1.4.5.2. Elementary polarization singularities resulting from interference of circularly polarized beams

Obviously, the elementary polarization structures result from a superposition of orthogonally circularly polarized beams also. Remember, -point is associated with the vortex of one, to say the right, component, and the smooth in phase orthogonal (left) component, and vice versa.

Remember also that the complex amplitudes of the linearly polarized components decomposed in the basis of circularly polarized beams have the form:

(1.109)

In terms of the vibration phase and the polarization azimuth, Eqs. (1.109) take the form:

. (1.110)

Assume that some additional phase difference between the interfering beams, , appears under superposition of the beams. It follows from analysis of Eqs. (1.109) and (1.110) that position and characteristics of -points (the topological charge and index) do not change under such conditions of superposition. It also follows from these equations that both the azimuth of polarization, , and the vibration phase, , are changed by the quantities corresponding to .

For the points of a field at -contour, for which , Eqs. (1.109) and (1.110) are transformed to the form:

, (1.111)

and

. (1.112)

One can see from these equations that changing the total phase difference between the beams does not lead to changing of -contour, and the field’s vector at each point of the contour rotates at the angle . The direction of rotation of the azimuth of polarization (+ or – ) depends on the sign of and on what is the beam (clockwise or counterclockwise), in which such phase difference is introduced. It follows also from these equations that changing the phase difference by is accompanied with the changing of the azimuth of polarization by , i.e. the field turns out into initial state.

Thus, the difference between the ways of formation of the elementary polarization structures is reduced to the following. Changing the phase difference between the orthogonally linearly polarized components of a field leads to changing the form of -contours and positions of -points. On the other hand, changing the phase difference between the orthogonally circularly polarized components of a field leads to the rotation of the azimuth of linear polarization along -contour and to the corresponding rotation of the polarization ellipses inside the area limited by such a contour. Positions and characteristics of -points remain unchanged.

1.4.5.3. Experimental mOdeling of elementary polarization singularities

Experimental modeling of elementary polarization domains formed through interference of linearly polarized beams is performed in the arrangement shown in Figure 1.43 [80,81]. Linearly polarized laser beam enters the double Mach-Zehnder interferometer. A plate is placed in front of the beam-splitter 2 to transform the linearly polarized beam in the circularly polarized one. A computer-synthesized hologram 3 generated following the technique described in Refs. [38,39] is introduced in a one leg of the interferometer. Diffraction of a Gaussian beam at such hologram results in reconstruction of the singly charged vortex beam

at one of the first diffraction orders.

S

Figure 1.43. Arrangement for experimental modeling of the elementary polarization domains. 1 – -plate, 2,7,17 – beam-splitters, 3 – computer-generated “vortex” hologram, 4,5,6 and 14,15,16 – beam expanders, 8,12 – polarizers, 9 – piezo-mirror, 11,13 – mirrors, 18 – analyzer.

preading and filtration of such a beam using a collimator 4-6 results in formation of circularly polarized, almost isotropic vortex. This beam enters the interior interferometer . The crossed polarizers 8 and 12 are placed in the legs of the interferometer. In such a manner, the orthogonally linearly polarized vortex beams are formed. Piezo-mirror 9 provides fine control of the phase difference between the beams within the limits . Using a beam-splitter 10 and mirrors 9 and 11, one can change the distance between the centers of the vortex beams and control collinear propagation of them. The resulting field is mixed with the reference beam at beam-splitter 17, and one observes an interference pattern at the plane P. To visualize the vortex motion along -contour, the beam-splitter 17 is followed by the analyzer 18. A diffraction pattern from the rectangular aperture is used as the mark to fix the position of an interference forklet.

The experimental results are represented in Figure 1.44, where the interferograms of the linearly polarized projections of the resulting field are shown. Orientations of the axis of maximal transmittance of the polarizer are spaced by .

O

Figure 1.44. Experimental modeling of the elementary polarization domains limited by the closed -contours. (a)-(d), (e)-(h) correspond to different phase differences between the vortex beams; (a)-(d) illustrate the motion of the polarization projection’s vortex along -contour for any unknown phase difference motion ; (e)–(h) correspond to the phase difference approaching ; (i),(j) – -contours reconstructed from the experimental data.

ne can see from Figure 1.44 that only one vortex moves along the closed -contour, when the orientation of the analyzer is changed. The size and the form of this contour change following to the changing of the phase difference between the beams. Trajectories of the vortex for the each specified azimuth of the analyzer are shown in Figures 1.44(i),(j). Deviation of the form of -contour from a circle is caused by residual aberrations and, accordingly, by complicated form of the equiphase lines of the field of phases of the vortex beams. The same -contours are shown in Figures 1.44(a),(d).

The elementary polarization structures arising under superposition of circularly polarized beams can be also obtained in the arrangement shown in Figure 1.43, if the plate 1 is placed at the reference leg of the outer interferometer, to say immediately behind the mirror 13, and the polarizers 8 and 12 are replaced by the plates whose orientation provides formation of the clockwise and the counterclockwise circularly polarized beams. The computer-synthesized hologram 3 is introduced at one of the legs of the interferometer, to say behind the mirror 11, rather than behind the beam-splitter 2. In this case, the circularly polarized vortex beam is formed at the lower leg of the interferometer, while the orthogonally polarized smooth beam is formed at the upper leg.

T

Figure 1.45. Results of modeling of the elementary polarization structures by superposition of circularly polarized beams.

he results of modeling are represented in Figure 1.45. Figures 1.45(a)-(d) illustrate shifting the polarization projection’s vortex along -contour resulting from rotation of the analyzer 18. Figure 1.45(e) shows an interference forklet indicating the position of -point, when the channel of the interior interferometer forming a smooth beam is blocked. -point and -contour reconstructed from the experimental data are shown in Figure 1.45(f). One can see that -contour is not circular, and the position of -point does not coincide with its center, what is explained by residual aberrations of the beams and deviation of the intensity distribution of the input beam from Gaussian one.