1.4.2. Vortices of phase difference. Sign principle for a vector field

1.4.2.1. Field decomposition into orthogonally linearly polarized components

It has been shown above that polarization singularities and disclinations are interconnected. Disclinations occur at -contours. Moving disclinations pass the each point of the -set [3,63]. At the same time, topology of disclination depends on the presence and distribution of -points at the area circumscribed by -contour [75]. One can expect that just disclinations govern the behavior of polarization singularities. Thus, disclination and polarization singularities are strongly interconnected, being mutually determined by each other. However, as it was mentioned above, disclinations are unobservable directly in optical experiment due to too rapid temporal changing the field.

For this reason, it is expedient to consider the system of any stationary singularities (analogous to disclinations), which would replace the system of temporal point defects of the field, being connected in the same manner with the - and - singular sets as the set of disclinations.

In other words, to construct a new system of singularities one must change the temporal characteristics by the spatial ones, as it is made in classical optics in description of elliptical polarization in terms of amplitudes of orthogonal components and phase difference of these components [28,78]. A new system of singularities must be connected with the systems of singularities of orthogonal components, which are analyzed within the scalar approach. As it was mentioned above, such system of scalar vortices is constituted by the net of phase vortices that comprehensively determine topological structure of the components, which is their turn determine behavior of amplitude and phase. As the each of orthogonal components is characterized by its own vortex net, one can expect that the vector field may be characterized by a superposition of them, i.e. by the “complete” system of phase vortices. We will refer to an analysis of such system as the vortex analysis of a vector field. The aim of this analysis is to establish an interconnection among the complete system of vortices and the polarization singularities and disclinations of a vector field. Emphasize again, in contrast to disclinations, the phase vortices of the orthogonal components are admissible to direct observation by use, for example, interferometric techniques [8-10,36-38].

Let us start from demonstration that the vector field in a far zone can be characterized by the common net of new optical vortices, namely the vortices of phase difference. Such approach leads the vector problem to the scalar one.

1.4.2.2. Principle of the vortex analysis of vector fields

Let us show that the vortices of phase difference unambiguously determine topology of a vector field, and that it provides feasibilities for establishing the interconnection among the complete system of phase vortices, on the one hand, and the polarization singularities of a vector field, on the other hand.

Consider electromagnetic monochromatic wave free propagating along -axis. It is known, the electrical vector of the field can be represented as [28]

, (1.57)

where is the complex vector:

. (1.58)

At far field, one can neglect -component of the electrical field and reduce the consideration to the analysis of the transverse field’s component . are smooth functions of two coordinates, , and is described as

, (1.59)

where are the unitary vectors; , are the complex amplitudes of the orthogonal field components; and are the phases and amplitudes, respectively.

Propagation of the field along -axis is reduced (within to ) to the scaling with any coefficient associated with the distance from the scattering object [54,92]. Note, the far-field approximation is not of fundamental nature, and one can easily extend this consideration to the case, when all three Cartesian coordinates are relevant.

Let us specify any Cartesian coordinates at the observation plane and introduce the evaluation function:

, (1.60)

where , .

Eq. (1.60) determines amplitude of some scalar field, whose phase coincides with the phase difference of the orthogonal components. The field (1.60) can be implemented, for example, holographically. The vortices of the field are at the same points as the amplitude zeroes of the orthogonal field components, as at these points, then or vanishes. As a consequence, the phase difference at these points is undetermined, i.e. these points are the singular points of the phase difference. Moreover, these points can be regarded as the vortices of the field . Thus, topology of a phase of the field determines comprehensively topology of the phase difference of the vector field, .

The condition of existence of amplitude zeroes of the field , obviously, takes the form

, (1.61)

quite similarly to the case of a scalar field.

The solutions of the first equation determine the lines along which or ., i.e. the lines along which the field is linearly polarized. So, these solutions correspond to the s-contours. The curves associated with the second equation we will refer to as “ -contours”. Along these lines, equals , and at the points, where , -points occur. Hereinafter, we refer any curve along which the phase difference is constant as the contour of such phase difference.

Note, the phase difference along -contour jumps by by crossing the vortex. Similarly to the wave front dislocations, the phase difference vortices can be characterized by two types of the topological indices, namely, by the Poincare index, , and by the topological charge, . Naturally, , similarly to the case of phase vortices.

The magnitude of the topological charge of the phase difference vortices is connected with the magnitude of the topological charge of the vortices supporting by the orthogonal field components. Let the phase difference to be calculated as , i.e. the phase of the -component is subtracted from the phase of the -component. Then, this interconnection is determined as

. (1.62)

It follows from Eq. (1.62) that the topological charge of the phase difference vortex coincides with the charge of the vortex associated with the orthogonal component, if the phase difference vortex produced by the vortex of the -component, and it is of the opposite sign in respect to the component vortex, if it is produced by the vortex of the -component.

Hereinafter, we will indicate the component vortices as ( and , respectively), and the phase difference vortices as (* and ~ being associated with the topological charges +1 and –1, respectively).

Note, the saddle points of the field of phase difference are the crossings of the only pair of the line of the same phase difference (topologically stable structure); as a consequence, and .

Applying the approach used in subsection 1.3.5 for analyzing the field of phase difference, one can establish the following regularities regarding conservation of the topological invariants:

1. The number of the phase difference vortices with the topological charge of any sign equals the number of such vortices of the opposite sign:

. (1.63)

2. The number of the saddle points of the field of phase difference is related to the number the extrema of phase difference and vortices as

. (1.64)

Note, the number of the extrema of the field of phase difference is much less than the number of vortices and saddles, as it immediately follows from the properties of the evaluating function and interpretation of it as a scalar field. (The number of vortices (saddles) of a scalar field, on the one hand, and the number of the phase extrema, on the other hand, relate as , see subsection 1.3.5.

From interpretation the function as a scalar field, one can also conclude that the sign principle proves to be justified for its vortices. Correspondingly, one can formulate the sign principle for the field of phase difference, which is analogous to the sign principle governing the behavior of phase vortices [80,81]:

a) there are even number of the field of phase difference at any closed -contour; for that, the adjacent vortices are of the opposite signs of the topological charge (*, ~);

b) if the adjacent phase difference vortices are produced by the phase vortices of the only component, these phase vortices are of the opposite signs (+, –).

The sign of any phase difference vortex determines the sign of all other vortices; in part, changing the sign of any vortex results immediately into changing the signs of all phase difference vortices.

Positions of - and -contours can be determined, alternatively, proceeding from the idea of superposition of the vortex nets associated with the orthogonal field components.

It was shown above (see subsection 1.3.1) that in the scalar case zeroes of the real and imaginary parts of the complex amplitude as the functions of spatial coordinates determine into observation plane two sets of continuous curves (the lines , , ). Cross-sections of these lines fix the net of vortices of the field. Thus, one have two nets of the phase vortices into the vector field, namely, one net for the each orthogonal component and .

Superposition of such nets is illustrated in Figure 1.25(a). The nets of vortices associated with - and -components are shown by bold and thin solid lines, respectively. Solid lines and dashed lines with crosses (lines and ) determine the set of points where the real and the imaginary parts of the complex amplitudes of the orthogonal field components vanish. The lines and for the each component of the electrical field divide the observation plane into the areas within which a phase changes within (in Figure 1.25(a), these areas are indicated by roman numerals). These are just the equiphase lines with a phase multiple of .

Phase difference of the complex amplitudes of the orthogonal components =0 is only achieved, where the regions are superimposed, , , , (in Figure 1.25(a), these crossings are indicated by unsaturated gray), and the points of the field with phase difference can only occur at the crossings of the areas, | , , , (such regions are show by saturated gray). In the same manner one can determine the areas where a phase difference is achieved (in Figure 1.25(a), these regions

Figure 1.25. Superposition of the nets of the component vortices and the net of phase difference vortices.

a) superposition of the vortex nets for the orthogonal field amplitudes. Solid bold lines are the lines , similar lines with crosses – lines ; solid thin lines are the lines and similar lines with crosses – lines ; are - and -vortices, respectively. The boundaries of phase changing within the regions limited by the lines , are depicted by roman numerals. The correspondence among the phase changing and roman numerals is shown in the table at the figure insert. The areas where the phase difference for the orthogonal field components vanishes are depicted by unsaturated gray; the areas where this phase difference reaches are depicted by saturated gray; and the areas where this phase difference reaches are depicted by white. Dashed lines and dashed-dotted ones correspond to - and -contours, respectively. b) the net of phase difference vortices, - and -contours. The areas with clockwise and counterclockwise polarization are depicted by gray and white, respectively. Double arrows indicate qualitatively the behavior of the azimuth of polarization along -contour. Phase difference along -contour at arcs between the phase difference vortices is indicates as and .

a re shown by white).

O

Figure 1.26. Changing the - and -contours resulting from changing the decomposition basis.

bviously, positions of the points where the phase difference of the complex amplitudes associated with the orthogonal field components is constant (within to ) constitute, due to the field continuity, the system of closed lines, namely the contours of a phase difference. The lines where a phase difference vanishes or equals is none other than -contours (in Figure 1.25(a), thin dashed lines), and the lines where are -contours (in Figure 1.25(a), thin dashed-dotted lines).

Let us emphasize that the crossings of the lines associated with the components of the same name (crossings the lines and , or and ) fix the points of the field where the phase difference vanishes or equals rigorously, i.e. -contours pass these points. Crossings the lines of the opposite names (crossings the lines and , or and ) fix the set of points where the phase difference is ; -contours pass these points.

In Figure 1.25(b), - and -contours are also depicted by dashed and dashed-dotted lines, respectively. The areas with clockwise and counterclockwise polarization are shown by white and gray, respectively.

Taking into account the sign principle and the fact that a phase difference along -contours changes by jump by passing the vortex, one can describe in qualitative manner the behavior of the azimuth of linear polarization along -contour into the chosen basis of field decomposition, [28]. Such behavior of the field vector is illustrated in Figure 1.25(b) by bold double arrows.

Let us analyze some peculiarities of the nets produced by the phase difference vortices. Not that the evaluation function depends on the choice of the basis of decomposition of a vector field into orthogonal components, see Figure 1.26. At the same time, the position of -contours is stable and does not depend on the choice of the decomposition basis. A simple dashed-dotted line and lighter vortices correspond to the initial decomposition basis. A dashed-dotted line with two points correspond to the changed basis. Positions of -points are indicated by the letter . Into arbitrary basis, -contours pass these points. -contours corresponding to two different decomposition bases are shown in Figure 1.26 by dashed and dashed-dotted lines, respectively.

As orientation of the decomposition basis is changed, new crossings of - and -contour can occur, i.e. the phase difference vortices can arise. For that, the number of additional vortices is always even, and the signs of topological charges obey the sign principle.

Let us consider another consequence following from the properties of the evaluation function . Let the complex amplitude of a scalar field is changed by some factor, , that is constant for a whole field. Vector field corresponding to such changing of the characteristic function can be determined, for example, by placing the object kind of a quarter-wave plate into the beam. For that, a structure of the phase difference is not changed, but - and -contours are shifted in correspondence with . So, if , then - and -contours replace to each other.

Thus, one can conclude that the new system of singularities of a vector field of the net of phase difference vortices has been established, and the main properties of such system are determined.

Further we will try to establish an interconnection among this system of singularities and disclinations and polarization singularities.