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.

  

Figure 1.10: Explanation of random errors and their correlation length (L), for Gaussian hat-like deformations

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 $\leq$ 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.(5)), and

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

where

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

In these equations is D 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].

  

Figure 1.11: Beam pattern measured on the IRAM 30-m telescope. The beam consists of the diffraction beam (also called main beam) and a combined, extended error beam (solid line).