2. Two-wave superposition for changeable degree of coherence of one component
An analysis of the spatial distribution of the averaged Poynting vectors shown in Fig. 1b
reveals the periodicity of this distribution, where the absolute magnitudes of the vectors are proportional to the lengths of lines shown in the figure. The lines corresponding to the singularities of the Poynting vector are shown in this figure by the indicated set of points [7
7. I. Mokhun and R. Khrobatin, “Shift of application point of angular momentum in the area of elementary polarization singularity,” J. Opt. A, Pure Appl. Opt. 10(6), 064015 (2008). [CrossRef]
9. A. Y. Bekshaev and M. Soskin, “M.S.Soskin Transverse energy flows in vectorial fields of paraxial light beams,” Proc. SPIE, 6729, 67290G, 67290G-12 (2007). [CrossRef]
A comprehensive explanation of the mechanisms for formation of such a distribution follows from considering both in statistical terms and with respect to the dynamics with the corresponding comments, which are formulated below:
2. The spatial distribution of the averaged Poynting vectors (Fig. 1b
) depicts the directions (trajectories) of energy transfer.
3. The yellow stripes in Fig. 1b
indicate the regions where the Poynting vector magnitudes are close to zero. In points along the vertical line of the yellow stripes, the Poynting vector magnitude equals zero so that they form a singular manifold for the Poynting vector field. In these singular points the optical field intensity is non-zero but the energy does not moves. This follows from the fact that vector H vanishes due to interference in this arrangement (90°-superposition of plane waves) where strictly coaxial vectors of equal amplitudes and with opposite phases are associated with two superimposed plane waves
4. Homogeneous intensity distribution and periodical spatial modulation of the Poynting vector simultaneously realized in the analyzing area have previously been discussed within the framework of Refs [10
10. 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]
11. O. V. Angelsky, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “Feasibility of estimating the degree of coherence of waves at the near field,” Appl. Opt. 48(15), 2784–2788 (2009). [CrossRef] [PubMed]
]. Spatial polarization modulation in the analyzing area is caused by superposition of the
field components with a change of the phase difference from point to point (Fig. 1a
). The photodetector only registers the intensity
. The sum of the average squared amplitudes of the electrical fields is constant in the analyzing area, though the state of polarization changes. The Poynting vector is defined by the vector product
. One observes the dependence of the result on the phase relation between vectors
through the vector magnitude and its direction. This relation changes from point to point in the analyzing area and results in polarization modulation. An obvious explanation for this follows from consideration of the vector product of the components of vector
components) with vector
. Both the magnitudes of projections
and their phases change from point to point in the observation plane. As a consequence, the vector product as well as the Poynting vector changes.
Simulation of particle motion was carried out for conducting particles affected by the Poynting vector, in correspondence with Ref [1
1. M. V. Berry, “Optical currents,” J. Opt. A, Pure Appl. Opt. 11, 094001 (2009). [CrossRef]
]. Two kinds of conducting particles were considered, viz.
completely absorbing and completely reflected ones. The difference between absorbing and reflecting particles has been taken into account in computing the forces acting on a particle surface. For absorbing particles, the force acting on the surface point coincides in direction with the Poynting vector and is proportional it to its modulus. In contrast, for reflecting particle if
is the incident Poynting vector and
is the reflected Poynting vector, the force is directed along
and its numerical value is proportional to
. The influence of the particles on the field itself and other fine effects were not taken into account due to complexity of the corresponding computations at this stage.
If one knows the forces affecting the surface of spherical particle at points measured from the center of mass, then its motion can be decomposed into sliding motion of the center of mass and rotational motion around the axis passing this center. The corresponding equations have the form:
are the mass of the spherical particle and its radius, respectively, η
is the viscosity of the liquid,
is the velocity vector of sliding motion of the mass centre, and
is the vector of angular velocity of the spherical particle. Initial conditions are
= 0 and
= 0. The second term on the right side of Eq. (1)
is the force due to the Stokes viscid friction. The second term on the right side of Eq. (2)
is the momentum of force of viscid friction due to rotation.
Rotational motion of a spherical particle does not affect essentially its sliding motion. That is why it is not taken into account in simulation. The tracking of the particle motion was divided into small time intervals for which the resulting force was considered constant. Elementary motion of a particle was computed for these intervals using Eq. (1)
. The sum of elementary motions of a particle results in the trajectory of the particle. Interaction of colliding particles was assumed be elastic.
The results of simulating the motion of particles embedded in the field of the considered distribution of the Poynting vector is shown in Fig. 2
Illustration of velocities of microparticles in a 2D distribution of the averaged Poynting vectors formed in arrangement Fig. 1a
for completely mutually coherent waves. Red and green axes (0.5 µm of length) correspond to x- and z-coordinates, respectively. Media 3
(see also Media 3
). We have here 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, as seen in Fig. 2
and Media 3
If the degree of mutual coherence of the superimposed waves equals 0.2, the spatial distribution of the averaged Poynting vectors becomes more homogeneous, the modulation depth decreases considerably, and the velocities of the microparticles become almost identical as shown in Fig. 3
Illustration of microparticles motion velocities in a 2D field distribution of the averaged Poynting vectors formed by the arrangement in Fig. 1
for a degree of mutual coherence of the components equal to 0.2. Red and green axes (0.5 µm of length) correspond to x- and z-coordinates, respectively. Media 4
and Media 4
When the degree of mutual coherence equals 0.5, the relative velocities of the microparticles along the same trajectories are half compared with velocities in case of full mutual coherence of the superposed waves, see Fig. 4
Illustration of microparticles motion velocities in the 2D field distribution of the averaged Poynting vectors shown in Fig. 1
for the degree of mutual coherence of the components equal to 0.5. Red and green axes (0.5 µm of length) correspond to x- and z-coordinates, respectively. Media 5
and Media 5
. One observes the dependence of the coherent properties of the superimposed waves on the motion of the microparticles with constant size and shape in media with constant viscosity. These differences in velocities of motion of microparticles may be explained physically in the following manner: Increasing the amount of incoherent radiation in the resulting field distribution causes a decrease of the modulation depth of the Poynting vectors spatial distribution, as well as in a decrease of the resultant force magnitude along the lines of energy transfer, which eventually induces the motion of the microparticles.
3. The superposition of four waves for changeable degree of coherence of one component
In the case of the superposition of four mutually coherent waves involving two sets of counter-propagating plane waves of equal intensities, linearly polarized in the plane of incidence and oriented at an angle of 90° with respect to each other, cf. Figure 5a
(a) Arrangement of superposition of four plane waves. (b) 2D distribution of the averaged Poynting vectors resulting from the superposition of four waves shown in Fig. 5a
. . Red and green axes (0.25 µm of length) correspond to x- and z-coordinates, respectively.
, the electric and magnetic vectors of the resulting field are written in the form:
are x, y and z unit normal vectors, respectively, i
= 1,2,3,4 is the number of the wave, j = x,y,z
are the vector components,
are amplitudes of the corresponding components of electrical and magnetic vectors, respectively,
are the wave vector and initial phase of the corresponding plane wave, and
is the position vector.
We consider two configurations:Initial phase of all four waves equals zero:
Initial phase of waves one to three equals zero, while the initial phase of the wave four isπ
Partial coherence of the beams is taken into account through the modulus of the complex degree of coherence [6
6. O. V. Angelsky, S. G. Hanson, C. Yu. Zenkova, M. P. Gorsky, and N. V. Gorodyns’ka, “On polarization metrology (estimation) of the degree of coherence of optical waves,” Opt. Express 17(18), 15623–15634 (2009). [CrossRef] [PubMed]
An example of the spatial distribution of the averaged Poynting vectors is shown in Fig. 5b
The 2D periodical distribution of the Poynting vectors is evident. As in the previous case, the lengths of the arrows in Fig. 5b
are proportional to the local Poynting vector 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.
In order to compare the influence of the temporal and spatial parameters of coherence on the motion of the microparticles, we have analyzed maps of the averaged Poynting vector with a superposition of four plane waves over a large area. During this, we have tracked the microparticles’ motion. This reveals the dependence of the microparticles’ velocities on the phase difference of the superposing beams. So, in the case of pair-by-pair four opposite-in-phase superimposed beams, particles become motionless, cf. Figure 6
Spatial distribution of the averaged Poynting vectors for four-wave superposition of components with initial phases from Eq. (9)
, illustrating a steady-state situation for the resulting field. (Time in sec.). Red and green axes (0.5 µm of length) correspond to x- and z-coordinates. Media 6
. Here, “opposite-in-phase” describes the situation when the wave phases are described by Eq. (10)
. This follows from the presence of the minimum of the modulation depth in the spatial distribution of the Poynting vector. Gradually decreasing the degree of coherence of one of the superimposed beams causes revival of the particle motion with increased velocity as the degree of coherence of one of the beams decreases, see Fig. 7
Illustrating the movement of a microparticle when of one of four waves with a mutual degree of coherence of 0.5 with other three waves takes part in the forming of a spatial distribution of the averaged Poynting vectors. (Time in sec.). Initial phases defined from Eq. (9)
. Red and green axes (0.5 µm of length) correspond to x- and z-coordinates, respectively. Media 7
If the phase relations between four superimposed 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. This conclusion is supported by two video fragments corresponding to complete coherence of the beams (Fig. 8
Illustration of moving microparticles, when the spatial distribution of the averaged Poynting vector is formed by four mutually coherent waves with phases defined from Eq. (10)
. Time in sec.). Red and green axes (0.5 µm of length) correspond to x- and z-coordinates, respectively. Media 8
) and the situation when one of the beams is incoherent (Fig. 9
Illustration of moving microparticles, when the spatial distribution of the averaged Poynting vector is formed by four beams, one of which is incoherent with respect to the other three. (Time in sec.). Initial phases defined from Eq. (10)
. Red and green axes (0.5 µm of length) correspond to x- and z-coordinates, respectively. Media 9
). This situation is realized when two orthogonal systems of the standing waves are such that their maxima coincide. This case we refer to as co-phasing of 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 averaged Poynting vectors on the phase relation of superposing waves as illustrated in Figs. 6
. It is here assumed that the phase relation between the superimposed waves is described by Eq. (9)
. Thus, the velocities of particles in such fields depend on the depth of modulation of the distribution of the averaged Poynting vector, as it is seen in Fig. 7
. 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, which can be used for transfer (transporting) as an entity of the set of periodically positioned microparticles to a desired zone, see Fig. 10
Fig. 10 “Cellular” distribution of the potential traps for microparticles in the case of superposition of four waves.
Here we represent the results of a more profound investigation of the peculiarities of motion of microparticles in order to reveal the coherent characteristics of the waves constituting certain spatial polarization distributions.
The use of strongly reflecting spherical test particles provides the means for more realistic notion on movement of particles in a field modulated in polarization in the incidence plane. Accordingly, the test particles are concentrated in zones (planes) of minima of the averaged Poynting vector and move along these planes, cf. Figure 11
Illustration of the direction and mechanism of energy transfer for superposition of two linearly polarized waves of equal intensity; here the waves are polarized in the plane of incidence, meeting at right angle. (Time in sec.). Red and green axes (0.5 µm of length) correspond to x- and z-coordinates, respectively. Media 10
. This situation reflects in the most adequate manner the processes of particle movement in spatially modulated fields modulated with respect to polarization.
4. Experimental technique and results
Direct experimental verification of the results of computer simulation is rather difficult. The spatial period of the polarization distribution resulting from orthogonal superposition of plane waves is less than a wavelength of the laser radiation in the visible range. In this case, diagnostics of optical currents presumes the use of 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.
For verifying the results of the computer simulations, 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. The scheme for forming a controllable distribution is shown in Fig. 12
Fig. 12 Arrangement for forming the desired spatial interference distribution. Polarization of both crossed beams is linear, perpendicular to the figure plane.
. Two parallel, linearly polarized beams converge at the focus of microobjective. According to [12
12. M. Born, and E. Wolf, Principles of optics. New York: Cambridge University Press (1999)
], two quasi-plane waves are formed at the focus. To provide a right angle between the beam axes, we use an immerse oil microobjective 90x with NA 1.25. If the electrical vectors of two beams are parallel, the intensity distribution as a set of interference maxima and minima is formed in the area of superposition of such beams. We used radiation from a semiconductor laser RLTMRL-III-635 (λ = 635 nm). Therefore, the period of the interference pattern is 449 nm. For investigating the influence of this field distribution on the particles, the particles’ size must be much less than the mentioned period. We have used spherical particles of hydrosol of gold with an approximate diameter of 40 nm. Hydrosol have been obtained following the standard technique [13
13. J. Turkevich, P. C. Stevenson, and J. Hillier, “A study of the nucleation and growth processes in the synthesis of colloidal gold // J,” Discuss. Faraday Soc. 11, 55–75 (1951). [CrossRef]
], by mixing of chloroauric acid (H[AuCl4]) and sodium citrate (Na3Cyt).
Periodical intensity and Poynting vector distributions cause movement of the particles and formation of a periodical distribution of concentration of particles in the planes coinciding with the interference minima of the intensity distribution. These planes can be regarded as the analogy of crystallographic planes. Direct visualization of particles and their currents is hampered due to the small particle size. However, at planes of dense packing of particles self-diffraction takes place. We have observed this phenomenon for crossing angles of the two beams less than 40°. For right crossing angle between the beams, each of the self-diffracted beams propagate both along of and contrary to the propagation direction of the two superposing beams. Thus, it is impossible to discriminate between the initial and the self-diffracted beams. That is why, taking into account the Bragg law, we use a test laser beam with a different wavelength, λ = 532 nm for diagnostics of the periodical distributions of particles. To form the same interference distribution (with the period 449 nm) with this wavelength, the crossing angle of the two beams should be 72.6°. So, the angle of incidence of the probing beam must be 36.3° with respect to the bisector of the writing beams. In this case, the Bragg law is strictly fulfilled for the probing beam. The mentioned angles are the angles of propagation in the light-scattering media, in our case water.
The experimental arrangement is shown in Fig. 13
Fig. 13 Experimental setup: L1, L2 - lasers, TS1, TS2 - telescopic systems, M1, M2, M3, M4 - mirrors, PW1, W2 - half-wave plates for λ = 635 nm, PP - plane-parallel plate, MO1, MO2, MO3 - microobjectives, C - cuvette with gold hydrosol, IF - interference filter at λ = 532 nm, D1, D2 - diaphragms, S - opaque screen, PD - photodetector, A - amplifier, ADC - analog-to-digital converter, PC - computer.
. The radiation emitted from lasers L1 and L2 is linearly polarized in the plane perpendicular to the figure plane and passes through the telescopic systems TS1 and TS2, respectively, each of which are formed by two objectives with a micron-size diaphragms at the focuses. Two parallel beams of diameter 0.6 mm are formed in this arrangement. A beam-splitter BS divides a beam of a red laser into two beams, and the system of mirrors M2-M3-M4 provides strict parallelism between them. An intensity distribution with a set of planes of interference maxima and minima is formed at the back focus of microobjective MO1. Two slightly shifted beams from a green laser propagate parallel to the beams of the red laser. The external green beam is stopped by the screen S, while the inner probing beam passes the microobjective MO1 and falls at an angle 36.3° into the area of the interference extrema. The diameters of the focused beams for the red and green lasers are approximately 12 µm and 10 µm, respectively. Glass cuvette C with gold hydrosol is placed in the area of the interference pattern. The thickness of the cuvette walls is 0.15 mm, and the thickness of the swept volume is 18 µm. Oil immersion with index of refraction 1.515 is placed between microobjectives MO1 and MO2. Microobjective MO2 is used for adjusting the optical arrangement and transmit the light onto the periodical distribution of gold particles. This radiation passes the interference filter IF with the maximal transmittance at 532 nm (excluding the radiation of the red laser), 0.7mm-diam diaphragm D1, microobjective MO3, microdiaphragm D2, and eventually falls on the photodetector PD. The signal from the photodetector is sent through an amplifier A and an ADC to a PC.
The optical path lengths of the two legs in the interferometer BS-M2-M3-M4-MO2 are strictly identical. As a result, the two beams from the red laser are mutually coherent and interfere at the focus of the micro-objective MO2. Placing a perfect plane-parallel plate PP of thickness 19 mm into one leg in the interferometer leads to disappearance of interference, as the corresponding optical path difference exceeds the coherence length of the red laser. It is important that the introduction of the plate PP must not be accompanied with shifting beams in the interferometer. In this manner, one can control the appearance and disappearance of interference extrema at the focus of MO1.
The experimental technique used here is as follows. If radiations from lasers L1 and L2 are linearly polarized in the plane perpendicular to the figure plane, one records the signal from the photodetector PD, cf. Figure 13
. We used a photomultiplier with and without a plane-parallel plate PP mounted as previously described. In the first case, the periodical interference planes are absent; so, the diffracted light is absent also. The signal from a PD is caused by Bragg scattering of the probing beam from the homogeneous distribution of gold particles. Removal of the plate PP results in an increased signal from the photodetector. This proves the formation of a periodical spatial distribution of gold particles and appearance of the diffracted probing beam (Fig. 14a
Fig. 14 Relative signal for 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 diffracted signal appears for the radiation power of the red laser exceeding 2 mW. However, for radiation power exceeding 50 mW, non-linear effects occur in the light-scattering medium. As a result, gold particles absorb radiation and the heated environment will act as a thermal lens. Thus we have carried out our experiments with radiation power of 5 mW for the red laser. The power of the probing beam was 0.5 mW, thus not affecting the gold particles. The experimental results shown in Fig. 14a
, where radiations of lasers L1 and L2 are polarized in the plane perpendicular to the figure plane, support the computer simulations.
If two beams form the red laser are linearly polarized, but one of them is polarized in the figure plane while another one is polarized perpendicularly to this plane (a half-wave plate PW2 for λ = 635 nm is inserted), the diffracted probing beam is absent, Fig. 14c
. This shows that at the focal point where the beams from the red laser are superimposed, the periodical distributions of gold particles are absent. This is supported by computer simulation, see Fig. 15
Illustration of the directions of oscillations of the vectors
for superposition of two waves of equal intensity one of which is linearly polarized in the plane of incidence and another perpendicularly to this plane. The crossing angle between the two beams equals 90°. Red and green axes (0.25 µm of length) correspond to x- and z-coordinates, respectively. Media 11
. This media file shows that discrimination of any periodical structure of spatial concentration of particles is absent in the averaged Poynting vector causing the modulation of spatial particle concentration. In other words, there are no ordered optical currents being noticed by optical diagnostics, as it has been evident in the previous case.