Many antennas, such as horns or paraboloid reflectors, can be analyzed simply as apertures. We replace the incident fields in the aperture with a combination of equivalent electric and magnetic currents. We calculate radiation as a superposition of each source by using the vector potentials. Most of the time, we assume that the incident field is a propagating free-space wave whose electric and magnetic fields are proportional to one another. This gives us the Huygens source approximation and allows the use of integrals over the electric field in the aperture. Each point in the aperture is considered to be a source of radiation. The far field is given by a Fourier transform of the aperture field:

  (1)

This uses the vector propagation constant

where   is the pattern in k-space. We multiply the Fourier transform far field by the pattern of the Huygens source:

  (2)

When apertures are large, we can ignore this pattern factor. In Eq. (1),   is a vector in the same direction as the electric field in the aperture. Each component is transformed separately. The far-field components  and  are found by projection (scalar products) from  times the pattern factor of the Huygens source [Eq. (2)].

If we have a rectangular aperture in which the electric field is expressed as a product of functions of x and y only, the integral reduces to the product of two single integrals along each coordinate. The Fourier transform relationships provide insight into pattern shape along the two axes. Large apertures radiate patterns with small beamwidths. An antenna with long and short axes has a narrow-beamwidth pattern in the plane containing the long dimension and a wide beamwidth in the plane containing the short dimension. This is the same as the time and frequency dual normally associated with the Fourier transform.

We draw on our familiarity with signal processing to help us visualize the relationship between aperture distributions and patterns. Large apertures give small beamwidths, just as long time pulses relate to low-frequency bandwidths in normal time–frequency transforms. The sidelobes of the pattern correspond to the frequency harmonics of an equivalent time waveform under the Fourier transform and rapid transitions in the time response lead to high levels of harmonics in the frequency domain. Rapid amplitude transitions in the aperture plane produce high sidelobes

(harmonics) in the far-field response (Fourier transform). Step transitions on the aperture edges produce high sidelobes, while tapering the edge reduces sidelobes and we relate the sidelobe envelope of peaks to the derivative of the distributions at the edges. To produce equal-level sidelobes, we need Dirac delta functions in the aperture that transform to a constant level in the pattern domain. Another example is periodic aperture errors that produce single high sidelobes. When we discuss aperture distribution synthesis, we see that the aperture extent in wavelengths limits our ability to control the pattern.

A uniform amplitude and phase aperture distribution produces the maximum aperture efficiency and gain that we determine from the following argument. An aperture collects power from a passing electromagnetic wave and maximum collectible power occurs at its peak amplitude response. If the amplitude response somewhere else in the aperture is reduced from the maximum, that portion will collect less power. The amplitude response can be reduced only by adding loss or reflecting power in reradiation. The antenna with the highest aperture efficiency reflects the least amount of power when illuminated by a plane wave. Similarly, if the phase shift from the collecting aperture to the antenna connector is different for different parts of the aperture, the voltages from the various parts will not add in phase. Gain is directly proportional

to aperture efficiency [Eq. (3)]:

Therefore, a uniform amplitude and phase aperture distribution has maximum gain. All this assumes that the input match on various aperture distribution antennas is the same.