*Z*of which satisfies the condition , where

*W*=120π — wave impedance of the free space, in the entire working area.

This article researches the possibility of using an array of rectangular waveguides of small electric sizes with dielectric filling to obtain necessary impedance properties in the space near the aperture.

Fig.1 Period of AA composed of rectangular waveguides with dielectric coating and sought surface impedance *Z*.

Let’s consider an infinite periodical AA, each period of which contains of rectangular half-infinite waveguides with shared ideally conducting flange. In the general case, the waveguides with the same period have different sizes and dielectric filling and AA – dielectric coating with thickness *t*. Let the array be incident to a flat electromagnetic wave of arbitrary polarization from the half-space z > 0 in the negative direction of the OZ axis, tangential electric and magnetic vectors of which near the waveguides can be represented as:

(1) |

where — given wave amplitude, — vector zeroth Floke harmonics [2] (*p* = 1 corresponds to *H*−harmonic, *p* = 2 — *E*−harmonic), — wave conductivity of the zeroth Floke harmonics [2], — transfer ration of the zeroth Floke harmonics from the homogenous area above the array to the area (fig.1), — Kronecker symbol, — longitudinal wave number, , — wavelength in vacuum, — angle between the OY axis and , vector, determined when = = 0°, * j*^{2} = -1.

Let’s refer to the secondary (diffractional) electromagnetic field as , . Then the boundary problem of electrodynamics for AA can be formulated as follows: found the electromagnetic field , , satisfying

— homogenous Maxwell equations;

— condition of continuity of tangential electric and magnetic fields in the connection apertures;

— condition of absence of secondary waves coming from infinity;

When these conditions are satisfied, the problem has unique solution [3].

Applying the Floke theorem [2], we can, by an analogy to the work [1] construct the transversal magnetic tensor Green function for the Maxwell equations, which for the homogenous area incident to the screen has the following form:

(2) |

где — sign of dyadic product of vectors, , — reflection coefficient for the *i*-th Floke harmonics from the border z = *t* (specified in [2]), — reflection coefficient for the *i*−-th Floke harmonics from the border z = 0 (in this case = -1), *i* — generalized Floke harmonics index [2], — radius−vector of the observing point, — radius−vector of the source point, the tangent magnetic field of the partial waves are related with the vector Floke harmonics:

the index "*-i*" corresponds to the flat wave, propagating with an angle of -, (, — angles of propagation of the wave with index "*i*"), and for the ambiguous function according to the radiation conditions a branch is selected for which .

According to the equivalence theorem [3], we will replace connection apertures with magnetic currents , on an ideally conducting screen and by an analogy with [2], write a system of operator equations in relation to these currents:

(3) |

where — area of the i−th aperture of the connection, — tensor Green functions for which in waveguide representation (2) vector Floke harmonics are replaced with proper functions of the waveguides, coefficient is zero, and = -1.

To solve the resulting system, we can use for example, Galerkin method [2] and project (3) to the linear capsule functions . After finding the unknown currents , tangential component of the diffractional field created by the connection apertures (CA) can be found using the following relations

where — expansion coefficients of currents for the selected in Galerkin method full system of basis functions, — transfer ration of the i−th Floke harmonics from the area to the uniform area above the array,

* — complex conjugation sign.

Then the full field above the array according to the superposition principle will be equal to:

where vectors , correspond to the primary wave reflected from the “surface−screen” structure and the sought surface impedance can be determined using the relation:

where *Z* is a matrix in general case.