1.4.6.3. Computer simulation of the vector field’s parameters

It has been noted above that our consideration is performed at far zone, and the paraxial approximation is valid, while the field is analyzed within small solid angle. Under these conditions, correlation length of the field does not depend on the structure of the scattering object, being depending on the relation of the field-of-view of the object (for the specified scattering angle) and the wave length alone.

To consider the problem in general setting, one must provide the algorithm of computer simulation allowing formation of arbitrary set of Stokes parameters. Note, however, that the second Stokes parameter equals zero, if the condition (1.114) is satisfied. Remember also, the first normalized Stokes parameter equals zero by definition. That is why the requirements to the simulation algorithm are reduced to the only requirement of formation of arbitrary third and fourth Stokes parameters.

At far zone, for the specified scattering angle and under assumption of statistical homogeneity of the field, the direct connection between the characteristics of the scattered field and the scattering object is lost. In part, it is not important, what specific object forms the field with the measured Stokes parameters. The same statement is true for the structure of the each of the orthogonally linearly polarized components of the scattered field. Thus, the simulation algorithm must provide obtaining arbitrary magnitude of the correlation coefficient, , for two components, what requirement is the equal strength as specifying the desirable magnitudes of the third and the fourth Stokes parameters.

Under paraxial approximation, the far-zone field, , is the Fourier transformation from the boundary object field, [92]. Taking into account that the field structure at far field is not connected directly with the object’s structure (excluding the requirement for the field to be random), one can form the field for the orthogonal components as the set of randomly distributed point sources with the unitary amplitudes and random phases:

, (1.140)

where is the total number of point sources, , are phase and coordinates of the th source. The only difference in forming the “input” sampling of point sources of the orthogonal components is that the positions and the phases of sources are quite different for two samplings associated with the orthogonally polarized components.

In this case, the components of equal intensities are formed at far zone, and the correlation coefficient for them is determined by the following simple ratio:

, (1.141)

where is the number of point sources with the same characteristics in both orthogonal components.

T he degree of “integral depolarization“, , is the polarization parameter characterizing the averaged polarization characteristics of a vector field. Note, the field remains to be completely polarized at the each point.

It is known that if the condition (1.114) is fulfilled, then integral depolarization is connected directly with the correlation coefficient of the orthogonal components, [28]:

. (1.142)

F

Figure 1.46

or this reason, in our case characterizes the level of polarization in assumption that the condition (1.114) is fulfilled. The ratios of the distance between the adjacent component vortices, , to the correlation length, – , have been obtained, as well as the maps of phase difference and the averaged Stokes parameters, for different levels of integral depolarization.

Figure 1.46 shows dependence of the phase difference dispersion, , on a mean distance between the adjacent component vortices .

Figure 1.47 illustrates the maps of a phase difference for the magnitudes of the prevailing phase difference, , 0 and . Note, the character of behavior of a phase difference does not depend on , while the phase differences in both cases are different by a constant at arbitrary point of the

a b

Figure 1.47. Maps of the phase difference between the orthogonal components of an optical field for 40% depolarization; effective phase differences are (a) and =0 (b). For clearness, the phase difference is given within to . The phase differences different by are depicted by the same level of grey. The boundaries of white and black colors are -contours; the points of gathering of the lines of all colors correspond to the phase difference vortices; the points at the centers of the like regions are the saddle points of phase difference.

а b

c d

Figure 1.48. Intensity distribution of non-uniformly polarized field. Intensity distributions for 5% (а), 10% (b), 30% (c), 50% (d) depolarization of the field (correlation coefficient of the orthogonal components is 0.95, 0.9, 0.7, 0.5, respectively).

field. It results in changing of the form, size and localization of -contours, while the saddle points and the phase difference vortices do not change their positions. One can see from the figure that -contours are small and almost all are concentrated within the area of figure, when = .

The results of computer simulation of the parameters of a vector field for different correlation coefficients are represented in Figures 1.48 and 1.49. Circular polarization was chosen as the prevailing one. In this case, the areas with rapidly changed polarization coincide with the size of -contours [81,82].

Intensity distribution of the field slightly differs for various degrees of depolarization, cf. Figure 1.48). One can see in Figure 1.49 that both the size of -contours and a mean distance between the adjacent vortices of the same sign grow, as the level of depolarization increases. -contours are small in comparison with an average speckle size, being the closed domains with one type of polarization (clockwise or counterclockwise), when the correlation coefficient exceeds 0.5.

а b

c d

Figure 1.49. Maps of a phase difference between the orthogonal components. Phase difference between the orthogonal components for 5% (а), 10% (b), 30% (c), 50% (d) depolarization of the field (correlation coefficient of the orthogonal components is 0.95, 0.9, 0.7, 0.5, respectively); – -component vortices, – -component vortices.

These domains are located just in the vicinity of the component vortices. The size of -contours grows and the localization of the domains with large polarization changes becomes random, when the correlation coefficient is less than 0.5.