Near-field coherent effects at thermal microwave radiation receiving on coupled linear wire antennas

Figs.4, 5, and 6 show that one can actually get the random electric dipole source centre inside the -plane of a single receiving linear wire antenna (Fig.3), via scanning this antenna along the - axis on the biological object boundary surface  = 0 and defining the HWHM of antenna interference function  maximum at  = 0. As this takes place we obtain the source centre seeming depth , with knowing the object absorption and source extension. Now we intend to define the real depth  of the source centre in the centre of  position inside the -plane, via antenna scanning along the -axis on biological object boundary surface.

Fig.9 depicts schematically scanning the random electric dipole extended source with centre on the -plane by shifting the single receiving linear wire antenna along -axis to position with coordinate  The interference function  of the shifted single antenna is obtained from Eq.(66) by setting  and has a form

(68)

with  where. The quantity  is function of shifted antenna position and given (Fig. 8) by relation

(69)

where ,  is projection of vector  on the -axis. All quantities in Eqs.(68) and (69), having dimension of length, are normalized to antenna half length .

The interference function of the shifted single antenna in Eq.(68) has evidently maximum at  where the scanning antenna is brought in nearest position to the random electric dipole extended source, with distant  between source centre and antenna centre becoming equal to the real depth  of the source centre. The two quantities  and  are connected between them by relation . Therefore one can get the real depth of the source centre , provided one knows from scanning experiment the antenna position  where antenna interference function has maximum. We have also possibility to determine the real depth of the source centre by study the maximum peak of interference function in Eq.(68). The Taylor expansion for small values of  gives a representation similar to one in Eq.(67) and written as

(70)

Here  is maximum value of . The quantity  is HWHM of the Taylor expansion in Eq. (70) defined by

(71)

where quantity  and function is defined according to

(72)

Eq. (71) shows that the HWHM  is related to the real depth of the source centre  directly and becomes smaller with absorption growing. Bearing in mind an asymptotics  as, one can conclude also that the HWHM  increases with taking into account a small extension of the source.

Fig.9 presents the normalized interference function from Eq.(68) versus to normalized shift  of scanning along -axis antenna,. with no absorption taking into account. Fig.9 shows that normalized interference function from Eq.(68) shift towards negative -axis direction with decreasing azimuth angle  according to above position of the normalized interference function maximum. Meanwhile the normalized interference function becomes wider with growing the azimuth angle , that calls for growing the real depth of the source centre, and with growing the source extension also. Fig.10 presents the HWHM of normalized interference function from Eq.(68) versus to the varied azimuth angle  at fixed the normalized seeming depth but for a set of source extension, with no absorption taking into account. Fig. 10 shows again that the normalized interference function becomes wider with growing the source extension.

Fig. 11 generalizes content of Fig.10 on the subject to take into account the absorption, showing that the normalized interference function from Eq.(68) becomes narrower with growing absorption and wider with growing the source extension. Fig.12 presents the final dependence of normalized real depth of centre of random electric dipole extended source versus the HWHM of normalized interference function from Eq. (68), with taking into account absorption. The curves in Fig.12 show a competition between effects of absorption and source extension that make the HWHM smaller and bigger, respectively. As a result of such competition it is seen a crossing, in particular, of two curves (1 and 2”) in Fig. 12.

Before going above to study the maximum peak of interference function in Eq.(68), we had mentioned that one can get the real depth of the source centre if one knows from scanning experiment the antenna position on -axis where antenna interference function has maximum. In this case the three points' set consisting of antenna centre origin position, the just mentioned shifted antenna position and the source centre position inside the -plane (see Fig.8) form a rectangular triangle. Let us note in order to generalize such kind forming a triangle that one can consider two positions of two single antennas 1 and 2 centers along -axis of the  coordinate system (Fig. 13) when a many-side triangle is formed inside the -plane by centers of these two antennas and random electric dipole source centre projection on the -plane. The lengths  and  of formed triangle two sides can be determined separately via scanning the antennas 1 and 2 along - axis on biological object boundary surface  and defining the HWHM of these antennas' interference functions  and  maximums at , in accordance with described study the interference function in Eq.(66). After that one can determine the real depth  of random electric dipole source centre by resolving the triangle , with knowing its two mentioned sides as two seeming depths of source centre from two antennas centers and knowing distant between antennas. A symmetrical position of random electric dipole source relatively antennas, which transforms the above many-side triangle into isosceles one, is especially interesting for the case when effects of antennas' coupling are taking into account. This case of two coupled tuned receiving linear wire antennas' exciting by the random electric dipole source is our next task.

In the framework of using random electric dipole source model the - component  of the incident random electric field along as coupled vibrator-dipole antenna 1 in Fig. 13 is given as along single vibrator-dipole antenna in Fig.3 by Eq.(58). Analogous expression for - component  of the incident random electric field along coupled vibrator-dipole antenna 2 in Fig. 13 is obtained from Eq.(58) by replacing  to . The incident random electric fields  and  excite along two coupled antennas some current distributions with random amplitudes  and  given by Eqs.(41) and (42), respectively, with replacing  to  and  to  in the RHS integrands. Writing now equations similar to Eqs.(59) and (60) and supposing the tuned vibrator-dipoles to be of length equal to add whole number of half wavelengths lead us to generalization of Eq.(61) in the form

(73)

and

(74)

Here sums are taken over index  two values , with distances  and  being defined similar to the case of single antenna in Eq.(61) and related to antennas 1 and 2, respectively, in Fig. 13. According to Eqs.(73) and (74) the two spherical waves are propagated from a point of random electric dipole source towards receiving vibrator-dipole 1 ends as well as two another spherical waves are propagated from the same point of source towards receiving vibrator-dipole 2 ends. Bearing in mind the reciprocity between receiving and transmitting antennas one can say also that four spherical waves are propagated from vibrator- dipoles 1 and 2 ends towards the random electric dipole source and interfere on the source area. Eqs.(57) jointly with Eqs.(73) and (74) enables us to get for the fluctuations' spectral densities  of current distributions' along coupled receiving antennas 1 and 2 amplitudes caused by random electric dipole source thermal radiation the following equalities

(75)

which generalize Eq.(62) written for a single antenna. Functions  in the RHS of these equalities, with indices  related to coupled antennas 1 and 2, have form of integral averaging along random electric dipole source extension in Eq.(63), with integrands  being presented as

(76)

and

(77)

Functions  here coincides in physical sense with interference functions for point random electric dipole source exciting a single antenna 1 or 2 and are defined accordingly to Eq.(64) by

(78)

where . While the functions in Eq.(78) we call the auto-interference functions of single antennas 1 and 2, the functions  and  need being called the cross-interference functions of coupled antennas because the last two functions take into account interference between a couple of spherical waves (see Fig. 13), one of which propagates from the point of random electric dipole source towards a receiving vibrator-dipole 1 end and another propagates to a receiving vibrator-dipole 2 end, as it is confirmed by equations

(78)

and where  is phase of the coupling factor  defined by . The sum in the RHS of Eq.(78) is taken over both indices  and  two values , with including four terms. The written equations for the cross- interference functions of antennas include an addition shift  caused by antennas coupling side by side with phase shift  equal to paths differences of two spherical waves in the biological object medium.

Consider dependence of antennas interference functions , defined in Eqs.(76) and (77) for the case of point random electric dipole source, on the antennas coupling factor . We see this dependence in a simple form of scaling factors  and  as well as in a complicate form of the addition phase shift in expressions for the cross-interference functions of coupled antennas. Nevertheless the complicate phase shift comes into  and  with opposite signs and hence transforms into a scale factor in the sum of these cross-interference functions

(79)

Summing next Eqs.(76) and (77), with accounting Eq.(79), gives

(80)

Summing at last the basic Eqs. (75) defined the fluctuations' spectral densities  and reminding definition of functions  give us

(81)

With

(82)

The obtained three Eqs.(80), (81) and (82) show that sum of the fluctuations' spectral densities of current distributions' along coupled receiving antennas 1 and 2 amplitudes caused by random electric dipole source thermal radiation has dependence on antennas coupling factor  in a simple form of scaling factors  and  only.

Although the RHS of Eq.(81) has a simple dependence on antennas coupling factor, the RHS of Eq. (80) leaves some difficult for study via complicate structure of four term sum, with each term describing interference of a waves' couple propagated from a point source towards different antennas' ends. Therefore we return to general Eq.(78) for the cross-interference function of coupled antennas and consider a mentioned above special case of symmetrical position of random electric dipole source relatively antennas (Fig. 13) when a many-side triangle, formed inside the -plane by centers of antennas and random electric dipole source centre projection on the -plane, becomes isosceles one. In this case of source symmetrical position we have relations , with index , that simplifies Eq.(78) immediately as

(83)

where  and  is given by Eq. (64). The final physically transparent result consists in equations

(84)

where  is presented in Eq.(62). Thus in the special case of source symmetrical position relatively antennas the both cross-interference functions of coupled antennas become equal to auto- interference function of single antenna accurate to scaling factor  as well as the fluctuations' spectral densities of current distributions' along coupled receiving antennas 1 and 2 amplitudes become equal to fluctuations' spectral densities of current distributions' along single antennas accurate to scaling factor . Ultimately one can reduce the scanning problem of random electric dipole source via two coupled tuned receiving linear wire antennas to considered already such scanning problem via a single antenna, provided one is able to place the scanning source into symmetrical position relatively antennas and move the antennas 1 and 2 together, not separately, along -axes on biological boundary surface  (Fig. 13).

Conclusions

Theory of electromagnetic wave multiple scattering by ensemble of dielectric and conductive bodies has been applied to study coupled receiving antennas. A basic exact system of Fredholm's second kind integral equations for electric currents excited inside antennas is derived and written in terms of the electric field tensor T-scattering operator of a single antenna, the electric field retarded Green tensor function of a background and the incident on antennas electric field. In this equations' system an antenna is a body with given complex dielectric permittivity on some frequency, and electric current excited inside the antenna means sum of volume conducting and displacement electric currents. The background medium can be inhomogeneous one with some complex dielectric permittivity. The kernels of derived integral equations' system are not singular for the case of no overlapping antennas, although the background electric field Green tensor function is singular in the origin. Such kind of three - dimensional singularity has been met really at study the wave integral equation for electric field inside single antenna in the homogeneous background, by verifying consistence this integral equation with boundary conditions on the antenna surface. As was demonstrated, one has to take into account two sorts of the homogeneous background electric field Green tensor function strong singularity: (i) electric field Green tensor function decomposition into a delta Dirac function term and principal part, and (ii) rule to bring out the second derivative outside the three-dimensional singular integral.

The derived integral equations' system for electric currents excited inside coupled antennas has been applied to study near field coherent effects caused by thermal microwave radiation incident electric field distribution along single or two coupled linear wire perfectly conducting receiving antennas in the form of thin vibrator- dipoles placed at heated biological object boundary surface and tuned to half wavelength in the object. After having been neglected wave interaction of antennas with biological object boundary surface and used asymptotic method of “big logarithm”, the T-scattering operator of single antenna in the form of tuned vibrator-dipole has become a separable wire T- scattering operator that lead to analytic evaluating the local total currents on two coupled receiving antennas and got a dimensionless antennas' coupling factor. Effects of homogeneous and local inhomogeneous biological object temperature components on receiving antennas have been considered. Homogeneous temperature component was treated as source for equilibrium thermal radiation with standard form of incident on receiving antennas electric field spatial correlation function. This treatment led to a generalized Nyquist formula for currents' fluctuations excited on coupled receiving vibrator- dipole antennas, with accounting the auto- correlation and cross- correlation functions of random electric field inside each antenna and on both antennas, respectively. More original results have been obtained at study effects of biological object temperature distribution local volume change in the model framework of random electric dipole source inside object absorption skin slab area, with dipole source being parallel to vibrator-dipole antennas parallel between themselves and placed on the object surface. In the case of single receiving vibrator-dipole antenna it was shown that two spherical waves are propagated from a point of random electric dipole source towards receiving vibrator-dipole ends. Bearing in mind the reciprocity between receiving and transmitting antennas one can say also that two spherical waves are propagated from vibrator- dipole ends towards the random electric dipole source and interfere on the source area. This physical interpretation led similarly with optics to single antenna auto-interference function. Extreme properties of single antenna auto-interference function depending on random electric dipole source extension and the source centre three-dimension position relatively receiving antenna on the biological object surface have been studied in details. It was shown, in particular, that the single antenna auto-interference function has maximum at source centre near antenna equatorial plane, with half width at half maximum (HWHM) becoming narrower and wider under biological object absorption and source extension growing, respectively. Ultimately a method of random electric dipole source scanning via single receiving vibrator- dipole antenna moving along the biological object surface was formulated. Two scanning most simple strategy was considered: (i) by defining via HWHM the two values of source centre seeming depth relatively two antenna positions with next evaluating the real depth of the source centre (many-side triangle strategy), and (ii) by getting a symmetrical position of source relatively two antenna positions that transforms the many-side triangle into a isosceles one (isosceles triangle strategy).

In the case of two coupled receiving vibrator-dipole antennas 1 and 2, as was shown two spherical waves are propagated from a point of random electric dipole source towards receiving vibrator-dipole 1 ends as well as two another spherical waves are propagated from the same point of source towards receiving vibrator-dipole 2 ends. Hence side by side with above single antennas 1 and 2 auto-interference functions, a cross-interference function of coupled antennas 1 and 2 has been introduced. The cross-interference function includes a complicate dependence on antennas coupling factor phase. Nevertheless, in the special case of source symmetrical position relatively antennas the cross-interference function of coupled antennas become equal to auto- interference function of single antenna, accurate to scaling factor equal to antennas coupling factor phase cosine. At the same time the fluctuations' spectral densities of current distributions' along coupled receiving antennas 1 and 2 amplitudes become equal to fluctuations' spectral densities of current distributions' along single antennas, accurate to scaling factor in the simple algebraic form of antennas coupling factor. Ultimately, the scanning problem of random electric dipole source via two coupled tuned receiving linear wire antennas has been reduced to such scanning problem via a single antenna, provided one is able to place the scanning source into symmetrical position relatively antennas and move the antennas 1 and 2 together, not separately, along biological object boundary surface.

Acknowledgments

This work was supported in part by Russian Foundation for Basic Research, Grant 09-02-00920-a, by the Russian Academy of Sciences project “ Passive multichannel human radio- and acusto-thermotomography in near zone” and “Investigations of new types of photonic crystals for development of optoelectronic elements of infocommunicationic nets”.

References

1. Godik E.E., Gulyev Yu.V. Functional imaging of the human body // IEEE Engineering in Medicine and Biology, 10, 21, 1991.