1.4.3. “Correlation” of intensity and polariszation of the vector field

Let us show that the behavior of the polarization characteristics of a field and its intensity are interconnected. This assumption follows from the fact that similar interconnection takes place for intensity and phase of the each orthogonal component considered as scalar field, see subsection 1.3.5.

It can be shown that the modulus of the intensity gradient of a vector field is described as

, (1.74)

where are the amplitudes and their partial derivatives for the clockwise and the counterclockwise polarization components. The squared modulus of gradient of the amplitude ratio for the clockwise and the counterclockwise polarization components is of the form:

. (1.75)

It is obvious that the following conditions must be fulfilled at the area of the field with slow changes of polarization:

, (1.76)

where is the phase difference of the clockwise and the counterclockwise polarization components. It follows from the first Eq. (1.76) and from Eqs. (1.74, 1.75) that the intensity gradient at the areas of the field with slow changes of polarization obeys the following relation:

. (1.77)

Let the average magnitudes of the amplitudes for the clockwise and the counterclockwise polarization components are difference with the coefficient . Then, taking into account that are the independent quantities with zero mean magnitudes, one can find an average gradient of intensity for a whole field:

, (1.78)

and for the areas with slow changes of polarization:

. (1.79)

It follows from Eqs. (1.78) and (1.79) that the ratio of these average gradients is

. (1.80)

It follows from Eq. (1.80) that modulus of the intensity gradient at the areas with slow changes of polarization is, in average, times larger than at other areas of the field [81]. So, for “completely depolarized” field (field with equal average magnitudes of intensity of the orthogonal components, ) . In other words, in vector fields slow changing of the state of polarization correspond to fast changing intensity.

T

Figure 1.31. “Correlation” of changes of intensity and polarization of the field. (a) – intensity distribution; (b) – distribution of the modulus of intensity gradient; (c) – overlapping the areas with small gradient of a phase difference and small gradient of the ratio ; (d) – correspondence between the areas with rapidly changing intensity and slow changing polarization (the state of polarization changes slowly within dark areas).

he results of computer simulation are represented in Figure 1.31. Figure 1.31(a) shows the intensity distribution of a random vector field with equal average magnitudes of intensities of the clockwise and the counterclockwise polarization components. The levels of grey correspond to the magnitude of intensity. Figure 1.31(b) shows behavior of the intensity gradient. Figure 1.31(c) illustrates the areas with slowly changed states of polarization. Dark areas are the areas of the field with small gradient of the ratio . Areas with mediate level of grey are the areas with small gradient of a phase difference. Areas with the lowest level of grey are the areas of overlapping of these regions, viz. the areas of slowly changing polarization. At last, Figure 1.31(d) illustrates the areas with slowly changing polarization (dark areas) and the areas with rapidly changing intensity. It is seen from Figure 1.31(d) that the areas of the field where polarization changes slowly are “attracted” by the areas with large intensity gradients.