1.4 The real Single-Dish Antenna

A telescope, however, is never perfect since mechanical, thermal, and wind-induced deformations of the structure occur, and the optics may be misaligned and/or have production imperfections, for one or the other reason. The resulting effect on the beam pattern is negligible if the corresponding wavefront deformations introduced by these imperfections are small compared to the wavelength of observation, generally smaller than $\sim \lambda/15$; the effect is noticeable and disturbing when the wavefront deformations are large compared to the wavelength ( $\sim 1/4 \lambda$ and larger). The wavefront deformations due to such imperfections may be of systematic nature, or of random nature, or both.

1.4.1 Systematic Deformations: Defocus, Coma, Astigmatism

There are three basic systematic surface/wavefront deformations (occasionally associated with pointing errors) with which the observer may be confronted, i.e. defocus, coma, and astigmatism (a transient feature on the IRAM 30-m telescope).

1. The most important systematic wavefront/beam error is due to a defocus of the telescope. This error is easily detected, measured, and corrected from the observation of a strong source at a number of focus settings. Figure 1.7 shows, as example, the beam pattern measured on Jupiter with the telescope being gradually defocused. Evidently, the peak power in the main beam decreases, the power in the side lobes increases, until finally the beam pattern has completely collapsed. To be on the safe side for observations, the defocus of the telescope should not exceed $\sim 1/10 \lambda$. A defocus does not introduce a pointing error.

   $\parbox{40mm}{ \caption{Effect on the b... ...ope (shifts of the subreflector in steps of $\lambda/4$, $\lambda$ = 3 mm).}}$

2. A telescope may have a comatic wavefront/beam error due to a misaligned subreflector, shifted perpendicular off the main reflector axis. Figure 8 shows, as example, a cross scan through a comatic beam of the IRAM 15-m telescope, especially produced by displacement of the subreflector. A comatic beam pattern introduces a pointing error. It may be useful for the observer to recognize this error, in particular if unexplained pointing errors occur in an observations. [The IRAM telescopes are regularly checked for misalignments, and correspondingly corrected.]

   $\parbox{50mm}{\caption{ Illustration of... ...Note the shift of the beam (pointing error) when the subreflector is shifted.}}$

3. A telescope may have an astigmatic wavefront/beam error, usually introduced by complicated mechanical and/or thermal deformations (a transient feature on the IRAM 30-m telescope). While this beam deformation is easily recognized by the observer from the difference in beam widths measured from in-and-out-of-focus cross scans, the improvement of the telescope usually is difficult, and out of reach of the observer. A focused astigmatic beam does not introduce a pointing error. Figure 1.9 shows the focused beam pattern measured on a telescope which has a strong astigmatic main reflector (amplitude of the astigmatism $\sim$0.5mm).

$\parbox{50mm}{ \caption{Illustration of an astigmatic beam pattern; well focused.}}$

The beam deformation of systematic wavefront deformations occurs close to the main beam, and the exact analysis should be based on diffraction calculations. A convenient description of systematic deformations uses Zernike polynomials of order (n,m) [Born & Wolf 1975]. Without going into details, the Zernike-type surface deformation $\delta_{\rm n,m} = \alpha_{\rm n,m} R_{\rm n}(\rho) \cos(m\theta)$ [with ( $\rho,\theta$) normalized coordinates of the aperture, and R special polynomial functions] with amplitude $\alpha_{\rm n,m}$ has a quasi rms-value $\sigma$ = $\alpha_{\rm n,m}/\sqrt{\rm n+1}$ and introduces a loss in main beam intensity of

$${\epsilon}_{\rm sys}/{\epsilon}_{o} {\approx} {\rm exp[-(4{\pi}{\alpha}/{\lambda})^{2}/(n+1)]}$$ (1.16)

For primary coma n = 1, for primary astigmatism n = 2. Although the beam deformation may be very noticeable and severe, the associated loss in main beam intensity may still be low because of the reduction by the factor (n+1).

1.4.2 Random Errors

Besides systematic surface/wavefront deformations explained above (mainly due to misalignment of the optics), there are often permanent random deformations on the optic surfaces like ripples, scratches, dents, twists, misaligned panels, etc., with spatial dimensions ranging from several wavelengths to significant areas of the aperture. These deformations introduce identical deformations of the wavefront, which cannot be expressed in mathematical form (as the Zernike polynomials used above). Nevertheless, the effect on the beam pattern of this type of deformations can be analyzed in a statistical way and from a simple expression, the RUZE equation. This equation is often used to estimate the quality of a telescope, in particular as function of wavelength. The values obtained from this equation are directly related to the aperture efficiency, and beam efficiency, of the telescope, and hence are important for radiometric measurements (see Sect.1.5).

As illustrated in Figure 1.10, there are two parameters which allow a physical-optics description of the influence of random errors, i.e. the rms-value (root mean square value) $\sigma$ of the deformations, and their correlation length L.

$\parbox{50mm}{\caption{ Explanation of ... ... \cite{ruze66}, Copyright: @ 1966 IEEE, reprinted by permission of IEEE, Inc.}}$

Random errors occur primarily on the main reflector; the other optical components of the telescope (subreflector, Nasmyth mirror, lenses, polarizers) are relatively small and can be manufactured with good precision. In order to explain the rms-value $\sigma$, we assume that the reflector aperture is divided into many elements (i = 1,2,...N), and that for each element [i] the deformation $\delta$(i) of the reflector is known with respect to a smooth mean surface. The rms-value of these random surface deformations is

$${\sigma} = {\sqrt{{\sum}_{\rm i=1,N} {\delta}({\rm i})^{2}/{\rm N}}}$$ (1.17)

The surface deformations $\delta$(i) introduce corresponding wavefront deformations $\varphi$(i), approximately two times larger than the mechanical deformations $\delta$ in case we are dealing with reflective optics. The rms-value $\sigma_{\varphi}$ of the corresponding phase deformations of the wavefront is

$${\sigma}_{\varphi} = 2 {\rm k} {\rm R} {\sigma}$$ (1.18)

again with k = 2 $\pi/\lambda$, and R $\approx$ 0.8 a factor which takes into account the steepness of the parabolic main reflector [Greve $\&$ Hooghoudt 1981].

A description of the wavefront deformation by the rms-value $\sigma_{\varphi}$ is incomplete since the value does not contain information on the structure of the deformations, for instance whether they consist of many dents at one part of the aperture, or many scratches at another part. A useful physical-optics description requires also a knowledge of the correlation length L of the deformations. L is a number (L $\leq \cal D$) which quantifies the extent over which the randomness of the deformations does not change. For example, the deformations of a main reflector constructed from many individual panels, which may be misaligned, often has a random error correlation length typical of the panel size, but also a correlation length of 1/3 to 1/5 of the panel size due to inaccuracies in the fabrication of the individual panels. A typical example is the 30-m telescope [Greve et al. 1998].

When knowing, by one or the other method, the rms-value $\sigma_{\varphi}$ and the correlation length L, it is possible to express the resulting beam shape in an analytic form which describes well the real situation. The beam pattern $\cal F(\Theta$) of a wavefront with random deformations ( $\sigma_{\varphi}$,L) [the telescope may actually have several random error distributions] consists of the degraded diffraction beam $\cal F_{\rm c}(\Theta$) and the error beam $\cal F_{\rm e}(\Theta$) such that

$${\cal F}({\Theta}) = {\cal F}_{\rm c}({\Theta}) + {\cal F}_{\rm e}({\Theta})$$ (1.19)

with

$${\cal F}_{\rm c}({\Theta}) = {\rm exp[}-({\sigma}_{\varphi})^{2}]{\rm A}_{\rm T}({\Theta})$$ (1.20)

where A $_{\rm T}(\Theta$) is the tapered beam pattern (Eq.1.14), and

$${\cal F}_{\rm e}({\Theta}) = {\rm a exp[-}({\pi}{\Theta}{\rm L}/{\lambda})^{2}]$$ (1.21)

where

$${\rm a} = ({\rm L/{\cal D}})^{2}[1 - {\rm exp(-}{\sigma}_{\varphi}^{2})]/{\epsilon}_{o}$$ (1.22)

In these equations, $\cal D$ is the diameter of the telescope aperture, $\lambda$ the wavelength of observation, $\Theta$ the angular distance from the beam axis, and $\epsilon _{o}$ the aperture efficiency of the perfect telescope. In the formalism used here the beam is circular symmetric. The error beam $\cal F_{\rm e}(\Theta$) has a Gaussian profile of width FWHP) $\Theta_{\rm e} = 0.53 \lambda$/L [radians], i.e. the smaller the correlation length (the finer the irregular structure), the broader is the beam width $\Theta_{\rm e}$. The random errors of panel surface deformation and panel alignment errors may have large error beams (up to arcminutes in extent) which can pick up radiation from a large area outside the actual source. A knowledge of the structure and of the level of the error beam(s) is therefore important when mapping a source and making absolute power measurements. Figure 1.11 shows the diffraction beam and the combined error patterns measured on the 30-m telescope at various wavelengths. The smaller the wavelength of observation, the smaller is the power received in the main beam and the larger the power received in the error beam. Due to its particular mechanical construction, this telescope has three error beams [Greve et al. 1998].

$\parbox{50mm}{ \caption{ Beam pattern m... ...beam ($\approx$ main beam) and a combined, extended error beam (solid line).}}$