1.4.6.2. Analysis of the averaged parameters for decomposition of the field into linearly polarized components

It is known [28,78] that can be written in terms of the coherence matrix elements:

. (1.117)

The coherence matrix is Hermitian one, viz. . Proceeding from Eqs. (1.116, 1.117), can be represented using the averaged components of the coherence matrix:

, (1.118)

where correspond to . Remember [28,78] that

, (1.119)

where designates temporal averaging.

For completely coherent, completely polarized light, temporal averaging is omitted, and , where and are the complex amplitudes of the orthogonal components, and , being the amplitudes and the phase difference of the orthogonal components. Note, expression for coincides with one for the evaluation function introduced in subsection 1.3.2.2, cf. Eq. (1.60). That is why, the phase difference vortices can be regarded to as the vortices of non-diagonal elements of a coherence matrix. Zeroes of coincide with the component zeroes, which are unambiguously connected with the polarization singularities ( -contours and -points).

The averaged element of a coherence matrix,

, (1.120)

can be interpreted as the maximal magnitude of the correlation function of complex amplitudes associated with the orthogonal components [28]. Note, is the rapidly changing factor. For that, the most rapid changing of takes place in the vicinity of zeroes of (in the vicinity of the components vortices), where phase of the component changes from to within very small region including the vortex center. Within the same regions, reaches minimal magnitude (either or tends to zero). As so, the contributions from such regions in is negligible, while the main contribution is provided by the region of the stationary points of , where the derivative from the phase difference changes slowly. While the number of the phase difference extrema at far field is 15 to 20 times smaller than the number of the phase difference saddles (see subsection 1.4.2.2), we will refer to just the saddle points as the stationary points of a phase difference.

Within the area of averaging, can be represented in the form:

, (1.121)

where

, (1.122)

being the area containing one stationary point of a phase difference.

Let us apply the method for approximation of Eq. (1.122), which is close to the stationary phase method [28,54].

As it is known, the stationary phase method permits us to obtain the approximate expression of the following integral:

, (1.123)

if magnitude is relatively large.

Note, the expression preceding the final formulae of the stationary phase method has the form [28]:

, (1.124)

where are the values of the second derivates of in the stationary point .

In our case value has specific construction:

. (1.125)

As a result, all second derivates of the phase difference are also the corresponding differences:

etc. (1.126)

At the same time:

i Phases of the orthogonal components are the “strongly” correlating quantities (at least when the degree of polarization is more than ~50%). In part, this fact is confirmed by the data of computer simulation, which will be presented below.

ii The amplitude of the component is small in the areas of vortices (regions of most phase changes) and phase is practically constant (within significantly less than ) in the areas, where the orthogonal component has significant magnitude [1,17,18]).

Due to this, stationary points of the phase difference are positioned in the areas, where changes smoothly within the same quantity.

Thus, it can be stated that the argument of exponent in the integrand of (1.124) is small and this exponent may be approximated as unit. Correspondingly, integral in Eq. (1.124) is approximately equal to the square and Eq. (1.122) transforms to the form:

, (1.127)

which is satisfied when, at least, the polarization degree of a field is more than ~ 50%.

are determined from the conditions:

, (1.128)

Thus, can be approximated in the following manner:

, (1.129)

where are the magnitudes of these quantities at the points of the phase difference saddles.

In correspondence with our assumptions, one can conclude that and are random quantities obeying Gaussian distribution function within the plane of analysis.

can be centered: , where is the prevailing phase difference of the orthogonal components, and is distributed symmetrically, following Gaussian law in respect to :

, (1.130)

where is the phase difference dispersion at the saddle points. The characteristic function is of the form (cf. Ref.2):

. (1.131)

The sum (1.129) can be regarded to as the sum of random phasors [2].

Proceeding from the conditions of the above consideration (first of all, Eq. (1.113)), the field at the plane of a photodetector is random and statistically uniform within the area of analysis, and its orthogonally linearly polarized components can be characterized by the correlations length , if the condition (1.114) is fulfilled. In correspondence with the same condition, we choose the basis for decomposition of the vector field. Thus, , while the orthogonal components are statistically independent.

It is known [2] that for symmetrical distribution function

, (1.132)

being the real and the imaginary parts of the sum

. (1.133)

From Eqs. (1.131-1.133) one obtains:

. (1.134)

Тhus,

. (1.135)

In this case, Stokes parameters are of the form:

, (1.136)

where is the effective phase difference of the orthogonal components.

The normalized Stokes parameters are of the form:

. (1.137)

Naturally, similarly to the case of partially coherent radiation, the following inequality is justified:

. (1.138)

One can observe integral depolarization and the degree of polarization calculated accordingly the relation:

(1.139)

is smaller then unity.

Equality in Eq. (1.138) takes place, when =0, i.e. for uniformly polarized field. One can see from Eq. (1.137) that the averaged Stokes parameters are determined by dispersion of a phase difference at the saddle points of this quantity.

Note, is connected with the mean distance between the adjacent vortices of the same sign belonging to the orthogonal components. For uniformly polarized field (the orthogonal components are completely correlated), and =0, i.e. amplitude zeroes of two orthogonal components coincide. Increasing is accompanied with increasing distance between the component vortices. The limit case, = , corresponds to the completely depolarized (in integral sense) field. One can conclude that is some function of : . This dependence can be derived, for example, from the data of computer simulation.