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238 ANTENNA BASICS

Here = −cos. For simplicity, let us accept that all radiators are unpolarized. Then the
total electric field in far field zone is the simple scalar superposition

() = ∑ = ∑ (5.73)
Σ =− =−
Here = 1 for simplicity and , || > 0, is the magnitude or excitation coefficient of -
ℎ

0
radiator, complex in general. The common phase factor 0 is omitted as irrelevant. It is
customary to call the pattern () = () an array factor as illustrated in Figure 5.4.3b (top
Σ
picture) for assembly of 9 isotropic radiators excited equally.
Remember that the EM field structure near each array radiator can be quite complicated and the
far field component is only one element of this neighboring field presentation. Further, the

nearby fields of adjacent radiators overlap and thus could induce more or less additional
currents and voltages in each array radiator (effect of mutual coupling). If so, the far field
component in array environment is defined not only by array radiator itself but by its

surrounding too. Note only that the mutual coupling phenomenon complicates the analysis of
arrays and estimation of their critical characteristics as frequency bandwidth, radiation pattern,
scan performance and radiators input impedance. In general, a reliable analysis for large arrays
in particular must be based on sophisticated numerical techniques.

5.4.3 Pattern Multiplication

Suppose that all radiators in the array are identical (this occurs frequently) but directional with
the patterns defined by the same given function (). Eventually, we can adjust (5.73) as

() = () + () −(−1) + ⋯ + () + ⋯ + ()
Σ − 0 (5.74)
= () ∑ = ()()

=−
Therefore, the entire array pattern is the product of the array factor and element pattern.
Looking back at Figure 5.4.3b (top diagram) we see that in general, a linear array of isotropic
radiators produces two main beams. The simplest way to suppress one of them is to build an
array using directional elements like Huygens’ radiators. According to (4.86) for this
element () = (1 + sin). The bot tom pic ture in Fig ure 5.4.3b dem onstrates the
effectiveness of such an approach.

It is important to note, that the pattern () of some element measured in an array environment
and measured alone in free space can be quite different because of the mutual coupling we have
just mentioned. Strictly speaking, the patterns of all elements in an array would be more or less
different. Looking back at Figure 5.4.3a we can see that the radiator #0 has N neighbors directly
above and under while the radiator #1 has N – 1 neighbors directly above and N + 1 under and
so on. The top element #N does not have any elements above at all. Such inequality means the
different level of mutual coupling that might change the radiation pattern of elements.
5.4.4 Basics of Linear Array Synthesis

Fourier Synthesis. The series in (5.73) looks like Fourier series presentation. If so, the inverse
Fourier transform makes it possible to reverse the linear array analysis process into a procedure
called synthesis: estimate the array excitation coefficients to get the desired pattern ().
Σ

Indeed, all excitation coefficients in (5.73) can be found by simple integration