. M. Wright, W. M. Lee, and K. Dholakia, “Nonlinear optical dynamics in non-ideal gases of interacting colloidal nano-particles,” to be submitted (2009).

J. Junio, E. Blanton, and H. D. Ou-Yang, “The Kerr effect produced by optical trapping of nanoparticles in aqueous suspension,” Optical Trapping and Optical Micromanipulation IV. Edited by Dholakia, Kishan; Spalding, Gabriel C. Proceedings of the SPIE, Volume 6644, pp. 664408 (2007).

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Figures (5)

Fig. 1
Fig. 1

shows the experimental setup. Two oppositely directed and aligned identical single-mode optical fibers (SMF) that are inserted into the colloidal suspension and separated by a distance D, shown in the inset. The input fiber to the left launches a well defined Gaussian beam (wavelength, λ = 1090nm, spot size w0 = 3.4 μm) at variable input power (monitored through the second output from the 50/50 fiber splitter and power meter (PM)) directly into the colloidal suspension, and the collecting fiber to the right is used to measure the power coupled into the same single-mode beam profile after propagating the distance D using a photodetector (PD). An orthogonal differential interference contrast (DIC) imaging (with LP – linear polarizer, NP – Nomarski prism) system is used to detect subtle refractive index variations (from the accumulated nanoparticles) within the sample. An imaging microscope objective (imaging MO) and a tube lens (TL) relay the DIC image onto a high speed digital camera (DC)

Fig. 2
Fig. 2

Plot of power measured at collecting fiber with a photodetector versus the input power for the two concentrations indicated. Insets show the DIC image of two position when input power is 0 mW and 514 mW.

Fig. 3
Fig. 3

shows the variation of the scaled nonlinear index change [Δn(I)/n2KIc1] versus scaled intensity (I/Ic) for (B2/Vp) = 25 using both the non-ideal gas model in Eq. (6) (red line) and the exponential model in Eq. (8) (blue line).

Fig. 4
Fig. 4

shows the power measured by the collecting fiber versus the input power using the artificial Kerr medium model, both powers being normalized to the critical power for self-focusing PcrK. The red line is obtained using the VG model and the blue line is obtained using the BPM.

Fig. 5
Fig. 5

shows the power measured by the collecting fiber versus the input power using VG approach based on the non-ideal gas model. The blue line is for (B2/Vp) = 10, the red line for (B2/Vp) = 20, and the green line for (B2/Vp) = 30.

Equations (13)

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φz=i2k0nb(2x2+2y2)φ+α0(iβ12)rφ     ,
σ=128π5a2nb43(aλ0)4(m21m2+2)2
φ(x,y,0)=2Pinπw02e(x2+y2)/w02       ,
Pcoll=|2πw02dxdye(x2+y2)/w02φ(x,y,D)|2       ,
Δn(I)=(α0βk0)r(I)=(npnb)Vpρ0(1+r'I+r''2I2+....)                                                                                     =Δn(0)+n2I+n4I2+.....     ,
αI4kBT=ln(r)+2(B2Vp)f0(r1)+32(B3Vp2)f02(r21),
pkBT=ρ+B2ρ2+B3ρ3+...ρ(1+B2ρ+B3ρ2),
Δn(I)=(npnb)Vpρ0eI/Icn2KIc+n2KI,
n2=(n2K1+2(B2/Vp)f0+3(B3/Vp2)f02)   ,n4=n2K2Ic(13(B3/Vp2)f02[1+2(B2/Vp)f0+3(B3/Vp2)f02]3)         .
φ(x,y,z)=2Pinπw2(D)eiϑ(z)e((x2+y2)/w2(z)+ik(x2+y2)/2R(z))   ,
d2wdz2=4k2w3(1PinPcr8w029w2(n4Icrn2)(PinPcr)2)   ,   1R(z)=1wdwdz   ,
Pcr=1.8962λ24πnbn2   ,
Pcoll=|4Pinπ2w02w2(D)dxdye(x2+y2)/w02e(x2+y2)/w2(D)+ik(x2+y2)/2R(D)|2=4Pin(w02/w2(D))|1+(w02w2(D))ikw022R(D)|2   ,

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