During last decade, the process of development of radio electronics, radio location, radio navigation, and radio communication worldwide was characterized by the following basic tendencies: technical realization of enhanced physical effects and technical solutions, aspiration to accomplish transmission and information processing in real time with the broad use of computers, and the expansion of the applications solved by technology. As a consequence, despite micro-miniaturization of radio electronics facilities (REF), the volume occupied by such equipment on mobile and stationary objects is increasing [1,2]. The progression of these modern trends is vitally necessary, but it aggravates even more the serious problem of the provision in radio engineering complexes (REC) of electromagnetic compatibility, which is understood as the ability of REF and REC to function together with limited degradation of their own essential parameters and features.
Practically, it is often required to provide significant decoupling between the receiving and transmitting antennas, located on a common surface at a small distance from each other. One of the most well-known ways to reduce coupling between antennas is the application of electromagnetic bandgap (EBG) structures [3-8]. The EBG structures have received increased attention in recent years [
In this paper, we re-visit the bandgap structure and present another interesting mathematical model for designing the structure and suppressing surface waves on metals. In addition, we examine the possibility of reducing the coupling between antennas located on the plane, using an inhomogeneous synthesized impedance. In particular, we investigate the design problem of the impedance surface when an infinite thread of in-phase magnetic current is located above the plane at a certain height, and also the case with its location right on the impedance surface. Finally, the behaviors of the complete field on the impedance surface and the decoupling level between antennas are also investigated.
The paper is organized as follows: in Section 2, we consider a solution to the problem of synthesis of an inhomogeneous impedance plane by a fixed reflected field.
A solution to the problem of coupling of antennas on an impedance plane is given in Section 3, and numerical results are discussed in Section 4. Finally, Section 5 is devoted to conclusions.
First, we consider a solution to the two-dimensional design problem for the arrangement shown in 
, there is an infinite thread of in-phase magnetic current 
located at the height
. On the surface
, the boundary impedance conditions of Shukin-Leontovich are fulfilled:
where 
is the unit normal to the 
plane, 
is the surface impedance, 
is the electric field, and 
is the magnetic field.
It is necessary to determine the dependence of the passive impedance 
on the surface S. Once Z(x) is obtained, the complete field in the upper space is found, and then the degree of decoupling between antennas can be obtained.
where 
and 
are reflected fields, 
and
. Here, 
is the ze-
roth-order Hankel function of the second kind, 
is the wave number, 
is the wavelength, 
is the imaginary unit, 
is the characteristic resistance of free space, 
, and 
is the first order Hankel function of the second kind.
The reflected field can be written as a sum of the reflected field 
in the fixed horizontal direction and the mirror-image field 
with unknown amplitude [
and
where 
and 
are field vector components of an imaginary mirror source, 
and 
are field vector components of the given reflected fields. The solution to the design problem given in this paper differs from the solution in [
, since the solution of reducing coupling between antennas in close proximity is of primary importance. The sense of this representation will become clear with further observation.
We now consider the analytical presentation of the distributed field on the 
plane. As long as the amplitude of the plane wave does not vary along the direction of its distribution, then for the reflected field in the direction
, it is possible to write:
where 
is the distribution of the scattered field on the surface
. We represent the mirrorimage field on the impedance plane as the following way:
where 
is the constant amplitude. Then, the summative magnetic field on the surface 
can be written
where
. From the first Maxwell equation, neglecting the derivative multiplier 
and
, we obtain for 
normalized on
:
where
. As a result, for the required impedance normalized on
, we also have:
where 
and
. In the general case, the resulting correlation gives the dependence of the passive impedance which gives a real part that can acquire positive as well as negative values.
Next, let us consider the design problem of the purely reactive impedance
. Presenting the correlation of Equation (6) as a real and imaginary part, we can obtain the condition of feasibility of purely reactive impedance:
An additional degree of freedom in the form of a mirror-image field 
gives an opportunity to realize the impedance structure with 
[
where 
and 
is an angle of reflection. When the source of the field in Equation (6) is located right on the impedance surface
, which provides a completely normal (at the angle
) reflection of the incident wave (without a mirror-image,
), the required impedance can be expressed:
where 
and
. From the condition of purely reactive impedance feasibility
it is not difficult to find the variation of the wave reflected from the inhomogeneous impedance plane,
:
where
. In this case, the impedance can also be found from a straightforward expression:
where
, 
and 
are the zeroth and first-order Bessel functions, respectively, and 
and 
are the zeroth and firstorder Neumann functions, respectively.
, 
, and
, shown in 
), the radiating waveguide (region
), and the receiving (region
) waveguide [
and
) by taking into account the boundary conditions on the surface of the impedance flanges and the equality of the tangential field components in the openings of the waveguides
in
;
in
):
, 
, and 
are solutions of the nonuniform Helmholtz equations for complex amplitudes of the vector potentials for regions
, 
, and
, respectively. In this way, the fields in the opening of the antennas and on the impedance part of the flange can be found. From this, the minimum level of coupling between the two antennas can then be determined.
and on the numerical solution of a system of integral equations through the KrylovBogolyubov method [
on the impedance surface as a function of its dimensions and the parameters, 
and
. In 
and 
(solid line), 
(dashed line), 
(dotted line). We see in 
. For
, the reactance remains zero for most of the interval, and for
, the reactance is only slightly below zero. At the end of the interval [0, 0.66λ], the curve is undefined (it becomes infinite) for
.
, normalized relative to the field 
above an ideal conducting plane for fixed 
and various angles:
(solid line), 
(dashed line) and 
(dotted line); and for the fixed angle 
with various values of the parameter, 
(solid line), 
(dashed line) and 
(dotted line), respectively. The length of the impedance structure is equal to 
for both cases. The results of calculations show that the best data (greatest decoupling) are obtained with the parameters 
and 
when the impedance acquires the greatest capacitive value near the source of radiation. The greatest decoupling level is obtained with 
and
. The synthesized impedance which gives appropriate results for increased decoupling should be taken into account, because it has a large negative value of the reactive part in close proximity to the antenna. This leads to the fact that the impedance practically creates an anti-phase field relative to the ideal conducting surface. As a result, all the energy of the electromagnetic field transfers into the energy stored around the antenna. The structure turns into a resonator without losses (for the reactive impedance), including radiation. As an example, 
of the antenna located above the ideal conducting surface (dashed line) and the impedance surface (solid line) with the parameters: 
and
. From the graph, it is apparent that decoupling is provided with reduction of the radiation field by 35 dB; i.e., a reduction of the main lobe of the radiation pattern.
and
), except for a sharp Fano-type variation at
. The reactance reaches its minimum point with a capacitive reactance of 60, and sharply transitions to a maximum inductive value of 60. The behavior