1.5. Singularities of the Poynting vector and the structure of optical fields

Natural question occurring to the reader of this book is: what are the reasons to consider here the singularities of the Poynting vector? Let us answer this question proceeding from the notions of “conventional” optical singularities, such as phase vortices, polarization singularities, etc.

First of all, the field within the area of an optical singularity is absolutely smooth, without any discontinuities, being strictly obeying the Maxwell’s equations. So, at the center of a scalar phase singularity (vortex), undeterminicity of a phase is, generally speaking, meaningless, while amplitude is zero.

S

Figure 1.55

imilar considerations can be taken into account in respect of the polarization singularities, viz. -contours and -points. Indeed, whereas the rotation of the axes of polarization ellipses at some distance from -point characterizes the difference in polarization characteristics of the field, ellipses in the nearest vicinity of the singularity ( -point) negligibly differ from a circle, cf. Figure 1.55, and the notions of azimuth, vibration phase, as well as the notion of a phase at the center of a vortex, are illegible. In other words, such parameters as the vibration phase and azimuth are unnecessary for description of the field at -point, and the direction of rotation of the field’s vector occurs to be “superfluous” characteristic for the points of -contour. Besides, temporal behavior of the field’s vector at -point (at -contour) and near this elements of the field is almost the same. Conventional optical measurements do not provide discrimination the points belonging to the singular set and the points lying in the nearest vicinity of this set. These areas are schematically depicted in Figure 1.55 as the regions and . Moreover, an optical singularity can be detected by using indirect interferometric techniques alone, by analyzing the field resulting from a superposition of the singular structure of interest with any reference field [8-10,36-38,82,83].

It seems, these considerations leads to the obvious conclusion that the only reason for study of optical singularities consists in their role in formation of the structure of a light field, both scalar and vector.

On the other hand, the presence of singularity of any parameter of a field inavoidably leads to some physical peculiarities of the field in its vicinity. The question arises: what is the physical manifestation of the optical singularities, which is characterized by specific behavior of the physical system, on which an electromagnetic wave influences?

It is known, for scalar fields such manifestation is connected with the orbital angular momentum of the electromagnetic field existing in the vicinity of an optical vortex [95-98]. Such a momentum arises due to specific, viz. helicoidal phase surface in the vicinity of a vortex and, generally speaking, due to peculiar temporal behavior of the field. In other words, the physical manifestation of a scalar singularity is reflected in peculiar temporal behavior of the components of an electromagnetic field.

In the end, polarization singularities must also be considered as the temporal peculiarities of a field: the azimuth of polarization, vibration phase and the direction of rotation of the field vector determine both spatial and temporal behavior of the field. It is clear, the physical manifestation of the vector singularities must be connected with temporal behavior of the field, which, in the end, is reduced to specific magnitude of the angular momentum of the of an electromagnetic field in the vicinity of the singularity and to behavior of this characteristic within the mentioned region different from its behavior in other areas of the field.

It is known, cf. for example Ref. 97,98 that the “direct force” (or “energetic”) effect of an electromagnetic wave on some physical system is associated with the Poynting vector. In any case, the spatial distribution of the parameters of this vector, such as its magnitude and orientation, is one of the main factors determining the influence of the wave on the system. Besides, the Poynting vector is directly connected with the angular momentum, cf. Refs. [95,96]. However, in contrast to the angular momentum, the Poynting vector is not “tied” to the point of applying, i.e. to the momentum axis. At the same time, information on behavior of this vector provides analysis of the angular momentum itself at arbitrary region.

Naturally, the characteristics of the Poynting vector’s components (including the modulus and orientation of its transversal component) can be considered for the field of general form as some spatially distributed parameters of the field. In general, such distributions possess singularities. Similarly to the conventional optical singularities, the Poynting vector’s singularities can be connected into nets, which must determine, at least in qualitative manner, behavior of the Poynting vector anywhere, in other words, must form the field’s skeleton and determine the regularities governing the spatial distributions of the field parameters.

Similarly to the above considered distributions of phase and intensity, and polarization ad intensity, see subsections 1.3.5 and 1.4.3, the characteristics of such singular sets and the behavior of the Poynting vector are expected to be connected with the characteristics of nets of the conventional singularities.

Taking into account these considerations, one can conclude that the analysis of the Poynting vector’s singularities and establishing the corresponding topological regularities are the highly relevant tasks. In addition, the applied aspects of such consideration are directly connected with the area of research and developments concerning so-called optical tweezers, which is among the hot topics of modern optics Ref. 99.