Antenna beams

Meaning of a "beam"

The concept of an antenna beam is intrinsic to all astronomy. The beam is simply the portion of the sky observed by the detector at a given time (Fig. 5). For example, in a non-focusing detection system, mechanical collimators might restrict the field-of-view to a circular region on the sky of 0.7° radius. The detector would be said to have a 0.7° beam (half width) or 1.4° (full width) that views ~^(0.7)2 = 1.5 deg2 of the sky.

A parabolic radio antenna is a classic example of a focusing system. If this antenna were broadcasting (rather than receiving), the power would be emitted more or less into a cone of angles, the antenna beam, with the power per unit solid angle at a maximum on the view axis and falling off at increasing angles from it. The power would not be emitted in a perfectly parallel beam (i.e., to a point at infinity). This is due to the phenomenon of diffraction that arises from the limited diameter of the antenna.

The same antenna in the receiving mode receives radiation from this same cone of angles; any celestial source within it would be detected, with efficiency (sensitivity) depending on the source location relative to the view axis. If several point-like sources lie in this region, they would be confused or "unresolved" (Fig. 5a). The

Antenna Beam

Two point sources, . unresolved X by beam \

Antenna Beam

Two point sources, . unresolved X by beam \

Radio antenna bandwidth d^

Celestial sphere

Radio antenna bandwidth d^

Celestial sphere

Diffuse or extended source on sky

Diffuse or extended source on sky

Side lobes (Power leaks into antenna)
Lighttools Sphere Cone Beam

Figure 5.5. Antenna observing (a) two adjacent point sources that are not separated (resolved) by the beam and (b) a diffuse source that has greater angular extent than the beam. The beam includes side lobes wherein small amounts of power from unwanted directions can enter the receiver. (c) Power diagram; the beam is drawn so that the radial distance from the origin (antenna) to the solid line represents the effective sensitivity or area of the telescope in that direction. (d) Power received by the telescope as a function of the angular displacement of the source 0' from the telescope center line. The full width between the half-power points is the full width at half maximum (FWHM) beam size, or equivalently, the half power beam width (HPBW).

Figure 5.5. Antenna observing (a) two adjacent point sources that are not separated (resolved) by the beam and (b) a diffuse source that has greater angular extent than the beam. The beam includes side lobes wherein small amounts of power from unwanted directions can enter the receiver. (c) Power diagram; the beam is drawn so that the radial distance from the origin (antenna) to the solid line represents the effective sensitivity or area of the telescope in that direction. (d) Power received by the telescope as a function of the angular displacement of the source 0' from the telescope center line. The full width between the half-power points is the full width at half maximum (FWHM) beam size, or equivalently, the half power beam width (HPBW).

angular resolution of the telescope system is comparable to the angular size of the beam.

Each small portion (resolution element) of the film or CCD in a camera can be thought of as a detector that views, say, a 1" x 1" portion of the sky. Adjacent elements view adjacent portions of the sky. Thus a camera or focusing telescope is in effect a multiple-beam instrument. Such an imaging system is able to record the signal coming from different sky positions simultaneously, whereas a single-beam system must study adjacent portions of the sky sequentially. Examples of single-beam systems are the parabolic radio antenna with a single detector at the focus and an optical telescope with a single small hole in the focal plane. In the latter case, all the light from the star of interest passes through the hole and the photon number may be measured with an electronic device called a photomultiplier tube which we describe in Section 6.2; see Fig. 6.1.

Point spread function

When abeam has a small angular size, closely spaced sources can be better resolved. Even if only one source is in the region, a narrow beam gives less contamination from background radiation from directions adjacent to the source. On the other hand, a broader beam is more efficient if one is searching a large portion of sky for sources or if one is studying diffuse radiation from the sky. However, it might confuse or wash-out bright spots in the emission pattern.

At the focus of a telescope, the beam size is the portion of sky that one single pixel in the focal plane can "see". Blurring by the telescope, the atmosphere, or by diffraction means that a faint point-like source will appear as a blurred (enlarged) image in the focal plane. Pixels in this region thus detect photons from the source even though they are not at the exact image position of the point source. Each pixel thus sees a larger portion of the sky than it would in the absence of blurring; the beam of each pixel, and hence of the telescope, is thus increased by any blurring (defocusing). In an optical telescope at ground level, this blurring may be primarily due to variable refraction in the atmosphere. (Diffraction is more important only for the smallest telescopes; see below.) Typically atmospheric blurring of 1" means the telescope beam size is effectively 1".

The analysis of images of the sky requires knowledge of the response of the telescope to a point source, in particular the x , y distribution of deposited energy in the image plane. This is a two-dimensional function f (x,y), the point spread function (psf). This multi-pixel image maps the single-pixel beam shape.

A typical psf will be peaked in the center with the deposited energy falling off with distance. For modestly bright objects recorded on photographic film, the film saturates; it reaches maximum blackness at the center of the image. But light scattering on the film surface enlarges the exposed region. Thus on photographs of the sky, the brighter stars appear bigger, not brighter. The psf in this case is flat and broad; the effective beam is larger for the brighter sources.

The number of photons collected (in a given time) is a strong function of the angular position of the source relative to the center of the beam as noted above. The effective area of a telescope is different for different parts of the antenna beam; an object directly on the axis of a radio telescope will deliver more of its energy to the focus than will an off-center one. Sometimes, a source far off to the side can be weakly detected if the beam has undesirable side lobes (Figs. 5b,c). A strong source in a side lobe will be indistinguishable from a weak source in the main beam.

An antenna beam can be drawn as shown in Fig. 5c. The radial distance from the origin (antenna) to the lobe boundary at a chosen angle is proportional to the effective area of the antenna for radiation arriving from that angle. Typically, the greatest efficiency is in the forward direction.

The quoted angular width of the beam depends on how the edge is defined. A definition often used is the full width half maximum (FWHM; Fig. 5d). If 01/2/ is the angle from beam center at which the power is reduced to 1 /2 its maximum value, the FWHM angle is 201/2/. Radio astronomers refer to this as the half-power beam width response (HPBW). Although the power received at Q1/2 is 1/2 the power in the center of the beam, the total power enclosed within this angle can be substantially greater than (or less than) 50% of the total power over all angles.

A good telescope beam will have a highly peaked response function which will include >90% of the received power within the FWHM limits, but a poor beam can have a response with large "wings" that result in only a small portion of the power falling within the FWHM angles. Another useful definition of beam width is the (half or full) angle that encloses 90% of the power. A beam with large wings would have a large value of this angle and vice versa.

Diffraction

One reason a telescope beam may not be as narrow as one might wish is an interference phenomenon known as diffraction. If a parallel beam from an infinitely distant source is incident upon an antenna, there is interference between the different parts of the incoming wavefront, called wavelets in the Huygen method of describing wave propagation. If the telescope has a limited diameter (i.e., it is not infinitely large), the interference will produce a blurred (enlarged) image of size that depends upon the diameter of the telescope.

Fraunhofer diffraction

A formal derivation of diffraction sums the effect of wavelets originating at each imaginary segment of an aperture such as that shown in Fig. 6a. The aperture could be the aperture of the primary mirror of a telescope. The wavelets are in phase at the aperture if they originate in a plane wave. They interfere with one another as they propagate downward from the aperture. The resultant propagation directions deviate from the vertical because the blocked parts of the original wavefront are missing.

Fraunhofer diffraction is the special case of diffraction where the distance to the image plane is large compared to the diameter d of the aperture. A segment of a plane wave of light leaving the aperture at a given angle 9 (Fig. 6b) thus will illuminate one part of the distant image plane, and the light leaving at a different angle will illuminate another portion. In this manner, two stars will appear as two spots on the distant focal plane. Alternatively, one could insert a thin lens just below

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