The concept of boundary waves provides obvious results and allows performing of quality analysis of boundary effects. Besides, separation of boundary waves provides the possibility to analyze the causes of distortion of characteristics of antenna arrays, which takes place while scanning.

The main factors in effect in finite antenna arrays can be conveniently separated when analyzing characteristics of the simplest antenna array, that is the array of infinite narrow parallel slots. Some characteristics of boundary waves in such an array are examined in [2].

Equations (4), (5) prove that boundary waves are excited by fictitious sources, which complement the regular part of the currents of the finite array to currents of the infinite array which correspond to the partial excitation .

Fig 1,a shows relation between the regular part of the equivalent magnetic currents of the slots of a semi-infinite antenna array and the direction of phasing (curve 1). The shape of the curve is typical for an infinite phased array. The amplitude of the current is reduced to zero when the diffraction maximum passes through the border of real and imaginary angles. At the same figure (curve 2) there is a relation between the amplitude of the fictitious sources field and the boundary wave excited by them (curve 3). As the graphics show, when the selected step value (T/=0,7) in one of the phase directions where the array is “blinded” and the regular part of the current is reduced to zero, the amplitude of the field of fictitious sources is at its maximum and the amplitude of the boundary wave is also at maximum. In the other direction, the regular part of the current, as well as the amplitude of the boundary wave and the field of fictitious sources is reduced to zero. The first direction corresponds to the deviation of the beam from the normal to the side of fictitious sources where the diffraction maximum of the radiation field of fictitious sources is directed along the screen in the direction of the slot array.

The second direction correspond to the deviation of the beam from the normal in the direction of the slot array when the diffraction maximum of the field of fictitious sources is also directed along the screen but in the opposite direction — in the direction opposite to that of the slot array. Thus, when phasing the finite antenna array in the blindness direction of the corresponding infinite array, the regular part of the current is reduced to zero, and only currents of boundary waves are excited in the array.

As it was shown in [2], the boundary wave has stable characteristics regardless of the excitation type of the array. Fig 1,b shows plots, which characterize relation of the current of the boundary wave and the distance to the border of the semi-infinite radiation curtain when feeding the edge (1) and tenth (2) slot, as well as in the uniformly excited array (3) when phasing out of some angle sector near the “blindness” direction. Comparison of the plots shows that in all of these cases, the same boundary waves appear.

Boundary effects in a finite slot array are the result not only of the interference of the boundary waves and the regular part of the current, but also the interference of the boundary waves, excited by the opposite edges of the radiating curtain. Boundary waves, when reaching the border of the radiating curtain, are reflected and the resulting boundary waves in finite antenna arrays are the superposition of waves that appear because of multiple re-reflections.

Fig 1,c shows the process of re-reflection of the boundary wave excited by one of the edges of the array with the number of radiators N=2 (curve 1), N=5 (curve 2) и N=10 (curve 3). The charts show that the reflection coefficient of the boundary value has almost zero relation to the number of radiators in the array and does not exceed 0,2 modulo.

In an array with the number of radiators N>5 difference in the amplitude of the boundary wave and the corresponding value in the semi-infinite array does not exceed 4%. Therefore, in such arrays, re-reflection of boundary waves can be ignored. Amplitude of the boundary wave quickly decreases when moving off the edge of the array and when N>5, there is almost no interference of boundary waves excited by the opposite edges of the array.

Direction diagram of the radiator in the array, elements of which are loaded on matched loads is the sum of direction diagrams of the regular part of the current and the current of boundary waves. Since in an array without dielectric coating, phase speed of the boundary wave is equal to the speed of light in the medium above the array, the direction of maximums of radiation is defined by the expression

(12)

When the array step T/<0,5, the only radiation maximum of the boundary wave is directed along the array plane. In arrays with step 0,5<T/<1 radiation of the boundary wave has two maximums (fig 2,a, curve 1), one of them is directed along the array plane, and the other is directed angularly, with the angle

(13)

relatively to the normal, equal to the direction of “blinding” of the infinite array. Signs before the right parts of equations (12), (13) correspond to the boundary waves excited before each of the two edges of the finite slot array. The regular part of the current in the semi-infinite array corresponds to the asymmetric direction diagram, asymmetry of which is decreased as the distance between the edge and the excited radiator is increased. Fig 2,a shows direction diagram of the regular part of the current, inducted in the slot semi-infinite array when the edge radiator is excited (curve 2) and direction diagram of the edge element (curve 3), which includes radiation of the boundary wave. Comparison of the charts show that the boundary wave slightly affects the direction diagram of the edge element in the dip area and direction of the screen plane. This effect quickly decreases as the distance between the edge of the radiating curtain and the excited element of the array increases. Therefore, direction diagram of the radiator in the array is quite precisely defined by the regular part of the current.

Fig 2,b (curve 1) shows direction diagram of the array consisting of 11 elements, maximum of which is deviated from the normal to angle =18,5°. Distribution of amplitudes of waves, which excite slot arrays, is defined by the function

(14)

where — the number of radiators in the array; k — number of the radiator. Existence of the boundary wave affects only some details of the structure of the direction diagram. Primarily, radiation of the boundary waves leads to bleeding zeros. However, the direction diagram of the regular part of the currents (curve 2) is significantly different from the direction diagram of the array with amplitude distribution (14). Level of side lobes is -26 dB instead of -32 dB. This is caused by the fact that the partial excitations, which form the amplitude distribution (14)

(15)

where

(16)

have different phase distributions. Mismatch of phase distributions results in different reflection coefficients in the infinite array and changes of the relation between the inducted currents as compared to (16). Changes in phase between the adjacent radiators, which correspond to partial excitations (16), are defined by the expressions:

(17)