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Distance between foci

fc

6.177 m

Throughout this text, we shall illustrate the discussion with numerical examples. As the basic antenna for these we take the element antenna of the Atacama Large Millimeter Array (ALMA). The basic geometrical parameters of this Cassegrain antenna are assembled in Table 2.1. [Mat.2.1]

■ 2.2. Geometry of aberrations

For an optimal performance of the antenna, it is important that the system is properly focussed. For a prime focus reflector this means localizing the phase center of the feed in the focus of the primary paraboloid. In a Cassegrain system, the most important criterion is the coincidence of one of the foci of the hyperboloidal secondary reflector with the primary focus of the paraboloid. The location of the feed in the secondary focus is far less critical, as we shall show below. In this section we develop the geometrical formulae, which describe an axial or lateral defocus, that is a deviation of the feed or secondary reflector from the focal point along the axis of symmetry or perpendicularly to it. Later we shall use these to calculate the effects of defocus on the beam characteristics and the pointing of the antenna.

The general case of an arbitrary shift of the feed from the true focal position (the defocus d) can be separated in two components: a shift along the reflector axis (axial defocus) da and one perpendicular to the axis (lateral defocus) dl. We call y the angle between the reflector axis and a ray from the focus to a point at the surface at radius r. We now want to calculate the path-length difference between such a ray and the central, on axis, ray for both lateral and axial defocus. We can then incorporate the resulting phase error function in the basic radiation integral to calculate the radiation pattern and beam parameters of the defocused system. It is obvious that axial defocus-ing will cause a pathlength error which is independent of the azimuthal coordinate c of the reflector aperture, while the pathlength due to lateral defocus in the azimuthal plane c = C will be proportional to cos (c- c). In most of what follows we shall assume that the defocus is small with respect to the focal length of the reflector, i.e. d / f << 1, so that normally we can neglect terms of order (d / f )2 and higher.

2.2.1. Lateral defocus

We treat first the case of a lateral defocus of the feed, or the secondary reflector, from the focal point. The situation is illustrated in Fig. 2.3; the defocus is denoted d, omitting the subscript for lateral. Remember that the pathlength error will be dependent on the azimuthal aperture coordinate c and we assume that the feed is moved in the plane where c = 0. We shall derive below the full pathlength error, but later concentrate our calculations of the beam characteristics in the plane of defocus, where the effects are of course most pronounced. Applying the cosine rule to the triangle PFF', where P is a point on the reflector surface at radius r and the angle PFF' = p/2 - y, we have p'2 = p2 +d2 - 2dp siny cosc

Fig. 2.3. The geometry of the lateral defocus in the plane of feed translation.

Fig. 2.3. The geometry of the lateral defocus in the plane of feed translation.

and using the series development for the square root sf(1 + x) = 1 + - x- + , we obtain ii d ■ , , d2 d2 ■ 2 , 2 \ p' = P J1 - -pp siny cos/C+ ypr - ypr sin y cos2, where we ignore terms of order higher than (d / p)2. Thus the pathlength difference A, is d2 d2 2 2 Dl ' -r = - d sin y cos ^ + - -Tj— sin y cos2 ^. (2.21)

Using several of the formulae for the description of the parabola in Section 2.1, we can eliminate both p and y from Eq. (2.21) to obtain the pathlength error DL over the aperture as function of r, resulting from a lateral defocus d in the plane ,c=0,

Dl = - f r cos x + f r3 cos -f r2 - -f r2 cos2 ^ + -j-r-f. (2.22)

The terms in this equation represent some of the well known aberrations in optical instrument theory (see e.g. Born & Wolf, 1980). The first term is the distortion, which radio engineers call beam tilt or squint. It causes a shift of the beam maximum to an off-axis angle without disturbing the beam shape. The second term is the coma effect, also linear in d but proportional to r3. It causes a beam shift in the opposite direction by a smaller amount than the first term and moreover introduces an asym metric beam distortion with a strong sidelobe on one side, the coma lobe. The third term is the field curvature, quadratic in r and independent of c It is reminiscent of an axial defocus, but the term's influence is not identical to that, because it is of second order in d and inversely proportional to f3 instead of f2(see below). The fourth term is called astigmatism, characterised by features in the beam which are four-fold over the aperture. Astigmatism is sometimes an important aberration and we shall give it some attention later. Finally, the last term is independent of the integration variable and can be dropped. For a full description of these aberrations, together with pictorial illustrations, we refer to the standard work by Born and Wolf (1980, Ch. 5 and 9).

It is useful to return shortly to the field curvature term. It essentially reflects the fact that the surface in the focal region on which the image is most sharp (to borrow a term from optics) is not flat but curved. This surface is known as the Petzval surface, who derived the radius of this surface (see Born & Wolf, 1980, Ch. 5.5.3). Thus the optimum location of a laterally displaced feed will involve a correction to the axial position. Ruze (1965) gives the following corrective formula for the axial feed displacement 8a needed to place a laterally displaced feed (by dl) on the Petzval surface:

Thus the optimum locus of an off-axis feed lies on a paraboloid with a "focal length" half of that of the main reflector and the off-axis feed must be moved slightly away from the paraboloid's vertex.

Returning to the discussion of pathlength error, we note that the pathlength error of the central ray is p' - p = d2 /(p' + p) ° d2 / 2 p , which is equal to the second term in Eq. (2.21). Thus the phase error over the aperture, being proportional to the difference of the pathlength p'(r) and p'(0), is normally approximated by the first term of Eq. (2.21) only. From the geometry relations we obtain siny= f = 2f (1 - cos y) = f (l+d^) = 7 (T+wW).

As we assume that (d / f) << 1, the terms in d2 and higher can safely be neglected and we are left with a practical formula for the pathlength error due to lateral defocus of the form

This function is illustrated in Fig. 2.4 [Mat.2.2] for the geometry of the earlier example (Table 2.1). We see that the error steadily increases from the center to reach a value almost as much as the lateral defocus at the edge of the reflector. It is easy to see that for a deep reflector with f / d = 0.25, where the focus lies in the aperture plane, the pathlength difference towards the edge is just equal to the defocus.

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