The beam pattern for a square antenna is derived here. It is shown that the beam pattern is basically the Fourier transform of the aperture. This is a general result that extends beyond the current illustration. Returning to a geometry that may remind the reader of Young’s double slit experiment, Fig. A1.2 shows a cut in the plane for a stylized phased-array antenna. Each of the small elements on the left hand side represents the source of a radar pulse, which then propagates to the right. The array elements are separated by a distance d, in the y direction.
Figure A1.2 Each of the array elements is the source of a spherically expanding wave, which reaches the "screen" at the right after traveling a distance that depends on y.
Each array element is responsible for an electric field element defined by the equation for a spherically expanding wave:
where
component due to
array element,
magnitude defined by the sources (all the same in this case),
contribution from the array element, here with the dimensions of the length squared, 
for the
array element,
from
element to observation point, and
is the wavenumber.
A classic principle of electricity and magnetism says that the total electric field can be obtained by adding up all of the components, keeping track of the phase, of course.
The total field from all the radiators is the sum of the elements:
Though accurate, this form is difficult to deal with analytically. Therefore, we will make some assumptions that allow us to simplify it. The primary one is that the observation point is a large distance from the array,
We can then make use of the fact that the variation in amplitude with distance varies slowly in the y direction,
compared to the relatively rapid changes in phase. For simplicity, we’ll take
constant, and
(the emitters are all in phase). This leads to the simplified equation
where we have factored out the slow inverse square variation in amplitude. Now we play some tricks with the complex term inside the summation, addressing the question of how the exponent varies as n varies.
If we take
to correspond to the n = 0 element, we get for r
The above form is exact, the trick is to factor out the ro term from the rn terms. We can do this because d is small. First we expand the term inside the square root:
Without any special tricks—yet—we expand and divide out:
Now come the tricks—first taking the third term as very small, and then using an approximation for the square root, where the second term is small:
(Reader exercise: check that the third term is small by plugging in some typical numbers: d = 1 cm,y = 500 m, x = 2000 m). Note that we can now turn to the familiar polar form
and simplify the remaining terms:
This defines the zeroes in the beam pattern. For a continuous antenna element, the sum is replaced by an integral over an antenna of length L = nd:
where the sum over the elements has been replaced by an integral over
to L/2, and the amplitude factor has been put back in for a moment, along with an inverse length to go with the integration variable. The integral on the right side is simply the Fourier transform of the square aperture of length L:
The power at any particular location will then be proportional to the square of the electric field strength. The resulting function is then proportional to the square of the sinc function:
