# 9.7 Radiation by Large-Aperture Antennas

For wire antennas, the sources of radiation are the infinitesimal current elements comprising the current distribution along the wire and the total radiated field at a given point in space is equal to the sum, or integral, of the fields radiated by all the elements. A parallel scenario applies to aperture antennas, except that now the source of radiation is the electric-field distribution across the aperture. Consider the horn antenna shown in Fig. 9-20. It is connected to a source through a coaxial transmission line with the outer conductor of the line connected to the metal body of the horn and the inner conductor made to protrude through a small hole partially into the throat end of the horn. The protruding conductor acts as a monopole antenna, generating waves that radiate outwardly toward the horn’s aperture. The electric field of the wave arriving at the aperture, which may vary as a function of x a and y a over the horn’s aperture, is called the electric-field aperture distribution or illumination, E a(x a, y a). Inside the horn, wave propagation is guided by the horn’s geometry, but as the wave transitions from a guided wave into an unbounded wave, every point of its wavefront serves as a source of spherical secondary wavelets. The aperture may then be represented as a distribution of isotropic radiators. At a distant point Q, the combination of all the waves arriving from all of these radiators constitutes the total wave that would be observed by a receiver placed at that point.

# Figure 9-20 A horn antenna with aperture field distribution E a(x a, y a).

Figure 9 20 Full Alternative Text

The radiation process described for the horn antenna is equally applicable to any aperture upon which an electromagnetic wave is incident. For example, if a light source is used to illuminate an opening in an opaque screen through a collimating lens, as shown in Fig. 9-21(a), the opening becomes a source of secondary spherical wavelets, much like the aperture of the horn antenna. In the case of the parabolic reflector shown in Fig. 9-21(b), it can be described in terms of an imaginary aperture representing the electric-field distribution across a plane in front of the reflector.

# Figure 9-21 Radiation by apertures: (a) an opening in an opaque screen illuminated by a light source through a collimating lens and (b) a parabolic dish reflector illuminated by a small horn antenna.

(a) Opening in an opaque screen

Figure 9 21-a Full Alternative Text

(b) Parabolic reflector antenna

Two types of mathematical formulations are available for computing the electromagnetic fields of waves radiated by apertures. The first is a scalar formulation based on Kirchhoff’s work, and the second is a vector formulation based on Maxwell’s equations. In this section, we limit our presentation to the scalar diffraction technique in part because of its inherent simplicity and also because it is applicable across a wide range of practical applications.

A distinctive feature of such an antenna is its high directivity and correspondingly narrow beam, which makes it attractive for radar and free-space microwave communication systems. The frequency range commonly used for such applications is the 1 to 30 GHz microwave band. Because the corresponding wavelength range is 30 to 1 cm, respectively, it is quite practical to construct and use antennas (in this frequency range) with aperture dimensions that are many wavelengths in size.

The x a–y a plane in Fig. 9-22 (denoted plane A) contains an aperture with an electric field distribution E a(x a, y a). For the sake of convenience, the opening has been chosen to be rectangular in shape, with dimensions l x along x a and l y along y a, even though the formulation we are about to discuss is general enough to accommodate any two-dimensional aperture distribution, including those associated with circular and elliptical apertures. At a distance z from the aperture plane A in Fig. 9-22, we have an observation plane O with axes (x,y). The two planes have parallel axes and are separated by a distance z. Moreover, z is sufficiently large that any point Q in the observation plane is in the far-field region of the aperture. To satisfy the far-field condition, it is necessary that

where d is the longest linear dimension of the radiating aperture.

# Figure 9-22 Radiation by an aperture in the x a–y a plane at z=0.

Figure 9 22 Full Alternative Text

The position of observation point Q is specified by the range R between the center of the aperture and point Q and by the angles θ and ϕ (Fig. 9-22), which jointly define the direction of the observation point relative to the coordinate system of the aperture. In our treatment of the dipole antenna, we oriented the dipole along the z axis, and we called θ the zenith angle. In the present context, the z axis is orthogonal to the plane containing the antenna aperture. Also, θ usually is called the elevation angle. The electric field phasor of the wave incident upon point Q is denoted E˜(R,θ,ϕ). Kirchhoff’s scalar diffraction theory provides the following relationship between the radiated field E˜(R,θ,ϕ) and the aperture illumination E˜a(x a, y a):

E˜(R,θ,ϕ)=j λ e−j k R R h˜(θ,ϕ),(9.74)

where

h˜(θ,ϕ)=∬−∞∞E˜a(x a, y a)⋅exp[j k sin θ(x a cos ϕ+y a sin ϕ)]d x a d y a.(9.75)

We shall refer to h˜(θ,ϕ) as the form factor of E˜(R,θ,ϕ). Its integral is written with infinite limits with the understanding that E˜a(x a, y a) is identically zero outside the aperture. The spherical propagation factor (e−j k R/R) accounts for wave propagation between the center of the aperture and the observation point, and h˜(θ,ϕ) represents an integration of the exciting field E˜a(x a, y a) over the extent of the aperture, taking into account [through the exponential function in Eq. (9.75)] the approximate deviation in distance between R and s, where s is the distance to any point (x a, y a) in the aperture plane (see Fig. 9-22).

Also, the power density of the radiated wave is given by

S(R,θ,ϕ)=|E˜(R,θ,ϕ)|2 2 η 0=|h˜(θ,ϕ)|2 2 η 0 λ 2 R 2.(9.76)