Antennas notes.

We start our discussion of aperture antennas by imagining a perfectly opaque plane sheet of absorber with an arbitrarily shaped hole cut in it. This is illuminated from the back side by a plane electromagnetic wave at normal incidence (ie travelling at right angles to the plane of the absorber).

The part of the wave passing through the hole in the absorbing sheet travels on, having been diffracted at the edges of the hole, and as we follow it further on its travels, it spreads out to form a "far field radiation pattern" of a "uniformly illuminated aperture antenna" having the shape of the hole.

If we consider a circular aperture of diameter D, then much of the energy passing through the aperture is diffracted through an angle of the order alpha = lambda/D radians from its original propagation direction. When we have travelled a distance R from the aperture, about half of the energy passing through the opening will have left the cylinder made by the geometric shadow if D/R = alpha. Putting these formulae together, we see that the majority of the propagating energy in the "far field region" at a distance greater than the Rayleigh distance 2D^2/lambda will be diffracted energy. In this region then, the polar radiation pattern consists of diffracted energy only, and the angular distribution of propagating energy will then no longer depend on the distance from the aperture.

We recall that the definition of a "plane wave" is that the wave has the same instantaneous electric and magnetic field values (size and directions) everywhere on any plane taken at right angles to the direction of propagation. If you like, the illumination across the aperture has the same amplitude and phase angle everywhere within the aperture.

There is a simple generalisation of "uniform plane wave illumination" whereby the wave incident on the back side of the aperture starts from a point, from which it expands spherically. Clearly the amplitude and/or phase of the illumination will not now be constant across the plane aperture. Such an expanding wave is said to have originated from a "phase centre". The number of wavelengths from any point on a spherical shell centred on this "phase centre" to the phase centre does not depend on the position we have chosen for the original point on the shell.

Clearly a phase centre can exist where the illumination amplitude in different directions has different values. We can also envisage a line source instead of a point source as a "phase centre". In some ways, the concept of a phase centre is rather artificial as all radiation has to originate from an oscillating current source which is necessarily spatially extended. Also, as we have seen elsewhere in these notes, there is no such thing as an "isotropic radiator" for which the radiation is uniform in all directions on the sphere.

Suppose the aperture is illuminated by a converging wave, rather than a diverging or plane wave. Then on the far side of the aperture there may be regions of greatly enhanced field strength known as "caustics" or "cusps". Such patterns are observed, for example, at the bottom of a swimming pool illuminated by sunlight refracted through the surface waves on the pool. In this discussion we do not consider caustics and cusps further; mathematically the phase of a wave can be discontinuous on propagation through such a feature.

In the most general case, the illumination of our (arbitrarily shaped) aperture has to be specified by giving the amplitude and phase of the illuminating wave everywhere within the aperture. We of course assume the amplitude is zero everywhere outside the aperture, on the plane of the aperture. There are continuity constraints imposed by the radiation physics on the possible distributions we can realise across the aperture.

Given an aperture in the x-y plane, of arbitrary shape, we can take the spatial Fourier transform (in 2 dimensions) to get the Kx and Ky components of the Fourier transformed function with angular spatial frequency Kx along x and Ky along y. See any standard signal analysis textbook for the mathematical details.

The direction normal to the aperture plane is clearly along the z axis. The magnitude of the propagation wave-vector (let us call this K) can be written |K| = sqrt[Kx^2+Ky^2+Kz^2] and of course the value of |K| is just (2 pi)/lambda where lambda is the free space wavelength.

Thus, if we choose a spatial direction [Kx,Ky,Kz]/|K| the amplitude of the radiated field strength is determined by the size of the Fourier coefficient for Kx,Ky. Thus, in this sense, the far field radiation pattern is determined completely by the Fourier transform of the field distribution across the aperture.

Apart from the fact that the Fourier transforms are in 2 dimensions rather than 1 dimension, we see that the design of aperture antennas (shape of aperture and illumination profile) has much in common with the design of electronic filters in the time-frequency domains; certainly this is true so far as the mathematics is concerned.

For the case of uniformly illuminated apertures of arbitrary shape, a very good idea of the far field patterns can be obtained by photographing the diffraction of a laser beam by a similarly shaped aperture, suitably scaled in wavelength.

Here is a circular aperture in an opaque screen

A CIRCULAR APERTURE, UNIFORMLY ILLUMINATED.

THE FAR FIELD PATTERN OF A CIRCULAR APERTURE

Now here are two such circular apertures placed one over the other....

TWO CIRCULAR APERTURES

FAR FIELD FROM TWO APERTURES, SHOWING INTERFERENCE BARS

If we rotate the two apertures and alter their spacing, the bars change separation and orientation....

TWO APERTURES, ROTATED.

TWO APERTURES, FAR FIELD PATTERN

For illustration, here is a triangular aperture with an internal blocking triangle. We note the extent of the sidelobes, and the symmetry of the far field pattern.

A TRIANGULAR APERTURE

FAR FIELD, SHOWING SIX-FOLD ROTATIONAL SYMMETRY FROM TRIANGULAR APERTURE

Suppose the phase of the illumination is constant across the aperture. Then, apart from an arbitrary constant phase factor (which can always be removed by redefining the origin of time) the illumination can be represented completely by a real function of x and y. The Fourier Transform is then an even function of Kx and Ky, that is, there is no difference if we replace Kx by -Kx or Ky by -Ky. That means that the far field radiation pattern of an arbitrary shaped aperture has the symmetry of the aperture plus inversion symmetry. For example, a triangular aperture has a hexagonal radiation pattern with six-fold rotational symmetry about boresight, even though the original aperture has only three-fold rotational symmetry about boresight.

To remove this symmetry it is necessary to have phase taper across the aperture. It is easy to see why this works, for a uniform phase taper from one side to the other can be realised by illuminating the aperture with a plane wave which is not at right angles to the plane of the aperture. In this case, boresight is tilted to lie along the direction of the travel of the illuminating plane wave.

Now we have an insight into electronic beam steering. Just as we can steer the beam of a phased array antenna by shifting the phase of the drive between successive elements, so we can steer the beam of an aperture antenna by using appropriate phase taper across the aperture.

The principles of array antenna design using pattern multiplication of element and array patterns can be used to design and analyse arrays of apertures. Here we assume the apertures are all the same shape and size, and are orientated in the same direction in the x-y plane. However, the Fourier Transform method works directly with the whole array, regarded as separated sections of the same aperture. There is a somewhat arbitrary distinction between what are called "grating nulls" and "aperture nulls". The nulls of an array of isotropes centred on the individual elements of the aperture array are called "grating nulls", whereas the nulls due to the diffraction pattern of an individual aperture are called "aperture nulls".

One gets into some possibility of confusion with arrays of differently spaced, sized and orientated apertures. It is of course always possible to construct a similar "grating" of isotropes, suitably weighted for the sizes and strengths of the aperture illumination, but the process of pattern multiplication no longer is applicable and so the advantage of the array formulation is lost.

Suppose we consider a circular aperture having diameter 2a, with radius from the centre r (r less than a). Then if the amplitude distribution across the aperture is of the form exp[-(r/pa)^2] where p is some numerical constant, then within the aperture, the amplitude distribution is Gaussian. It is truncated at the rim of the aperture; but since the Fourier Transform of an un-truncated Gaussian is also a Gaussian we may expect that such an aperture antenna has no (or minimal) sidelobes. This gives an insight into the process known as apodisation or "edge taper"; by reducing the illumination intensity towards the perimeter of the aperture we have a commonly used method for suppressing sidelobes. The trade-off here is that the main beam becomes wider, since the effective diameter of the aperture is somewhat less.

In wave optics there is a principle that apertures and obstacles have similar diffraction properties. This is called "Babinet's Principle". Clearly we can regard a plane diffractor having absorption for x<0 and non-obstruction for x>0 as either a half-plane aperture or a half-plane blockage. Viewed either way, the diffraction pattern is the same.

Suppose we have an aperture antenna with an obstacle within the aperture. Such an obstacle might be a horn feed, or the support struts of a sub-reflector, or the sub-reflector itself. Such an obstacle will generate a diffraction pattern, which, since the obstacle is necessarily smaller than the aperture in which it is located, extends over a wider range of angles than the original beam.

As an example of the far field patterns produced by blockage, consider the ring antenna shown in the figure here

A common method of avoiding blockage in a reflector aperture antenna fed from the front is to offset the feed so that the radiation reflected misses the feed on its way out into space. Of course, this makes the profile design of the antenna more difficult. The effect of offsetting the feed is to put a uniform phase taper across the reflector dish.

Clearly then, if we construct a plane reflector antenna the reflected wave will also be plane apart from the diffraction effects providing the incident illumination is in the form of a plane wave. However, in practice the illumination is supplied by some kind of horn feed or dipole feed, which generates an expanding wavefront. If we regard the radiation from the feed as expanding spherically from a point or focus, then the ideal profile for a reflector aperture antenna is parabolic.

There may be profile errors in constructing a parabolic dish reflector. It is possible to apply some fairly advanced mathematics to relate the r.m.s. profile errors to the performance parameters of the antenna (gain, beamwidth, sidelobes performance).

In practice a horn or dipole feed does not act precisely like a point source, and so it may be necessary to adjust the profile of the reflector to compensate for this.

It is also possible to construct a multiple-patch antenna, with patches having differing sizes and resonant frequencies. The phase shifts on reflection from the different sizes of patch will be different. Thus it is in principle possible to construct a reflector antenna on a flat or plane surface, which has a phase profile approximating a parabolic reflector. Such antennas are much cheaper to mass-produce. Again, they may be designed for an offset-feed arrangement to avoid the blockage problems. Malibu Research at the URL http://maliburesearch.com/ have patents in this area. This site has some nice pictures and diagrams of various antenna arrangements.

For circular apertures which have diameter much smaller than a wavelength, the radiation coupled through is proportional to the sixth power of the radius. Thus for a small hole in a conducting screen, halving the hole diameter will reduce the coupled energy by a factor 64. This is why you can make a good punctured screen for a microwave oven, through which you can see the contents. The reference to the theory is

H.H.Bethe, "Theory of diffraction by small holes" Physical Review volume 66 pages 163-182, 1944

Credit for red-bordered pictures to Dr Michael Willis

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Copyright D.Jefferies 1997, 1999, 2002, 2005, 2007.

D.Jefferies email 6th June 2007.