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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 7 — Apr. 8, 2013
  • pp: 8922–8938
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Self-diffraction of continuous laser radiation in a disperse medium with absorbing particles

O. V. Angelsky, A. Ya. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova  »View Author Affiliations


Optics Express, Vol. 21, Issue 7, pp. 8922-8938 (2013)
http://dx.doi.org/10.1364/OE.21.008922


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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.

© 2013 OSA

1. Introduction

Interaction of laser radiation with a medium containing subwavelength absorptive particles is accompanied by a set of non-linear phenomena characterized by the square-law dependence on the electromagnetic field amplitudes. They originate from diverse physical mechanisms: electro- and magnetostriction, thermal effects due to light absorption, concentration-deformation mechanism, temperature gradients, etc [1

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]

]. All the mentioned phenomena manifest themselves by induced spatial inhomogeneity of the complex refraction index and by the emergence of acoustic waves.

The striction mechanism is associated with the variations in medium density caused by modulated light flow and predominantly occurs in weakly absorbing media at high modulation frequencies [2

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

]. If the light absorption coefficient α > 1 cm–1, and the modulation frequency does not exceed 100 MHz, the striction mechanism can be neglected in comparison to the thermal one [3

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

]. The concentration-deformation mechanism connected with the medium density variations owing to the concentration of photoexcited charge carriers [4

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]

] plays a noticeable role only when the carriers’ life time is higher than the characteristic time scale of the light intensity modulations. This preferably takes place in semiconductors exposed to picosecond laser pulses.

A suitable way for studying various non-linear phenomena employs the development of amplitude and phase gratings due to heating and bleaching of absorbing solutions [10

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).

]. Additionally, in dispersed systems, the amplitude gratings can appear due to regular displacements of the suspended particles. Typical optical-field ponderomotive factors (gradient force and scattering force) act differently on dielectric and absorbing particles [11

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

,12

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]

], respect to the field intensity gradient and the field transverse momentum (energy flow) directions. In a lot of real situations with a medium containing absorptive particles, the particles are pushed off the high-intensity field regions.

An important question is the stability of the field-induced gratings. In usual cases when short and powerful laser pulses are used, the temperature gradient and the particles’ Brownian motion have no time to destroy the grating. In typical experiments exploiting continuous-wave lasers, the energy absorbed within a high-intensity region may efficiently flow away to adjacent regions via the particles’ motion and due to the heat conductivity, making the induced grating disappear. However, one can expect that for certain conditions a sort of equilibrium is established and the energy outflow from the high-intensity regions is compensated by continuous supply of absorbed energy.

Of course, this requires rather efficient light absorption and dispersed systems with absorptive particles. For example, usually thermal modulation of the refraction index in water is only available at very high power of the laser radiation because of minor absorption in the visible range. In the presence of small absorptive particles, the thermal grating is formed with essentially lower laser power. The aim of the present work is to analyze, both theoretically and with experiment, the main regularities of thermal grating formation in water-based disperse systems and possible manifestations of the gratings via the incident beam self-diffraction.

2. Grating formation and self-diffraction

In this paper, our main object is the water-filled cell with suspended absorptive particles whose sizes are small enough with respect to the light wavelength. Initially, the particles are chaotically distributed and the whole cell with the particles can be treated as an optically homogeneous absorptive medium with the effective refraction index n0 and energy absorption coefficient α.Let us consider a superposition of two coherent plane waves propagating within a thin layer of such a medium, Fig. 1
Fig.1 General model for the superposition field configuration formed in the cell.
. The total field amplitude distribution in the medium is described by the relation
A=A1exp(iφ1)exp(iΦ1)exp(αz2cosθ1)+A2exp(iφ2)exp(iΦ2)exp(αz2cosθ2),
(1)
where A1 and A2 are (real) amplitudes of the electric vectors of each wave and φ1,φ2 initial phases of the waves whose propagation axes make angles θ1 and θ2with the normal to the layer surface,
Φ1=k0(xsinθ1+zcosθ1),Φ2=k0(xsinθ2+zcosθ2),
(2)
k0=n0k is the medium wavenumber for the radiation with frequency ω,cis the velocity of light, k=ω/c is the vacuum wavenumber. In the following consideration we restrict ourselves to the symmetric case
θ1=θ2=θ;
(3)
then the energy density distribution of the field with amplitude (1) can be represented in the form
I(x,z)=gn02|A(x,z)|2=I0(1+VcosΦ)exp(αzcosθ),
(4)
where φ=φ1φ2, g is the numerical coefficient equal to g=(8π)1 in the Gaussian system of units,
Φ=Φ1+φ1Φ2φ2=2kxsinθ+φ,
(5)
and
I0=gn02(A12+A22),V=2A1A2A12+A22.
(6)
According to Eq. (4), the light energy is inhomogeneously distributed over the medium volume and the interference grating is formed with period
Λ=πksinθ.
(7)
Now we proceed with the grating formation. Its source is the inhomogeneous distribution of the absorbed light energy; the density of the energy absorbed during the time t equals to
q(x,z,t)=q0(t)cosθ(1+VcosΦ)exp(αzcosθ),q0(t)=αcn0I0t.
(8)
This absorption entails inhomogeneous heating of the medium which, in turn, induces a change of the refraction index, which instead of basic (zero-intensity) value n=n0is modified to n=n0+Δn. For the thermal mechanism of the refraction index modulation, we can express this in the form
Δn=Δn(x,z,t)=(dndT)q(x,z,t)Cρ.
(9)
In this equation, (dn/dT) is the refractive index temperature coefficient, C is the specific heat capacity (per unit mass of the medium) and ρis the medium mass density; for simplicity, here we neglect the absorbed energy redistribution due to the heat conductivity. Subsequently, Eqs. (8) and (9) together with (4) and (6) determine the spatial distribution of the induced medium inhomogeneity.

According to Eq. (9), the spatial profile of the refraction index explicitly depends on time. Of course, this is only correct for the initial stages of the field-medium interaction; with growing temperature gradient, the process of absorbed energy accumulation slows down due to the energy leakage to low intensity regions, and finally a stationary state with equilibrium between the absorbed and outflowing power is established. Analysis of this process is a complicated problem requiring the heat conductivity, diffusion and convection arguments and will not be conducted here. Now we note that in this stationary state, each small volume of the medium contains certain extra energy (with respect to the initial state without laser radiation) that can be called “stationary absorbed energy” (SAE). In the first approximation, it is distributed similarly to the quantity q(x,z,t) so we can preserve the notation and use the function q(x,z) and parameter q0 to characterize the SAE in the same manner as Eq. (8) characterizes the transient regime of the energy absorption.

A thin layer of the medium with inhomogeneous refraction index can be considered as a phase transparency; if the layer width equals to d, its coordinate-dependent transmission, within a constant phase factor, is
T(x)=Hexp(ik0dΔndz)exp(αdcosθ)Hexp(αd2cosθ)exp[ikdcosθ(dndT)q0CρVcosΦ],
(10)
where H is an inessential constant phase factor, |H|=1, that will be omitted in the following analysis. The field-induced inhomogeneity of transmission (10) is a source of the self-diffraction. The spatially periodic transmission distribution can be represented via the Fourier series [13

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]

]
T(x)=m=Tmexp(imΦ),
(11)
where each term represents a certain diffraction order and coefficients Tm describe the diffracted beams’ amplitudes. Because of the known relation [14

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

]exp(iBcosΦ)=m=m=imJm(B)exp(imΦ),where Jm(B) is the Bessel function of the first kind, and in view of Eq. (10), the coefficients of the series (11) are determined by the equation

Tm=imJm[kdcosθ(dndT)q0CρV]exp(αd2cosθ).
(12)

The relative efficiency for the m-th order diffraction is characterized by the quantity Dm=|Tm|2. Under assumptionΔn/n0<<1, for the most interesting zero-order and first-order diffracted beams we obtain

D0={112[kdcosθ(dndT)q0CρV]2}exp(αdcosθ),
(13)
D1=14[kdcosθ(dndT)q0CρV]2exp(αdcosθ).
(14)

In particular, Eq. (14) testifies that the first-order diffracted intensity grows with increasing incidence angleθ. This happens due to increasing the light pathway in the self-induced grating. On the other hand, longer light pathway causes the reverse influence associated with growth of the diffracted light absorption (see the exponential multiplier in Eqs. (13) and (14)). However, in many practical situations the first factor prevails. For example, consider the following experimental conditions (Table 1

Table 1. Experimental Conditions

table-icon
View This Table
).

Figure 2
Fig. 2 First-order diffraction efficiency vs. the incidence angle for the SAE density q0 = 30, 40 and 50 J/cm3.
illustrates the incidence-angle dependence of the diffracted beam intensity for the case ofA1=A2=A. It represents the first-order diffracted intensity normalized by the half incident intensitygn02|A|2, for three values of the SAE density. One can see that the self-diffraction efficiency is approximately proportional to the square of q0, in agreement with Eq. (14); besides, it really grows with the incidence angle and reaches maximum for θ0.9 radwhen further increase of the diffracted power is suppressed by the radiation absorption.

3. Experimental study of the self-diffraction

3.1. Experimental technique and equipment

In the experiments, we used a water suspension of the pigment ink 10BP designed for inkjet printers. The ink particles are polymer spheres 0.2 μm in diameter with a coating made of carbon-based absorptive resin. The self-diffraction efficiency was quantitatively estimated via the quantity D1 as a ratio of the first-order diffracted power to the power of one of the identical incident beams.

Measurements were performed with the experimental setup illustrated in 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.
. The semiconductor laser CASIX LDS-1500 generates radiation with wavelength λ = 532 nm, power controlled to 100 mW, and linearly polarized within the plane orthogonal to the figure plane. The radiation traverses the telescopic system TS (two objectives with common focal plane where a micrometer-size pin-hole is placed) that produces a parallel beam with diameter 0.7 mm. A beam-splitter BS divides the laser beam in two, and the 90° angular reflectors AR1 and AR2 redirect the beams strictly backward but with a certain transverse shifts. Angular reflectors facilitate transverse displacements of the beams at the beam-splitter output without changing their path difference and make the beams interfere.

In the micro-objective focus, the intensity distribution is formed with parallel planes corresponding to the interference maxima. The diameter of the focused beams was approximately 40μm. In the interference region, a cell of fused silica-C with the investigated medium is placed. The thickness of the cell walls is 1 mm and the working volume thickness is 10 μm. After passing through the cell, the radiation hits the screen S, if the intensity is high, or directly onto the CCD camera at low laser intensities. The plane-parallel plate PP is employed to change the path difference between the two beams by an amount greater than the coherence length of the laser radiation. This enables switching between the coherent (well-developed interference pattern) and incoherent (no interference pattern) regimes. Figure 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.
shows the fields observed in the cell plane in the two cases.

In 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.
examples are presented of the self-diffraction pattern observed at a distance 50 mm behind the cell. The panel (a) illustrates the self-diffraction suppression when the plane-parallel plate PP (Fig. 3) is introduced into the beam path; the other panels 5(b)–5(e) represent the diffracted beams’ appearance and qualitatively characterize their intensities. As the thermal grating is a phase grating, the number of observable orders should be high; however, only the first diffraction order is really observed (Figs. 5(b)5(e)). This testifies that the grating cannot be considered thin, and the Bragg spatial-filtering mechanism effectively works in the rather thick disperse-medium layer.

The phase grating modulation contrast depends on the particles’ concentration and the temperature gradient. The use of a thin grating would be the simplest solution for the self-diffraction modeling; however, in this case the number of particles interacting with the radiation is relatively small, and to obtain the necessary diffraction efficiency with a thin grating one should employ a higher temperature gradient than in a thick grating. Simultaneously, high temperature leads to the undesirable medium transformation, up to local phase transitions and cavitation effects. In view of this, the optimal grating thickness of 10 μmwas chosen.

3.2. Results and discussion

The main subject of the study was the first-order self-diffracted beam, its power and spatial characteristics. In general, there are two beams propagating in the directions of the first-order self-diffraction (dashed lines in Fig. 3). E.g., the left dashed line corresponds to the first-order diffraction of the right incident beam and to the second-order diffraction of the left incident beam. However, the second-order intensity was much lower, and could be neglected in the actual experimental conditions.

The first-order diffraction efficiency D1 was measured as a ratio of the diffracted beam power to the power of a single incident beam. Figure 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.
illustrates the observed dependence of D1 on the incident beams’ convergence angle2θ. In contrast to the simulation results (Fig. 2), increase of the angle θ leads to decrease of the self-diffraction efficiency. This result can be caused by two effects: (i) dynamical characteristics of the induced grating formation and peculiarity of the dynamic equilibrium between the energy accumulation and dissipation in the inhomogeneously illuminated medium and (ii) partial spatial filtering of the self-diffraction orders by the thick grating. The first reason is associated with the Brownian motion of the suspended particles and will be discussed in Sec. 4.2 below; the role of the second one is determined by the parameter of the grating thickness

Q=2πdλnΛ2.
(15)

In our experiments, the interference pattern period Λvaried from 2.5 to 10 μm so that for d=10 μm the parameter Q was within the limits 0.25 to 4. This testifies that the condition of thick Bragg grating Q>5 is not fulfilled and the grating cannot be considered thick. However, certain influence of the grating thickness was yet perceptible. One can notice an inessential decrease of the first-order intensity due to the partial space-time frequency filtering and the total filtering of the higher diffraction orders with growing diffraction angles when the grating thickness approaches the “thick grating” conditions.

4. Motion of the particles and the grating dynamics

The experimental results demonstrate that as long as the incident radiation is continuous, the observed diffraction pattern is temporally stable and the self-diffraction process possesses a stationary character. However, this stable behavior is a result of dynamical equilibrium: the temperature distribution is maintained inhomogeneous and intensities of the diffraction orders do not vary while the energy of the incident radiation is permanently absorbed and dissipated.

Besides, possible motion of the particles, both field-induced and stochastic, plays an important role in the thermal grating formation, maintenance and degradation. This is supported by our experimental observations: When the interference pattern period is comparable with the mean free-path length of the Brownian particles or with the effective thermal diffusion length, the grating is destroyed and the self-diffraction efficiency vanishes.

The process of spatial redistribution of the suspended particles due to the mechanical action of the interference optical field was studied in many details by Rubinov [16

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

]. However, he dealt with rather coarse non-absorbing particles whose sizes were comparable with the grating period. In such conditions, the thermal effects in the cell volume as well as the role of the particle’s Brownian motion were negligible. The full theoretical analysis of the grating dynamics and degradation processes is rather sophisticated. In this Section, we consider the main factors involved in our conditions and qualitatively discuss their relative weights and meaning.

4.1. Field-induced mechanical action

Along with absorption of the light energy, the particles experience a ponderomotive action due to the inhomogeneous optical field. We may estimate this action analytically in the dipole approximation which is acceptable if the particle radius a satisfies ka<<1. In this case the electromagnetic field exerts on a non-magnetic particle with electric polarizabilityαethe dipole force [15

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]

18

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.

]
Fe=14gn02Re(αe)I+ωgIm(αe)pOe,
(16)
where II(x,z) is the field energy density - in our case described by relation (4) - and
pOe=g2ωIm[1μE*()E]
(17)
is the electric part of the field orbital momentum [18

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

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

], E being the electric vector of the field. The polarizability αeis characterized by the relations [18

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

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

]
αe=αe01i23εk3αe0αe0+i23εk3|αe0|2,
(18)
αe0=εa3εpεεp+2ε,
(19)
where ε=n02 and εpare the medium and particle permittivity, respectively. In the field formed by interference of two plane waves withA1=A2, with symmetrical incidence (Eq. (3) holds), the orbital momentum pOe is directed exactly along the z-axis [20

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]

] and the second term of (16) makes no contribution to the transverse motion of the particles. A possible z-directed motion is prevented by the cell structure and plays no important role in the experimental behavior.

Therefore, the only transverse action is that associated with the gradient force expressed by the first term of Eq. (16). Without loss of generality we may consider a particle situated near the cell surface (z=0), where the force can be evaluated via the equation
Fe(x)=I0ka32gRe(εpεεp+2ε)sin(2kxsinθ+φ)sinθ,
(20)
which follows directly from Eqs. (16), (17), (18) and (4), (5) and coincides with the gradient force used in [16

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

]. A numerical estimation requires knowledge of the particle dielectric properties. The pigment particles used in our experiments are nearly spherical with mean radius a=0.1μm and contain ~20 nm absorptive coating of carbon-based resin. Assuming the complex refraction index of the coating to be close to that of the atmospheric soot, we accept the following values of the particle and medium (water) parameters [21

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

]:

np= 1.82 + 0.74i,εp=np2= 2.76 + 2.69i,ε=1.3321.77.
(21)

Another reason for the particle transverse motion, not taken into account in Ref [16

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

], is the photophoresis force owing to inhomogeneous heating a particle by the absorbed radiation. Mechanisms for the photophoretic action and means for their analysis are mainly developed for gaseous media [22

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

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]

]; the theory of photophoresis in liquids is not well elaborated and only a few attempts were made in this field [24

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

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]

]. For rough estimations, we use a simplified calculation scheme based on combination of approaches presented in Refs [23

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]

]. and [25

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]

]. We start with Eq. (17) of Ref [25

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]

]. which gives the photophoresis velocity Vp of a particle and define the corresponding photophoretic force Fp via Stokes formula
Fp=6πaηVp,
(22)
where η is the water viscosity. Then we take the asymmetry factor J1 [23

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]

,25

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]

] in the form J1=(2/3)aαp [23

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]

], where
αp=2kIm(np)
(23)
is the light absorption coefficient of the particle, and assume the particle to be non-hydrophobic (the ‘slip length’ Ls in [25

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]

]). After some manipulations, the photophoretic force can be represented by the expression
Fp=κπa2P,
(24)
where
κ=29βTAHr02v0αpχp+2χ
(25)
is an analog of the coefficient κ introduced by Eq. (8) of Ref [23

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]

], which characterizes the momentum transfer from the liquid molecules to the particle with given properties, P=(c/n0)I is the energy flow density in the incident light field supposed to be a plane wave in [25

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]

]. In Eq. (25) βT is the cubic thermal expansion coefficient of the medium, AH is the Hamaker constant, r0 is the radius of the medium molecule, ν0 is the specific molecular volume, χ and χp are the thermal conductivities of the medium and the particle, respectively. Then, following [23

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]

], the result (24) is generalized to the case of the transverse force in an inhomogeneous incident field via the replacement
πa2PS+raP(r)dS,
(26)
where the integration is performed over the projection of the particle side exposed to the incident radiation, r is the 2D radius-vector originating from the particle center projection, P(r)(c/n0)I(x,z) is the incident beam power density (the explicit z-dependence reflects change of the incident radiation with the cell depth but for rough estimates we still assume z=0).

Again, we employ expressions (4) and (5) with V=1 and, introducing polar coordinates in the transverse plane and takingz=0, evaluate integral (26) as follows:
S+raP(r)dS=can00a02πr(cosϕsinϕ)I(r,ϕ)rdrdϕ=can0I00ar2dr02π(cosϕsinϕ)cos(2krsinθcosϕ+φ)dϕ.
(27)
This result is in the form of a column vector whose elements represent the x- and y-components of the force. Due to the symmetry, the y-component vanishes and the whole force (24) is directed along the x-axis. As a result, Eq. (24) and elementary transformations of (27) yield the final expression for the photophoretic force
Fp(x)=πa2I0cn0κsin(2kxsinθ+φ)J2(2kasinθ)kasinθπ2I0cn0ka3κsin(2kxsinθ+φ)sinθ,
(28)
where J2 is the symbol of the Bessel function. The second line of the equation represents an approximation valid when the particle is small with respect to the period of the interference pattern (kasinθ<<1).

For latex particles with carbon coating suspended in water, βT=2.57104K1, AH=5.01020J, r0=1.54Å, ν0=30 Å3, χ=0.61 W/(m K) [25

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]

] and χp=0.55 W/(m K) [26

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

], and after evaluating αp from Eqs. (20) and (22) one finds κ = 2.1⋅10−8 s/m. The spatial dependence of the forces Fp and Fe for a = 0.1μm, θ = 3.0° and φ = 0 (see Eqs. (20) and (28)) is illustrated by 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)).
. It expectedly shows that both forces are comparable in magnitudes but opposite in sign: the gradient force pulls the particle towards the interference maxima while the photophoretic one promotes the particles to move to the low-intensity regions. The ‘net’ field-induced mechanical action is dictated by the prevailing photophoretic force (note that possible hydrophobic properties of the particles may only increase the photophoretic effect [25

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]

]).

4.2. Stability and degradation of the self-induced grating

The above-described mechanisms of formation and evolution of the self-induced grating show that the grating instability as well as its stationary pattern is essentially influenced by the heat conductivity and by the motion of the pigment particles from ‘hot’ regions to ‘cold’ ones. Along with the chaotic diffusion-like redistribution of particles caused by molecular mechanisms, the electromagnetic field-induced ponderomotive action provides regular drift-like displacements of the particles that contribute to the creation of the additional amplitude grating.

During the simulation, the step duration was assumed to be τ=1μs, and the behavior of 105 particles was calculated; initially (i=0) the particles were uniformly distributed over the modeled volume. The results are presented in 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).
. It is seen that the particles are, indeed, concentrated near the interference minima thus forming the amplitude grating which may act together with the phase of thermal origin. Note that under the accepted conditions, the mean value of (Δxi)reg is 5.51010μm or 111010μm for the incident beam power 40 and 80 mW (the interference pattern peak intensity 6⋅107 W/m2 and 12⋅107 W/m2), respectively, while (Δxi)rand=2.1103μm, which satisfies the requirement for the validity of Eq. (29) for the presumed grating period Λ=4μm. The regular constituent of the particle motion is obviously much smaller than the stochastic fluctuations but the tendency to form the regular periodic distribution of particles is nevertheless quite perceptible. However, the Brownian motion permanently acts against this tendency, and limits available contrast of the amplitude grating. In the equilibrium state that is established after more than 2⋅105 steps (which corresponds to 0.2 s), the particles’ density varies within 5 to 10 percents, according to the incident power (see Figs. 8(c), 8(c'), 8(d) and 8(d')).

It is interesting to compare the simulated patterns of Fig. 8 with the averaged distribution of particles calculated analytically by means of the one-dimensional Nernst – Plank equation [27

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

]
Nt=Df2Nx21kBTx(NF).
(32)
Here N is the particles’ concentration,
F=Fe+Fp=F0sin(2kxsinθ+φ)
(33)
is the total transverse force and, according to Eqs. (20) and (28)F0=π2I0ka3sinθ[cn0κ8Re(εpεεp+2ε)].

The stationary solution of this equation corresponding to zero time derivative in Eq. (32) can be easily found in the form [16

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

]
N(x)=N0exp[F0cos(2kxsinθ+φ)2kBTksinθ]
(34)
(the normalization factor N0 is determined from the condition that the total number of particles N(x)dx is always the same and equal to the initial number of particles). The average distribution of Eq. (34) is illustrated by yellow lines in Figs. 8(d) and 8(d'). The obvious difference between the simulated and analytical results testifies for another important property of the field-induced particle distribution: generally, it is not stationary and experiences strong random deformations due to stochastic fluctuations. This fact, which invoked no important consequences in conditions of Ref [16

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

] because large particles were rather regularly arranged within the interference fringes, is generally responsible for the non-stationary and fluctuating character of the field-induced amplitude grating. Absence of perceptible fluctuations in the experimental self-diffraction patterns serves as an additional indirect confirmation of the relative weakness of the amplitude-grating effects.

In general, Fig. 8 along with the discussion of Sec. 4.1 confirms our conclusion that in real experimental conditions the role of the amplitude grating was not essential. Its single reliable manifestation was a slightly suppressed absorption of the incident light in case of well-developed interference pattern, compared to the case of incoherent superposition of the two beams (see Sec. 3.1). In full agreement with the simulation results, experimentally we could not detect a noticeable contribution of the amplitude grating to the diffraction efficiency up to beam powers of 50 – 100 mW. On the other hand, the data of Fig. 8 and Eq. (34) suggest that the amplitude grating could be more perceptible at higher intensities but in reality the beam power of 100 mW or more appeared to be inappropriate because under such circumstances the processes of the pigment melting, local boiling of water and cavitation effects become essential. In the whole range of conditions available in our experiments, the amplitude grating was rather weak. This conclusion is additionally supported by the self-diffraction pattern (Fig. 5) where higher-order diffracted beams are absent and by the negligible level of additional fluctuations in the diffracted beam profiles caused by the Brownian motion of the particles. In all cases, the first-order intensity distribution appears to be similar to the zero-order intensity, which confirms the predominantly phase character of the grating.

All the mentioned facts clearly indicate that the field-induced ponderomotive action cannot cause a noticeable distortion or destruction of the self-induced thermal grating described and analyzed in Sec. 2. Essential questions of the thermal grating development, consistency and degradation should be considered based on kinetic processes including the heat exchange, convection and thermal conductivity [27

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

] that can be properly studied via observing the transient phenomena accompanying the grating emergence and vanishing after the laser radiation is turned off. However, important kinetic characteristics of these processes can be extracted from analysis of the stationary gratings with special attention to their relations with the grating spatial characteristics.

The main elementary kinetic process involved in the thermal grating formation is the energy exchange between a particle and the surrounding water. Since the specific heat capacity of water is much higher than the heat capacity of the particle, the latter remains ‘hot’ during certain time interval t0 after the particle has absorbed a portion of the light energy. This time interval correlates with the time of the thermal relaxation. It is difficult to estimate this time directly but it can be determined by using the kinetic properties of the whole disperse system. In fact, the time t0 regulates the diffracted intensity dependence on the grating spatial frequency [28

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]

]:
Dm=Dm0exp(2Dfq2t0),
(35)
where q=2π/Λ is determined by the grating period Λ, Dm0 is the ‘initial’ m-order efficiency for a high grating period Λ>>2πDft0 and the diffusion coefficient Df is given by Eq. (29). Relations (29) and (35) explicitly connect the diffraction efficiency with molecular and kinetic characteristics of the Brownian motion of particles, which destroys the grating. In essence, they show that the grating can exist until the diffusion time Λ(2πDf)1 exceeds the thermal relaxation time, and the thermal grating disappears for small enough grating periodΛ.

The experimental dependence D1(Λ1) complies with Eq. (35) and the best fitting procedure (solid curve in Fig. 6) yields t0=80ms (other parameter values areT=300K, η=1.0103Pa·s, aH=0.1μm). A high degree of correlation between the experimental data and approximating curve supports the validity of the model chosen.

Conclusion

We have demonstrated the possibility of producing controllable phase gratings in disperse liquid media with suspended absorptive sub-wavelength particles. The grating prototype is formed by interference between two identical laser beams obliquely incident onto the medium. The light energy absorbed by the particles is transmitted to the medium thus forming the inhomogeneous temperature distribution and associated thermal grating due to the temperature dependence of the refraction index. Importantly, the light-induced grating structures are obtained with low-power continuous-wave laser radiation. We have found that the grating life time, contrast (the refraction index modulation) and, consequently, the self-diffraction efficiency are determined by the degree of coherence of the beams involved.

Mechanism and dynamical characteristics of the thermal grating formation and degradation are analyzed, and their relations with the kinetic and thermal properties of the medium and particles are discussed. The roles of the ponderomotive field action (gradient and photophoretic forces) and of the stochastic fluctuations in the medium (Brownian motion) are studied. The developed experimental approaches and the results obtained can be used for the recording and diagnostics of rapid processes in particle-medium interactions.

Finally, we would like to outline possible directions for further development of the proposed approach. First, it can be extended to multiple-beam interference configurations. This may contribute to the sharper interference fringes with corresponding enhancement of the optical field inhomogeneity. As a result, enhancement of the non-linear behavior could be expected. However, according to Section 4.2, the sharper fringes are subject to faster degradation which may neutralize the anticipated growth of the self-induced effects. On the other hand, the use of smaller suspended particles, up to the nanometer size, may be promising in view of decreasing thermal relaxation time. This will improve the grating dynamical characteristics with probable increase of its controllability and accuracy. These features could be especially useful in the study of non-stationary gratings induced in media with absorbing particles when the interfering coherent waves have unequal frequencies. In such situations, the controllable evolution of the interference pattern will offer a new degree of freedom for the induced grating management, with a lot of interesting and promising applications.

Acknowledgments

The authors are grateful to the anonymous referee for the useful suggestions on the meaning and possible ways for further development of the approach proposed. Steen G. Hanson acknowledges the financial support from the Danish Council for Technology and Innovation under the Innovation Consortium LICQOP, grant #2416669.

References and links

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]

OCIS Codes
(260.2160) Physical optics : Energy transfer
(260.5430) Physical optics : Polarization
(350.4990) Other areas of optics : Particles
(350.4855) Other areas of optics : Optical tweezers or optical manipulation

ToC Category:
Physical Optics

History
Original Manuscript: February 4, 2013
Revised Manuscript: March 20, 2013
Manuscript Accepted: March 23, 2013
Published: April 4, 2013

Virtual Issues
Vol. 8, Iss. 5 Virtual Journal for Biomedical Optics

Citation
O. V. Angelsky, A. Ya. Bekshaev, P. P. Maksimyak, A. P. Maksimyak, S. G. Hanson, and C. Yu. Zenkova, "Self-diffraction of continuous laser radiation in a disperse medium with absorbing particles," Opt. Express 21, 8922-8938 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-8922


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References

  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. Express19(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. A27(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. Express18(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. A86(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. Express17(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. Express17(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. Express18(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. Express14(12), 5755–5764 (2006). [CrossRef] [PubMed]

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