## RECTANGULAR APERTURES

There are many kinds of antennas for which the radiated electromagnetic fields emanate from a physical aperture. This general class of antennas provides a very convenient basis for analysis and permits a number of well-established mathemat­ical techniques to be applied that provides expressions for the distant radiation fields.

Horns or parabolic reflectors, in particular, can be ana­lyzed as aperture antennas. Incident fields are replaced by equivalent electrical and magnetic currents. With use of vec­tor potentials, the far fields are found as a superposition of each source. Generally one can assume that the incident field is a propagating free-space wave, the electrical and magnetic fields of which are proportional to each other. This will give the Huygens source approximation and allow us to use inte­grals of the electric field in the aperture. Each point in the aperture is considered a source of radiation.

The first step involved in the analysis of aperture antennas is to calculate the electromagnetic fields over the aperture due to the sources on the rearward side of the infinite plane and to use these field distributions as the basis for the predic­tion of the distant fields in the forward half-space. The elec­tromagnetic fields in the aperture plane cannot be determined exactly but approximation distributions can be found by many different methods, which are dependent upon the antenna. One can find the far-field radiation pattern for various distri­butions by a Fourier-transform relation.

For instance, consider a line source of length Lw using the coordinate system as illustrated in Fig. 7. Assume that the

 Figure 7. Coordinate system used to analyze a linear aperture of length Lw.

 L

 w

 where k = 2’n/A. For real values of в, —1 < sin в < 1, the field distribution represents radiated power, while outside this re­gion it represents reactive or stored power (18). The field dis­tribution E(sin в), or an angular spectrum, refers to an angu­lar distribution of plane waves. The angular spectrum for a finite aperture is the same as the far-field pattern, E(e). Thus, for a finite aperture the Fourier integral representation of Eq. (6) may be written (8):

 source is positioned in a ground plane of infinite extent. This model is simple and yet the analysis gives results that illus­trate the main features of the most practical of the two-di­mensional apertures. The line-source distribution does have a practical realization, namely, in a long one-dimensional array that has sufficient elements to enable it to be approximated to a continuous distribution. The applicable transform pair is (7,17)

 This sin(x)/x distribution is very important in antenna theory and is the basis for many antenna designs. It has a first side – lobe level of —13.2 dB. Another popular continuous aperture distribution is the co­sine raised to power n,

 where —Lw/2 < x < Lw/2. This is shown in Fig. 8 for n = 0, 1, 2, and 3. To make a relative comparison of the various distri­butions, we must first normalize to the transmitted power of the uniform case. To do this, we multiply the pattern function

 Note that Eq. (8) is a relative relation. For example, consider a uniform distribution for which 1

 С-Lw/2 E (0) = E (x)ejkx sin 0 dx J+Lw/2

 E (sin 0) = j E(x)ejkx sin 0 dx

 E (x) = cos" ( j—x Lw

 E(x) = — Lw

 0d(sin 0)

 E (0) =

 and

 X

 30

 The field distribution pattern can be found by incorporating this into Eq. (8):

 We complete the straightforward integration to get the final result:

 (10)
 by the normalization constant:

 1

 Ср =

 (13)

 Lw/2

 і

 E2 (x) dx

 (15)

 Lw/2

 where 0 < p < 1. This is a combination of a uniform plus a cosine type distribution. The triangular distribution is popu­lar:

 E (x) = 1 +

 x

 Lw/2
 (14)
 Figure 9. Radiation patterns of line sources for three different aperture distri­butions (Lw = 1m, A = 3 cm).
 0, and
 (16)

To demonstrate the principles, we computed the antenna ra­diation pattern of a 1 meter long line-source antenna for cosine0 (uniform), cosine1, and cosine2 distributions. The op­erating wavelength is 3 cm. The resulting patterns are shown in Fig. 9. These data indicate that the more heavily tapered illuminations result in decreased side-lobe levels, but at a penalty of main beam peak gain.

Many distributions actually obtained in practice can be ap­proximated by one of the simpler forms or by a combination of simple forms. For example, a common linear aperture dis­tribution is the cosine on a pedestal p:

f n x

2£(x) = p + (1 – p) COS I —

VLw

for – Lw/2

x

E (x) = 1 —

Lw/2

for 0 < x < Lw/2.

In practice, the rectangular aperture is probably the most common microwave antenna. Because of its configuration, the rectangular coordinate system is the most convenient system to express the fields at the aperture. The most common and convenient coordinate system used to analyze a rectangular aperture is shown in Fig. 10. The aperture lies in the x-y plane and has a defined tangential aperture distribution E(x, y). In keeping with the equivalence principle we shall assume

 Figure 10. Coordinate system used to analyze rectangular aperture of dimensions Aw, Bw.

the x-у plane is a closed surface that extends from — oo to + oo in the x-у plane. Outside the rectangular aperture bound­aries we shall assume that the field distribution is zero for all points on this infinite surface. The task is to find the fields radiated by it, the pattern beam widths, the side-lobe levels of the pattern, and the directivity.

Note that a horn of aperture size Aw by Bw, with Aw/A > 1 and Bw/A < 1, can be analyzed as a continuous line source. If these conditions are not met, the pattern must be obtained by the integral (19):

Г-Bw/2 Г-Aw/2

E(0, ф) = / E(x, y)ej(kxx+kyy) dxdy (17)

JBw/2 JAw/2

where

kx = k sin в cos ф ky = k sin в sin ф

These are the x and у components of the propagation vector k (20).

For many types of antennas, such as the rectangular horn, the x and у functions are separable and may be expressed by the form

For nonseparable distributions, the integration of Eq. (17) is best carried out on a PC computer using numerical meth­ods. Figure 11 is a listing of a simple program written in Ba­sic that can be run on any PC computer.

In running the program, ф = 0 corresponds to the principal plane pattern in the x-z plane while ф = 90° is the principal plane pattern in the y-z plane. For example, consider an ap­erture with Aw = 75 cm, Bw = 125 cm, and A = 3 cm. Assume cosine distribution in each plane. The principal plane pat­terns in the x plane and у plane and the pattern in the inter­cardinal plane (в = 45°) that result are shown in Fig. 12.

We applied the computer code to compute the secondary pattern characteristic produced by uniform, cosine raised to power n, cosine on a pedestal p, and triangular aperture dis­tributions. The results shown in Table 1 compare the gain, beam width, and the first side-lobe levels. All gain levels are compared with the uniform illumination case.

A uniform line-source or rectangular aperture distribution produces the highest directivity. However, the first side lobe is only about —13.2 dB down. Thus, aperture distributions used in practice must be a trade-off or a compromise between the desired directivity (or gain) and side-lobe level.