Computing Reflector Antenna Radiation Patterns

Reflector antenna radiation patterns are determined
by a number of factors, but the most important ones are the radiation
pattern of the feed antenna and the shape of the reflector. Parabolic
reflectors have the unique feature that all path lengths from the
focal point to the reflector and on to the aperture plane are the same.
As shown in Figure 19.1,

since the parabola is described in polar form by,

When the reflector dimensions are large compared to the wavelength, geometrical optics principles can be used to determine the power distribution in the aperture plane. If the feed pattern is azimuthally symmetric, then the normalized far-field radiation pattern of reflector depends on

- , where is the radius of the aperture, , and is the angle subtended by the far-field point with respect to the parabola's focal axis
- The feed taper, [4],[5], which is defined as the amplitude of the feed radiation pattern at the rim of the parabolic reflector relative to the maximum value (assumed to be along the parabola axis). (Note that in standard power plots of radiation patterns (in dB), the edge taper is related to C by ).
- The focal length which determines how the power from the feed
is spread over the aperture plane. If
is the
radiation pattern of the feed, is distance in the aperture
plane, and is the power density in the aperture plane, then
we have

(19.4.4)

(19.4.5)

In Chapter 3 we saw that the far field is in general the Fourier transform of the aperture plane distribution. In the case of azimuthally symmetric distributions, this can be written as

where is the far field pattern, is a normalized distance in the aperture plane, , is the feed's pattern projected onto the aperture plane as discussed above. A convenient parameterization of the feed pattern in terms of the taper, is

The aperture illuminations corresponding to different values of the parameter are shown in Figure 19.2. The case corresponds to a uniform aperture distribution.

For uniform illumination the far field pattern is given by

Simple closed-form expressions are available for integer values of . If
the above expression is denoted as , (since ) the
general form for any integer is given by

where,

Table 19.1 gives the halfpower beamwidth (HPBW), the first sidelobe level and the taper efficiency (see Section 19.4.1) for various edge tapers and shape parameter .

Edge Taper | |||||||

HPBW | Sidelobe | HPBW | Sidelobe | ||||

(dB) | C | (rad.) | level (dB) | (rad.) | level (dB) | ||

-8 | 0.398 | -21.5 | 0.942 | -24.7 | 0.918 | ||

-10 | 0.316 | -22.3 | 0.917 | -27.0 | 0.877 | ||

-12 | 0.251 | -22.9 | 0.893 | -29.5 | 0.834 | ||

-14 | 0.200 | -23.4 | 0.871 | -31.7 | 0.792 | ||

-16 | 0.158 | -23.8 | 0.850 | -33.5 | 0.754 | ||

-18 | 0.126 | -24.1 | 0.833 | -34.5 | 0.719 |

From Table 19.1 (see also the discussion in Chapter 3) we find that as the edge-taper parameter decreases, the HPBW increases, the first sidelobe level falls and the taper-efficiency also decreases. Note that has to be less than unity since we have assumed a radiation pattern which decreases monotonically with increasing angle from the symmetry-axis (Eqn 19.4.6, Fig 19.2).