**3. Investigation of optical currents in completely coherent and partially coherent vector fields**

In this section we present the computer simulation results on the spatial distribution of the Poynting vector and illustrate motion of micro and nanoparticles in spatially inhomogeneously polarized fields.

The Poynting vector **S** , is defined, in its simplest form, as the vector product of the vectors of electric field, **E** , and magnetic field, *viz.* **H** , **S EH** . This form is referred to as the Abraham form of the Poynting vector. By definition, the Pointing vector *for a plane wave* is perpendicular to vectors **E** and **H** , being representing the energy flux (in W/m2) of an electromagnetic field. As it will be seen from the following consideration, in light fields with complex wave fronts and with variable in space (inhomogeneous) state of polarization the Poynting vector can exhibit much more sophisticated behavior, being also changing from point to point at the beam cross-section and leading to new applications of light.

To facilitate understanding the material of the following sections, let us briefly remind the main light beam characteristics involved in our consideration. The key notion of *geometrical optics* is a ray; a bundle of rays constitutes an optical beam. The only information provided by rays is the direction of propagation of light. More diverse, more fundamental and more informative parameters of light are derived within the framework of wave optics. There are wave amplitude and phase (both quantities are unobservable directly though lying in the basis of theory [17]); polarization that is determined by the amplitude ratio and phase difference of orthogonal components of a beam; the degree of polarization is determined by correlations between these components; coherence parameters characterizing concordance of light disturbances at two (or more) spatialtemporal points of light field. These base characteristics defined in seminal book [17] as well as in numerous other handbooks in optics, are added by the Poynting vector determining the direction of propagation of light beams in terms of electrodynamics, and by more complex polarization parameters than simple azimuth of polarization and ellipticity inherent in elementary wave optics. In our study, much attention will be paid to the Stokes parameters, polarization parameters intrinsically connected with both energy and coherent characteristics of light beams. These notions will be specified in the

Organization of the review is not quite usual. The most of review papers written up to now on the topic of interest are of theoretical nature, sometimes with valued simulation background. Relatively less attention is paid to experimental aspects of the problem. Partially, this gap is filled by recent books and book chapters written by the authors of this review [37, 38]. But the scope of experimental results rapidly grows, and this outlook does not repeat our previous papers. So, we represent here several independent experimental current views on the problem of metrology of coherent and polarization properties of optical fields, with especial accent on singular optical prerequisites and consequences of our experimental approaches. In spite of forced incompleteness of this consideration, the authors hope that it would stimulate further experimental investigations in this field and lead to broadening of practically significant applications of *Singular Optics of Partially* 

**3. Investigation of optical currents in completely coherent and partially** 

point to point at the beam cross-section and leading to new applications of light.

In this section we present the computer simulation results on the spatial distribution of the Poynting vector and illustrate motion of micro and nanoparticles in spatially

The Poynting vector **S** , is defined, in its simplest form, as the vector product of the vectors of electric field, **E** , and magnetic field, *viz.* **H** , **S EH** . This form is referred to as the Abraham form of the Poynting vector. By definition, the Pointing vector *for a plane wave* is perpendicular to vectors **E** and **H** , being representing the energy flux (in W/m2) of an electromagnetic field. As it will be seen from the following consideration, in light fields with complex wave fronts and with variable in space (inhomogeneous) state of polarization the Poynting vector can exhibit much more sophisticated behavior, being also changing from

corresponding sections of the chapter.

*Coherent and Inhomogeneously Polarized Optical Fields*.

**coherent vector fields** 

inhomogeneously polarized fields.

The use of small particles for diagnostics of microstructure of light is widely used approach [25 - 30], but mainly in approximation of complete coherence of an optical field. Here, the influence of phase relations and the degree of mutual coherence of superposing waves in twowave and four-wave configurations on the characteristics of the microparticle's motion is analyzed. The possibility of diagnostics of optical currents in liquids caused by polarization characteristics of an optical field alone is demonstrated using nanoscale metallic particles. We also discuss the prospects of studying temporal coherence using the proposed approach.

There is the motivation of this study. Experimental investigation and computer simulation of the behavior of small spherical particles embedded in optical fields provide a deeper understanding of the role of the Poynting vector for description of optical currents in various media [24, 36]. So, interference between waves polarized in the plane of incidence has been shown to be effective in creation of polarization micro-manipulators and optical tweezers. On the other hand, this is a vital step in optimal metrological investigation of optical currents in vector fields [39 - 42]. Besides, the study of spatial and temporal peculiarities of the motion of particles embedded in optical fields with various spatial configurations and with various scale distributions of the Poynting vector leads to new techniques for estimating the temporal coherence of optical fields [43].

Computation of the spatial distribution of the time-averaged Poynting vector determining the forces affecting microparticles and their movement is performed following the algorithm proposed by M. Berry [24] who has shown that the vector force affecting a small particle in an optical field is proportional to the time-averaged Poynting vector. We will show that the study of the motion of microparticles in inhomogeneously polarized fields provides reconstruction of the spatial distribution of the time-averaged Poynting vectors, *viz.* the optical currents.

#### **3.1 Two-wave superposition for changeable degree of mutual coherence of the components**

Superposition of two plane waves of equal amplitudes polarized in the plane of incidence (Fig. 2a) results in the distribution of the Poynting vector shown in Fig. 2b. Such distribution arises when the interference angle is equal to 90°, and the only periodical polarization modulation of the field (in the absence of intensity modulation) takes place in the plane of observation [44].

The coherency matrix 1 2 *W t* ( , ,) **r r** describes the coherence properties of vector optical fields, being characterising the correlation of two fields at two different spatial points 1**r** and 2**r** [45, 46], and is determined as

$$\mathcal{W}(\mathbf{r}\_1, \mathbf{r}\_2, t) = < E^{(1)}{}\_i(\mathbf{r}\_1, t) E^{(2)}\_{\phantom{1}}(\mathbf{r}\_2, t) > 0$$

where *i, j* = *x, z*. Within the framework of such approach the degree of mutual coherence of the field is defined as [15]

$$\eta\_{\vec{\eta}}(\mathbf{r}\_1, \mathbf{r}\_2, t) = \frac{\mathcal{W}\_{\vec{\eta}}(\mathbf{r}\_1, \mathbf{r}\_2, t)}{\sqrt{\text{tr}[\mathcal{W}(\mathbf{r}\_1, \mathbf{r}\_1, 0)] \cdot \sqrt{\text{tr}[\mathcal{W}(\mathbf{r}\_2, \mathbf{r}\_2, 0)]}}} = \frac{\mathcal{W}\_{\vec{\eta}}(\mathbf{r}\_1, \mathbf{r}\_2, t)}{\sqrt{\sum\_{\vec{\eta}} \mathcal{W}\_{\vec{\eta}}(\mathbf{r}\_1, \mathbf{r}\_1, 0) \mathcal{W}\_{\vec{\eta}}(\mathbf{r}\_2, \mathbf{r}\_2, 0)}}\tag{3}$$

The distribution of the time-averaged density of the energy current in space determines the current magnitude at different points of the plane of observation, being unambiguously determined by the degree of coherence of the superposing waves. The direction of the resulting current is set by the directions of the Poynting vectors of these waves.

Fig. 2a. Superposition of plane waves of equal amplitudes linearly polarized in the plane of incidence having an interference angle of 90°. Periodical spatial polarization modulation takes place in the plane of incidence.

Fig. 2b. Spatial distribution of the time-averaged Poynting vectors resulting from superposition of two orthogonally linearly polarized waves with an interference angle of 90°.

An analysis of the spatial distribution of the time-averaged Poynting vectors shown in Fig. 2b reveals the periodicity of this distribution, where the lengths of lines shown in the figure are proportional to the absolute magnitudes of the vectors. The lines corresponding to the singularities of the Poynting vector are shown by the indicated set of points [47 - 49].

The distribution of the time-averaged density of the energy current in space determines the current magnitude at different points of the plane of observation, being unambiguously determined by the degree of coherence of the superposing waves. The direction of the

Fig. 2a. Superposition of plane waves of equal amplitudes linearly polarized in the plane of incidence having an interference angle of 90°. Periodical spatial polarization modulation

Fig. 2b. Spatial distribution of the time-averaged Poynting vectors resulting from

superposition of two orthogonally linearly polarized waves with an interference angle of

singularities of the Poynting vector are shown by the indicated set of points [47 - 49].

An analysis of the spatial distribution of the time-averaged Poynting vectors shown in Fig. 2b reveals the periodicity of this distribution, where the lengths of lines shown in the figure are proportional to the absolute magnitudes of the vectors. The lines corresponding to the

takes place in the plane of incidence.

90°.

resulting current is set by the directions of the Poynting vectors of these waves.

Spatial distribution of the time-averaged Poynting vectors, cf. Fig. 2b, shows the trajectories of energy transfer. The points at the map of the time-averaged Poynting vectors (Fig. 2b) correspond to the areas through which energy transfer is absent, showing: (i) the loci of singularities of the Poynting vector; (ii) the points (with constant intensity) forming the directions along which light energy is non-vanishing, but is conserved; (iii) the points where the vector *H* vanishes due to interference, while in this arrangement (90°-superposition of plane waves) superposition of strictly coaxial vectors *H* of equal amplitudes associated with two superimposed plane waves takes place.

On the other hand, the instantaneous magnitude of the electric (magnetic) field strength's vector of the resulting distribution, which is formed in the plane of observation, is written as

$$\mathbf{E} = \left| \mathbf{E}^{(1)} + \mathbf{E}^{(2)} \right| \cos(\alpha t + \delta\_c) \mathbf{a}\_c \quad \text{(or} \ \mathbf{H} = \left| \mathbf{H}^{(1)} + \mathbf{H}^{(2)} \right| \cos(\alpha t + \delta\_h) \mathbf{a}\_h \text{ )}\_{\mathbf{H}} $$

where, *<sup>e</sup>* **a** , *<sup>h</sup>* **a** are the unit vectors in the direction of propagation of the electric (magnetic) components for the resulting field in the plane of observation; () *e h* is the phase difference of the electric (magnetic) field components of superposed waves. In this case, the instantaneous magnitude of the Poynting vector is

$$\mathbf{S}\_{inst} = \mathbf{E} \times \mathbf{H} = \left| \mathbf{E} \right| \cdot \left| \mathbf{H} \right| \cos \left( \alpha t + \delta\_e \right) \cos \left( \alpha t + \delta\_h \right) \left( \mathbf{a}\_e \times \mathbf{a}\_h \right) \,, \, \mathbf{x}$$

and the time-averaged magnitude of the Poynting vector is

$$\mathbf{S}\_{\rm av} = \frac{|\mathbf{E}| \cdot |\mathbf{H}|}{2} (\mathbf{a}\_{\varepsilon} \times \mathbf{a}\_{h}) \cos(\delta\_{\varepsilon} - \delta\_{h}) = \frac{1}{2} (\mathbf{E} \times \mathbf{H}) \cdot \cos(\delta\_{\varepsilon} - \delta\_{h}) \,. \tag{4}$$

Because the phase difference of the electric field changes from point to point (polarization modulation), the time-averaged magnitude of the Poynting vector is modulated in space taking the maximum (minimum) at different points of the plane of observation, as it is seen from Eq. (4).

Homogeneous intensity distribution and periodical spatial modulation of the Poynting vector simultaneously realized in the observation region have previously been discussed within the framework of Refs [50, 51]. Spatial polarization modulation at the plane of observation is caused by superposition of the *Ex* and *Ez* field components with changing the phase difference from point to point, cf. Fig. 2a. A photodetector registers only intensity, 2 2 *x z IE E* . The sum of the squared amplitudes of the electrical field components is constant at the plane of observation, though the state of polarization changes. The Poynting vector is defined, as mentioned above, by the vector product, *S EH* . One observes the dependence of the result on the phase relation between vectors **E** and **H** through the vector magnitude and its direction. This relation changes from point to point in the plane of observation and manifests itself in polarization modulation. An obvious explanation for this follows from consideration of the vector product of the components of vector **E** ( *Ex* and *Ez* components) with vector **H** . Both the magnitudes of projections *Ex* and *Ez* and their phases change from point to point in the observation plane. As a consequence, the vector product changes as well as the Poynting vector. The result of modulation is shown in Fig. 3.

The results of simulating the motion of particles embedded in the field of the considered distribution of the Poynting vector are shown in Fig. 4. We have here tacitly assumed the particles to be absorbing and of size 0.1 µm. One observes that in the case of the distribution resulting from superposition of completely mutually coherent waves, the velocities of particle motion along the lines of maxima and zeroes of the Poynting vector are considerably different from one another.

The particle size is here comparable with a half-period of the corresponding distribution; however, the resultant force giving rise to the particle motion along the lines close to the Poynting vector maxima exceeds the resultant force for lines close to the zeroes of the Poynting vector. The results of modulation of particle movement velocity along the peaks and zeroes of the averaged field of the Poynting vector are shown in Fig. 4a and Fig. 4b, respectively.

If the degree of mutual coherence of the superposed waves equals 0.2, the spatial distribution of the averaged Poynting vectors becomes more homogeneous, the modulation depth decreases considerably, and the velocities of microparticles become almost identical.

When the degree of mutual coherence equals 0.5, the relative velocities of the microparticles along the same trajectories are lower in comparison with velocities in case of complete mutual coherence of the superposed waves and lie in the vicinity of the average magnitudes for coherent and incoherent cases [52]. One observes the dependence on the coherent properties of the superimposed waves for the motion velocities of microparticles with constant size and form in media with constant viscosity [52]. When analyzing the motion of test particles in the region of distributed magnitude of the Poynting vector, the influence of the parameters of superposing fields on the character of particle motion can be determined, cf. Fig. 4a, 4b.

Fig. 3. The polarization distribution in the registration plane is marked by thin lines. The direction and magnitude of the Poynting vector are marked by bold lines. The point at the end of the vector determines the energy transfer direction. The modulation of the Poynting vector takes place according to the polarization modulation at the plane of observation.

The results of simulating the motion of particles embedded in the field of the considered distribution of the Poynting vector are shown in Fig. 4. We have here tacitly assumed the particles to be absorbing and of size 0.1 µm. One observes that in the case of the distribution resulting from superposition of completely mutually coherent waves, the velocities of particle motion along the lines of maxima and zeroes of the Poynting vector are

The particle size is here comparable with a half-period of the corresponding distribution; however, the resultant force giving rise to the particle motion along the lines close to the Poynting vector maxima exceeds the resultant force for lines close to the zeroes of the Poynting vector. The results of modulation of particle movement velocity along the peaks and zeroes of the averaged field of the Poynting vector are shown in Fig. 4a and Fig. 4b,

If the degree of mutual coherence of the superposed waves equals 0.2, the spatial distribution of the averaged Poynting vectors becomes more homogeneous, the modulation depth decreases considerably, and the velocities of microparticles become almost identical. When the degree of mutual coherence equals 0.5, the relative velocities of the microparticles along the same trajectories are lower in comparison with velocities in case of complete mutual coherence of the superposed waves and lie in the vicinity of the average magnitudes for coherent and incoherent cases [52]. One observes the dependence on the coherent properties of the superimposed waves for the motion velocities of microparticles with constant size and form in media with constant viscosity [52]. When analyzing the motion of test particles in the region of distributed magnitude of the Poynting vector, the influence of the parameters of superposing fields on the character of particle motion can be determined,

Fig. 3. The polarization distribution in the registration plane is marked by thin lines. The direction and magnitude of the Poynting vector are marked by bold lines. The point at the end of the vector determines the energy transfer direction. The modulation of the Poynting vector takes place according to the polarization modulation at the plane of observation.

considerably different from one another.

respectively.

cf. Fig. 4a, 4b.

Fig. 4. The change of the particle motion velocity with time obtained for different magnitudes of the degree of coherence of superposing waves in the case of particles moving along the peak (*a*) and the minimum (*b*) of the field of time-averaged Poynting vector magnitude: curves 1, 2, and 3 correspond to the degree of coherence, which equals 1, 0.5, and 0.25, respectively.

When the degree of mutual coherence equals 0.5, the relative velocities of the microparticles along the same trajectories are lower in comparison with velocities in case of complete mutual coherence of the superposed waves and lie in the vicinity of the average magnitudes for coherent and incoherent cases [52]. One observes the dependence on the coherent properties of the superimposed waves for the motion velocities of microparticles with constant size and form in media with constant viscosity [52]. When analyzing the motion of test particles in the region of distributed magnitude of the Poynting vector, the influence of the parameters of superposing fields on the character of particle motion can be determined, cf. Fig. 4a, 4b.

As it was shown in papers [44, 51, 52], the degree of coherence of superposing waves determines not only the visibility of an interference distribution, but also the structure of the polarization field, *viz.* it determines the distribution of the Poynting vector. Under the same other conditions of the wave superposition, *changing the degree of coherence results in changing motion velocity of the test particles, what can serve as an estimating parameter for determining the coherence properties of superposing waves*. These differences in velocities of motion of microparticles are explained physically in the following manner: Increasing the share of incoherent radiation in the resulting field distribution causes a decrease of the modulation depth of the Poynting vector's spatial distribution, as well as a decrease of the resultant force magnitude along the lines of energy transfer, which induces the motion of the microparticles. The increase of the degree of coherence brings about an accelerated particle motion in the field of averaged energy magnitudes.

The following diagram (Fig. 5) shows the particle velocity distribution (in this case, 39 particles), embedded into the field formed by the averaged magnitudes of the Poynting vector in the case of superposition of completely mutually coherent waves. With time (~1.2 sec) practically all particles gain equal velocity magnitudes (see column 7); redistribution of particles in the direction of the resultant force and uniform motion along the zero value of the Poynting vector take place.

Fig. 5. The diagram of particles velocity distribution with time.

#### **3.2 Superposition of four waves for changeable degree of mutual coherence of the components**

In the case of superposition of four waves, see Fig. 6a, involving two sets of counterpropagating plane waves of equal intensities, linearly polarized in the plane of incidence and oriented at an angle of 90° with respect to each other, the spatial distribution of the time-averaged Poynting vectors is formed as shown in Fig. 6b.

The 2D periodicity of the Poynting vector's distribution is evident. As in the previous case, the lengths of the time-averaged Poynting vectors are proportional to their magnitudes. The nodal points in this distribution correspond to zero magnitudes of the Poynting vector, i.e. singularities of the Poynting vector. In the following simulation, the diameters of the particles are changed to be comparable with a half-period of the corresponding spatial distribution of the Poynting vector.

Fig. 6a. Arrangement of superposition of four plane waves.

**3.2 Superposition of four waves for changeable degree of mutual coherence of the** 

In the case of superposition of four waves, see Fig. 6a, involving two sets of counterpropagating plane waves of equal intensities, linearly polarized in the plane of incidence and oriented at an angle of 90° with respect to each other, the spatial distribution of the

The 2D periodicity of the Poynting vector's distribution is evident. As in the previous case, the lengths of the time-averaged Poynting vectors are proportional to their magnitudes. The nodal points in this distribution correspond to zero magnitudes of the Poynting vector, i.e. singularities of the Poynting vector. In the following simulation, the diameters of the particles are changed to be comparable with a half-period of the corresponding spatial

Fig. 5. The diagram of particles velocity distribution with time.

time-averaged Poynting vectors is formed as shown in Fig. 6b.

Fig. 6a. Arrangement of superposition of four plane waves.

**components** 

distribution of the Poynting vector.

This follows from the presence of the minimum of the modulation depth at the spatial distribution of the Poynting vector. If the phase relations between four superposed beams are such that the modulation depth of the spatial distribution of the Poynting vector is maximal, the particle velocities will depend on the degree of mutual coherence between the interfering beams, see Fig. 7.

In order to compare the influence of the temporal and spatial parameters of coherence on the motion of the microparticles, we have analyzed the maps of the time-averaged Poynting vector with a superposition of four plane waves over a large area. During this, we have tracked the microparticles' motion. The dependence of microparticles' velocities on the phase difference of the superposing beams has thus been revealed. So, in the case of pair-bypair four opposite-in-phase superposed beams, particles become motionless. For that, the "opposite-in-phase" configuration covers the situation where two sets of mutually orthogonal standing waves are characterized by the fact that their nodes strictly coincide.

Fig. 6b. 2D distribution of the time-averaged Poynting vectors resulting from the superposition of four waves shown in Figure 6a.

Fig. 7. The variation of motion velocity of a test particles in an averaged field of distributed Poynting vectors with the change of the degree of mutual coherence of the waves (four superposing waves are in phase): curve 1 – one of the waves is incoherent; curves 2, 3, 4 correspond to the degree of coherence 0.25, 0.5, and 0.75, respectively.

Increasing the degree of mutual coherence of the waves sets a more uniform velocity magnitude of moving particles. The magnitude of the resultant force causing this motion under increasing the degree of coherence, practically, does not change with time, see Fig. 8. The maximum depth of modulation for coherent equiphase waves determines the stable position of particles. *The chaotic state and the average particle velocity value can be taken as a possible guideline in estimating the degree of coherence of superposing waves.*

Fig. 8. The change of the resultant force of the test particle motion in the time-averaged field of distributed Poynting vectors with the change of the degree of mutual coherence of the waves (four superposing waves are in phase): curve 1 – one of the waves is incoherent with all other waves; curves 2, 3 and 4 correspond to the degree of coherence of the waves 0.25, 0.5, and 0.75, respectively.

Fig. 9. "Cellular" distribution of the potential traps for microparticles in the case of superposition of four waves.

It is worth emphasizing two issues for the case of superposition of four plane waves. First, the dependence of the depth of modulation for the distribution of the time-averaged

Increasing the degree of mutual coherence of the waves sets a more uniform velocity magnitude of moving particles. The magnitude of the resultant force causing this motion under increasing the degree of coherence, practically, does not change with time, see Fig. 8. The maximum depth of modulation for coherent equiphase waves determines the stable position of particles. *The chaotic state and the average particle velocity value can be taken as a* 

Fig. 8. The change of the resultant force of the test particle motion in the time-averaged field of distributed Poynting vectors with the change of the degree of mutual coherence of the waves (four superposing waves are in phase): curve 1 – one of the waves is incoherent with all other waves; curves 2, 3 and 4 correspond to the degree of coherence of the waves 0.25,

Fig. 9. "Cellular" distribution of the potential traps for microparticles in the case of

It is worth emphasizing two issues for the case of superposition of four plane waves. First, the dependence of the depth of modulation for the distribution of the time-averaged

*possible guideline in estimating the degree of coherence of superposing waves.*

0.5, and 0.75, respectively.

superposition of four waves.

Poynting vectors on the phase relation of superposing waves. It is here assumed that changing the phase relation between the superposed waves causes a transition (in the observed pattern) from the situation when the maxima of two systems of mutually orthogonal standing waves coincide to the case when the nodes of two such systems coincide. Thus, the velocities of particles in such fields depend on the depth of modulation of the distribution of the time-averaged Poynting vector, as it is seen in Fig. 8. Second, the superposition of four waves linearly polarized in the plane of incidence results in forming so-called "cellular" structure in the resulting field distribution, see Fig. 9, which can be used for transfer (transporting) of the set of periodically positioned microparticles *as an entity* to a desired zone.

One considers a future deeper investigation of the peculiarities of motion of microparticles to reveal the coherent characteristics of the waves constituting certain spatial polarization distributions.

The use of strongly reflected test spherical particles provides obtaining more realistic notion on movement of particles in the field modulated in polarization in the incidence plane. So, the test particles are concentrated in zones (planes) of minima of the time-averaged Poynting vector and move along these planes. This situation reflects in the most adequate manner the processes of particle moving in the fields spatially modulated in polarization.

### **3.3 Experimental technique and results**

Direct experimental verification of the results of computer simulation is rather difficult. Spatial period of the polarization distribution resulting from superposition of plane waves meeting at right angle is less than a wavelength of the laser radiation of the visible range. In this case, diagnostics of optical currents presumes the using test particles (preferably spherical) of size much less than the period of polarization distribution. That is why, direct visualization and diagnostics of such particle currents is hampered.

Fig. 10. Experimental setup: L1, L2 - lasers; TS1, TS2 - telescopic systems; M1, M2, M3, M4 - mirrors; PW1, PW2 - half-wave plates for λ=635 nm; PP - plane-parallel plate; BS - beam-splitter; MO1, MO2, MO3 - microobjectives; C - cuvette with gold hydrosol; IF - interference filter at λ=532 nm; D1(0.7-diam), D2 - diaphragms; S - opaque screen; PD - photodetector; A - amplifier; ADC - analog-to-digit converter; PC - computer.

For verifying the results of above consideration and computer simulation, we have studied experimentally the influence of the field resulting from superposition of two plane waves meeting at right angle with various combinations of their states of polarization on the test particles. Experimental arrangement is shown in Fig. 10. To provide right angle between the beam axes, we use immerse-oil-microobjective 90x with NA 1.25. Two parallel linearly polarized beams converge at the focus of microobjective 1. If the electrical vectors of two beams are parallel, the intensity distribution as a set of interference maxima and minima is formed at the area of superposition of such beams. We used radiation from a semiconductor laser RLTMRL-III-635 (λ=635 nm). For that, the period of an interference pattern is 449 nm. For investigation of the influence of the field distribution with such period on the particles, particle's size must be much less than the mentioned period. We have used spherical particles of hydrosol of gold with diameter 40 nm, approximately. Hydrosol have been obtained following the standard technique [53], by mixing of chloroauric acid (H[AuCl4]) and sodium citrate (Na3Cyt). Let us describe the experimental conditions in more details. Weight concentration of gold particles in water was 5·10-6kg/m3. Experiment was carried out at room temperature, humidity 65% and air pressure 741 mm Hg. Note, our observation show that modest changing of the experimental conditions against ones mentioned here do not influence appreciably the results of experiment. As a matter of fact, we perform experiment with low-power of laser radiation (not exceeding 5 mW). In this case, one can neglect acoustic waves arising due to thermal action of laser radiation, which would become very important when one operates with high-power impulse laser beams. On these reasons, special precautions, such as use of anechoic chamber, are not undertaken in our study.

Periodical intensity distribution causes movement of the particles and formation of the periodical distribution of concentration of particles as the planes coinciding with interference minima of the intensity distribution at the area of superposition of two beams. These planes can be regarded as the analog of crystallographic planes in crystals. Direct visualization of particles and their currents is hampered due to small particle size. However, at planes of dense packing of particles self-diffraction takes place. We have observed this phenomenon for angles of meeting of two beams less than 40°. For right angle of meeting of the beams, the each self-diffracted beam propagates *along of* and *contrary to* the propagation direction of other of two superposing beams. Thus, it is impossible to discriminate the initial and self-diffracted beams. That is why, taking into account the Bragg law, we use, for diagnostics of periodical distributions of particles, the test laser beam with another wavelength, λ=532 nm. To form the same interference distribution (with the period 449 nm) with such wavelength the angle of meeting of two beams could be 72.6°. So, the angle of incidence of the probing beam must be 36.3° in respect to bisector of the writing beams. In this case, the Bragg law is fulfilled strictly for the probing beam. The mentioned angles are the angles of propagation in light-scattering media, in our case in water.

Two some shifted beams from a green laser (marked by thin line in Fig. 10) propagate in parallel to the beams of red laser (marked by thick line). The external green beam is stopped by the screen S, while the inner probing beam passes the microobjective MO1 and falls at the angle 36.3° into the area of interference extrema. Diameters of the focused beams of red and green lasers are approximately 12 µm and 10 µm, respectively. Glass cuvette C with gold hydrosol is placed at the area of interference pattern. Thickness of the cuvette walls 0.15 mm, and thickness of the swept volume is 18 µm. Oil immersion with refraction index 1.515 is placed between microobjectives MO1 and MO2. Microobjective MO2 is used for adjusting the optical arrangement and output the radiation diffracted on periodical distribution of gold particles.

For verifying the results of above consideration and computer simulation, we have studied experimentally the influence of the field resulting from superposition of two plane waves meeting at right angle with various combinations of their states of polarization on the test particles. Experimental arrangement is shown in Fig. 10. To provide right angle between the beam axes, we use immerse-oil-microobjective 90x with NA 1.25. Two parallel linearly polarized beams converge at the focus of microobjective 1. If the electrical vectors of two beams are parallel, the intensity distribution as a set of interference maxima and minima is formed at the area of superposition of such beams. We used radiation from a semiconductor laser RLTMRL-III-635 (λ=635 nm). For that, the period of an interference pattern is 449 nm. For investigation of the influence of the field distribution with such period on the particles, particle's size must be much less than the mentioned period. We have used spherical particles of hydrosol of gold with diameter 40 nm, approximately. Hydrosol have been obtained following the standard technique [53], by mixing of chloroauric acid (H[AuCl4]) and sodium citrate (Na3Cyt). Let us describe the experimental conditions in more details. Weight concentration of gold particles in water was 5·10-6kg/m3. Experiment was carried out at room temperature, humidity 65% and air pressure 741 mm Hg. Note, our observation show that modest changing of the experimental conditions against ones mentioned here do not influence appreciably the results of experiment. As a matter of fact, we perform experiment with low-power of laser radiation (not exceeding 5 mW). In this case, one can neglect acoustic waves arising due to thermal action of laser radiation, which would become very important when one operates with high-power impulse laser beams. On these reasons, special precautions, such as use of anechoic chamber, are not undertaken in our study.

Periodical intensity distribution causes movement of the particles and formation of the periodical distribution of concentration of particles as the planes coinciding with interference minima of the intensity distribution at the area of superposition of two beams. These planes can be regarded as the analog of crystallographic planes in crystals. Direct visualization of particles and their currents is hampered due to small particle size. However, at planes of dense packing of particles self-diffraction takes place. We have observed this phenomenon for angles of meeting of two beams less than 40°. For right angle of meeting of the beams, the each self-diffracted beam propagates *along of* and *contrary to* the propagation direction of other of two superposing beams. Thus, it is impossible to discriminate the initial and self-diffracted beams. That is why, taking into account the Bragg law, we use, for diagnostics of periodical distributions of particles, the test laser beam with another wavelength, λ=532 nm. To form the same interference distribution (with the period 449 nm) with such wavelength the angle of meeting of two beams could be 72.6°. So, the angle of incidence of the probing beam must be 36.3° in respect to bisector of the writing beams. In this case, the Bragg law is fulfilled strictly for the probing beam. The mentioned angles are

Two some shifted beams from a green laser (marked by thin line in Fig. 10) propagate in parallel to the beams of red laser (marked by thick line). The external green beam is stopped by the screen S, while the inner probing beam passes the microobjective MO1 and falls at the angle 36.3° into the area of interference extrema. Diameters of the focused beams of red and green lasers are approximately 12 µm and 10 µm, respectively. Glass cuvette C with gold hydrosol is placed at the area of interference pattern. Thickness of the cuvette walls 0.15 mm, and thickness of the swept volume is 18 µm. Oil immersion with refraction index 1.515 is placed between microobjectives MO1 and MO2. Microobjective MO2 is used for adjusting the optical arrangement and output the radiation diffracted on periodical distribution of gold particles.

the angles of propagation in light-scattering media, in our case in water.

Optical lengths of two legs of an interferometer BS-M2-M3-M4-MO2 are strictly identical. So, two beams from red laser are mutually coherent and interfere at the focus of the microobjective MO2. Placing a perfect plane-parallel plate PP of thickness 19 mm into one interferometer leg leads to disappearance of interference, while the corresponding optical path difference exceeds the coherence length of red laser. What is important, introducing the plate PP must no be accompanied with shifting beams into interferometer. In such a manner, one can control appearance and disappearance of interference extrema at the focus of MO1.

The change of the position of the plate leads to the change of the photodetector signal. Тhus, in the case of superposition of radiations from lasers L1 and L2 which are linearly polarized at the plane perpendicular to the figure plane removal of the plate PP results in increasing signal from a photodetector. It shows forming periodical spatial distribution of gold particles and appearance of the diffracted probing beam. The diffracted signal appears for radiation power of red laser more than 2 mW. However, it has been observed that for radiation power exceeding 50 mW, non-linear effects occur in light-scattering medium. So, gold particles absorb radiation and heat environment, acting as a thermal lens. That is why we have carried out our experiment for radiation power of red laser 5 mW. Radiation power of the probing beam was 0.5 mW, so that it can not affect gold particles.

If two beams of red laser are polarized in the figure plane (half-wavelength plate PW1 for λ=635 nm is inserted) and their convergence angle is equal to 90°, only polarization modulation takes place in the incidence plane. In this case the diffracted probing beam is present as well. The signal at the photodetector output with and without plane-parallel plate is shown in Fig. 11 b. The diffracted probing beam is present, but is approximately of half the intensity in comparison with the case illustrated in Fig. 11 a. This experimental result is also in accordance with the result of computer simulation. The spatially modulated in polarization field (in the plane of incidence) is correlated with concentration of the test particles at the planes of minima of the time-averaged magnitude of the Poynting vector, and particles move along these planes.

If two beams form red laser are linearly polarized, but one of them in the figure plane while another one perpendicularly to this plane (a half-wave plate PW2 for λ=635 nm is inserted), the diffracted probing beam is absent, cf. Fig. 11 c. This shows that at the focal plane where the beams from red laser superpose, the periodical distributions of gold particles are absent. This experimental result is also in agreement with earlier computer simulation [42]. In other words, there are no any ordered optical currents being liable to optical diagnostics, as it has been made in previous case.

Thus, temporal and space peculiarities of particle's motion in optical fields without intensity modulation, but only due to polarization modulation causing the spatial modulation of the time-averaged Poynting vector (depending on the degree of mutual coherence of superpose waves) opens up new feasibilities for the use of such field characteristics and the parameters of microparticles motion for estimating the temporal coherence of the tested field. Here we have demonstrated a possibility of influence of only the polarization factor on formation of optical currents in liquids by the use of the principles of spatial polarization modulation in the observation plane. Besides, we have shown the possibility of diagnostics of optical currents using test particles of nanoscale.

Fig. 11. Relative signal of a photodetector (a plate PP is inserted on 2 sec and then removed) in the case when radiation of red laser is linearly polarized: (a) both beams are polarized in the plane perpendicular to the figure plane; (b) both beams are polarized in the figure plane; (c) one beam is polarized at the figure plane, while another one is polarized perpendicularly to this plane.

The explained metrology of microstructure of optical fields may be extended, to all appearance, on polychromatic waves. The initial steps in this direction have been recently made in studies [52, 54 – 60].

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Fig. 11. Relative signal of a photodetector (a plate PP is inserted on 2 sec and then removed) in the case when radiation of red laser is linearly polarized: (a) both beams are polarized in the plane perpendicular to the figure plane; (b) both beams are polarized in the figure plane; (c) one beam is polarized at the figure plane, while another one is polarized perpendicularly

The explained metrology of microstructure of optical fields may be extended, to all appearance, on polychromatic waves. The initial steps in this direction have been recently

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