One of important problems of the theory of antennas is the research of boundary effects in finite antenna arrays. Two methods are used to determine characteristics of such arrays. The first one is based on applying direct numerical methods to solve the corresponding boundary value problems of electrodynamics. This method can be applied to arrays containing a small number of elements. The second method is based on using an infinite array as a model of the finite array. This method allows determining the characteristics of radiating elements situated in the central part of the array containing a large number of such elements. Also, combination methods are used that consider the antenna array to be a part of an infinite array where nonexcited radiating elements are loaded with matched load. The above methods have an essential drawback: they do not reveal the physics of boundary effects and do not allow using the mathematical apparatus effectively. The attempt to use formal mathematical approach to researching properties of radiating elements in the boundary area by analyzing the resulting set of algebraic equations does not succeed because of the complexity of such a problem.

This work suggests another method of analyzing characteristics of finite antenna arrays, free of drawbacks listed above. The method is effective because it is based on the adequate model of physical processes that take place in the boundary areas of finite arrays. The main point of this model is considering currents in the radiating elements of a finite antenna array to be a superposition of unperturbed currents of an infinite array that contains the finite antenna array as a part, and a boundary wave that propagates from the boundary area to the center of the array. In this case, changes in amplitude and phase of currents of radiating elements in the boundary area is the result of interference of unperturbed currents of the infinite array and the currents of the boundary wave. As the research shows, the boundary wave has quite stable characteristics. For example, in linear arrays only the amplitude of the boundary wave is changed while scanning, but the speed of the wave in the array and the changes in the amplitude from the boundaries to the center remain almost unchanged. This allows formulating a qualitative representation of changes in current distribution in the array while scanning. Understanding of the physics of boundary effects allows using the mathematical apparatus most effectively.

The theory of finite antenna arrays, based on the concept of boundary waves can be derived using various methods. At first, it is necessary to show that boundary waves do exist. It is enough to consider the simplest model of a semiinfinite antenna array, which allows excluding the interaction of the opposite boundaries. As an antenna array that satisfies this condition, the semiinfinite flat slotted array of narrow infinite parallel slots, connected to feeder lines exciting the reacting elements, through matching circuit is chosen. The matching circuit consists of an ideal transformer and a reactance. The characteristics of these elements are chosen to match the slot radiator in the infinite antenna array. As an operator equation the energy balance equation is used

where S — closed surface containing the radiating elements and the reactive elements of the matching circuit; P — complex power, accumulated in the reactive elements. Let , , , n be voltage, current strength, impedance of the reactive element and transformation coefficient of the ideal transformer respectively. Then the equation (1) can be rendered as follows:

where — magnetic-field strength of the p-th slot with unit voltage on the k-th slot.

After representing the current and voltage in the semiinfinite antenna array as a sum of the corresponding elements in the infinite array and the boundary wave

and inserting (3) into (2), we receive the set of equations for determining the voltage and current of the boundary wave at the input of the matching circuits of the radiating elements and the equations for determining the current and voltage in the infinite antenna array:

where the v value is expressed by the array step, wave length and phase angle .

If the matching of the array is performed for the phasing direction, defined by the parameter, the characteristics of the elements of the matching circuit are determined by the following relations:

where — impedance of the feeder line. The formal solution of the set of equations (4) can be found using the Viner-Hopf method [1,2]. However the implementation of factorization procedure in this case involves the numerical integration methods. Thus, it is worthwhile considering the possibility of using other methods.

The analytical solution can be found in the extreme case when the slot step tends to zero. It can be shown that in this case the set of equations (4) is transformed into the Fredholm integral equation of the second kind with weak polar nucleus and semiinfinite integration limits

where — voltage at the input of the radiating elements of the infinite array.

The theory of such equations is developed in [3]. Using this theory it is possible to solve the equation (8). In particular, when the arrays is phased in the direction of the tangent to the screen and the voltage in the array is equal to the voltage of the boundary wave, the integral equation above is transformed into the shore effect problem, solved in [3]. The analysis of the solution to this problem directly characterizes the boundary wave in a small step array.

The analysis of the set of equations (4) shows that the boundary wave excitation is equal to the effect of fictitious sources as a set of parallel magnetic current threads, situated on a part of the screen free from slots. The radiation field of the fictitious sources has a strong impact on the boundary wave strength and structure. Basic properties of the fictitious source wave can be determined using asymptotic methods. Asymptotic evaluation shows that the fictitious sources field is a cylindrical wave propagating among the semiinfinite array. When the phasing direction is changed, the fictitious sources field does not change the characteristics of the cylindrical wave, only its amplitude is changed. This results in the stability of the boundary wave field structure that changes only in amplitude when scanning. The only exception is small phase angles near the directions where the diffraction lobes cross the boundary between the real and imaginary angles or when phasing the array in the direction parallel to the array plane. In such cases the field of the flat homogenous wave propagating among the array plane prevails in the fictitious sources field. The fictitious sources wave field does not decrease when moving from the array boundaries to the center. Because of this, the boundary wave amplitude decreases more slowly as compared with other phasing angles.

When the ray deviates from the normal to the part of the screen that contains no slots, the maximum of the fictitious sources radiation field is directed opposite to the array and the amplitude of the fictitious sources wave is decreased. If the maximum of the radiation field of the array field deviates to the array, the amplitude of the fictitious sources wave is increased. According to the changes in amplitude of the fictitious sources wave, the amplitude of the boundary wave also changes. When the resonant effect is observed, the voltage in slot radiating elements is equal to the voltage of the boundary wave as the voltage in a finite array is equal to zero.

In the simplest cases the set of equations for the boundary wave can be solved using direct numerical methods. The example is the semiinfinite slotted antenna array described above. However, these methods become ineffective when finding the boundary wave in arrays with more complex radiating elements. To find the boundary wave in such arrays the following iterative procedure is suggested.

When finding currents that are excited by the fictitious sources field, it is assumed that the finite array is extended to infinity by radiating elements, loaded with matching load. The found solution is corrected by subtracting the value that the radiating elements of the additional array add to the boundary wave. Then it is again assumed that the finite array is extended to infinity and so on.

As an example let’s consider a semiinfinite slotted array. The solution to the set of equations (4) is represented as a sum

where the summands are the solutions to the following sets of equations:

The first set of equations determines an approximate solution and the adjustment to the approximate solution of the set of equation (4) on the п-th step of the iterative procedure and describes the process of excitement of the infinite antenna array.

The equation (11) corresponds to the source semiinfinite array and its solution extends the approximate solution to the set of equations (4)

to the exact (9).

The set of equations (10) is solved using Fourier transform. The choice of the number of steps of the iterative procedure is done to make sure that the inaccuracy of the approximate solution (12) does not exceed the accepted value. This negates the necessity to solve the set of equations (11) for the semiinfinite array. To prove the convergence of the iterative procedure, it is proved that as the number of iterations increases, the vector of free terms of the set of equations (10) tends to zero and the conditions of the determinant of the set being not equal to zero are determined.

Asymptotical estimates of the fictitious sources field and the numerical experiment show that each correction to the approximate solution and the approximate solution of the set of equations for the boundary wave itself are described by almost the same functions with the amplitude coefficients being the only difference. This allows significantly simplifying the iterative procedure.

The possibility for simplification arises because the relation almost does not depend on the number of iteration. Therefore, the solution to the set of equations (4) can be rendered as: где , T — can be determined on the first step of the iterative procedure if the side boundary wave excited in the radiating elements of the auxiliary array are determined simultaneously with the boundary wave. In case of a finite slotted array containing N radiating elements, the set of equations describing the boundary waves, using the method described above, can be rendered as a collection of two independent sets of equations each of them defining the boundary waves excited by one of the edges of the array:

According to (13), (14) the fictitious sources, excited the boundary waves , , are situated on the opposite edges of the finite array. Boundary effects in the finite array are determined using the sum of the solutions to the sets of equations (13), (14)

A solutions to each of the sets of equations (14), (15) can be found as a series, which members are the boundary waves that appear as a result of the multiple re-reflections of the first boundary wave from the edges of the array.

The numerical results of the research of the wave characteristics of the fictitious sources in the semiinfinite slotted array are shown at fig. 1 — 2. Normalized dependencies of the amplitude of the fictitious sources wave on the number of the radiating element and the angle of scanning for an array with step Т = 0,7 are shown at fig. 1a,b. Phasing angles that the curves 1,2,3 shown at fig 1,a correspond to, are shown at fig. 1,b. Comparing the charts shows that the character of dependence of the amplitude of the fictitious sources wave on the distance to the edge of the array remains constant for a wide sector of scanning angles with the exception being sectors near the phasing direction that correspond to the transition of the diffraction maximums over the boundary between real and imaginary angles. The corresponding boundary waves (fig. 1,c) have similar properties. The fig. 1,c also shows the dependence of the boundary wave phase on the distance to the edge of the array, which shows that the phase velocity of the boundary wave is equal to the velocity of light in the medium above the slotted screen. The curves 1, 2, 3 at fig. 1,c, 2 correspond to the curves 1, 2, 3 at fig. 1,a and dots , , at fig. 1,b.

The curves shown at fig. 2, а, characterize the dependence of the voltage in the slots of a semiinfinite array on the number of the radiating element that is a result of interference of the boundary wave and the voltage in the slots of the infinite slotted array. Changes in period and amplitude of oscillation of voltage in the slots of the boundary area when changing the phasing direction completely agree with the qualitative considerations above.

Fig. 2,b shows the directivity chart F of the edge slot radiating elements and the directivity chart of the boundary wave . The directivity chart of the boundary wave has two maximums, one of which is directed along the plane of the array and another is directed with the diffraction maximum angle in the array with step T = 0,7, for the phasing direction in the array plane. It can be shown that radiation of the boundary wave results in «diffusing» of the zero trough of the chart that shows directivity of the edge area radiating elements directed away from the array (to the left from the maximum F at fig. 2,b).

Thus, the proposed method allows creating of a constructive theory of finite antenna arrays.