We now place an optical instrument (a mirror, lens, telescope etc.) in the beam between and with the intention, for instance, to form an image of a star. Optical instruments are invented and developed already since several centuries; however, the physical-optics (diffraction) understanding of the image formation started only a good 200 years ago. Thus, speaking in mathematical terms, the telescope (T) manipulates the phases (not so much the amplitudes) between the points () of the aperture plane () and the points () of the image plane () by the phase transfer function , so that the wavefront converges in the focal point. The receiver(R)/detector introduces an additional modulation of the amplitude , as described below. Using this information, the field distribution in the focal plane () of the telescope becomes
This equation says that the field distribution E() in the focal plane of the telescope is the Fourier transform () of the receiver-weighted field distribution A() in the aperture plane. Since E()E for a realistic optical instrument/telescope with limited aperture size, we arrive at the well known empirical fact that the image of a point-like object is not point-like; or; with other words, the image of a star is always blurred by the beam width of the antenna , with the diameter of the reflector.
To close the argumentation, we need to show that the telescope manipulates the incident wave in the way given by Eq.1.7). To demonstrate this property in an easy way, we consider in Figure 1.2 the paraxial rays of a parabolic reflector of focal length F. From geometrical arguments we have
(1.9) |
(1.10) |
The fundamental Eq.1.8 can be used to show that an interferometer is not a single dish antenna, even though one tries with many individual telescopes and many telescope positions (baselines) to simulate as good as possible the aperture of a large reflector. If we assume for the single dish antennas that A( and , then the power pattern P() (beam pattern) in the focal plane of the single antenna is
(1.11) |
where J is the Bessel function of first order (see [Born & Wolf 1975]). The function [J(u)/u] is called Airy function, or Airy pattern. The interferometer does not simulate a continuous surface, but consists of individual aperture sections , , .... of the individual telescopes, so that its power pattern P ) (beam pattern) in the focal plane is
(1.12) |
The single telescope selects a part of the incident plane wavefront and 'bends' this plane into a spherical wave which converges toward the focus. This spherical wavefront enters the receiver where it is mixed, down-converted in frequency, amplified, detected, or correlated. The horn-lens combination of the receiver modifies the amplitude of the spherical wavefront in a way expressed by the function . This function, called taper or illumination function of the horn-lens combination, weighs the wavefront across the aperture, usually in a radial symmetric way. Figure 1.3 shows, schematically, the effect of a parabolic taper as often applied on radio telescopes, and expressed as
Figure 1.4 shows as example a two-dimensional cut through the calculated beam pattern A of the IRAM 15-m telescope at = 3mm, once without taper (i.e. for ), and for a -10 dB edge taper, i.e. when the weighting of the wavefront at the edge of the aperture is 1/10 of that at the center (see Figure 1.3). As seen from the figure, the taper preserves the global structure of the non-tapered beam pattern, i.e. the main beam and side lobes, but changes the width of the beam (BW: ), the position of the first null ( ), and the level of the side lobes. The effect of the taper depends on the steepness of the main reflector used in the telescope, as shown in Figure 1.5. The influence of several taper forms is given in Table 1.1 [Christiansen and Hogbom 1969].
|
The complete telescope, i.e. the optics combined with the receiver, has a beam pattern A (in optics called point-spread-function) with which we observe point-like or extended objects in the sky with the intention to know their position, structural detail, and brightness distribution B as function of wavelength. The telescope thus provides information of the form
(1.15) |
When we point the antenna toward the sky, in essence we point the beam in the direction of observation. If, for instance, we observe a point-like source it is evident that the peak of the main beam should point exactly on the source which requires that the pointing errors ( ) of the telescope should be small in comparison to the beam width. The loss in gain is small, and acceptable, if the mispointing . Since modern radio telescopes use an alt-azimuth mount, this criterion says the mispointing in azimuth ( ) and elevation ( ) direction should not exceed 1/ this value. The pointing and focus (see below) of the IRAM antennas are regularly checked during an observation, and corrected if required. The corresponding protocol of an observing session at Plateau de Bure, using 5 antennas, is shown in Figure 1.6.