1.3.6. Vortex nets. Phase skeleton of a scalar field

1.3.6.1. Reconstruction of the field’s phase on the basis of shifted vortex nets

Let the field is characterized by some amplitude and phase modulation. Amplitude modulation means that some “image” is determined at the plane . A random field formed at far field is determined as the Fourier transformation of the field .

It is known that the field (or image close to ) can be reconstructed using the Shannon sampling arranged from the readings of the field at the plane , if is the function with a finite spectrum, or if the sampling of readings is performed within the domain bearing the most part of energy of the field [see, for example, ref. 59]. The maximal spacing between the readings coincides, within the framework of the optical interpretation of the Shannon theorem, with the correlation length of the field, , at the plane . The image close to is reconstructed even in the case, when the spacing between readings is not constant, being undergoing random changes, but the mean magnitude of the spacing equals [59].

Assume that the area, where transfers the most part of energy, contains amplitude zeroes of a field. It is known [9,12] that in most cases average distance between amplitude zeroes equals .

Consider some sampling of the point sources within this area with random coordinates . For that, amplitude and phase of these sources, , equals amplitude and phase of the field at points , and the number of sources equals the number of vortices. We will refer to such a sampling as random one. It can be described in the following form:

. (1.37)

Taking into account that an average distance between the readings is (as the number of them equals the number of vortices), one expects that the sampling (1.37) will reconstruct the field close to .

T

a b c

Figure 1.17. (a) – original image of the test-object; (b) – reconstruction of the original field using the Shannon sampling; (c) – reconstruction of the original field using a “random” sampling.

he results of computer simulation of the process of reconstruction of the test-field are shown in Figure 1.17. We use as a test-field the set of randomly located point sources of the unitary amplitude, whose phase is randomly changed within the limits much exceeding . Loci of the sources are specified in such a manner that an “image” resembles three vertical bars, cf. Figure 1.17(а). Figure 1.17(b) and Figure 1.17(c) illustrate, respectively, the reconstruction using the Shannon sampling, and the reconstruction using a random sampling.

Spatial spacing of readings was specified in the following manner. In the case of the Shannon sampling, spacing of the readings, , does not exceed the correlation length of the field, , in agreement with the Shannon theorem [59]. In the case of “random” sampling, the coordinates of readings are determined in random manner. An average spacing of the readings, , is also equals . Correct reconstruction of the initial field is possible in this case [59].

Assume that the positions of point sources of a random sampling are shifted in respect to the positions of the vortices at . For determinicity, assume that the direction of shift coincides with the -axis.

W

Figure 1.18

e refer to such a sampling as the “shifted” sampling. While the field is random, then the positions of vortices, , are random also. Thus, such a sampling does not differ in statistical parameters from the sampling (1.37). Following, one can expect that such a sampling will provide the image of the field similar to the image shown in Figure 1.17(c). It is not quite true, however [59]. First of all, the net of vortices of the field consists, in general, from anisotropic vortices. Nevertheless, the system of such vortices can be replaced, within accuracy , by the system of isotropic vortices, cf. subsection 1.2.1. Such replacing is similar to the situation, when a transparent plane-parallel plate with slightly changed thickness in the areas of small amplitude is placed in the beam, see Figure 1.18. Obviously, the field behind such a plate leaves almost unchanged, and the inverse Fourier transformation will result in reconstruction of the image of .

Assume to be of small magnitude, and amplitude obeys linear approximation in the vicinity of amplitude zeroes. In this case

, (1.38)

where is the amplitude factor determining the behavior of amplitude, viz. the gradient of amplitude, in the vicinity of the -th zero into -direction. The field formed by the reconstructing sampling at the image plane is the Fourier image of the field , being described as

, (1.39)

where are the coordinates of zeroes, are the numbers of positive and negative vortices, and is the initial phase of the -th vortex. It can be seen from Eq. (1.39) that , within the constant factor , does not depend on . If , then does not depend on also.

Let us replace the sampling (1.37) by the following system of elementary fields:

, (1.40)

where is the averaged over the vicinity of the -th vortex magnitude of , and (in polar coordinates with the origin at the vortex center) is of the form:

, (1.41)

where is the “vortex” amplitude function, which equals zero at . In this case, Eq. (1.39) is transformed to the form:

, (1.42)

where , , , and approaches , if the spatial domain is small (see Appendix 1.2). Note that the representation of the reconstructing field by Eq. (1.37) can be interpreted as some sampling from the field (1.39).

Thus, intensities of the reconstructed fields are in both cases modulated by the function that is analogous to . Such “vortex” modulating function vanishes at and grows monotonically as point of observation removes from .

The results of computer simulation of the image reconstruction using a shifted sampling are represented in Figure 1.19. One can see that, indeed, the field reconstructed on such a sampling is modulated by the function kind of . To emphasize the radial symmetry of the modulating function, a white ring is inserted in the reconstructed image, cf. Figure 1.19(c). Figure 1.19(b) explains the principle of formation of the shifted sampling. The shift of sampling coordinates is performed in directions of both the -axis and the -axis. One can see in Figures 1.19(c)-(e) that the reconstructed field remains almost unchanged, if the shift does not exceed 0.5 . Even if 0.75 , cf. Figure 1.19(f), dark area is observed at the center of the image. Only when the shifts exceed , cf. Figures 1.19(g),(h), the field reconstructed using the shifted sampling becomes similar to the field reconstructed by the random sampling, i.e. the influence of the vortices on the field

Figure 1.19. Reconstruction of the test-field using the shifted sampling. (a) – original image; (b) – Illustrating the principle of formation of the shifted sampling; – grey points correspond to the positions of amplitude zeroes, black points indicate the positions of the readings of a sampling, – shift of the readings in respect to the vortex coordinates; (c)-(h) correspond to various magnitudes of : (c) – 0.1 ; (d) – 0.25 ; (e) – 0.5 ; (f) – 0.75 ; (g) – ; (h) – 2 .

structure becomes negligible, and the shifted sampling is transformed to the random one. In other words, the reconstructed field “remembers” that it has been formed by the point sources whose phases are determined by the phase helicoids of the vortex structure, if the shift of the reconstructing sampling does not exceed .

Thus, the analysis of the results of field reconstruction using the shifted sampling enables to conclude that the “phase” influence of individual vortex is extended over the distanceof the order of magnitude of the correlation length of a field.