Computing Antenna Patterns

Figure 3.12: Aperture illumination for a parabolic dish.

The next step is to understand how to compute the power pattern of a given telescope. Consider a parabolic reflecting telescope being fed by a feed at the focus. The radiation from the feed reflects off the telescope and is beamed off into space (Figure 3.12). If one knew the radiation pattern of the feed, then from geometric optics (i.e. simple ray tracing, see Chapter 19) one could then calculate the electric field on the plane across the mouth of the telescope (the `aperture plane'). How does the field very far away from the telescope lookslike? If the telescope surface were infinitely large, then the electric field in the aperture plane is simply a plane wave, and since a plane wave remains a plane wave on propagation through free space, the far field is simply a plane wave traveling along the axis of the reflector. The power pattern is an infinitely narrow spike, zero everywhere except along the axis. Real telescopes are however finite in size, and this results in diffraction. The rigorous solution to the diffraction problem is to find the appropriate Green's function for the geometry, this is often impossible in practise and various approximations are necessary. The most commonly used one is Kirchoff's scalar diffraction theory. However, for our purposes, it is more than sufficient to simply use Huygen's principle.

Huygen's principle states that each point in a wave front can be regarded as an imaginary source. The wave at any other point can then be computed by adding together the contributions from each of these point sources. For example consider a one dimensional aperture, of length $l$ with the electric field distribution (`aperture illumination') $e(x)$. The field at a point P(R, $\theta$) (Figure 3.13) due to a point source at a distance x from the center of the aperture is (if $R$ is much greater than $l$) is:

$$dE = { e(x) \over R^2} e^{-j {2 \pi x sin \theta \over \lambda }}$$

Figure 3.13: The far-field pattern as a function of the aperture illumination.

Where $x\sin\theta$ is simply the difference in path length between the path from the center of the aperture to the point P and the path from point $x$ to point P. Since the wave from point $x$ has a shorter path length, it arrives at point P at an earlier phase. The total electric field at P is:

$$E(R,\theta) = \int_{-l/2}^{l/2} { e(x) \over R^2}e^{-jk\mu x} dx$$

where $k = {2 \pi \over \lambda }$ and $\mu = sin \theta$ and x is now measured in units of wavelength. The shape of the distribution is clearly independent of R, and hence the unnormalized power pattern $F_U$ is just:

$$F_U(\mu) = \int_{-\infty}^\infty e_1(x) e^{-jk\mu x} dx$$ (3.5.12)

where

$$e_1(x) = e(x)~~~if~~~\vert x\vert \leq l/2~~~;~~~0 ~~~~ {\rm otherwise}$$

The region in which the field pattern is no longer dependent on the distance from the antenna is called the far field region. The integral operation in equation (3.5.13) is called the Fourier transform. $F_U(\mu)$ is the Fourier transform of $e_1(x)$, which is often denoted as $F_U(\mu) = {\bf F} \bigr[{e_1(x)}\bigl]$. The Fourier transform has many interresting properties, some of which are listed below (see also Section 2.5).

1. Linearity

   If $G_1(\mu) = {\bf F} \bigr[{g_1(x)}\bigl]$ and $G_2(\mu) = {\bf F} \bigr[{g_2(x)}\bigl]$ then $G_1(\mu) + G_2(\mu) = {\bf F} \bigr[{g_1(x) +g_2(x) }\bigl]$.

2. Inverse

   The Fourier transform is an invertible operation; if
   

   $$G(\mu) = \int_{-\infty}^\infty g(x) e^{-j 2 \pi \mu x} dx$$
   
   
   then
   
   $$g(x) = \int_{-\infty}^\infty G(\mu)e^{j 2 \pi \mu x} d\mu$$
   
   

3. Phase shift

   If $G(\mu) = {\bf F} \bigr[{g(x)}\bigl]$ then $G(\mu - \mu_0) = {\bf F} \bigr[{g(x) e^{-j 2 \pi \mu_ 0 x}}\bigl]$. This means that an antenna beam can be steered across the sky simply by introducing the appropriate linear phase gradient in the aperture illumination.

4. Parseval's theorem

   If $G(\mu) = {\bf F} \bigr[{g(x)}\bigl]$, then
   

   $$\int_{-\infty}^\infty \vert G(\mu)\vert^2 d\mu = \int_{-\infty}^\infty \vert g(x)\vert^2 dx$$
   
   
   This is merely a restatement of power conservation. The LHS is the power outflow from the antenna as measured in the far field region, the RHS is the power outflow from the antenna as measured at the aperture plane.

5. Area

   If $G(\mu) = {\bf F} \bigr[{g(x)}\bigl]$, then
   

   $$G(0) = \int_{-\infty}^\infty g(x) dx$$
   
   

With this background we are now in a position to determine the maximum effective aperture of a reflecting telescope. For a 2D aperture with aperture illumination $g(x,y)$, from equation (3.4.10)

$$A_e^{max} = {\lambda^2 \over \int P(\theta,\phi) d\Omega} = {\lambda^2 \over \int \vert F(\theta,\phi)\vert^2 d\Omega}$$ (3.5.13)

But the field pattern is just the normalized far field electric field strength, i.e.

$$F(\theta,\phi) = {E(\theta,\phi) \over E(0,0)}$$

where $E(\theta,\phi) = {\bf F} \bigr[{g(x,y)}\bigl]$. From property (5)

$$E(0,0) = \int g(x,y) dxdy'$$ (3.5.14)

and from Parseval's theorem,

$$\int \vert E(\theta,\phi)\vert^2 d \Omega= \int \vert g(x,y)\vert^2 dxdy$$ (3.5.15)

substituting in equation (3.5.14) using equations (3.5.15),  3.5.16 gives,

$$A_e^{max} = {\lambda^2 \bigr\vert \int g(x,y) dxdy \bigl\vert^2 \over \int \vert g(x,y)\vert^2 dxdy}$$

For uniform illumination

$${A_e^{max} \over \lambda^2} = { A_{g}^2 \over A_{g}} = A_{g}$$

Note that since $x$ and $y$ are in units of wavelength, so is $A_{g}$. $A_e^{max}$ however is in physical units. Uniform illumination gives the maximum possible aperture efficiency (i.e. 1), because if the illumination is tapered then the entire available aperture is not being used.

As a concrete example, consider a 1D uniformly illuminated aperture of length $l$. The far field is then

$$E(\mu) = \int_{-l/2}^{l/2} e^{-{j 2 \pi x \mu \over \lambda}} dx$$

$$= {\lambda \sin(\pi l / \lambda \mu) \over \pi \mu}$$

and the normalized field pattern is

$$F(\mu) = {\sin(\pi l / \lambda \mu) \over (\pi l/ \lambda \mu)}$$

This is called a 1D sinc function. The 1st null is at $\mu =\lambda/l$, the 1st sidelobe is at $\mu = 3/2 (\lambda/l)$ and is of strength $2/(3 \pi)$. The strength of the power pattern 1st sidelobe is $(2 /3 \pi)^2 = 4.5\%$. This illustrates two very general properties of Fourier transforms:

1. the width of a function is inversely proportional to width of its transform ( so large antennas will have small beams and small antennas will have large beams), and
2. any sharp discontinuities in the function will give rise to sidelobes (`ringing') in the fourier transform.

Figure 3.14: Power and field patterns for a 1D uniformly illuminated aperture.

Figure 3.14 shows a plot of the the power and field patterns for a 700 ft, uniformly illuminated aperture at 2380 MHz.

Aperture illumination design hence involves the following following tradeoffs (see also Chapter 19):

1. A more tapered illumination will have a broader main beam (or equivalently smaller effective aperture) but also lower side lobes than uniform illumination.

2. If the illumination is high towards the edges, then unless there is a very rapid cutoff (which is very difficult to design, and which entails high sidelobes) there will be a lot of spillover.

Another important issue in aperture illumination is the amount of aperture blockage. The feed antenna is usually suspended over the reflecting surface (see Figure 3.3) and blocks out part of the aperture. If the illumination is tapered, then the central part of the aperture has the highest illumination and blocking out this region could have a drastic effect on the power pattern. Consider again a 1D uniformly illuminated aperture of length l with the central portion of length $d$ blocked out. The far field of this aperture is (from the linearity of fourier transforms) just the difference between the far field of an aperture of length $l$ and an aperture of length d, i.e.

$$E(\mu) \propto {sin(\pi l \mu / \lambda) \over \pi \mu} - {sin(\pi d \mu / \lambda ) \over \pi \mu}$$

or the normalized field pattern is:

$$F(\mu) = {\lambda \over (l -d )} \bigl[ {sin(\pi l \mu / \lam... ...er \pi \mu} - {sin(\pi d \mu / \lambda) \over \pi \mu} \bigl]$$

Figure 3.15: Effect of aperture blockage on the power pattern.

The field pattern of the ``missing'' part of the aperture has a broad main beam (since $d < l$). Subtracting this from the pattern due to the entire aperture will give a resultant pattern with very high sidelobes. In Figure 3.15 the solid curve is the pattern due to the entire aperture, the dotted line is the pattern of the blocked part and the dark curve is the resultant pattern. (This is for a 100ft blockage of a 700 ft aperture at 2380 MHz). Aperture blockage has to be minimized for a `clean' beam, many telescopes have feeds offset from the reflecting surface altogether to eliminate all blockage.

Figure 3.16: The Ooty radio telescope.

As an example of what we have been discussing, consider the Ooty Radio Telescope (ORT) shown in Figure 3.16. The reflecting surface is a cylindrical paraboloid ( $530 m \times 30 m$) with axis parallel to the Earth's axis. Tracking in RA is accomplished by rotating the telescope about this axis. Rays falling on the telescope get focused onto the a line focus, where they are absorbed by an array of dipoles. By introducing a linear phase shift across this dipole array, the antenna beam can be steered in declination (the ``phase shift'' property of Fourier transforms). The reflecting surface is only part of a paraboloid and does not include the axis of symmetry, the feed is hence completely offset, there is no blockage. The beam however is fan shaped, narrow in the RA direction (i.e. that conjugate to the $530m$ dimension) and broad in the DEC (i.e. that conjugate to the $30m$ dimension).

Figure 3.17: Turret positioning error. Ideally the feed should point at the vertex of the reflecting surface, but if the feed turret rotation angle is in error then the feed points along some offset direction.

Aperture blockage is one of the reasons why an antenna's power pattern would deviate from what one would ideally expect. Another common problem that affects the power pattern is the location of the feed antenna. Ideally the feed should be placed at the focus, but for a variety of reasons, it may actually be displaced from the focus. For example, as the antenna tracks, the reflecting surface gets distorted and/or the feeds legs bend slightly, and for these reasons, the feed is displaced from the actual focal point of the reflector. In an antenna like the GMRT, there are several feeds mounted on a cubic turret at the prime focus, and the desired feed is rotated into position by a servo system (see Chapter 19). Small errors in the servo system could result in the feed pointing not exactly at the vertex of the reflector but along some slightly offset direction. This is illustrated in Figure 3.17. For ease of analysis we have assumed that the feed is held fixed and the reflector as a whole rotates. The solid line shows the desired location of the reflector (i.e. with the feed pointing at its vertex) while the dashed line shows the actual position of the reflector. This displacement between the desired and actual positions of the reflector results in an phase error (produced by the excess path length between the desired and actual reflector positions) in the aperture plane. From the geometry of Figure 3.17 this phase error can be computed, and from it the corresponding distortion in the field and power patterns can be worked out. Figure 3.18[A] shows the result of such a calculation. The principal effect is that the beam is offset slightly, but one can also see that its azimuthal symmetry is lost. Figure 3.18[B] shows the actual measured power pattern for a GMRT antenna with a turret positioning error. As can be seen, the calculated error pattern is a fairly good match to the observed one. Note that in plotting Figure 3.18[B] the offset in the power pattern has been removed (i.e. the power pattern has been measured with respect to its peak position).

Figure 3.18: [A] Calculated beam pattern for a turret positioning error. [B] Measured beam pattern for a turret positioning error. The offset in the pattern has been removed, i.e. the power pattern has been measured with respect to its peak position.

Further Reading

1. Antenna Theory Analysis and Design , Constantine A. Balanis, Harper & Row, Publishers, New York.

2. Radio telescopes, second edition , W. N. Christiansen & J. A. Hogbom, Cambridge Univ. Press.

3. Microwave Antenna Theory and Design, Samuel Silver (ed.), IEE

4. Reflector Antennas, A. W Love (ed.), IEEE press, Selected Reprint Series.

5. Instrumentation and Techniques for Radio Astronomy, Paul F. Goldsmith (ed.), IEEE press Selected Preprint Series.