Investigation of the Field Scattered by Phased Equidistant Arrays Based on Asymptotic Methods of Electrodynamics

It is suggested that the asymptotic method of saddle point be used for solution of  integral electrodynamic equations of the electromagnetic field scattered by phased equidistant antenna arrays. This makes it possible to study the regularities of the re-radiated field in two arbitrary planes.The obtained expressions will contribute to design of new antenna systems, which will reduce electromagnetic field scattering. This will as well improve the electromagnetic compatibility of electronic equipment installed in the same or in adjacent objects equipped with antenna systems.The research results can be used to design algorithms for detection, recognition and identification of radar targets.


Introduction
The study of electromagnetic field scattering characteristics (re-radiation) caused by air and ground objects is important for radio detection and navigation. Sometimes it means "radar visibility"of the objects.
The antenna systems of air and ground objects are the largest contributors to their radar visibility.
They are the main sources of follow-up radiation or re-radiation of electromagnetic waves from the probing radar stations (RS). This phenomenon increases radar visibility of such objects in addition to negative impact on electromagnetic compatibility of radio electronic devices installed therein.
According to a number of sources [1][2][3][4][5], the contribution of antenna systems can be up to 98% of the total effective scattering surface (ESS) of the objects. This is especially true of aperture antennas and phased antenna arrays (PAA) of RS [5][6][7].

Analysis of recent research and publications
Analysis of the causes and regularities of PAA scattering proves complexity of its elimination. Special design and application of special coatings that reduce the level of reflected signal from the objects are not always acceptable for their antenna systems.
Typically, such improvements result in degradation of other main characteristics of antenna systemsgain, directional operation, etc. Therefore there is a need to optimize them according to the "efficiency-visibility"characteristic [4][5][6][7].
Unfortunately, it is not always appropriate. The problem is that each radio system is a source of radiation and any antenna scatters more than half of the incident energy [5].
Moreover, if the antenna system does not scatter any energy, it and does not receive it. It means that it is impossible to avoid such scattering (re-radiation) completely, but the scattering can be significantly reduced [2][3][4][5].
So it is essential to study the electromagnetic field and scattered PAA in order to develop methods reducing it.
The scattering properties of any objects are usually described by effective surfaces (widths, areas) the scattering is to be described by integral ( Σ ) differential ( ) surfaces and the scattering matrix (M) [8].
When studying scattering or wave re-radiation, PAA antenna system waves shall be considered a group of radiators, which represent an ensemble of shiny spots.
In this case, the problem of finding integral and differential scattering surfaces is reduced to the calculation of ESS group of radiators, with surface current brought upon each and electromagnetic field strength amplitudes excitation [5][6][7].
The complex strength amplitude of the electromagnetic field scattered by the reflector, which is at a distance from the observation point shall be calculated according to expression [7]: where ⃗ is a complex wave strength amplitude on the reflector; ⃗ 1 is ESS for one radiator; = 2 / is wave number; is wave length.
On the basis of the superposition principle for a linear equidistant array from radiators [7,9], the reradiated signal at the receiving end shall be created by the interference of the signals reflected from all the radiators located along axis ( fig. 1).
The complex field strength amplitude ⃗ Σ for a linear equidistant PAA with emitters in [7] shall be calculated according to the expression proposed in [7]: where ⃗ is a complex strength amplitude for the wave falling on radiator; ⃗ is ESS of radiator; is a distance from observation point to radiator. Fig. 1. Parameters of equidistant rectangular PAA for calculation of the scattered field at an incidence angle for a plane electromagnetic wave normally polarized to the incidence plane The expression for back scattering Σ ( , ) (of an integral single-position ESS) for a rectangular PAA shown in Fig. 1, shall be as follows [7]: where ( , ) is a back scattering diagram for radiator; , is a reflection coefficient; ∆ , ( , ) is a travel path deviation for a plane equidistant antenna array; , ( , , ) is a phase distribution of the incident signal as a function of the spatial and temporal aperture coordinates; is a number of radiators along axis ; is a number of radiators along axis . The study of the causes and regularities of secondary emission show that the main parameters of the system "PAA -probing RS"that affect the level of the reflected signal, are the angles , . They characterize the aspect of PAA re-radiation. The re-radiated signal level is influenced by a complex of PAA characteristics: orientation diagram ( , ), number of radiators along axis and along axis , the distance between them and and the electromagnetic waves travel path deviation.
A possibility to reduce ESS of the antenna system by means of effective phase distribution of the incident signal , ( , , ) as a function of spatial and temporal PAA aperture coordinates was considered in [7].
( , ), , , , are a priori known design characteristics of PAA class. They are practically unchanged during the service life of RS [7].
Angles , vary depending on PAA spatial position relative to the probing RS and PAA operating mode [7].
Consequently, we can reduce ESS of an equidistant PAA by changing design properties of all radiators, or by changing distances between them proportionally , .
For this purpose, we must study the amplitudes in the waveguide aperture of radiator (Fig. 2). The amplitudes of the field, which is excited on a linear equidistant array (Fig. 3) were considered in [10].
However, the final expressions concerned determination of the amplitudes, excited during aperture of such array only if the incident wave is normally polarized to the incidence plane.
In other free sources [7,9,11], the excited amplitudes and re-radiation field expressions are simplified and rather approximate. They do not allow to study the total field scattered from PAA with necessary accuracy in the case it is exposed to the wave normally polarized to the incidence plane at an angle from arbitrarily selected in order to reduce it.  Fig. 3. Characteristics of horn-type radiator line used for calculation of the scattered field with incident plane electromagnetic wave normally polarized to the incidence plane Against this background, the purpose of the article is to study the field re-radiated by a linear equidistant PAA, which is formed by an incident wave normally polarized to the incidence plane at an angle , and to determine causes and regularities of this phenomenon in order to reduce it.

Presentation of the basic material
Let the electromagnetic wave ⃗ , ⃗ which is formed by external currents distributed with density ⃗ and ⃗ be incident to the PAA aperture consisting from horn-type radiator linear arrays (see Fig. 1).
It is important to find a field scattered by such antenna.
Solution. We shall number the radiators from the center to the edges so that the central one is a zero one, while the extreme radiators are ( − 1)/2 as shown in (see Fig. 1). The total number of the elements in the array shall be odd and equal to .
The strength of the field , formed by a pair of radiators symmetrically located about the center shall be written as where + is the strength of radiator field = 1; − is the strength of field = -1.
In this case, the total field scattered by the linear radiator array shall be: where 0 is the strength of the field, which is formed by the central radiator.
In order to determine 0 it is necessary to investigate the wave amplitude in the aperture.
Let an independent source be placed inside the central radiator in the transmission mode (see Fig. 1) and form a field, which shall be marked ⃗ 0 , ⃗ 0 inside and outside.
Inside the horn such a field shall be usually considered eigenfunction and marked -. However, due to the fact that index is already in use, we shall propose a field with a unit amplitude as ⃗ ± , ⃗ ± . It is reflected from the aperture with a reflection coefficient where (− , ) is the number of standing semiwaves, which fall on the sides of the cross-section and extend from the neck to aperture; (+ , ) is the number of standing semi-waves propagating from the aperture to the neck.
Consequently, on the surface of the antenna aperture from the inner side, a full field can be presented by an eigenfunction expansion [9,10]: where ± are eigenfunction amplitudes; ⃗ + , ⃗ + are eigenfunctions, which spread from the aperture to the neck; ⃗ − , ⃗ − are eigenfunctions spreading from the neck to the aperture; − -coefficient of eigenfunction reflection from internal inhomogeneities in the horn.
In order to study the field ⃗ scattered by aperture of one or -th radiator of the antenna, we shall use asymptotic methods with application of Lorentz lemma.
In order we could determine such a field, it is necessary to implement strict boundary conditions, i.e. continuity and tangential components of the total field ⃗ and ⃗ to aperture . Let's place coordinate origin in the center of the array ( = 0) (see Fig. 3).
For this case we have: where ⃗ , ⃗ are the strengths of electric and magnetic fields incident on the aperture and excited by currents beyond the horn; ⃗ , ⃗ are the strengths of the electric and magnetic fields from the aperture (4).
A component of the field ⃗ scattered throughout the space may be presented as a continuous spectrum of plane waves [9]: where ⃗ ( , ) is a spectral function of the complex amplitudes of plane waves; , , are projections of the wave vector to the axes , , , which are connected by formula 2 = 2 + 2 + 2 .
After we plug (4) and (6) into (5), we get: where + is an amplitude of the excited wave.
A similar expression will be used for field components.
Let's multiply both sides of the equation (7) by ( + ) and integrate them over and on the aperture surface .
We shall write the Lorentz lemma [9] for the volume limited by an infinitely distant from the antenna surface and a bounded surface (see Fig. 2) as: and then complete the integration from to infinite limits provided ( = 0) = 0 beyond the surface . If we use the relation obtained in [9]: we shall have: Let's plug (10) into (7) The magnetic components of the scattered field can be obtained from the Maxwell's equations: In order to calculate the scattered field according to [5] we shall apply the cross-section method. We must take into consideration that in addition to the waves excited in the aperture and reflected from the internal inhomogeneities, there are so-called "parasite"waves.
The field scattered by -th radiator shall be written as: where is the aperture integration surface of -th radiator (see Fig. 2) from the inner side; ⃗ is a unit electromagnetic field strength vector; ⃗ + is a spectral function of the complex amplitudes of the plane waves excited at the aperture; + is the strength vector of the electromagnetic field incident on the -th radiator; , , are the projections of the wave vector on axes , , ; ( = 0) is the strength vector of the electromagnetic field after taking into account the boundary conditions and integration ad infinitum beyond the surface ; − the internal inhomogeneities reflection coefficient of the -th radiator.
For the case of normal polarization of the wave in the plane of incidence, after plugging eigenfunctions [9] into (13) and their integration over , the tangential component of the field ⃗ ⊥ will be: where and are the dimensions of the rectangular aperture of the -th horn radiator (see Fig. 3).
We have to decide on the most effective or appropriate method studying the field ⃗ ⊥ in order to solve the problem.
Generally, the theory of wave propagation suggests a limited number of problems that allow exact solution.
In those few cases where strict relations are known, they are quite complicated and do not allow to reveal physical nature or cause of the process regularities even with the help of advanced software packages.
We can understand the attention to the approximate methods of the wave theory, particularly asymptotic methods [10,11] in recent years.
They are still relevant. In order we could apply the above methods, we shall rewrite (14) as: .
The functions 1 ( ) and 2 ( ) under the integrals in the expression (15) depend upon several parameters characterizing the system.
In this case, it will be the method of asymptotic evaluations helping us both take the integral and obtain the explicit dependence from the parameters specified in arbitrary planes of the incident wave.
The method of saddle point (steepest descent) is one of such asymptotic methods. It gives us an adequate accuracy and has a wide application for the study of different wave phenomena: acoustic, electromagnetic, etc [10].
In order to apply the above method, we shall present the expression (15) in a spherical coordinate system: = sin cos , = sin sin , = cos .
We get the following dependence: Using the method of saddle point, we solve the equation (21): In order we could identify the physical nature of the phenomenon, let us consider a field scattered in some planes.
Usually we use a plane = 3 /2 and = .
In contrast to the existing approaches, the expression (27) allows to calculate a field both for linear and equidistant rectangular PAA.
The expression (27) differs from (21) because it takes into account the number of radiator and the distance between them .
The difference is also in use of (27) for more accurate expression for the amplitude ⊥ +0 and additional multipliers exp (− ) and exp ( ( sin П − )).  The studies were carried out for different values of the reflection coefficients.
The wave amplitudes (lines 1, 2, 3) in Fig. 5 are almost identical with the sensing angle wave amplitudes in a polar coordinate system (lines 1, 2, 3) in Fig. 6, which were obtained using a simplified expression.
We can check the reliability of the obtained mathematical expressions using asymptotic methods for solution of integral equations.
To reveal the causes and regularities of re-radiation from the aperture of the equidistant PAA for the specified arbitrarily selected provided a normal polarization of the incident wave in the incident plane, we shall use again the asymptotic method of saddle point in the plane = 3/2 and = .  Fig. 6. How the wave amplitude depends from the side-scan remote sensing angle in a polar coordinate system After we take an integral (16) the field scattered by the antenna array shall be as follows in the plane = 3/2 : = /2 Thus, the field scattered in the plane = 3/2 shall be: In the plane = we get: Taking into account (33), (34) the expression for the scattered field in the plane = shall be:

Conclusion
The application of asymptotic methods of electrodynamics allows us to determine the field scattered from the horn aperture in the case of normal polarization of the incident wave to the plane of incidence, and in the case of coincidence of the plane of incidence and wave polarization.
The electromagnetic field scattered by phased equidistant RS antenna arrays can be reduced by application of asymptotic methods for solution of integral equations as evidenced from the simulation according to the obtained expressions.
We have proved that in order to determine the scattered field it is advisable to apply the method of saddle point.
The wave amplitude diagrams as functions of the side-scan remote sensing angle shown in Figure 4, 5, 6 demonstrate that an improvement of adjustment in the antenna path shall result in an increase in the maximum amplitude of the signal in the transmission mode. According to the antenna reciprocity principle, such adjustment will improve absorption of the top type waves at the aperture of an individual radiator or an equidistant antenna array.
This will reduce the voltage standing wave ratio and the side lobe level.
Consequently, the probing RS will receive a reduced re-radiated signal, which will improve reconnaissance protection of PAA [1,9].
The above expressions (6) for an individual radiator and (16) for an equidistant antenna array have computational, practical, and methodological value. Their consistent development and physical interpretation will allow us to estimate their use in the study of the scattered (re-radiated field) for PAA and other antenna systems with pyramidal horns used as exciters.
The results obtained apply both to the development of electrodynamics theory and to the improvement of calculation methods [12][13][14][15][16]. They can be applied in the development of algorithms for detection and recognition of radar targets.