1.3.6.3. Some remarks on the field reconstruction by the use of nets of the intensity sTationary points

Let us summarize the established facts:

1. The nets of phase and intensity are connected (at least in statistical sense).

2. The saddle points of intensity are mainly positioned in the field areas, where the changes of a phase are enough fast, i.e. in the areas, where density of the equiphase lines is maximal. Moreover, in the most cases, the adjacent vortices of opposite signs may be connected by the line of gradient current, passing via the intensity saddle.

3. The phase changes within in the area of a phase speckle. Therefore, any line, constructed on the relatively small distance from the true field equiphase line may be used as this field line with enough accuracy.

Due to that the following hypothesis may be formulated – gradient current line passing the saddle point of intensity may be identified with one of the equiphase line connecting the adjacent vortices of opposite signs of the topological charge.

In other words, the pairs of adjacent vortices of opposite signs (within ) may be identified by the analysis of characteristics of stationary points set and gradient current lines of intensity. Such vortices may be connected by gradient current lines, which are identified as equiphase lines. As a result, the phase net that is close to the true one (or close to the net of conjugate field) may be constructed.

Naturally, such statement is true only with some probability and obviously it is not satisfied for all field regions. However, the following statement is justified completely – the set of areas, where such operation is possible may be specified in the field.

The main problem of the integration of such “local” phase nets in the global field net brings up due to the fact that the local nets are determined within the sign .

Thus, if one can establish the correspondence between the signs of vortices combined in the local nets, then the global phase net (may be excluding some field areas) close to the true field net may be obtained and field phase may be reconstructed.

As it is known, cf. subsection 1.2.3, information on the vortices signs may be obtained only by the analysis of the results of interferometric experiment. At the same time, as a rule, the formation of regular reference beam is impossible, due to the lack of knowledge of the pre-history of the analyzed field.

Nevertheless, such interferometric information may be obtained, for example, from the analysis of the results of the shift interferometry.

Thus, the represented analysis provides a new insight in the physical sense of the currently used techniques for reconstruction of the field’s phase on the intensity distribution, and can be put in the basis of the development of advanced algorithms for solving such “inverse problems” in optics [61].

1.4. Singularities of a vector field

1.4.1. Disclinations. Polarization singularities

Similarly to the case of scalar field, one can specify some sets of singular points for the vector electromagnetic field, whose characteristics are spatially distributed, and such points can constitute peculiar nets. As in the scalar case, the characteristics of such nets provide information (at least qualitative) on the field behavior at any its point. It directly follows from the properties of a natural space. In other words, one can construct a topological skeleton for the vector field that governs the changes of the field characteristics from point to point.

At the same time, situation for a vector field differs considerably from the scalar case. So, in contrast to the scalar field, stationary amplitude zeroes are absent into vector field [3,66-70]. Really, presence of a stationary amplitude zero in a vector field presumes simultaneous presence of zeroes for all three orthogonal components of the field at any its point. Probability of such event is negligible. Moreover, existence (if even supposed) of such zero is doubtful, while infinitesimal disturbance results into shift of the components zeroes from its initial position. Thus, a stationary singularity of this kind may be only regarded from the point of view of model notions, when the distance between the component zeroes is too small to be reliably detected in experiment [66-69]. In this case, behavior of the field in the vicinity of such “model” singularity will be the same as in the case of a true zero of the vector field.

B esides, owing to the vector nature of electromagnetic field, one must distinguish carefully the field of general form from the beam-like field obeying the paraxial approximation. It has been shown [3,71] that the sets of singularities are quite different in these two cases.

H

Figure 1.22

ereinafter we consider only the beam-like fields. For beam-like field one can specify temporal singularities, disclinations, i.e. the sets of points where the field components vanish at specified instant [3,62-64,75,76]. Existence of such sets can be imagined using Figure 1.22.

Let us assume that the prevailing direction coincides with -axis, and (due to the paraxial approximation) -component of the field can be neglected. Let the field be linearly polarized at any point . We decompose the field into orthogonal linearly polarized components, , within the basis shown in figure. It is seen that -component identically equals zero. So, one obtains the vortex for the component at point . At the same time, the component twice along the vibration period vanishes. Just at these instants the total field’s amplitude equals zero, and one observes a disclination at point .

Note that at the point with elliptical polarization the total field never vanishes. Thus, one can conclude that disclinations can occur only at the points where the field is linearly polarized. It can be easily seen that due to continuity of the field these points constitute closed one-folded lines (surfaces, in three-dimensional space) dividing the areas with opposite directions of polarization, see Figure 1.23. These lines are referred to as the -contours ( -surfaces, in three-dimensional space) [3].

E

a b

Figure 1.23. Structure of -contours: (a) – structure of “Matryoshka”-like -contours; (b) – forbidden structure of -contours for random field («islands into the ocean»).

xistence of a disclination at -contour does not impose any limitations on the azimuth of polarization at any point of this contour. Moreover, one can conclude that the azimuth of polarization gradually changes along -contour, while otherwise one must assume a possibility that for some of polarization projections of the field the edge dislocation occurs (not only at the observation plane, but also into three-dimensional space), that is impossible, as was shown in [47,48]. Generally, the length of -contours can change from zero (degenerative case when -contour reduced to a point) to infinity, and the mean length of -contour depends on prevailing polarization of the vector field, as it will be shown below.

In part, the only structure of the -contours shown in Figure 1.23(a) is admissible for a random field (globally depolarized field with Gaussian statistic [72-74]), when the area with any specified direction of polarization (or several of such areas) are inserted into the area with the opposite direction of polarization, where the rotation direction of the electrical vector of the field is altered again, and so on. Figuratively speaking, the structure of -contours kind of “Matryoshka” (the world-known Russian version of the “toy-in-toy”) is realized. For that, the maximal size of the largest of contours is infinitely large. Another conceivable type of -contours («islands into the ocean»), Fig 1.23(b), is never realized into random field, as soon as to provide this possibility (even for equal areas covered by the oppositely polarized scraps) one must presume preference of right-hand or left-hand polarization.

Thus, -contours as the loci of disclinations constitute the peculiar structure of a vector field. On the other hand, the notion of disclination as the singularity is the relevant one for only electromagnetic waves of radio-frequency domain. For optics, these singularitys of the field are not “observables quantities” due to rather rapid changes of a field hampering the direct measuring the amplitude of oscillations.

In optical domain, stationary or “polarization” singularities of a vector field are the only relevant. Really, the electrical vector at the each point of a space circumscribes a polarization ellipse whose parameters (the azimuth of polarization and the direction of rotation of the vector ) as the functions of the spatial coordinates can also possess singularities.

For the beam-like fields (the paraxial approximation) one can specify two main types of polarization singularities [3,63-65]:

1. The mentioned above -contours ( -surfaces, in three-dimensional space); the rotation direction is undetermined at these contours. The connection among -contours and disclinations is not surprising, being the manifestation of the fact that all singular sets of any physical quantity are interconnected (the same is true for a scalar field, see section 1.3.4).

2

Figure 1.24

. -points ( -lines, in three-dimensional space) are the points (lines) of circular polarization (see Figure 1.24), where the polarization ellipse degenerates into the circle, and the direction of the large axis (azimuth) of the polarization ellipse becomes undetermined, as well as the magnitude of the so-called vibration phase [3,78] determining the position of the electrical vector regarding to the large axis of the ellipse. Naturally, -point as the topological element can be characterized it two ways [3,79]:

(i) by the topological charge of the vibration phase of the -point,

, (1.47)

(ii) and by the topological charge of the azimuth singularity,

, (1.48)

which coincides with the Poincare index of -point. Hereinafter, we refer to the charge of -point as , and the index of this point as .

Note that in contrast to the charges and indices of the scalar singular points, these quantities take the values . In part, under circumference the С-point over closed loop, the azimuth of polarization turns at the angle multiple of [3,81,82].

Note, the existence of -points, for circumference of which the axes of ellipses accomplish more number of rotations, is not forbidden, but such field structures are topologically instable. Hereinafter, we believe that that the index and the charge of -point take magnitudes alone.

Let us establish also the interconnection among the magnitudes and . Such a consideration is straightforward, as one use a decomposition of the field into orthogonally circularly polarized components. Note, the coordinates of -point occurring in the area with any specified polarization direction (clockwise or counterclockwise) coincide with the coordinates of the vortex of the orthogonally polarized field component. Its amplitude vanishes just at the vortex center, and the total field is completely circularly polarized.

The following relations are justified for the phases of counterclockwise (left-hand) and clockwise (right-hand) circularly polarized field components [3,78]:

, (1.49)

where is the vibration phase, and are the phases of the clockwise and counterclockwise circularly polarized field components, respectively.

Let us assume that -point is observed at the area with clockwise polarization. Then, the coordinates of this point indicate the vortex of counterclockwise component. As so, the clockwise field component is “smooth” (singularity-free) within this area, while the orthogonal component supports a phase singularity at -point. Following [3], the following relations take place:

(1.50)

where – topological charge of the vortex associated with the counterclockwise component.

While , it follows from Eqs. (1.47)-(1.50) that

. (1.51)

Taking into account that the charge of a vortex , one obtains:

. (1.52)

If -point is into the area with counterclockwise polarization, Eqs. (1.51) are replaced by the following:

, (1.53)

and for the area with right-hand polarization one obtains:

. (1.54)

Eqs (1.52) and (1.54) can be written in a common form by introducing a so-called handedness factor , which equals for the areas with clockwise polarization and for the areas with left-hand polarization polarization. In this case, one can re-write Eqs (1.52) and (1.54) as follows [79]:

. (1.55)

Let us remember that the total topological index within any area equals the sum of the elementary indices and is found from the relation:

, (1.56)

where is a one-folded closed loop circumscribing all singularities.

Let us choose any -contour. Then, it follows from Eq. (1.56) that the interconnection among - and -singularities consists in that the total number and direction of rotation of the vector of linear polarization along -contour equals the total index of -points, circumscribed by this contour. So, for example, in Figure 1.24 the total topological index of -points within the area bounded by the -contour is that corresponds to the rotation of the field vector at the angle under counterclockwise circumference.

In other words, polarization situation within the area of interest reflected homeomorphically to the -contour, and the parameters of - and -singularities govern the behavior of the field at any its point.

Thus, one can conclude that, similarly to the case of a scalar field, the system of stationary (polarization) singularities forms a skeleton of the vector field, which governs behavior of the field at any its point.