1.3.5. Topological indices of the field of intensity. Extrema of phase and intensity. “correlation” of phase and intensity

Let us emphasize that while the topological charges are intrinsic to the vortex-bearing fields alone, the Poincare index, N, can be introduced for the gradient fields as well [25]. Let us introduce the topological indices for the field of intensity and keep up with the changing of directions of the current lines for the intensity gradient in the vicinity of the stationary points. The extrema, including amplitude zeroes as the absolute minima of the field intensity, possess the index N=+1, and the saddles of intensity possess the negative index N=1. Accordingly, birth (annihilation) of the extremum must be accompanied by appearance (disappearance) of the saddle of intensity providing topological connection among the extremum and other field structures, as well as other extrema. Though the vortices and the maxima of intensity do not differ in the magnitude of the index N, nevertheless one can derive some conclusions concerning the numbers of them. It follows from simple physical considerations that if the extremum of any physical quantity exists in any area of definition of this quantity, then at least one opposite extremum must exist within the same area. Applying the approach close to the one described in [11], one concludes that the minimal number of the maxima and the saddles of intensity obeys the relation:

, (1.24)

where is the minimal number of maxima, is the number of vortices, and is the minimal number of the saddles of intensity.

Now we consider the question: how many extrema of phase and intensity might be generally?

In assumption that a complex amplitude of the field in a far zone is described by the relation

` (1.25)

(i.e., the dependence of a complex amplitude on is determined by the constant phase factor alone), the following system of equations has been derived [11,24]:

, (1.26)

where is the amplitude, ...etc. are the partial derivatives of the first and the second orders from the amplitude and phase, and in the rite sides of these equations .

It is easily to see that the conclusion on impossibility for existence of the phase extrema in a far zone follows from the second equation (1.26). Indeed, this equation is transformed for the points of the phase extrema to the form:

, (1.27)

which is wittingly not true, as the second derivatives at the points of extrema are always of the same sign.

However, the analysis of the first equation (1.26) leads to much more surprising conclusion, viz., that the maxima of intensity are impossible in a far field. Indeed, consider the possibility of appearance of the maxima of intensity. At these points , and . In this case, the sum of two terms of the first equation (1.26) is always negative, while the quantity in the parentheses is always positive. Thus, the left side of the first equation (1.26) does not equal zero at the points of intensity maxima. This is obvious contradiction, as the presence of amplitude zeroes presumes, at least, the presence of the same number of intensity maxima. Following, the assumption (1.25) is not acceptable in the case of interest for analysis of the field in a far field.

The results of computer simulation of random fields [17,18] reliably confirm this conclusion. The authors of the cited papers state that phase extrema occur even in a far field (though in not great number). So, one additional phase extremum is on 14 to 20 vortices.

Let us attempt to cancel the contradiction following from the Eqs. (1.26). In general, the strength of the electrical field and the associated wave function are the vector quantities. As such a case, the scalar treatment of the problem is appropriate under the paraxial approximation alone. The Fourier approximation is too rough for analyzing a fine structure of the field, and we prefer use a more general Fresnel approximation (see for example ref. 53):

Let us write a complex amplitude of the field, , at any distance from the input plane within the domain corresponding to the Fresnel diffraction:

, (1.28)

where is the boundary field.

To find the explicit form of , we assume the area within which change being much less than the area within which change. In other words, we assume the area of analysis being much larger than the transverse size of the scattering object. These assumptions are true for the most practical situations. In this case, one can neglect the factor , and Eq. (1.28) is reduced to the form:

, (1.29)

where , and designates Fourier transformation.

Then, neglecting the terms of the second order of magnitude one obtains, instead Eqs. (1.26) the following system:

. (1.30)

For the case of a phase extremum ( ), the Eqs. (1.30) is reduced to the form:

. (1.31)

One can see from comparison of Eqs. (1.30) and (1.26) that the Eqs. (1.31), in contrast to the Eqs. (1.26) allows the existence of the phase extrema and asymmetric saddles.

For the case of stationary points of intensity ( ) Eqs. (1.30) takes the form:

. (1.32)

It follows from Eqs. (1.32) that the stationary points of intensity of all kinds (without any exclusion) exist.

One can estimate the number of the additional stationary points in comparison with the minimal number using the following considerations:

1. The correlation length of the field, , is the universal characteristics of the field at far zone determining the spatial scale of fluctuations for all other parameters of such field [40].

2. Mean size of a speckle in determined by the same quantity, , whose magnitude coincides with an average distance between the vortices [9,12,24,55-58]. Within the field of a phase, four topological elements are associated with an “average” speckle, viz. two vortices of opposite signs and two phase saddles. That is why, probability for a speckle to possess additional topological elements, such as the extremum and the associated with it saddle, is small.

3. Mean size of a speckle of intensity is also determined by the magnitude of .

4. The minimal number of topological elements within the field of intensity is determined by Eq. (1.24), i.e. it is the same as for the field of a phase.

5. Birth of two topological elements accompanies appearance of the additional extremum within the field of intensity, similarly to the field of a phase.

Thus, one can conclude that probability of appearance of the additional maxima or minima of intensity must be no exceeding probability of appearance of the phase extrema.

Accordingly, one can neglect such topological elements as rather rare ones and consider that the field of intensity at far zone possesses the “main” maxima, absolute minima of intensity (phase vortices) and saddles alone.

I

Figure 1.15. Nets of the singular points of intensity and phase produced by the current lines of the intensity gradient and by the lines . Poincare indices for the intensity maxima, phase vortices – the absolute minima, and saddle points of intensity ( ) designated by circles.

t is obvious, the extrema of the some kind (maximum, maximum and minimum, minimum) are connected to each other by the saddles of intensity. In other words, the stationary points of intensity, similarly to the singular points of a phase, form the corresponding net. Figure 1.15 illustrates an example of such a net. While the elements of the net of a phase and the net of intensity are connected, intensity and a phase, to all appearance, can not change independently [17-26].

Let us analyze the interconnection of the vortex net characterizing the phase behavior and the net of stationary points of intensity. To do this, we find out average magnitudes of the phase gradient and the intensity gradient in the stationary points of intensity and phase, respectively [25,26].

Modula of the gradients of intensity and phase are determined by the following equations:

, (1.33)

, (1.34)

where J and R are the imaginary and the real parts of the complex amplitude U, and ,… are the partial derivatives of them, .

Modula of these gradients at the stationary points are of the form:

1. at the stationary point of intensity:

; (1.35)

2. at the stationary point of a phase:

. (1.36)

It is obvious, the average magnitudes of and and of their derivatives equal zero. We can assume that the statistics of these quantities is the same. Then, analyzing Eqs. (1.33, 1.34) and (1.35, 1.36), one can show that modulus of a mean phase gradient at the stationary point of intensity is times larger than in any other point of the field (excluding the vortex zones). The same is true for the stationary points of intensity. Namely, modulus of a mean intensity gradient at the stationary point of a phase is times larger than in any other point of the field. Thus we have established that in statistical sense, slow changes of a phase correspond to fast changes of intensity, and vise versa, intensity changes slowly within the areas where a phase changes rapidly [20,21,25.26].

Figure 1.16. Interconnection between the behavior of a phase and intensity of scalar random field. (a) – intensity distribution; (b) – modulus of the intensity gradient; squares designate the vortices of the field; (c) – a phase map; phase magnitude corresponds to the level of grey; (d) – loci of the zones with large modulus of the phase gradient ( ) and with small modulus of the intensity gradient ( ); (e) – the stationary points of intensity and their loci in respect to the field zones with large modulus of the phase gradient.

Note, the intensity maxima occur in the centers of speckles, near the saddle points of a phase [17,18]. Thus, the saddle points of intensity occur at zones with large magnitude of the phase gradient.

The results of computer simulation of scalar random field are shown in Figure 1.16. Fragments Figure 1.16(a),(b),(c) illustrate the behavior of intensity, and the modula of intensity gradient and phase gradient, respectively. Loci of zones with large magnitude of the modulus of the phase gradient and small magnitude of the intensity gradient are shown in Figure 1.16(d). One can see from Figure 1.16(d) that localizations of such zones are equal, in quite agreement with the above consideration. The regions of the field with small magnitude of the intensity gradient, where the intensity maxima are located, are the exclusions. This circumstance is additionally illustrated in Figure 1.16(e), where zones with considerable magnitude of the phase gradient and loci of the intensity maxima and saddle points of intensity are shown. One can see that almost all saddle points of intensity are in the areas of the field where a phase of the field changes rapidly.