The aim of this work is to suggest a variant of constructing of a receiving AA providing a wide band of operating frequencies and wide sector of operating angles simultaneously. Then, develop a mathematical model of the suggested AA. Based on the constructed mathematical model, conduct a series of numeric experiments to discover the main properties of the AA.
The AA in question is a multilayer periodic ribbon structure located in an environment with planelayer dielectric filler above a metallic surface (fig. 1).
Into gaps of the metallic ribbons of the AA, active elements can be directly included, such as optoelectronic converters, the first link of the input device of the receiving module of the AS. In this work, the specific implementation of the active elements is not studied and the AA is considered loaded on the lumped resistances, which are the input resistances of these elements.
Mathematical model of the multilayer antenna array. The analysis of characteristics of the multilayer AA in this work comes to solving of an external boundary value problem of electrodynamics. In the process of solving, the boundary value problem turns into the integral equation relative to current on the metallic elements of the construction, which then after representing the current in a specific basis, comes to a system of linear algebraic equations relative to the amplitudes of the basis current functions. The antenna array is periodic with periods Тx and Тy on X and Y axis respectively. It is excited by a plane homogenous wave propagating in the negative direction of the Z axis. The direction of wave propagation is defined by the θ and φ angles of the spherical coordinate system corresponding to the source Cartesian one. In the general case, the extraneous field can belong to the neither E-waves, nor H-waves. However, it can always be decomposed into E and H components and solve the problem separately for each excitation type. Thus, we will consider that the structure is excited either by E, or H-wave and in the process of scanning, the polarization of the extraneous field remains unchanged. Orientation of the scanning plane is defined by the azimuth angle φ0, the angle of scanning in this plane is angle θ0.
Fig.1
The following expressions relate the excitation parameters to angles, which characterize the direction of propagation of the corresponding waves.
H-waves:  |
(1) |
E-waves:  |
(2) |
The expressions (1) and (2) describe the field of the incident wave in the free space. To account for the planelayer dielectric filler, we will use (to describe the combination of dielec-tric layers in the space waveguide) the irregular long line model. The irregular line in question is composed of several homogenous parts, wave resistances of which are equal to the characteristic resistances of Floke harmonics in the corresponding dielectric layers. Let’s consider a dielectric filler composed of N+1 layers with known dielectric permittivitties εi. N+1 layer is free space on which the extraneous field is given in the form of a plane homogenous wave incident to the structure. If the planelayer dielectric filler is present, the relation between the extraneous field in layer M and the extraneous field in free space is defined by the following expression:
 |
(3) |
where 
is the extraneous field in free space; 
is the element of the transmission matrix, describing the layers with numbers greater than М; R is the reflectivity coefficient for the layer М-1; z' is the coordinate counting from the joint between layers М+1 and М.
Then we shall describe the method of representing electric currents on the AA ribbons. As basis functions, we shall use functions, exponentially changing along the ribbons and having features on the edges of the ribbons. The parameters of the exponent describing the changes in current along the ribbon correspond to the excitation parameters. The feature of the current on the edges of the ribbon has power 
. Let’s write expressions for basis functions of the current:
— for ribbons parallel to the X axis:
 |
(4) |
— for ribbons parallel to the Y axis:
 |
(5) |
Solution for the diffraction problem in this configuration consists of determining the amplitudes of the basis functions of the current and then determining the scattered field. To determine the amplitudes of the basis current functions, we shall use the power balance equation, which in our case can be written as follows:
 |
(6) |
where 
, S is the surface, which includes elements with unknown currents; Pr is the power on the inputs of receiving modules. Using the thin plane approximation [1], transform (6) to the following:
 |
(7) |
To determine 
we shall use the Green tensor function representation for the space waveguide with planelayer dielectric filler. By integrating (7), we get the quadratic form, which is then reduced to the system of linear algebraic equations relative to the amplitudes of basis functions of the current
 |
(8) |
where Zij is the mutual resistance of basis functions with numbers i and j.
We shall designate the basis current function paralles to the X axis “belongs to the X class” and the basis current function parallel to the Y axis “belongs to the Y class”. For mutual resistances Zij the following expressions are valid:
 |
(9а) |
 |
(9б) |
 |
(9в) |
 |
(9г) |
where
When calculating the proper resistance of the basis function we must take into account the input resistance of the corresponding receiving module
 |
(10а) |
 |
(10б) |
Elements of the free term column in (8) are defined by the incident wave field and can be calculated as follows:
— extraneous field: H-wave (Е-polarization)
 |
(11) |
— extraneous field: E-wave (H-polarization)
 |
(12) |
where Fip is defined as follows:
 |
(13) |
Solution of the system (8) defines amplitudes of the basis functions of the currents for the given excitation. When calculating the scattered field, the depolarization of the incident E or H-wave is taken into account. To do it, we consider excitation of the structure by a combination of two plane homogenous waves, with one of them being E-wave and the other being H-wave. Let both waves propagate in the negative direction of the Z axis by angles θ и φ, and the intensity of the electric field modulo for both waves is equal to 1. Then the transversal components of vectors 
of these waves are defined by expressions (1) and (2). By solving the diffraction problem separately for each type of wave we get the values for transverse components of the back-scattered field in the far-field region
 |
(14) |
In both these cases the full back-scattered field will be related to the incident wave by the fol-lowing relations:
 |
(15) |
where S is the scattering matrix, which characterizes the reflective properties of the structure. Since, after solving the diffraction problem we know the left parts of the expressions (15), they can be considered equations relative to the elements of the S matrix
 |
(16) |
By transposing the matrix in the system (16) we get the following expressions for the elements of the matrix [S]:
 |
(17) |
Results of numeric modeling. We shall estimate the effectiveness of the receiving ultrawideband AA with wide-angle scanning by using the reflectivity coefficient of the incident wave. All specified sizes are determined relative to the unit frequency. To determine main properties of multilayer plane antenna arrays, we have conducted a series of numeric experiments, which determined the influence of various parameters of the structure on the frequency and angle dependencies of the reflectivity coefficient modulo. The scanning plane and polarization of the incident wave were selected to ensure that no depolarization of the scattered field takes place in the entire angle sector. It was assumed that the input resistance of the receiving modules does not depend on the frequency.
Fig.2
Fig. 2 shows the relation between the reflectivity coefficient and frequency for three scanning angles: 0°, 30° и 60°°; the number of layers is 1; the width of ribbons is 0.1; the direction to the screen is 0.25; the array period is 0.25; the lumped resistance in this case is 377 Ohm. As we can see in the charts, with these parameters of the structure are assumed with frequency 1.09 for normal incidence, good matching is provided. The width of the operating wideband of the single layer AA with reflectivity coefficient equal to -20 dB is 20%. When the phasing angle is changed, the frequency, on which the reflectivity coefficient module is minimal, is sliding and the matching quality becomes lower.
Fig.3
Fig.4
Fig 3 and 4 display the same dependencies for structures with higher amount of layers(5 for fig.3 and 10 for fig 4.) As we can see that as the number of layers and the frequency band is increased, the reflectivity coefficient becomes less susceptible to changes of the phasing angle. The ribbon AA with 5 layers in the 120% band frequency has the reflectivity coefficient not worse than -18 dB with normal wave incidence. When the number of layers is increased to 10 in the same frequency band, the reflectivity coefficient does not exceed -30 dB and on the level of -20 dB the frequency band is 180% for normal phasing and 120% in the sector of angles ±60°.
Fig.5
Fig 5. shows relation between the maximum and minimum operating frequency (coeffi-cient of bandness) for the level of -20 dB of a multilayer ribbon AA, and the number of layers for various laws of changing the lumped resistances of ribbons from layer to layer. As the charts show, the increase in the number of layers from 5 to 10 does not affect the characteristics of the structure very much and 4…5 layers are enough to provide near-optimum properties. Thus, multilayer scanning receiving antenna arrays provide high values for the bandness coefficient in wide sector of angles.
