According to the method given in [2], the equation for the regular part of the current and the boundary wave can be derived from the original equation of antenna array

(1) |

where *Z* — matrix of the system of linear algebraic equations; — vector of unknown coefficients in factorizations of radiator currents by basis functions; — vector of the excitation of the finite array. To do this, the *Z* matrix is complemented to the matrix of an infinite antenna array:

(2) |

and the vector is represented as a sum of the regular part of the current and the current of the boundary wave

(3) |

By substituting (2), (3) to (1) we get the equivalent system of equations:

(4)-(5) |

where — partial excitation of the infinite array, which is the “continuous” continuation of to the radiator, which complements the finite array to the infinite one. Arbitrary excitation of the finite array can be represented as a superposition of partial feeds using the discrete Fourier transform. The equation (4) defines the regular part of the current, and (5) defines the boundary wave.

To find the regular part of the current we use the known method of solving periodical structure excitation problems. The boundary wave equation is the equation of the finite array excitation problem. To solve it, the iterative procedure [2] is used, which transforms the equation (5) like the original equation (1) to the equivalent system of equations:

(6)-(7) |

where vectors , related to by the relation

(8) |

The vector is the first approximation of the boundary wave, is the correction of . Equation (6) corresponds to the problem of the excitation of a finite array amounting to the infinite one, radiators of which are loaded to the matched load, and is solved similar to equation (4). To solve equation (7), the next step of the iterative procedure is performed and so on. At the *n*-th step we have

(9)-(10) |

The boundary wave is defined as the sum of solutions , received at each step of the iterative procedure

(11) |

The numerical experiment shows that in most cases, the error is reduced to less than 1% after the third step of the iterative procedure. The convergence of the iterative procedure is ensued from the asymptotic approximations of the right part of equation (9), which state that the coordinates of vector converge to zero when as the terms of the decreasing progression. The characteristic feature of the algorithm (6)…(11) is the fact that each step of the iterative procedure solves the problem of the excitation of a finite array amounting to the infinite one. This allows using of the existing programs for infinite antenna arrays with minor changes. The principal possibility of using this algorithm does not depend on the step and number of radiators in the array, existence or absence of dielectric coating and surface waves, which it causes.

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

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

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) |

The shorter the array is, the more different quantities (17) are. This means that as the number of radiators decreases, the amplitude distribution in the array corresponding to the regular part of the current significantly distorts as compared to the amplitude distribution of waves in feeder lines, which excite the radiators. This is the main cause of distortion of the direction diagram of the array with a small number of radiators, which becomes apparent in the increase of level of side lobes. As the size of the array increases, the quantities (17) converge to the same limit, equal to , and the relation between the amplitudes of slot currents, inducted by the partial excitations (16) converge to the relation between the partial excitations. That is why in arrays with large number of radiating elements, level of side lobes of the direction diagram for the regular part of the current converges to the value corresponding to the amplitude distribution of waves, which excite the radiators. Thus, distortions of the direction diagram of the phased array with a small number of radiators can be revealed by analyzing the interaction of radiators in infinite antenna arrays. This does not mean that boundary waves do not distort the direction diagram of a phased array with a large number of radiators. It can be shown that distortions of the direction diagram caused by the radiation of boundary waves in large arrays appear when the maximum of the direction diagram approaches the edges of the scanning sector. Analysis of the radiation field of the slot infinite array and the plots at fig 1,a show that when the beam is approaching the edge of the scanning sector, because of decreasing amplitude of the regular part of the current, the level of the primary maximum of the direction diagram as well as the level of maximum radiation of the boundary wave does not depend on the number of radiators of the slot. Therefore, in the direction of the maximum of radiation of the boundary waves, a side lobe appears, the level of which infinitely grows as the number of radiators increases, as compared to the level of radiation of the regular part of the current in the given direction. The ratio of the maximums of radiation of the regular part of the current to the boundary wave stays almost constant.

Fig 2,c shows relation between the level of maximum of the side lobe related to the radiation of the boundary wave and the level of side radiation of the regular part of the current in the direction of the maximum, and the number of radiators in a uniformly excited slot array, beam of which is deviated to the extreme angle, defined by the expression [3]

(18) |

The charts show that in large arrays, in the area of side lobes with level -30…-50dB, because of the radiation of the boundary wave, an additional side lobe appears, the level of which is -22.5dB, i.e. the level of side radiation in the direction of the maximum of the direction diagram of the boundary wave significantly increases. At fig 2,a (curves *4*, *5*), in the scale of the curve 1, envelopes of the side lobes of the regular part of the current of a uniformly excited slot array with a different number of radiators, beam of which is deviated from the normal to the extreme angle (18), are shown. Because of the significant width of the direction diagram of the boundary wave, an increase in the level of side radiation happens in the quite large angle sector.

Thus, boundary effects caused by the existence of boundary waves on the edges of the radiating curtain lead not only to oscillations of the current of radiators of the boundary area, but also to appearance of additional lobes in the “blinding” direction of the array. The distortions of the direction diagrams appear when the beam of arrays with a large numbers of radiators is deviated to the extreme angle in the sector of single-beam scanning. In arrays with low level of side radiation and small number of radiators, significant change in the level of side lobes is caused by changes of amplitude-phase distribution of the regular part of the current, which is described by the interaction of radiators in an infinite antenna array.