a Matlab package for phased array beam shape inspection

22 4 A PHASED ARRAY
where we used
D x D y M x M y = d xM x d y M y
λ 2
= L xL y
λ 2 = A λ 2 .
The factor diric()diric() is unity in the grating directions, so the maximum gain, the
antenna directivity D(û), to grating direction û, is
D(û) =
g E(û)
∑ g E (u g)
g cos θ g
4πA
λ 2 . (71)
The grating sum ∑ g is to be evaluated over all the grating directions u g which are
the periodic replicas of the given direction u, in the visibility circle, by period 1/D x in
x-direction and 1/D y in y-direction.
If D x < 0.5 and D y < 0.5, the grating sum contains only the single element u g = u.
For such a dense array, the directivity depends on the beam direction,
D(û) =
4πA cos(θ)
λ 2 , (72)
but does not depend at all on the element pattern, nor the element spacing (as long as
it is denser than the above limit). For instance, if the area of our benchmark 50 × 20
array of density D = 1.5 wavelength, would be covered with 0.5 wavelength spacing or
tighter, the broadside (θ = 0) gain would always be 4π × 50 × 20 × 1.5 2 = 44.5 dBi. The
result is geometrically natural, as it says that the effective area A eff = Gλ 2 /4π of the
antenna is just the area of the antenna as seen from the target, that is, is equal to the
projection of the area to the direction perpendicular the beam direction. Interestingly,
the result is analogous to the sampling theorem result for time-signals, which says that
if the signal is sampled fast enough, namely, so fast that there can be no aliazing, then
no information is lost. In the dense array case, what seems to happen is that when the
incoming plane way is sampled tightly enough in space—so tightly that no directional
aliasing can occurs—again “no signal is lost”, now in the sense that the element grid
behaves as it would cover without holes all of the area that is within its outer boundaries.
We cannot perhaps hope to do any better, and that might be why the element gain does
not effect the overall gain at all.
For less dense grids, the elements (non-normalized) gain, g E , affects the array beam
shape and directivity.
4.11. Gain model for the array element
We will assume a simple analytical form for the array element gain. We will assume
circular polarization, and will assume axially symmetric element power gain pattern. As
suggested by Kildar (p. 61), we will use the form
G E = G E (θ ′ ) = G e cos ne (θ ′ /2) , (73)
where θ ′ is the angular offset from the axis direction (φ E , θ E ) of the element. Normalization
of G E to 4π over the solid angle gives the relation
n e = 2 G e − 2 (74)