*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.

Below, results of numeric calculations on a PC using the "ArrayGuides Rectangular" software are given.

Let’s consider the case where the AA period contains of one waveguide, the wide sheath of which has the size *a* and is oriented along the OX axis, the narrow sheath — size *b*, and the flat electromagnetic wave is polarized along the OY axis. The fig.2 shows the family of curves which reflects the change in the frequency band of impedance properties of the surfaces located at the distance of 0.08 ( — wavelength corresponding to the bottom frequency of the range) from the array aperture in the point x = y = 0. There are no dielectrics, the wave is normally incident to the AA surfaces, the waveguides are located in the nodes of the rectangular array.

Fig.2 Behavior of the imaginary part of the surface impedance above the array of supercritical rectangular waveguides from frequency (a: curve 1 - = 0.21, curve 2 — = 0.22, curve 3 — =0.23; b: curve 1 — = 0.18, curve 2 — = 0.19, curve 3 — = 0.2; c: curve 1 — *a* = 0.2, curve 2 — *a* = 0.19, curve 3 — *a* = 0.18; d: curve 1 — *b* = 0.17, curve 2 — *b* = 0.16, curve 3 — *b* = 0.15; e: curve 1 — = 0.08, curve 2 — = 0.07, curve 3 — = 0.06).

The given curves show how the outcome is affected by various structure parameters: array period (fig.2а) and (fig.2b), size of the wide (fig. 2c) and narrow (fig.2d) waveguide sheath and distance from the analyzed surface to the array aperture (fig.2e) to the value of the imaginary part of *Z*. The array geometry: = 0.21, = 0.18, waveguide: *a* = 0.2, *b* = 0.17. Since the waveguide is supercritical in the entire frequency range, the real part of *Z* is zero. The computation error specified using the internal convergence of the numerical procedure does not exceed 1…3% when to describe the field in the aperture of the rectangular waveguide, basis functions corresponding to the waves , , , are used. (Later on, to describe the results of the numerical experiment, those proper waves of the rectangular waveguide are specified, accounting for which provides the specified precision). Analyzing the curves shown at fig.2, the following conclusion can be made:

— the most significant effect on the value of impedance has change of the wide sheath of the waveguide and distance from the surface to the AA aperture;

— change of impedance to increasing in the bottom part of the range leads in all cases to shift to lower frequencies of resonance and therefore lowering the useful frequency range where .

The dotted line shows impedance determined using the formula:

where *a* = 0.2, *b* = 0.17, = 0.08. The specified formula corresponds to the zero approximation, and in it , — conductivity of the wave of the rectangular waveguide.

Also, the effect of the infinitely thin diaphragm located in the aperture of the supercritical waveguide was researched. It was determined that usage of the diaphragm does not allow obtaining the necessary surface impedance *Z* in the wide frequency range either.

Fig.3 Dependency between the surface impedance (a, 1 - module, 2 – real part, 3 – imaginary part) and the input resistance of RS (b, 1 – real part, 2 – imaginary part) and the frequency . RS is located on the surface with impedance (a).

Fig.3a displays curves of dependency *Z* above the AA of subcritical waveguides and the frequency in point x = y = 0. Array geometry — = = 0.2, rectangular array. Waveguide size — *a* = *b* = 0.19, dielectric filling = 7.2 (which corresponds to the wave slice frequency and ~0.98). The surface is located at the distance of = 0.125. The flat wave is normally incident to the AA surface. The following waves were taken into account in the waveguide: , , , , , , , .

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.