Passage of higher type waves through the corresponding slice frequencies does not lead, unlike waves 
and 
, to sudden changes in the behavior of the electromagnetic field near the connection aperture, therefore the impedance module Z behaves rather smoothly and does not get lower than 660 Ohm. It has been also determined that the main contribution to the forming of the field is made only by the main waveguide wave and the nearest to it, whereas the contribution of the other waves (including supercritical ones) is negligible.
Fig.3b shows calculated on a PC behavior of the real and imaginary parts of the input resistance of RS AA, located at the distance of 0.125
from the waveguide with the sizes specified above. The computation is performed based on the condition that the impedance Z is distributed uniformly across the array period and is variable in frequency (fig.3a), using the action formulas [1] where 
= 0, 
. The radiator has length l = 
= 0.2, width 0.045 and is excited by 
−generator. The array is phased in the normal direction. The figure shows that in the frequency band with overlap 1.7 the radiator can be well-agreed with the feeder line.
As the numerical experiment showed, usage of more than one waveguide in the AA period does not allow improving the behavior of impedance Z significantly.
To find out the maximum abilities of the subcritical waveguide in obtaining the necessary surface impedance, optimization of the waveguide array was performed. As optimization parameters, dielectric permittivity 
of the waveguide and its size a, b and dielectric permittivity 
and width t of the dielectric coating were used. Also 
= t, and all magnetic permittivity values were set to one. As a goal function, the following function was used
 |
(4) |
to minimize which, the local variation method [4] was used. In the expression (4) 
— frequency in the i−th point of the range, 
— required impedance value, x = y = 0. For double frequency band when M = 10, = 900 Ohm, period AA = 
= 0.2 and normally incident flat wave the optimization results are follows: = 7.89, a = 0.19 , b = 0.2, = 1.247, t = 0.127. In this case the impedance Z was found in the point x=y=0. Behavior of the optimized structure in the frequency range is illustrated by fig.4. In the rectangular waveguides the following waves were taken into account: , , 
, 
, 
, 
, 
, 
.
Fig.4 Behavior of module (curve 1), real part (curve 2) and imaginary part (curve 3) of the surface impedance of the optimized structure "surface – array of rectangular waveguides" in the frequency range 
.
It is practically interesting to solve the problem of determining the characteristics of radiation and matching of RS located in the plane z = t (i.e. on the coating of the optimized waveguide array). To do this, the system of operator equations relative to the current 
on the RS and magnetic current 
in the connection aperture:
 |
(5) |
where 
— RS surface, 
— area of the connection aperture, 
— transversal electric Green tensor [1], tensor 
— is defined by the expression (2), and other tensors are equal to:
and the rot operator is applied to non−primed coordinates according to the rules of tensor analysis. The coefficient = 0, and 
— is determined from the solution of the boundary value problem for the i−th Floke harmonics in the plane z = t.
Fig.5 DD (a) and module of the reflection coefficient (b) of RS, located in the AA above the optimized impedance structure composed of a rectangular waveguide and dielectric coating in the frequency range (1 — f = 
, 2 — f = 1.25, 3 — f = 1.5, 4 — f = 1.75, 5 — f = 2).
For RS with length l = = 0.2, oriented along the OY axis, the fig.5a and 5b shows direction diagrams (fig.5a) and modules of the reflection coefficient (fig.5b) in the H−plane depending on the frequency. The radiators are fully matched in the direction of normal to the array in average frequency (curve 3). The used optimized impedance structure supports good efficiency for RS in the frequency band with 2:1 overlap and sector of angles ±55°, and as the fig.6 shows, the total active power passed to the rectangular waveguide does not exceed 0.33 from the excitation power of the RS.
Fig.6 Relation of the active power passed to the rectangular subcritical waveguide (
) to the excitation power of RS (
) in the sector of angles in H-plane of the radiator in the frequency range (1 — f = , 2 — f = 1.25, 3 — f = 1.5, 4 — f = 1.75, 5 — f = 2).
The following conclusion can be made:
— the magnetic tensor Green function of Maxwell equations has been constructed for an arbitrary area of the single cell of the periodic structure;
— mathematical model of RS as a component of the infinite AA and located above an arbitrary number of waveguides (which don’t have to be rectangular) with dielectric inserts and coating has been constructed;
— usage of the array of supercritical rectangular waveguides does not allow obtaining near the AA of a large modulo value of the surface impedance whatever geometry the array and waveguides have;
— when using the array of subcritical waveguides, the slice frequency of the main wave of which is equal to approximately 0.96, it is possible to obtain surface impedance providing for at least double frequency band and sector ±55° for RS located in this surface.