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

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

Y. N. Barabanenkov¹, M. Yu. Barabanenkov², V. A. Cherepenin¹

 



Abstract. We apply to study coupled receiving antennas a theory of electromagnetic wave multiple scattering by ensemble of dielectric and conductive bodies, with describing the excited currents inside bodies in terms of electric field tensor T-scattering operator. A system of equations for currents on surfaces of coupled perfectly conductive receiving antennas is written with the aid of a single antenna surface T-scattering operator. This system is resolved for the case of coupled linear wire receiving antennas in the form of thin vibrator-dipoles when asymptotic method of “big logarithm“ leads to separable wire T-scattering operator of single tuned vibrator-dipole. The separability simplifies analytic evaluating the local total currents on two (and more) coupled receiving antennas and get a dimensionless coupling factor. Our final aim consists in using the obtained analytic solution to study near field coherent effects caused by thermal microwave radiation incident electric field distribution along single or two coupled receiving vibrator-dipole antennas placed at a heated biological object boundary surface and tuned to half wavelength in the object. In the case of equilibrium thermal radiation we meet 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. In the case of local volume change of the biological object temperature distribution we reveal in the model framework of random electric dipole source inside object absorption skin slab area the interference-extreme properties for fluctuations of currents excited along antennas depending on relative positions of antennas and random electric dipole source. The reveled extreme properties are used as base to reconstruct the random electric dipole position via scanning the single antenna or two coupled ones along biological object boundary surface. electromagnetic wave multiple scattering

Key words: receiving coupled antennas, excited currents, T- scattering operator, thin vibrator antennas, biological object, temperature local variation, thermal radiation, interference receiving fields, scanning local random source.

Аннотация. Мы применяем теорию многократного рассеяния электромагнитных волн на ансамбле диэлектрических и проводящих тел для изучения взаимодействующих принимающих антенн, описывая возбужденные внутри антенн токи с помощью тензорного T -оператора рассеяния электрического поля. Система уравнений для токов на поверхностях взаимодействующих металлических принимающих антенн записывается с помощью поверхностного T- оператора рассеяния изолированной антенны. Записанная система решается для случая взаимодействующих линейных проволочных принимающих антенн в форме тонких вибраторов - диполей асимптотическим методом “ большого логарифма “, приводящим к сепарабельному линейному T- оператору рассеяния изолированного настроенного вибратора-диполя. Сепарабельность упрощает аналитическое вычисление локальных токов на двух ( и более) взаимодействующих антеннах и безразмерного параметра взаимодействия антенн. Наша окончательна цель состоит в использовании полученного аналитического решения к изучению ближнеполевых когерентных эффектов, обусловленных распределением падающего электрического поля теплового микроволнового излучения вдоль изолированного или двух взаимодействующих принимающих вибраторов-диполей, расположенных около граничной поверхности нагретого биологического объекта и настроенных на половину длины волны в объекте. При равновесном тепловом излучении мы приходим к обобщенной формуле Найквиста для флуктуаций токов, наведенных на взаимодействующих принимающих вибраторах-диполях, с учетом авто-корреляционной и кросс-корреляционной функций случайного электрического поля на каждом или обоих вибраторах-диполях, соответственно. Рассматривая локальное объемное изменение распределения температуры биологического объекта, мы обнаруживаем в рамках модели случайного электрического дипольного источника внутри поглощающего скин-слоя объекта интерференционно-экстремальные свойства возбуждаемых вдоль принимающих вибраторов - диполей флуктуационных токов в зависимости от относительного расположения вибраторов-диполей и случайного электрического дипольного источника теплового излучения. Обнаруженные экстремальные свойства используются как основа для восстановления расположения случайного электрического дипольного источника путем сканирования изолированным или двумя взаимодействующими принимающими вибраторами-диполями вдоль граничной поверхности биологического объекта.

Ключевые слова: принимающие взаимодействующие антенны, возбужденные токи, T- оператор рассеяния, тонкая вибраторная антенна, биологический объект, локальная вариация температуры, тепловое излучение, интерференция принимаемых полей, сканирование локального случайного источника.

Introduction

Measuring method of the temperature spatial distribution inside of a biological tissue object by recording its own thermal radiation in the microwave range is well known at the present time [1]. During the last twenty years, there has been realized that a contact antenna [2-4] or system of contact antennas [5] are situated in the area of a heated object thermal near fields, which have basically the form of evanescent electromagnetic waves exponentially decaying in perpendicular to the object boundary surface direction accordingly with Rytov's prediction [6].

Note, mean size  of a contact antenna aperture is rationally to be chosen in accordance with [2-5] as being much smaller than radiation wavelength  in free space and of the order of the radiation wavelength  inside a biological object, i.e., . Such inequalities suppose the real part  of biological tissue complex dielectric permittivity  in microwave range to be substantially greater the dielectric permittivity  of free space that is really the case for some tissues. For example, the dielectric permittivity of human head brain may be taken at frequency  = 780 MHz to be  [5] that gives  = 39 cm,  = 7 cm and absorption skin depth  = = 4 cm where  =  =  is the complex wave number inside brain medium, with  =  and  = . Under above inequalities putting on mean size aperture, the contact antenna receives strong decaying evanescent waves of thermal radiation in free space, which correspond to weak decaying evanescent waves inside biological object.

The wave theory [6,7,8] of an absorbing body thermal electromagnetic radiation having been published , Levin and Rytov [9] and Rytov et al [10] have considered some problems on current excitation in metallic antennas by thermal radiation fields, including the case of thin metallic antennas (linear wire antennas). To considered problems are related: (i) antenna excitation in equilibrium thermal radiation field; (ii) current excitation in antenna by thermal radiation of distant bodies; (iii) current excitation in coupled antennas, with phenomenological accounting of coupling effect by antenna mutual impedance.

The aim of our presented work is to consider the current excitation in coupled receiving antennas by incident electromagnetic field, with evaluating the coupling effect via analytic solution to system of equations for currents' distribution on coupled linear wire tuned antennas. Especially we intend to study the current excitation in such antennas by thermal radiation of small body placed in near field zone of antennas, as well as by equilibrium thermal radiation. Actually our work consists in the following.

Because electromagnetic mutual interaction (coupling) the antennas is included in theory of wave multiple scattering by ensemble of bodies [11], we write the volume current density inside a body in terms of the body electric field tensor T-scattering operator [12-17] and then derive the general system of equations for volume current densities inside all bodies of ensemble ones with the aid of Watson composition rule [12, 15] for scattering operators. A specific property of the derived system of equations for currents' densities inside bodies consists in that the system is written in terms of single body T-scattering operator. In particular case of antennas placed at a heated biological object boundary surface the derived general system of equations for volume currents' densities inside antennas takes into account also the coupling effects between antennas and object boundary surface (inhomogeneous object) that we do neglect in our concrete calculations. What is more, bearing in mind that mean size  of contact antennas conforms to wavelength  inside a biological object, we think them as were placed in the object boundary subsurface, with thickness being not more or order the absorption skin depth . On the way of such simplifications the mentioned general system of equations is transformed to a reduced system of equations for surface currents' densities on surfaces of coupled perfectly conducting receiving antennas written with the aid of a single antenna surface T-scattering operator. The reduced system supposes all coupled antennas to be placed in an effective unbounded medium, with wave number being equal to wave number  inside the biological object under study. The next our step consists in passing to linear wire antennas, with mean diametrical size of each wire being very small compared to wire length  and wavelength , ; . Passing to linear wire replaces the above single antenna surface T-scattering operator to single wire T-scattering operator that is one-dimensional kernel, which expresses the total current in a cross section of the wire through excited electric field component along wire. According to Levin and Rytov [9], the single wire T-scattering operator of tuned linear wire antenna,  ( is integer number), takes a simple separable form in asymptotic limit of “big logarithm” when parameter  has a small value. Theory of this asymptotic limit was elaborated by Leontovich and Levin [18] after Hallen paper [19] for analytic solution to Pocklington integral equation [20] that describes the current distribution along linear wire antenna depending on excited electric field component along the wire. We use the separable form of the single wire scattering operator to resolve the system of equations for currents' distributions along two coupled tuned linear wire antennas in terms of excited electric field components along both wires and their coupling factor, which is defined as ratio of antenna mutual impedance to self-impedance of a single antenna and evaluated directly. The mutual impedance of two wire antennas was studied earlier [21, 22] for transmitting antennas with the aid of more complicated method; convenient formulas for mutual impedance of antennas placed on big distant between them are considered in paper [23]. The obtained currents' distributions along coupled tuned linear wire antennas are applied to the case when exciting electric field is caused by thermal radiation of a biological object, with temperature being equal to sum of homogeneous component  and temperature local volume spatial variation. The homogeneous component of the object temperature  creates an equilibrium thermal radiation that is characterized by standard form [10] of electric field spatial correlation function as electric field tensor Green function imaginary part, with a scalar factor being in front of it. A local volume variation of the object temperature  gives rise to the thermal radiation from corresponding local random electric current density variation  (dipole) delta-correlated with respect to spatial position and polarization (orientation). Such electric current  can be thought as sum of three mutually perpendicular and statistically independent random currents (dipoles). On this way we come to a model of random electric dipole source inside a biological object oriented parallel to coupled tuned linear wire antennas. We suppose the random dipole to be placed in near wave zone of both antennas and evaluate the antennas' excitation via this dipole with the aid of method elaborated by Brilliouin [24], Pistolkors [25], and Bechmann [26] to study electromagnetic field in near zone of linear wire antenna (see also [27]). This model of random electric dipole source reveals the interference - extreme properties for fluctuations of excited in antennas currents depending on relative positions of antennas and the random dipole source. The revealed interference - extreme properties are analyzed from viewpoint to reconstruct the random dipole source position via special arrangement of two antennas on biological object boundary surface or via special scanning the single antenna along this boundary surface.

The organization of the paper is as follows. In Sec.2 the general equations' system for volume currents' densities inside coupled receiving antennas near biological object boundary surface is written in terms of a single antenna electric field tensor T -scattering operator. In Sec.3 the derived general system is transformed to reduced system of equations for surface currents on perfectly conducting antenna's surfaces written in terms of a single antenna surface T- scattering operator. In Sec. 4 a passing to system equations for currents along coupled linear wire antennas is described and these equations' system is written in terms of a single wire T-scattering operator. In Sec. 5 the equations' system for currents along two coupled turned linear wire antennas is resolved in asymptotic limit of the so-called “big logarithm”. In Sec. 6 the obtained currents' distributions along coupled tuned linear wire antennas is applied to the problem of antennas' exciting by a biological object thermal radiation via the object temperature homogeneous component. In Sec. 7 a local volume variation of the object temperature is modeled by corresponding random electric dipole source and the interference- extreme properties for fluctuations of excited in antennas' currents depending of relative positions of antennas and the random dipole source are considered. Conclusions are made in Sec. 8.

 Coupled receiving antennas at biological object boundary

We start with the Helmholtz vector wave equation for electric field  of a monochromatic electromagnetic wave of frequency  in a three -dimensional () inhomogeneous isotropic dielectric and conducting structure writing the equation as

 

(1)



Figure 1. Schematic showing of biological object (grey area; dotted line symbolically visualizes an absorption skin depth ) thermal radiation receiving on two coupled antennas whose length, diameter and the distance along the x-axis direction are denoted as , , and  respectively; . The designations , , and  stand for homogeneous and random local volume variations of the object temperature , and local electric current density variation related to the local temperature variation , respectively.

 

Our structure consists of homogeneous biological object occupying (Fig. 1) the left half space  > 0 of the Cartesian coordinate system  and receiving antennas placed at the object boundary surface = 0. Symbol  denotes the complex dielectric permittivity of the structure without antennas (bounded biological object) equal to dielectric permittivity  = 1 in free space  < 0 and  inside biological object  > 0, with  and  being real part of the complex dielectric permittivity and specific conductivity of the object, respectively. An effective “scattering potential” of antennas is defined by



(2)

where  is the complex dielectric permittivity of antenna, with  and  being its real dielectric permittivity and specific conductivity, respectively. The function  is the characteristic function of the -th antenna, equal to unity if point  belongs to the region occupied by the -th antenna and equal to zero otherwise. The antennas are assumed to be identical and centered at positions , . Each antenna is characterized by a single scattering potential , defined via Eq.(2). The magnetic permittivity is supposed to be  everywhere. The vector  denotes the electromagnetic field source volume current density, which can be random as in the case of biological object thermal radiation. The summation over repeated Greek subscripts is implied in the limits from 1 to 3, with 1, 2, 3 corresponding to the  axes. The Gaussian system of units is used and  denotes the light speed in the free space.

We are interested in the electric currents excited inside antennas. Let us introduce a vector

( 3)

that presents accurate with the factor  a sum  of volume conducting and displacement current densities inside the -th antenna. A complete current density  excited in all antennas can be evaluated, provided the electric field tensor -scattering operator  [16,17] of antennas' system is known . Really, denote  the electric field retarded Green tensor function of the bounded biological object, which satisfies the Helmholtz Eq.(1) without antenna scattering potential in the left-hand side (LHS) and with the delta-source term  in the right-hand side (RHS) of the equation and the radiation conditions in the infinity. The electric field Green tensor function  enables one to bring [15-17] a solution problem for the Helmholtz Eq.(1) to wave integral equation

(4)

Here  denotes the incident on antennas electric field given by

(5)

Solution to the wave integral Eq.(4) is presented [15-17] in terms of the electric field tensor -scattering operator of antennas' system by an equality that has a form

(6)

Comparison of this equality with integral Eq.(4) gives the Lippmann- Schwinger equation for the electric field - scattering operator of antennas' system



(7)

and a symbolic operator relation , which in details is written as

(8)

The obtained relation shows that one can indeed evaluate the complete current density excited in all antennas, having known the electric field tensor - scattering operator of antennas' system and the incident on antennas electric field.

The wave integral Eq. (4) can be derived with the aid of vector Green theorem and the boundary conditions on antennas surfaces and at infinity, with similar to [16] manipulating for the case of bodies' system in free space.

Side by side with the electric field -scattering operator of antenna's system it is convenient to consider also the electric field -scattering operator of a single -th antenna , that satisfies the Lippmann-Schwinger equation with the single scattering potential, . According to Watson composition rule [12,15] connection between - scattering operator of antenna's system and single scattering operators  of antennas is given by a system of operator equations

(9)

Here  has sense of - scattering operator of -th antenna coupled with all other antennas. Convolving Eq.(9) with the incident electric field  gives for current densities  inside antennas (3) a basic system of equations

(10)

written in terms of current densities inside of single antennas  and the single scattering operators  of antennas.

The basis system of Eqs .(10) for currents' densities inside antennas is too complicated, having accounting the coupling effect not only between antennas but also between antennas and bounded biological object. In the next section the system (10) is simplified via accounting the coupling effect between antennas only.

Coupled receiving perfectly conducting antennas inside the object boundary subsurface

Remind that in practice [2-5] the mean size of contact antennas conforms to wavelength  inside a biological object. We use that conformity to avoid the problem of coupling between antennas and biological object boundary, with replacing in all equations of the preceding section the electric field Green tensor function  of the bounded biological object to the Green tensor function  of unbounded biological object. Simultaneously one needs replacing in Eqs. (2) and (3) the permittivity  to .

We consciously use such rather rough approach to focus our attention on coupling effects between antennas at receiving the biological thermal radiation.

The above electric field Green tensor function of unbounded biological object is given by

(11)



where  denotes a scalar Green function. The basic Eqs.(10) for currents densities  inside two coupled antennas  takes a form

(12)

and

(13)

where indices 1 and 2 are related to antennas 1 and 2 (Fig.1), respectively, and integrations are performed inside volumes of these antennas.

The next problem consists in evaluating the single scattering operators  and  of antennas 1 and 2. Instead of direct solution to the Lippmann- Schwinger equation for a single scattering operator mentioned before a system of Eqs.(9), we remind an expression after Eqs.(10) for current density inside a single scattering antenna in terms of its single scattering operator and incident electric field, with rewriting the total electric field (6) everywhere around and inside antenna as

(14)

Resolving this equation with respect to current density  gives a convenient tool to find the single scattering operator Green theorem of perfectly conducting antenna. But previously to make so we need discussing some general properties of Eq. (14) for a body with finite dielectric permittivity.

The integral Eq. (14) for electric field, with replacing  to , is known in wave multiple scattering theory for a long period of time and often related to pioneering papers of Foldy [28] and Lax [29]. Nevertheless, an opinion exists [30] that one needs verifying consistence the Eq. (14) with boundary conditions on the body surface directly. In this connection we would like to make the two following remarks.

First, as was mentioned after Eq, (8), the Eq. (14) has been derived [16] with the aid of vector Green theorem and accounting the boundary conditions.

Second, Eq. (14) is a three- dimensional singular integral equation and demands some accuracy at handling with it. If point  is placed inside the body one has to take into account the strong singularity of the tensor Green function, Eq. (11), at , with decomposing [31-33] the one into a delta Dirac function term  and principal part  where a constant tensor  depends on the shape of the exclusion domain chosen to define the principal part. If the exclusion domain is an infinitesimal sphere the tensor  Substituting the decomposed tensor Green function into Eq. (14) gives after [32,34] an equation

(15)

where integration in the RHS is performed over body volume, with excluding an infinitesimally small spherical domain around point . A local electric field  and transformed scattering potential  are defined by



(16)

The strong singular term of the tensor Green function in Eq. (11) at  and  has a form  and becomes of zero value by averaging over unit vector  along vector  directions. Hence the integral in the sense of principal value in the RHS of Eq. (15) is good defined one and can be expressed according to general theory of many dimensional singular integrals [35] as sum of absolutely converged integrals. However, Eq. (15) does not show directly its consistence with boundary conditions. Therefore we transform this equation to a physically transparent form usual for theory of electromagnetism [27].

Let us apply a lemma (see, e.g., [36], appendix 50) as the rule to bring out the second derivative outside the three dimensional singular integral

(17)

where  is some scalar test function. This lemma enables one to transform Eq.(15) as follows

(18)

Here  denotes the Hertz vector that is written in terms of the body polarization vector  as



(19)

with omitting the limit symbol at the RHS integral. The scalar electric potential  is evaluated by applying the divergence operator under integral sign in the RHS of Eq. (19). Then similar to the case of dielectric polarization in electrostatics (see [27], paragraph 3.13) one can introduce a volume density  of body electric polarization charge for point  inside the body and a surface  density one for point  on the body surface where  is the outward unit normal vector to the surface. The surface charge is coursed by polarization vector discontinuity at the body surface crossing and leads to discontinuity of the electric field (see [27], paragraph 3.15) on the body surface according the equation

(20)

where the subscript plus and minus are related to regions out and inside the body at surface point , respectively. Eq. (20) means discontinuity the normal component of electric field at the body surface crossing according to relation  as well as continuity of electric field tangential component at the body surface crossing.

Having verified the consistence of Eq. (14) with the boundary conditions on the body surface, let us return to system Eqs. (12) and (13) for currents' densities inside two coupled antennas. In the limit of a perfectly conducting antenna the current density inside its volume becomes to be confined near the antenna surface . Therefore we write, e.g.,  where  and  are elements of antenna volume and surface, respectively, as well as  and  are antenna volume and surface current densities, respectively. We write similarly a relation  =  defined a surface - scattering operator . Substituting into Eqs. (12) and (13) gives

(21)

and

(22)

Here the indices 1 and 2 are related to antennas 1 and 2, respectively, but in deference from Eqs.(12) and (13) the integrations are performed now along the surfaces of antennas 1 and 2. Because the total electric field  on perfect conducting antenna's surface is equal to zero the Eq.(18) takes a form

(23)

if point  is placed on the antenna 1 surface. Resolving Eq.(23) with respect to the surface current density and bearing in mind a relation

(24)



one can get the desired single surface - scattering operator  of antenna 1.

Coupled receiving linear wire antennas

Consider the case of linear wire antennas in the form of thin vibrator- dipoles to be parallel to  axis and occupying the regions  as depicted in Fig.1. For a cylindrical vibrator with diameter  only tangential component  of the surface current density  exists and one can introduce a total current  in the cross-section  of the vibrator. Eq.(23) for the surface current density of a single antenna is transformed in the case of linear wire antenna under consideration to the Pocklington integral equation [20] (see also [37])

(25)

with vector potential -component  given by

(26)

Here  as well as the observation  and source  points are presented in cylindrical coordinates. The - component of vector potential (26) is evaluated asymptotically for thin vibrator near zone, where  and , as a sum of logarithm's and addition terms



(27)

with addition term  having a current functional form

(28)

Substituting the vector potential - component asymptotics (27) into Pocklington integral Eq.(25) leads to the Leontovich - Levin version [18] of Hallen integro-differential equation [19] (see also [37]) for current distribution  along single linear wire thin vibrator-dipole receiving antenna

(29)

where current functional  is defined by

(30)

Remind that  is small parameter in asymptotic limit of “big logarithm”.

Relation (24) between a single antenna surface current density and antenna single surface - scattering operator takes in the limit of linear wire thin vibrator -dipole antenna the form



(31)

with a single wire - scattering operator  defined by

(32)

Similarly, the system of Eqs. (21) and (22) for surface current densities of two coupled antennas passes to equation system for current  distributions along two coupled linear wire antennas  = 1,2 written in terms of single wire - scattering operators  as

(33)

and

(34)

Here  where  and  is the distant between antennas (Fig. 1). In the next section we present asymptotic solution to Leontovich - Levin-_Hallen Eq.(29) and to equation system (33) and (34).

Current distributions along two coupled tuned receiving linear wire antennas in the limit of “big logarithm”

Bearing in mind the small parameter  one can apply after [18] the perturbation method of solution to Eq.(29) for current distribution  along single linear wire antenna in the form of expansion  with result of substitution

(35)

(36)

This system of equations should be resolved successively. We restrict ourselves by simple case of tuned vibrator-dipole.

The length  of tuned vibrator - dipole is equal to the whole multiple of half wavelength  in the biological object,  or, with  being integer. Eq.(35) for zero approximation has a solution, where  is current amplitude and function  if  is odd and  if  is even. Current amplitude  is defined by orthogonality condition of zero approximation to RHS of Eq.(36) for the first approximation that after some algebra gives

(37)

The quantity  has sense of input impedance of single vibrator - dipole feeding in the current maximum point, when  and , and it is evaluated via formula



(38)

Here the functions  and  are defined by integrals

(39)

with  being the integral sine [38] and regular function  related to the integral cosine  and Euler constant  ? 0.5772 as .

Substituting current distribution  along turned single vibrator- dipole into Eq.(31) gives for single wire - scattering operator  of such vibrator- dipole a separable value defined by

(40)

The separability property of single wire - scattering operator makes the Eqs.(33) and (34) for current distributions along two coupled linear wire antennas to be exactly resolved. One can write out result of this resolution in a form similar to the case of a single linear wire antenna putting  where amplitudes  of current distribution along two coupled linear wire antennas are given by



(41)

and

(42)

Here the indices 1 and 2 are related to antennas 1 and 2, respectively, and integrations are performed along these two antennas. A specific coupling factor  where  is mutual impedance [37] of two vibrator - dipoles is defined by integral equality

(43)

Transforming double integral in this equality RHS by the method [25-26], with applying part by part integration, leads to a working formula for coupling factor evaluation

(44)

where a function  is expressed in terms of integral cosine and integral sine as  and parameters  and  have a form

(45)



Figure 2. Dependence of the real (solid line) and imaginary (dashed line) parts of the specific coupling factor  of two half wavelength  antennas versus normalized distance  between antennas.

 

An useful asymptotics for coupling factor  at small distances  between antennas reads

 

(46)

where input impedance  of single vibrator - dipole is defined by Eq. (38). Fig.2 represents the real and imaginary parts of the coupling factor  versus the dimensionless distant  between two half wavelength  vibrators.

In the next section we apply the presented currents' distributions along single tuned vibrator- dipole as well as along two coupled tuned vibrator - dipoles to the problem of antennas' exciting by a biological object thermal radiation via the object temperature homogeneous component.



Exciting single and coupled tuned receiving linear wire antennas by equilibrium thermal radiation

According to the general theory [6] the electromagnetic thermal radiation of a heated absorbing body is created by a random electric volume current density  with the spatial correlation function spectral density given by

(47)

where  denotes the body temperature multiplied by the Boltzmann constant. In this section we are interesting in effects of homogeneous component  of the biological object temperature. In this case one obtains from Eqs. (5), (11) and (47) the following expression for the incident on antennas random electric field spatial correlation function spectral density

(48)

that is a standard form [10] of equilibrium thermal radiation electric field spatial correlation function spectral density.

Apply first the expression in Eq. (48) to the case of a single tuned vibrator-dipole antenna exciting by a biological object thermal radiation via object temperature homogeneous component. Eq. (37) gives for the fluctuations' spectral density  of current distribution along receiving antenna amplitude an equality



(49)

where one sees in the RHS integrand the auto- correlation function of random electric field inside antenna. Using for this auto- correlation function the expression in Eq. (48) and transforming the double integral by abovementioned method [25-26] leads to

(50)

that is actually the Nyquist formula for thermal excitation in conductors [39].

Turn now to the problem of two coupled tuned vibrator-dipoles' exciting by a biological object thermal radiation via object temperature homogeneous component. On base of Eqs. (41), (42) and (48) the fluctuations' spectral density  of current distribution amplitude along, e.g., the antenna 1 is given by relation

(51)

The first double integral in the RHS of this relation takes into account electric field auto-correlations along single antenna 1 and single antenna 2, as in the case of the Nyquist formula in Eq. (49) derivation, and was considered actually in [9]. In deference, the second double integral takes into account cross-correlation of electric field fluctuations along antenna 1 and antenna 2 and was missed in [9]. Proceeding evaluation of double integrals in the Eq. (51) RHS leads to the following result

(52)

where factor  of equilibrium thermal radiation coherence between two coupled antennas has a real value and is defined by

(53)

A part of this factor, which accounting for the term  only in the Eq. (53) RHS square brackets, was obtained in [9] on a phenomenological level of consideration, that is in terms of coupled antenna 1 input impedance  =  and mutual impedance . Asymptotics in Eq. (46) for coupling factor  at small distances  between antennas shows that factor in Eq. (53) has a limit  as distant between antennas becomes much smaller their length.

Exciting single and coupled tuned receiving linear wire antennas by thermal radiation from modeled local temperature inhomogeneity

In the preceding section a problem was considered about single and coupled tuned vibrator-dipoles' exciting by a biological object thermal radiation via object temperature  homogeneous component . One can relate to this temperature homogeneous component a random electric volume current density component  in Eq. (47) LHS. Similarly we can connect the object temperature local volume spatial variation with a corresponding local random electric volume current density variation . Supposition about statistical independence from  leads to the spatial correlation function spectral density for local variation  in a form of Eq. (47), with replacing  to  in the RHS of this equation that is

(54)

Eqs. (5), (11) and (54) give expression for the incident on antennas random electric field  spatial correlation function spectral density caused by local spatial variation of the object temperature

 (55)

Detailing the idea about local random electric volume current density variation one can split this variation into sum  of three mutually perpendicular and statistically independent random currents (random dipoles)  where  are the unit vectors along the  axes, respectively to  The random currents  satisfy the Eq. (54) evidently. Correspondingly the incident random electric field splits into sum  of three statistically independent incident random electric fields defined by

(56)

Physically the random electric field  is created by random electric current (dipole source) .

 

Figure 3. Schematic showing of vibrator antenna in thermal radiation field of random electric dipole source  which is extended in limits  along the -axes and defined by a vector  in the - plane (gray area) with length  and azimuth angle .

We have come up closely to our basic model of random electric dipole source inside a biological object oriented parallel to single (Fig. 3) or coupled (Fig. 1) tuned linear wire antennas. In this model only the random electric field  created by random electric current  is taken into account. We think the last random electric current as being confined inside a thin and finite cylindrical region, which is extended in limits  along the -axes and defined by a vector  in the - plane as, e.g., in the case of the single vibrator- dipole antenna (Fig.3). Integrating the random electric volume current density variation  over the cylindrical region cross- section leads to a total random current variation  in chosen cross-section, with a longitudinal correlation function spectral density giving by

(57)

Here  is a volume of the random electric dipole source.

One started actually to consider a problem of exciting single receiving linear wire antenna by thermal radiation from local temperature inhomogeneity. In the framework of the introduced random electric dipole source model, the - component  of the incident random electric field along single vibrator- dipole antenna (Fig. 3) is given according to Eqs. (11) and (56) by relation

(58)

where a Green function  is obtained from  defined after Eq.(34) by replacing  to and  to . The incident random electric field in Eq.(58) excites along single tuned vibrator- dipole receiving antenna a current distribution with random amplitude  given by Eq.(37), with replacing  to  in the RHS integrand. Thus one gets

(59)

Applying to the RHS of this equation the integration part by part with respect to variable  , similarly with integral in Eq. (43), gives

(60)

For the case of tuned vibrator -dipole of length  equal to odd whole number  of half wavelengths Eq. (60) is transformed as

(61)

where  are defined by  (see Fig.4). According to obtained equation the 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. Eqs. (57) and (61) enable us to write out for the fluctuations' spectral density  of current distribution along receiving antenna amplitude caused by random electric dipole source thermal radiation a following equality



(62)

 

Figure 4. (a) The normalized interference function versus normalized height  (from the  plane) of random electric dipole point source at different values of the point source normalized seeming depths , with neglecting object absorption:  = 1 (curve 1), 1.5 (2), 2 (3), and 6 (4). (b) the normalized seeming depth  versus HWHM of the curves like ones in the panel (a) is delivered by curve 1 and calculated by Eq.(67) is given by curve 2.

A function in the RHS of this equality has a form of integral



(63)

where integrand  is defined by

(64)

with . Similarly to optics [36] the definitions in Eqs.(63) and (64) can be called the single linear wire receiving antenna interference functions for the cases of finite extended and point random electric dipole source, respectively.

The spatial integral averaging in the RHS of Eq. (63) along random electric dipole source extension we perform using an approximate formula

(65)

where a function . This formula is derived by writing  and approximate bringing the denominator  outside the integral in the middle point. Actually we needs spatial averaging the fast varying cosine term of  only that gives for an expression

(66)



As one sees the interference function  for finite extended random electric dipole source is different from such interference function  for point source on a -factor that defines a spatial coherence degree of extended random electric dipole source.

Our next task consists in study the interference function  extreme properties depending on random electric dipole source extension and  position defined by mentioned above coordinate  and vector . In this case the coordinate  gives height of random electric dipole source centre above () or under () the single linear wire antenna - plane (Fig. 3) while the vector  characterizes  position of random electric dipole source centre projection on the -plane that can be characterized also by the vector  length  and its azimuth angle  (see Fig. 3). The minimum value  of the vector  length corresponds to the azimuth angle  and gives us the real depth of random electric dipole source centre, with  being a seeming depth of this source centre. Henceforth the aim of the single receiving antenna scanning along biological object boundary surface  consists to get the random electric dipole source centre inside the -plane, first, and to define the source centre real depth by, e.g., placing this centre on the - axes, second. We intend to show that interference function  extreme properties can be a physical base to realize such kind of single receiving antenna scanning strategy.

The interference function  has extreme value - maximum at , at last. One can verify this statement easily in the simple case of point random electric dipole source, with neglecting effect of absorption when interference function  has for small  the Taylor expansion

(67)

Here  is maximum value of , with  being equal to  at  The quantity  is half width at half maximum (HWHM) of the Taylor expansion in Eq. (67). A constant  in the RHS of the second Eq.(67)

is defined as  for the half wavelength vibrator antenna.

Eq.(67) shows that HWHM of interference function  can be used to determine the seeming depth  of random electric dipole point source. Because the HWHM is experimentally measured quantity we study next the interference function  extreme points and their HWHM in details, with taking into account effects of absorption and random electric dipole source extension.

Fig.4(a) presents calculated by Eq.(64) dependence of the normalized interference function  on normalized height  of random electric dipole point source at a set of the point source normalized seeming depths  ranged from 1 up to 6 when the curves have only one peak and absorption is neglected. From this calculated curves we take HWHM and plot HWHM against the known normalized seeming depth  (Fig.4(b), curve 1). The curve 2 in the Fig.4(b) plots HWHW according to the Taylor expansion in Eq. (67) that is the quantity . The Fig. 4 shows that HWHM of interference function growths monotonously with growing the random electric dipole point source seeming depth when effect of biological object absorption is neglected. The curve 2 in Fig.4(b) predicts substantially smaller value for HWHM at given  compared with the curve 1 in Fig.4(b) since the Taylor expansion dealing with only the peak top of the curves in the Fig.5(a). In Fig. 5 we repeat the curve 1 from Fig.4(b) and test the role of absorption in the case of point source. The absorption is measured in the units of  where n = 1, 2, and 3. The units of absorption measurement are chosen, with taking into account that for tuned vibrator-dipole  and for the human head brain , according to Introduction.

 

Figure 5. The normalized seeming depth  versus HWHM of the dependences of normalized interference function calculated by Eq.(66) in the case of point source but accounting for an absorption  in the units of  = : n = 1 (curve 2), 2 (3), and 3 (4). The reference curve 1 (no absorption) is the curve 1 in Fig.5(b).

 

Fig.5 shows that the more absorption is the less HWHM corresponds to the given point source seeming depth. In another words, the maximum peak of interference function  at  becomes narrower with growing absorption. 

Figure 6. The normalized seeming depth  versus HWHM of the dependences of normalized interference function calculated by Eq.(66) with neglecting of absorption (a) and with absorption accounting (b) in the case of point source (curve 1) and extended source  = 1 (curve 2), 1.5 (3), and 2 (4). The absorption is equal to . The curves are enumerated by the figures in a manner that the lower is the curve the higher is its number.

Figs.6(a,b) depict HWHM of interference function against the normalized seeming depth  at a set of random electric dipole source normalized extensions , with neglecting absorption, Fig.6(a), and accounting absorption, Fig.6(b). It is seen from the panels (a) and (b) of Fig.6 that the absorption leads only to an increase in the slope angle of a “bundle” of the curves with respect to the HWHM axis. From the other hand, comparison of Fig.5 and Figs.6 demonstrates that absorption and source extension make change in different manner the HWHM dependence on source centre seeming depth. As one can note, the maximum peak of interference function  at  becomes wider with growing the source extension in both cases of neglecting and accounting absorption. In addition to noted the effects of absorption and source extension are more sensitive for relatively big and small source centre seeming depths, respectively. Fig.5 curves allow one as to determine the point source seeming depth at given absorption as Fig.6(b) curves permit one to get a source centre seeming depth, with knowing biological object absorption and the source extension. If in the last case the source extension is known approximately with some accuracy the source centre seeming depth is obtained also approximately with corresponding accuracy.

Having described Fig.4(a), we mentioned that the normalized interference function curves have only peak when source normalized seeming depth ranged from 1 up to 6 and absorption is neglected. Fig.7(a,b) shows that for smaller values of source normalized seeming depth the normalized interference function curves can have several peaks, with possibility for side peaks being not less the central peak (Fig.7, curve 4 has three equal peaks). The side peaks may be studied in manner similar to the central peak consideration. Nevertheless we will not study side peaks here.

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