1.4.6. Fine structure and averaged polarization characteristics of inhomogeneous vector fields

Polarization parameters characterizing a light field, such as a components of coherence matrix, Stokes parameters etc. [28,78], were introduced historically for description of incoherent, quasimonockhromatic, etc. beams. Determination of these parameters presumes certainly temporal and space integration. The question arises: is it possible to introduce similar local (in time and in space) characteristics of a field? The answer is positive, at least for inhomogeneous coherent fields [3]. Of course, interaction of light with a matter results in deterioration of the coherence characteristics of a field. Nevertheless, a wide class of light-scattering objects exists, which change the coherent characteristics slightly, so that decreasing of the coherence length, in comparison with the coherence length of a common laser, is negligible. There are: singly-scattered objects (thin films and other) [89], small pieces of multimode optical fibers [90], etc. The field interacted with such objects is a completely polarized speckle field, due to preservation of coherence, and the state of polarization changes from point to point, passing all states of polarization, from clockwise circular to counterclockwise circular [3]. At the same time, the characteristics of vector field, to say the components of coherence matrix and Stokes parameters being measured for the beam as a whole occur to be similar to such parameters for the case, when the object renders depolarizing action.

Thus, one can expect certain analogy between behavior of the traditional polarization parameters of partially coherent or incoherent light beams, on the one hand, and the corresponding averaged parameters characterizing polarization of any area of the coherent beam, where the state of polarization changes from point to point. This conclusion thrusts itself upon that fact that replacing the temporal averaging by the space one is adequate for the most physically realized situations [59]. As a consequence of linearity of such averaging, the averaged Stokes parameters, as well as the elements of a coherence matrix, are simple integrals from the relevant local parameters over the area of analysis. What is interconnection between the averaged polarization parameters with the peculiar structures of a vector field, such as polarization singularities, the areas of the field containing the saddle points of the polarization parameters, which form the field’s skeleton (see subsections 1.4.1-1.4.3) and determine the field’s behavior at each point?

1.4.6.1. Avaraged stokes parameters

Assume the laser beam with large enough coherence length illuminates a light-scattering object with randomly distributed optical characteristics, scattering centers, surface relief etc. Assume that the object belongs to the class of scatterers preserving the coherent characteristics into scattered field, and a random coherent speckle field is formed at far zone. A mean speckle size is determined by the wavelength of radiation, dimension of a cross-section of the light spot immediately behind the object and by the distance between the object and the observation point [40]. Polarization characteristics of the field determined by applying any commonly used technique are the characteristics averaged over the phototodetector area. To obtain the averaged data approaching the mathematical waiting, the number of speckles covered by the sensitive area of a photodetector must be large enough [91]. In other words, steady measuring presumes the sensitive area to be exceeding a mean speckle size by 20 to 100 times, regarding to the required measuring accuracy.

It is clear, randomly spatially distributed quantities, to say intensity, are statistically inhomogeneous being the function of the angle of illumination and the scattering angle. However, within a small solid angle determined by the sensitive area of a photodetector, one can regard the characteristics of the field as homogeneous and, in most cases, obeying Gaussian distribution along arbitrary direction.

Thus, the following relation is true:

, (1.113)

where is the photodetector area, and are the correlation lengths in directions of the axes and of the laboratory coordinates, respectively. In the case of symmetric light spot behind the object, the magnitudes and are approximately equal to each other for enough large scattering angles. Moreover, these magnitudes are identical for the laboratory coordinates where

, (1.114)

where and are intensities of the orthogonal components averaged over the photodetector area. Further consideration is performed for some fixed angle of scattering . As soon as Eq. (1.114) is fulfilled, we will use the only magnitude . Orienting the axis along the direction , one can conclude that (i) a paraxial approximation is justified, if the transversal sizes of a light spot just behind the scattering object and the sizes of the sensitive area of a photodetector at the plane of analysis are much less than the distance between the object and the plane of analysis; (ii) the state of polarization of the field at the plane of analysis orthogonal to -axis can be represented by the conventional Stokes parameters (both local and averaged).