1.5.3.2.2. Non-symmetrical distributions of amplitudes and phases of the interfering beams

Let us show that the positions of singular points of the averaged transversal component of the Poynting vector are shifted from the positions of -points, if the amplitude and phase distributions of the interfering beams are asymmetrical. This effect takes place for even the areas containing only one -point.

Singularity is shifted from -point, if the both interfering beams or even one of them are asymmetrical in the phase or amplitude distributions (hereinafter, simply: asymmetrical) in respect to the position of -point. The main factor determining the magnitude of a shift is the phase relations and amplitude ratio of the interfering beam, rather than the magnitude of asymmetry per se. Thus, to establish the regularities governing the magnitude of a shift of the Poynting vector’s singularity in respect of -point, it is sufficient to control a phase or intensity only for one of two beams. We will vary the parameters of the smooth beam. In this case, the averaged - and -components of the Poynting vector are represented as

, (1.178)

where are the derivatives on a phase of the smooth beam, is its amplitude, are the relative changes of the amplitude.

It follows from Eqs. (1.178) that the shift of zero of the transversal component of the Poynting vector is determined both by the gradient of a phase of the reference beam and by the gradient of the relative changes of its amplitude. The only difference is that the - and -changes of a phase affect the same component of the vector, while the changes of the amplitude affect the alternative component.

Assume the introduced changes be not too large, so that the linear approximation is valid for the amplitude of the beam and for its phase. This approximation lies in the basis of computer simulation.

Figures 1.80 to 1.82 illustrate the changes in the Poynting vector’s field corresponding to various magnitudes of asymmetry introduced in the reference beam. For the sake of comparability, the distributions of the

Figure 1.80. Shifting singularity of the transversal component of the Poynting vector resulting from the phase asymmetry in the reference beam. (a) – distribution of the modulus of the transversal component, (b) – distribution of the azimuth of the component;

the azimuth of the transversal component in fragments (c) coincides with the arrow directions, and the magnitude of the component’s module corresponds to the arrow length.

1 – «zero» asymmetry;

2 – «moderate» phase asymmetry of the smooth reference beam;

3 – «large» phase asymmetry of the smooth reference beam.

Figure 1.81. Shifting singularity of the transversal component of the Poynting vector resulting from asymmetry of the amplitude in the reference beam. (a) – distribution of the modulus of the transversal component, (b) – distribution of the azimuth of the component; the azimuth of the transversal component in fragments (c) coincides with the arrow directions, and the magnitude of the component’s module corresponds to the arrow length.

1 – «zero» asymmetry;

2 – «moderate» asymmetry of the amplitude of the smooth reference beam;

3 – «large» asymmetry of the amplitude of the smooth reference beam.

Poynting vector’s parameters of the field formed by symmetrical beams are represented in the row 1. Figures 1.80-1.82, (a) illustrate behavior of the modulus of the transversal component of the Poynting vector of the resulting field. Figures 1.80-1.82, (b) correspond to the behavior of the azimuth of the Poynting vector’s component. Figures 1.80-1.82, (c) illustrate the distributions of the modulus and of the azimuth. Orientation of the vector coincides with the arrow directions, and the arrow length corresponds to the magnitude of the component.

The influence of the phase asymmetry on the position of the Poynting vector’s singularity is represented in Figure 1.80. It follows from analysis of Figures . 1.80 (а)3-(с)3 that for some magnitude of asymmetry singularity of the transversal component can move from the position of -point and even cross -contour, viz. can move into the area with other type of elliptical polarization.

Figure 1.81 illustrates the shift of singularity resulting from amplitude asymmetry.

Figure 1.82. Shifting singularity of the transversal component of the Poynting vector resulting from asymmetry of both kinds. (a) – distribution of the modulus of the transversal component, (b) – distribution of the azimuth of the component; the azimuth of the transversal component in fragments (c) coincides with the arrow directions, and the magnitude of the component’s module corresponds to the arrow length.

1 – «zero» asymmetry;

2 – «moderate» asymmetry of the smooth reference beam;

3 – «large» asymmetry of the smooth reference beam.

Figure 1.82 represents the data of computer simulation verifying the shift of the Poynting vector’s singularity resulting from both kinds of asymmetry.

Thus, the phase and/or amplitude asymmetry arising at least in one of the interfering beams results in shifting the singularity of the transversal component of the Poynting vector and the point of applying the maximal averaged orbital momentum in respect of the position of -point. It is clear that such shift is intrinsic both to the elementary polarization cells and to the non-uniform in polarization fields of general form. Naturally, the corresponding asymmetry of the interfering beams can be introduced artificially, by forming the special smooth or vortex beams using, to say, the computer-synthesized hologram technique. Such holograms can be formed at any reversal carrier, such as spatial-temporal light modulator. In our opinion, this provides promising feasibilities for creation of bright non-uniform in polarization light trap with the controlled point of applying of the orbital angular momentum of a field.