Abstract

We study the self-action of light in a water suspension of absorbing subwavelength particles. Due to efficient accumulation of the light energy, this medium shows distinct non-linear properties even at moderate radiation power. In particular, by means of interference of two obliquely incident beams, it is possible to create controllable phase and amplitude gratings whose contrast, spatial and temporal parameters depend on the beams’ coherence and power as well as the interference geometry. The grating characteristics are investigated via the beams’ self-diffraction. The main mechanism of the grating formation is shown to be thermal, which leads to the phase grating; a weak amplitude grating also emerges due to the particles’ displacements caused by the light-induced gradient and photophoretic forces. These forces, together with the Brownian motion of the particles, are responsible for the grating dynamics and degradation. The results and approaches can be used for investigation of the thermal relaxation and kinetic processes in liquid suspensions.

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  1. S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10(5), 609–636 (1968).
    [Crossref]
  2. D. A. Hutchins, “Mechanisms of pulsed photoacoustic generation,” Can. J. Phys. 64(9), 1247–1264 (1986).
    [Crossref]
  3. V. E. Gusev and A. A. Karabutov, Laser Optoacoustics (AIP, 1993)
  4. S. M. Avenesyan, V. E. Gusev, and N. I. Zheludev, “Generation deformation waves in the process of photoexcitation and recombination of nonequilibrium carriers in silicon,” Appl. Phys., A Mater. Sci. Process. 40(3), 163–166 (1986).
    [Crossref]
  5. D. N. Auston, D. J. Bradley, A. J. Campillo, K. B. Eisenthal, E. P. Ippen, D. von der Linde, C. V. Shank, and S. L. Shapiro, Ultrashort Light Pulses: Picosecond Technique and Applications (Springer-Verlag, 1977)
  6. S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29(7), 642–647 (1986).
    [Crossref]
  7. V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22(9), 742–756 (1979).
    [Crossref]
  8. O. V. Angelsky, S. B. Yermolenko, C. Yu. Zenkova, and A. O. Angelskaya, “Polarization manifestations of correlation (intrinsic coherence) of optical fields,” Appl. Opt. 47(29), 5492–5499 (2008).
    [PubMed]
  9. O. V. Angelsky, M. P. Gorsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Investigation of optical currents in coherent and partially coherent vector fields,” Opt. Express 19(2), 660–672 (2011).
    [Crossref] [PubMed]
  10. E. V. Ivakin, I. P. Petrovich, and A. S. Rubanov, “Self-diffraction of radiation by light-induced phase gratings,” Sov. J. Quantum Electron. 3(52), (1973).
  11. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
    [Crossref]
  12. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
    [Crossref] [PubMed]
  13. A. Ya. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
    [Crossref]
  14. M. Abramovitz and I. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1964), Vol. 55.
  15. S. H. Simpson and S. Hanna, “Orbital motion of optically trapped particles in Laguerre-Gaussian beams,” J. Opt. Soc. Am. A 27(9), 2061–2071 (2010).
    [Crossref] [PubMed]
  16. A. N. Rubinov, “Physical grounds for biological effect of laser radiation,” J. Phys. D Appl. Phys. 36(19), 2317–2330 (2003).
    [Crossref]
  17. M. Nieto-Vesperinas, J. J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18(11), 11428–11443 (2010).
    [Crossref] [PubMed]
  18. A. Y. Bekshaev, “Subwavelength particles in an inhomogeneous light field: Optical forces associated with the spin and orbital energy flows” // arXiv:1210.5730 [physics.optics] 21 Oct 2012.
  19. A. Y. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
    [Crossref]
  20. A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
    [Crossref]
  21. L. S. Ivlev and Yu. A. Dovgaliuk, Physics of Atmospheric Aerosol Systems (Saint-Petersburg State University, 1999)
  22. V. G. Shvedov, A. S. Desyatnikov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Optical guiding of absorbing nanoclusters in air,” Opt. Express 17(7), 5743–5757 (2009).
    [Crossref] [PubMed]
  23. A. S. Desyatnikov, V. G. Shvedov, A. V. Rode, W. Krolikowski, and Y. S. Kivshar, “Photophoretic manipulation of absorbing aerosol particles with vortex beams: theory versus experiment,” Opt. Express 17(10), 8201–8211 (2009).
    [Crossref]
  24. N. V. Malai, “Effect of motion of the medium on the photophoresis of hot hydrosol particles,” Fluid Dyn. 41(6), 984–991 (2006).
    [Crossref]
  25. C. Y. Soong, W. K. Li, C. H. Liu, and P. Y. Tzeng, “Theoretical analysis for photophoresis of a microscale hydrophobic particle in liquids,” Opt. Express 18(3), 2168–2182 (2010).
    [Crossref] [PubMed]
  26. P. Gerstner, J. Paltakari, and P. A. C. Gane, “Measurement and modelling of heat transfer in paper coating structures,” http://www.tappi.org/Downloads/Conference-Papers/2008/08ADV/08adv26.aspx
  27. R. F. Probstein, Physicochemical Hydrodynamics: An Introduction (Wiley-Interscience, 2003, 2 ed.)
  28. Y. Wada, S. Totoki, M. Watanabe, N. Moriya, Y. Tsunazawa, and H. Shimaoka, “Nanoparticle size analysis with relaxation of induced grating by dielectrophoresis,” Opt. Express 14(12), 5755–5764 (2006).
    [Crossref] [PubMed]

2012 (1)

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[Crossref]

2011 (2)

2010 (3)

2009 (2)

2008 (2)

A. Ya. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

O. V. Angelsky, S. B. Yermolenko, C. Yu. Zenkova, and A. O. Angelskaya, “Polarization manifestations of correlation (intrinsic coherence) of optical fields,” Appl. Opt. 47(29), 5492–5499 (2008).
[PubMed]

2006 (2)

2003 (1)

A. N. Rubinov, “Physical grounds for biological effect of laser radiation,” J. Phys. D Appl. Phys. 36(19), 2317–2330 (2003).
[Crossref]

1986 (4)

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11(5), 288–290 (1986).
[Crossref] [PubMed]

D. A. Hutchins, “Mechanisms of pulsed photoacoustic generation,” Can. J. Phys. 64(9), 1247–1264 (1986).
[Crossref]

S. M. Avenesyan, V. E. Gusev, and N. I. Zheludev, “Generation deformation waves in the process of photoexcitation and recombination of nonequilibrium carriers in silicon,” Appl. Phys., A Mater. Sci. Process. 40(3), 163–166 (1986).
[Crossref]

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29(7), 642–647 (1986).
[Crossref]

1979 (1)

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22(9), 742–756 (1979).
[Crossref]

1973 (1)

E. V. Ivakin, I. P. Petrovich, and A. S. Rubanov, “Self-diffraction of radiation by light-induced phase gratings,” Sov. J. Quantum Electron. 3(52), (1973).

1970 (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[Crossref]

1968 (1)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10(5), 609–636 (1968).
[Crossref]

Akhmanov, S. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29(7), 642–647 (1986).
[Crossref]

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10(5), 609–636 (1968).
[Crossref]

Angelskaya, A. O.

Angelsky, O. V.

Ashkin, A.

Avenesyan, S. M.

S. M. Avenesyan, V. E. Gusev, and N. I. Zheludev, “Generation deformation waves in the process of photoexcitation and recombination of nonequilibrium carriers in silicon,” Appl. Phys., A Mater. Sci. Process. 40(3), 163–166 (1986).
[Crossref]

Bekshaev, A. Y.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[Crossref]

A. Y. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Bekshaev, A. Ya.

A. Ya. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Bjorkholm, J. E.

Bliokh, K.

A. Y. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Chantada, L.

Chirkin, A. S.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29(7), 642–647 (1986).
[Crossref]

Chu, S.

Desyatnikov, A. S.

Dziedzic, J. M.

Gómez-Medina, R.

Gorsky, M. P.

Gusev, V. E.

S. M. Avenesyan, V. E. Gusev, and N. I. Zheludev, “Generation deformation waves in the process of photoexcitation and recombination of nonequilibrium carriers in silicon,” Appl. Phys., A Mater. Sci. Process. 40(3), 163–166 (1986).
[Crossref]

Hanna, S.

Hanson, S. G.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[Crossref]

O. V. Angelsky, M. P. Gorsky, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, “Investigation of optical currents in coherent and partially coherent vector fields,” Opt. Express 19(2), 660–672 (2011).
[Crossref] [PubMed]

Hutchins, D. A.

D. A. Hutchins, “Mechanisms of pulsed photoacoustic generation,” Can. J. Phys. 64(9), 1247–1264 (1986).
[Crossref]

Ivakin, E. V.

E. V. Ivakin, I. P. Petrovich, and A. S. Rubanov, “Self-diffraction of radiation by light-induced phase gratings,” Sov. J. Quantum Electron. 3(52), (1973).

Karamoch, A. I.

A. Ya. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Khokhlov, R. V.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10(5), 609–636 (1968).
[Crossref]

Kivshar, Y. S.

Krolikowski, W.

Kukhtarev, N. V.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22(9), 742–756 (1979).
[Crossref]

Li, W. K.

Liu, C. H.

Maksimyak, A. P.

Maksimyak, P. P.

Malai, N. V.

N. V. Malai, “Effect of motion of the medium on the photophoresis of hot hydrosol particles,” Fluid Dyn. 41(6), 984–991 (2006).
[Crossref]

Moriya, N.

Nieto-Vesperinas, M.

Odulov, S. G.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22(9), 742–756 (1979).
[Crossref]

Petrovich, I. P.

E. V. Ivakin, I. P. Petrovich, and A. S. Rubanov, “Self-diffraction of radiation by light-induced phase gratings,” Sov. J. Quantum Electron. 3(52), (1973).

Rode, A. V.

Rubanov, A. S.

E. V. Ivakin, I. P. Petrovich, and A. S. Rubanov, “Self-diffraction of radiation by light-induced phase gratings,” Sov. J. Quantum Electron. 3(52), (1973).

Rubinov, A. N.

A. N. Rubinov, “Physical grounds for biological effect of laser radiation,” J. Phys. D Appl. Phys. 36(19), 2317–2330 (2003).
[Crossref]

Sáenz, J. J.

Shimaoka, H.

Shvedov, V. G.

Simpson, S. H.

Soong, C. Y.

Soskin, M.

A. Y. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

Soskin, M. S.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22(9), 742–756 (1979).
[Crossref]

Sukhorukov, A. P.

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10(5), 609–636 (1968).
[Crossref]

Totoki, S.

Tsunazawa, Y.

Tzeng, P. Y.

Vinetskii, V. L.

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22(9), 742–756 (1979).
[Crossref]

Vysloukh, V. A.

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29(7), 642–647 (1986).
[Crossref]

Wada, Y.

Watanabe, M.

Yermolenko, S. B.

Zenkova, C. Y.

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[Crossref]

Zenkova, C. Yu.

Zheludev, N. I.

S. M. Avenesyan, V. E. Gusev, and N. I. Zheludev, “Generation deformation waves in the process of photoexcitation and recombination of nonequilibrium carriers in silicon,” Appl. Phys., A Mater. Sci. Process. 40(3), 163–166 (1986).
[Crossref]

Appl. Opt. (1)

Appl. Phys., A Mater. Sci. Process. (1)

S. M. Avenesyan, V. E. Gusev, and N. I. Zheludev, “Generation deformation waves in the process of photoexcitation and recombination of nonequilibrium carriers in silicon,” Appl. Phys., A Mater. Sci. Process. 40(3), 163–166 (1986).
[Crossref]

Can. J. Phys. (1)

D. A. Hutchins, “Mechanisms of pulsed photoacoustic generation,” Can. J. Phys. 64(9), 1247–1264 (1986).
[Crossref]

Fluid Dyn. (1)

N. V. Malai, “Effect of motion of the medium on the photophoresis of hot hydrosol particles,” Fluid Dyn. 41(6), 984–991 (2006).
[Crossref]

J. Opt. (1)

A. Y. Bekshaev, K. Bliokh, and M. Soskin, “Internal flows and energy circulation in light beams,” J. Opt. 13(5), 053001 (2011).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Phys. D Appl. Phys. (1)

A. N. Rubinov, “Physical grounds for biological effect of laser radiation,” J. Phys. D Appl. Phys. 36(19), 2317–2330 (2003).
[Crossref]

Opt. Commun. (1)

A. Ya. Bekshaev and A. I. Karamoch, “Spatial characteristics of vortex light beams produced by diffraction gratings with embedded phase singularity,” Opt. Commun. 281(6), 1366–1374 (2008).
[Crossref]

Opt. Express (6)

Opt. Lett. (1)

Phys. Rev. A (1)

A. Y. Bekshaev, O. V. Angelsky, S. G. Hanson, and C. Y. Zenkova, “Scattering of inhomogeneous circularly polarized optical field and mechanical manifestation of the internal energy flows,” Phys. Rev. A 86(2), 023847 (2012).
[Crossref]

Phys. Rev. Lett. (1)

A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24(4), 156–159 (1970).
[Crossref]

Sov. Phys. Usp. (3)

S. A. Akhmanov, A. P. Sukhorukov, and R. V. Khokhlov, “Self-focusing and diffraction of light in a nonlinear medium,” Sov. Phys. Usp. 10(5), 609–636 (1968).
[Crossref]

S. A. Akhmanov, V. A. Vysloukh, and A. S. Chirkin, “Self-action of wave packets in a nonlinear medium and femtosecond laser pulse generation,” Sov. Phys. Usp. 29(7), 642–647 (1986).
[Crossref]

V. L. Vinetskii, N. V. Kukhtarev, S. G. Odulov, and M. S. Soskin, “Dynamic self-diffraction of coherent light beams,” Sov. Phys. Usp. 22(9), 742–756 (1979).
[Crossref]

Other (8)

E. V. Ivakin, I. P. Petrovich, and A. S. Rubanov, “Self-diffraction of radiation by light-induced phase gratings,” Sov. J. Quantum Electron. 3(52), (1973).

V. E. Gusev and A. A. Karabutov, Laser Optoacoustics (AIP, 1993)

D. N. Auston, D. J. Bradley, A. J. Campillo, K. B. Eisenthal, E. P. Ippen, D. von der Linde, C. V. Shank, and S. L. Shapiro, Ultrashort Light Pulses: Picosecond Technique and Applications (Springer-Verlag, 1977)

M. Abramovitz and I. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series (National Bureau of Standards, 1964), Vol. 55.

A. Y. Bekshaev, “Subwavelength particles in an inhomogeneous light field: Optical forces associated with the spin and orbital energy flows” // arXiv:1210.5730 [physics.optics] 21 Oct 2012.

L. S. Ivlev and Yu. A. Dovgaliuk, Physics of Atmospheric Aerosol Systems (Saint-Petersburg State University, 1999)

P. Gerstner, J. Paltakari, and P. A. C. Gane, “Measurement and modelling of heat transfer in paper coating structures,” http://www.tappi.org/Downloads/Conference-Papers/2008/08ADV/08adv26.aspx

R. F. Probstein, Physicochemical Hydrodynamics: An Introduction (Wiley-Interscience, 2003, 2 ed.)

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Figures (8)

Fig.1
Fig.1

General model for the superposition field configuration formed in the cell.

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Fig. 2
Fig. 2

First-order diffraction efficiency vs. the incidence angle for the SAE density q0 = 30, 40 and 50 J/cm3.

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Fig. 3
Fig. 3

Optical system for the self-diffraction investigation: (L) laser CASIX LDS-1500; (BS) beam-splitter; (СR1), (СR2) 90° angular reflectors; (Ob) micro-objective; (S) screen; (ССD) camera.

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Fig. 4
Fig. 4

Resulting field formed in the cell plane when the two beams are coherent and the interference pattern is well developed (a) and when the beams are mutually incoherent (b). The interference pattern period in (a) is 2.5 μm.

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Fig. 5
Fig. 5

Diffraction patterns observed in the screen: (a) pattern observed when the plate PP is introduced (no self-diffraction due to the lack of coherence) and (b) – (e) self-diffraction patterns at different angles of the beams’ convergence as indicated.

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Fig. 6
Fig. 6

First-order self-diffraction efficiency vs. the angle of the incident beams’ convergence: experimental points (squares) are confronted with the approximation solid curve obtained via Eqs. (28), (35) with t0 = 80 ms.

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Fig. 7
Fig. 7

Spatial dependence of the light intensity P (right scale), gradient Fe and photophoretic Fp force (left scale) within a single period of the interference pattern for θ = 3.0° and I0 = 1.33⋅10−1 J/m3 (peak intensity 6⋅107 W/m2). The particle parameters are described in the text (see also Eq. (20)).

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Fig. 8
Fig. 8

Histograms of the particles’ distribution along x-axis (integrated over the y- and z-directions) within a two-period fragment of the interference pattern (white line shows the light energy distribution) for the single-beam power 40 mW (left column) and 80 mW (right column): (a), (a') initial distribution of the total of 105 particles (homogeneous); (b), (b') after 2⋅104 steps (the amplitude grating is not yet developed); (c), (c') after 16⋅104 steps; (d), (d') after 30⋅104 steps (pattern corresponding to dynamical equilibrium). Total number of particles is 105, the step duration accepted τ = 1 μs, other conditions are the same as specified in Sec. 4.1 and Fig. 7. Yellow lines in panels (d) and (d') describe corresponding averaged distributions analytically calculated via Eq. (34).

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Tables (1)

Tables Icon

Table 1 Experimental Conditions

View Table

Equations (35)

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A= A 1 exp( i φ 1 )exp( i Φ 1 )exp( − αz 2cos θ 1 )+ A 2 exp( i φ 2 )exp( i Φ 2 )exp( − αz 2cos θ 2 ),
Φ 1 = k 0 ( xsin θ 1 +zcos θ 1 ),   Φ 2 = k 0 ( xsin θ 2 +zcos θ 2 ),
θ 1 = − θ 2 = θ;
I( x,z )=g  n 0 2 | A( x,z ) | 2 = I 0 ( 1+VcosΦ )exp( − αz cosθ ),
Φ= Φ 1 + φ 1 − Φ 2 − φ 2 =2kxsinθ+φ,
I 0 =g n 0 2 ( A 1 2 + A 2 2 ),  V= 2 A 1 A 2 A 1 2 + A 2 2 .
Λ= π ksinθ .
q( x,z,t )= q 0 ( t ) cosθ ( 1+VcosΦ )exp( − αz cosθ ),   q 0 ( t )=α c n 0 I 0 t.
Δn=Δn( x,z,t )=( dn dT ) q( x,z,t ) Cρ .
T( x )=Hexp( ik ∫ 0 d Δn dz )exp( − αd cosθ )≈ Hexp( − αd 2cosθ )exp[ i kd cosθ ( dn dT ) q 0 Cρ VcosΦ ],
T( x )= ∑ m=−∞ ∞ T m exp( imΦ ) ,
T m = i m J m [ kd cosθ ( dn dT ) q 0 Cρ V ]exp( − αd 2cosθ ).
D 0 ={ 1− 1 2 [ kd cosθ ( dn dT ) q 0 Cρ V ] 2 }exp( − αd cosθ ),
D 1 = 1 4 [ kd cosθ ( dn dT ) q 0 Cρ V ] 2 exp( − αd cosθ ).
Q= 2πdλ n Λ 2 .
F e = 1 4g n 0 2 Re( α e )∇I+ ω g Im( α e ) p O e ,
p O e = g 2ω Im[ 1 μ E * ⋅( ∇ )E ]
α e = α e 0 1−i 2 3ε k 3 α e 0 ≈ α e 0 +i 2 3ε k 3 | α e 0 | 2 ,
α e 0 =ε a 3 ε p −ε ε p +2ε ,
F e ( x )=− I 0 k a 3 2g Re( ε p −ε ε p +2ε )sin( 2kxsinθ+φ )sinθ,
n p = 1.82 + 0.74i,    ε p = n p 2 = 2.76 + 2.69i,   ε= 1.33 2 ≃1.77.
F p =6πaη V p ,
α p =2kIm( n p )
F p =−κπ a 2 P,
κ= 2 9 β T A H r 0 2 v 0    α p χ p +2χ
π a 2 P→ ∫ S + r a P( r )dS ,
∫ S + r a P( r )dS = c a n 0 ∫ 0 a ∫ 0 2π r( cosϕ sinϕ )I( r,ϕ )r dr dϕ = c a n 0 I 0 ∫ 0 a r 2 dr ∫ 0 2π ( cosϕ sinϕ )cos( 2krsinθcosϕ+φ ) dϕ .
F p ( x )=π a 2 I 0 c n 0 κsin( 2kxsinθ+φ ) J 2 ( 2kasinθ ) kasinθ ≈ π 2 I 0 c n 0 k a 3 κsin( 2kxsinθ+φ )sinθ,
D f = k B T 6πη a H
( Δ x i ) reg = F p ( x i−1 )+ F e ( x i−1 ) 6πaη τ.
x N = x 0 + ∑ i=1 M [ ( Δ x i ) rand + ( Δ x i ) reg ] .
∂N ∂t = D f ∂ 2 N ∂ x 2 − 1 k B T ∂ ∂x ( NF ).
F= F e + F p = F 0 sin( 2kxsinθ+φ )
N( x )= N 0 exp[ − F 0 cos( 2kxsinθ+φ ) 2 k B T ksinθ ]
D m = D m0 exp( −2 D f q 2 t 0 ),

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