# 9.8 Rectangular Aperture with Uniform Aperture Distribution

To illustrate the scalar diffraction technique, consider a rectangular aperture of height l x and width l y, both at least a few wavelengths long. The aperture is excited by a uniform field distribution (i.e., constant value) given by

E˜a(x a, y a)=E 0 for −l x/2≤x a≤l x/2 and −l y/2≤y a≤l y/2,0 otherwise.(9.77)

To keep the mathematics simple, let us confine our examination to the radiation pattern at a fixed range R in the x–z plane, which corresponds to ϕ=0. In this case, Eq. (9.75) simplifies to

h˜(θ)=∫y a=−l y/2 l y/2∫x a=−l x/2 l x/2 E 0 exp[j k x a sin θ]d x a d y a.(9.78)

In preparation for performing the integration in Eq. (9.78), we introduce the intermediate variable u defined as

u=k sin θ=2 π sin θ λ.(9.79)

Hence,

h˜(θ)=E 0∫−l x/2 l x/2 e j u x a d x a⋅∫−l y/2 l y/2 d y a=E 0 e j u l x/2−e−j u l x/2 j u⋅l y=2 E 0 l y u e j u l x/2−e−j u l x/2 2 j=2 E 0 l y u sin (u l x/2).(9.80)

Upon replacing u with its defining expression, we have

h˜(θ)=2 E 0 l y 2 π λ sin θ sin (π l x sin θ/λ)=E 0 l x l y sin(π l x sin θ/λ)π l x sin θ/λ=E 0 A p sinc (π l x sin θ/λ),(9.81)

where A p=l x l y is the physical area of the aperture. Also, we used the standard definition of the sinc function, which for any argument t is defined as

sinc t=sin t t.(9.82)

Using Eq. (9.76), we obtain the following expression for the power density at the observation point:

where S 0=E 0 2 A p 2/(2 η 0 λ 2 R 2).

This occurs when θ=0. Hence, at a fixed range R, S max=S(θ=0)=S 0. The normalized radiation intensity is then given by

F(θ)=S(R,θ)S max=sinc 2 (π l x sin θ/λ)=sinc 2 (π γ)(x–z plane).(9.84)

Figure 9-23 shows F(θ) plotted (on a decibel scale) as a function of the intermediate variable γ=(l x/λ) sin θ. The pattern exhibits nulls at nonzero integer values of γ.

# Figure 9-23 Normalized radiation pattern of a uniformly illuminated rectangular aperture in the x–z plane (ϕ=0).

Figure 9 23 Full Alternative Text

# 9.8.1 Beamwidth

The normalized radiation intensity F(θ) is symmetrical in the x–z plane, and its maximum is along the boresight direction (θ=0, in this case). Its half-power beamwidth β x z=θ 2−θ 1, where θ 1 and θ 2 are the values of θ at which F(θ,0)=0.5 (or −3 dB on a decibel scale), as shown in Fig. 9-23. Since the pattern is symmetrical with respect to θ=0, θ 1=−θ 2, and β x z=2 θ 2. The angle θ 2 can be obtained from a solution of

F(θ 2)=sinc 2 (π l x sin θ/λ)=0.5.(9.85)

From tabulated values of the sinc function, it is found that Eq. (9.85) yields the result

π l x λ sin θ 2=1.39,(9.86)

or

sin θ 2=0.44 λ l x.(9.87)

Because λ/l x≪1 (a fundamental condition of scalar diffraction theory is that the aperture dimensions be much larger than the wavelength λ), θ 2 is a small angle, in which case we can use the approximation sin θ 2≈θ 2. Hence,

A similar solution for the y–z plane (ϕ=π/2) gives

The first sidelobe level is 13.2 dB below the peak value (see Fig. 9-23), which is equivalent to 4.8% of the peak value. If the intended application calls for a pattern with a lower sidelobe level (to avoid interference with signals from sources along directions outside the main beam of the antenna pattern), this can be accomplished by using a tapered aperture distribution—one that is a maximum at the center of the aperture and decreases toward the edges.

The steeper the taper, the lower are the side lobes and the wider is the main lobe. In general, the beamwidth in a given plane, say the x–z plane, is given by

where k x is a constant related to the steepness of the taper. For a uniform distribution with no taper, k x=0.88, and for a highly tapered distribution, k x≈2. In the typical case, k x≈1.

To illustrate the relationship between the antenna dimensions and the corresponding beam shape, we show in Fig. 9-24 the radiation patterns of a circular reflector and a cylindrical reflector. The circular reflector has a circularly symmetric pattern, whereas the pattern of the cylindrical reflector has a narrow beam in the azimuth plane corresponding to its long dimension and a wide beam in the elevation plane corresponding to its narrow dimension. For a circularly symmetric antenna pattern, the beamwidth β is related to the diameter d by the approximate relation β≈λ/d.

# Figure 9-24 Radiation patterns of (a) a circular reflector and (b) a cylindrical reflector (side lobes not shown).

(a) Pencil beam

Figure 9 24-a Full Alternative Text

(b) Fan beam

Figure 9 24-b Full Alternative Text

# 9.8.2 Directivity and Effective Area

In Section 9-2.3, we derived an approximate expression [Eq. (9.26)] for the antenna directivity D in terms of the half-power beamwidths β x z and β y z for antennas characterized by a single major lobe whose boresight is along the z direction:

D≈4 π β x z β y z.(9.90)

If we use the approximate relations β x z≈λ/l x and β y z≈λ/l y, we obtain

D≈4 π l x l y λ 2=4 π A p λ 2.(9.91)

For any antenna, its directivity is related to its effective area A e by Eq. (9.64):

D=4 π A e λ 2. (9.92)