The term aberration, which originated in optics, refers to the imperfection of lens in reproduction of the original image. In antenna theory, the performance is measured in terms of the aperture amplitude and phase distributions. The phase distribution, however, is the most critical parameter and influences the far field significantly. It is therefore used in evaluating
_(n — 1 )2 (n cos 91 — 1) 
(n — cos 91 )3 
(22) 
1/2 
Elliptic lense with spherical surface S1 
Table 1. Amplitude Distributions for the Hyperbolic and Elliptic Lenses of Figs. 3 and 5

n = 1.6 er = 2.55 61max = 51.3° 
(24) 
/1 
(25) 
(26) 
/—1 
a «’“>=2 
the performance of aperture antennas such as lenses and reflectors. With a perfect lens and a point source at its focus, the phase error should not exist. However, there are fabrication tolerances, and misalignments can occur that will contribute to aberrations. Even without such imperfections, lens antennas can suffer from aberrations. Practical lens feeds are horn antennas and small arrays. Both have finite sizes and deviate from the point source (2). This means that part of the feed aperture falls outside the focal point, and rays emanating from them do not satisfy the optical relationships. Thus, on the lens aperture the phase distribution is not uniform. Similar situations also occur when the feed is moved off axis laterally to scan the beam. Again, aperture phase error occurs due to the path length differences. A somewhat different situation arises when the feed is moved axially, front or back. In this case the phase error is symmetric, because all the rays leaving the source with equal angles travel equal distances and arrive at the aperture at an equal radial distance from the axis—that is, on a circular ring. However, the length of the ray increases, or decreases, with radial distance on the aperture. The phase error is, therefore, quadratic on the aperture and reduces the aperture efficiency, while raising the side lobes.
The general aberration (i. e., the lens aperture phase error) can depend implicitly on both feed and lens coordinates and can be difficult to comprehend. However, like all other phase — errorrelated problems, it can also be represented as the path length difference with a reference ray. For rotationally symmetric rays, the natural reference is the axial ray. The path length difference can then be obtained by a Taylortype expansion of the general ray length in terms of the axial one. For small aberrations the first few terms in the expansion will be sufficient to describe the length accurately. In terms of the aperture polar coordinates p and ф the expansion becomes
L(p, ф) — Laxial + ap cos ф + вР2[1 + cos2 ф] + yp3 cos ф +—
(23)
where a, fi, and у are constants indicating the magnitude of each phase error. The leading term is linear in p and ф, then becomes quadratic, cubic, and so on, and the magnitude of each depends on the nature of imperfection causing the phase error. The even terms are caused either by an axial defocus — ing or by an axially symmetric error. The odd terms can be due to a lateral displacement of the feed or can be due to asymmetric errors.
The effects of each error can be investigated by its introduction in the aperture field and determining the far field using a Fourier transformation or diffraction integral. For onedimensional errors (i. e., p = x and ф = o) the effect can be understood easily and has been investigated by Silver (1). The first term is linear, and in a Fourier integral it shifts the transform variable. It thus causes a tilt of the beam, but the gain remains the same. Using Silver’s notation, if f(x) is the aperture distribution and g(u) is the far field (i. e., its Fourier transform with a linear phase error), one finds with no phase error
a f 1
8o(u) = 2 J f(x) exp[jux]dx
and with phase error
8(u)= 2 I f (x) exp[j(ux — ax)] dx — g0(u — a)
where u = (na/A)sin 6 and a is the aperture length. Equation
(25) shows that the beam peak is moved from the 6 = 0 direction to 60, calculated by
u — a = 0
or
aX
90 = sin I —
A quadratic phase error is symmetric on the aperture and does not tilt the beam, but reduces its gain. For small values of fi, it can be calculated analytically (1) and is given by
f (x) exp[j(ux — ^x2)] dx
(27)
= 2 feoO) +JP§o(u)]
where go(u) is the second derivative of g0(u). Due to this phase error the gain decreases progressively with increasing fi, and eventually the beam bifurcates and maxima appear on either
u = (nalX) sin в Figure 6. Effect of quadratic phase error on the farfield pattern. 
side of the axis. It also raises the sidelobe levels. Figure 6 shows typical pattern degradation due to this error.
The next important phase error is the cubic one, which has odd power dependence on the aperture coordinate. This error not only tilts the beam, but also reduces the gain and asymmetrically affects the sidelobes, raising them on one side while reducing them on the opposite side. Its effect is therefore a combination of that of the linear and quadratic phase errors. For small errors its far field is given by (1)
/ 
1
f(x) exp[j(ux — <5×3)] dx 1 (28)
= I ЬоО) +<^о"(и)]
where go'(u) is the third derivative of g0(u). For a few small phase errors the far fields of this phase error are shown in Fig. 7. They show clearly the beam tilt, the gain loss, and the rising of the sidelobes toward the beam tilt. They are known as coma lobes, after the corresponding aberration in optics. Also, because this phase error causes more severe pattern
u = (nalX) sin в 
degradation than others, it is desirable to eliminate it, especially that it manifests mostly in beam scanning. Feed lateral displacements to scan the beam can readily cause coma lobes. Fortunately, a number of lens surface modifications have been found to reduce the effects of this error (3).