1.5.3.2.3. Experimental proving of the existence of the orbital momentum in the vicinity of -point

Elementary polarization cell with -point at the center of the area bounded by the closed -contour was formed by a superposition of the circularly polarized vortex beam and the orthogonally circularly polarized reference wave with Gaussian intensity distribution (see subsection 1.4.5).

T he sign of the topological charge of the vortex did not coincide with the sign of the handedness factor of the vortex beam. In this case, the distribution of the polarization characteristics of the resulting field is similar to the distribution shown in Figure 1.83.

P

Figure 1.83. Distribution of the polarization characteristics within the elementary polarization cell of a field. Grey area corresponds to the region with counterclockwise polarization. The topological charge of C-point is = , and its sign is the same as the sign of the topological charge of the vortex beam.

roceeding from the consideration in subsection 1.5.3.2.1.b, one expects the orbital momentum of the field in the vicinity of -point.

As the experimental formation of the elementary polarization cell is easy, the main problem in verification of the theoretical predictions consists in a proper choice of the “indicator” of the existence of the orbital momentum in the tested field. Unfortunately, the choice of the practicable techniques for identification of the orbital momentum is rather restricted. The most practicable technique is based on the phenomena of transfer of the orbital momentum from the field to some mechanical system [97,98]. Such identification of the orbital momentum is based of the following facts:

1. It is known [99] that focusing of a laser beam results in formation of an optical trap, which is able to capture a microobject and to hold it.

2. The captured microobject is rotated, if the field possesses the spin or the orbital momentum [97]. The magnitude of such momentum is the main factor determining the rotation frequency, and the direction of rotation is determined by the sign of the momentum.

3. In general case, the focused beam possesses both the spin and the orbital momentum. The rotation frequency of a microobject is maximal, when the signs of both momenta coincide, and it is minimal (perhaps, frequency vanishes or is altered in direction), when these signs are opposite, i.e. the spin momentum compensates the orbital one.

Thus, focusing a non-uniform in polarization beam formed as a superposition of circularly polarized beams results in non-uniform in polarization optical trap, which is able to capture and rotate a micoobject. For that, the characteristics of rotation are controlled by changing of the parameters of the interfering beams, such as the sign of the vortex beam ar its handedness factor.

E

Figure 1.84. Experimental arrangement for observing the orbital momentum of the field of a polarization trap.

1 – He-Ne laser; 2,8,10 – beam-splitters; 3,6 – -plates; 4,5 – mirrors; 7 – vortex computer-synthesized hologram; 9 – analyzer; 11,14 – microobjectives; 12 – sample with microparticles; 14 – green filter; 15 – CCD-camera; 16,17 – illuminating system.

xperimental arrangement used for the identification of the orbital momentum of the field, see Figure 1.84, is similar to the arrangement described in subsection 1.4.5, being different from it in that the devices for the beam focusing and for observation of the affect of the field on the microparticle are introduced at the output of the interferometer used for the forming the elementary polarization singularity.

A linearly polarized beam from a He-Ne laser enters the Mach-Zehnder interferometer (elements 2 to 8). This beam is transformed into orthogonally circularly polarized beams using the -plates 3 and 6. One of them passes through the vortex computer-synthesized hologram 7; the circularly polarized vortex 4 is reconstructed by the hologram.

Non-uniform in polarization field containing -point is formed at the interferometer output.

Further, the resulting field is focused by the microobjective 11 into the plane of the sample with micrparticles 12. The result of affect of the beam on micoparticles is detected using the optical system 13 and 14 and CCD-camera.

We used for the forming of the optical trap a microobjective with the unitary aperture. The transversal size of the trap was 8 to 10 .

To determine the polarization characteristics of the resulting field, we choose the horizontal experimental arrangement. In this case, the acting beam only passes through the optical surfaces without reflection. So, the polarization structure of the beam at the plane of the sample is the same as at the output of the interferometer.

The sign of the orbital momentum (the direction of the field action at the transversal plane) can be easily controlled by choosing the diffraction order of the vortex hologram 7. It is known [37-39] (see also subsection 1.2.4) that the vortices formed in the positive and the negative diffraction orders differ in the sign of the topological charge.

Figure 1.85 illustrates the intensity distribution of the circularly polarized components of the resulting field. The analyzer 9 can be introduced in the interferometer to visualize the polarization modulation into the trap. The intensity distributions of the linearly polarized projections o f the resulting field for different orientations of the analyzer are represented in Figure 1.86. Dark spots at the trap boundary in all fragments correspond to the position of the vortex of the polarization projection (shown as white spots in fragments of 1.86).

T

a b

Figure 1.85. Intensity distributions of the circularly polarized component of the resulting field: (a) – circularly polarized vortex beam, (b) – smooth reference beam.

hese points identify also the coordinates of the points of the Nye disclinations moving along -contour due to temporal changes of the vector field. One can see from Figure 1.86 that intensities of the vortex and the smooth beams are chosen in such a manner that -contour occurs at the boundary of the trap. As so, the area with one magnitude of the handedness factor (correspondingly, with one “direction” of the spin moment) is realized anywhere within the trap. The coordinates of -point can be determined as the coordinates of the point inside the trap with the same intensity for all polarization projections.

T

Figure 1.86. Intensity distributions of various polarization projections of the trap’s field. The axis of analyzer rotates counterclockwise. Grey line corresponds to the experimentally found out -contour. White points at -contour indicate the positions of the vortices in the polarization projections. Angular spacing of rotation of the analyzer axis is 30 deg, approximately.

hus, we have implemented the polarization cell with the orbital momentum determined by the sign of the topological charge of the vortex beam, and with the one kind of “direction” of spin angular momentum over the area of the trap with significant intensity. Density of the spin angular momentum is decreased from -point to -contour as a consequence of diminishing eccentricity of the polarization ellipses. So, one can conclude that the area, within which the influence of the spin angular momentum is significant, is close to -point, and this area is mush smaller than the optical trap. Changing the sign of the orbital angular momentum must affect the characteristics of the trapping of a microobject comparable in size with the trap. On these reasons, we use the microparticles of in oil. Behavior of the captured particle is illustrated in Figures . 1.87-1.89.

One can see from Figure 1.87 that the captured particle rotates clockwise. The period of rotation is 4 to 5 sec.

Figure 1.88 illustrates the situation when the sign of the vortex forming the trap is altered, what corresponds to the changing of the sign of the orbital momentum. The particle rotates counterclockwise much more slowly. The period of rotation is 8 to 10 sec. This difference in rotation periods in two cases can be explained in the following way. In the first case, the spin momentum affects the particle in the same direction as the orbital momentum, while changing the topological charge of -point results in compensating the orbital angular momentum by the spin one.

a b c d

Figure 1.87. Rotation of relatively large transparent particle due to the orbital angular momentum of the field; clockwise rotation.

a b c d

Figure 1.88. Changing the direction and speed of rotation of the captured particle due to the changing of the sign of the orbital angular momentum.

a b c d

Figure 1.89. Rotation of a small absorbing (dark) particle captured by the diffraction ring encircling the polarization trap; clockwise rotation. The most bright part of the diffraction ring is depicted by the white arrow in fragment (b).

The result shown in Figure 1.89 illustrates the influence of the orbital angular momentum on a small absorbing particle captured by a dark diffraction ring encircling the polarization trap. Unfortunately, one can see in this figure only the part of a bright diffraction ring, while intensity at the center of the trap exceeds considerably intensity of the ring. Moreover, intensity is changed considerably along the ring. As a consequence, the dynamical range of the used CCD-camera did not provide reproduction of all levels of intensity, in correspondence with Figure 1.89.

One can see from Figure 1.89 that the particle rotates along the “main” trap area. Such character of the particle rotation can be explained only by the presence of the orbital momentum of the field. We used particles with the diameters from 2 to 4 , and the rotation period was changed from 0.5 to 1 seconds.

We would like to emphasize on the characteristics of the particle rotation illustrate in Figure 1.87. It follows from the figure that the center of the particle rotation is obviously non-coinciding with the center of the optical trap. In means that the Poynting vector’s singularity is shifted from the position of -point. Such shifting becomes understood from Figure 1.85, from which it follows that the interfering beams possess, at least, amplitude asymmetry. It follows from the consideration in subsection 1.5.3.2.2 that it must lead to the shifting of the point of applying of the maximal averaged orbital angular momentum.

The existence of the orbital angular momentum in the vicinity of -point is interesting not only from the fundamental point of view, but it can be also used in applications, viz. for creation of bright polarization traps with the controlled orbital momentum.