2 Terminology and theory

2.1 The structure of the beam pattern

The directional response, i.e. the beam $\cal F$ = $\cal F$$_{\rm c}$ + $\cal F$$_{\rm e}$ of a single dish radio telescope consists of the diffracted beam $\cal F$$_{\rm c}$, formed by coherently focused radiation, and the error beam(s) $\cal F$$_{\rm e}$, formed by radiation "scattered" towards the focal plane. The diffracted beam $\cal F$$_{\rm c}$ = $\cal F$$_{\rm mb}$ + $\cal F$$_{\rm sl}$ consists of the main beam $\cal F$$_{\rm mb}$ and sidelobes $\cal F$$_{\rm sl}$. The diffracted beam of a perfect, circular reflector of diameter D (with vertex hole, shadowing from the subreflector and its support, illumination taper of the receiver etc.) is a tapered Airy-type pattern $\cal A$$_{\rm T}$ with components $\cal A$$_{\rm T}$ = $\cal A$$_{\rm T,mb}$ + $\cal A$$_{\rm T,sl}$. For a shallow, perfect, full aperture, non-tapered reflector the diffracted beam is ${\cal A}(u) = [J_{1}(u)/u]^{2}$ with J₁ the Bessel function of first order and u the spatial coordinate of the focal plane (Born $\&$ Wolf 1980). The steepness of a radio reflector and the illumination taper preserves to a large extent the sidelobe structure of the Airy pattern (Minnett $\&$ Thomas 1968; Goldsmith 1987; see Sect. 5), however, the sidelobe levels are made by purpose significantly lower than those of the non-tapered beam. The width (FWHP) of the main beam ${\cal A}_{\rm T,mb}$is

$${\theta}_{\rm b} = { k} {\lambda}/{D}\ \ [{\rm rad}]$$ (1)

with k (1 1.4) a factor dependent on the illumination taper and blockage of the aperture (Christiansen $\&$ Högbom 1985; Kraus 1986). Measurements with the 30-m telescope and current SIS receivers of $\sim -13$ dB Gaussian edge taper give k = 1.16 for 0.8 mm $\lambda$ 3 mm (Kramer 1997).

We assume in the following that the beam is degraded by phase perturbations of the wavefront $\delta$$_{\varphi}$ $\approx$2(2$\pi$/$\lambda$)$\delta$ which are primarily due to deformations $\delta$ of the main reflector surface. For a good quality telescope we may assume also that the phase perturbations are small compared to the wavelength so that the resulting beam degradation is the sum of the individual degradations (see Shifrin 1971; Sect. 2.4). In addition we assume that the main reflector surface is constructed from a large number of panels. Large-scale surface deformations, which do not change significantly over several panel areas or a considerable fraction of the reflector surface, degrade the diffraction pattern but preserve, in general, the main beam and sidelobe structure. Small-scale wavefront deformations, which change significantly over single panel areas or panel sub-sections, produce the underlying error beams. Surface deformations which change over distances of wavelengths behave like rough surfaces, and are discussed in optical journals.

Different mathematical formalisms are used to calculate the beam degradation from spatially large-scale and small-scale wavefront deformations.

2.2 Large-scale deformations

Large-scale deformations, which often appear as systematic deformations, are described by combinations of low order Zernike polynomials Z _ij ( 10, 10)

$${\delta} = {\sum}_{ i,j}a_{ ij}{ Z}_{ ij} = {\sum}_{ i,j}{ a}_{ ij}{ R}_{ i}({\rho}){\cos} (j{\psi})$$ (2)

(with R _i functions of the normalized aperture radius $\rho$, $\psi$ the azimuth angle of the aperture, and a _ij the amplitude of component (ij)) (Born $\&$ Wolf 1980; Greve et al. 1996b), or other orthogonal functions for decomposition of wavefront deformations (Smith $\&$Bastian 1997). The corresponding, degraded, tapered beam pattern $\cal Z$$_{\rm c,T}$ has a main beam and sidelobes, and can be calculated exactly from diffraction theory. Well known examples are defocus ($\cal Z$₁₀, $\cal Z$₂₀), coma ($\cal Z$$_{\rm 31}$), and astigmatism ($\cal Z$$_{\rm 22}$). These deformations are sometimes due to misaligned components, for instance a misaligned subreflector or a misaligned receiver.

2.3 Small-scale deformations

Small-scale random deformations of the reflector surface are characterized by the (root mean square) rms-value $\sigma$ and the spatial correlation function $C_{\delta}(d)$ (0 $\leq d \leq D$). For a reflector surface sampled at a large number of positions (n = 1, 2, ..., K; with K a few hundred), the rms-value is

$${\sigma}^{2} = {\sum}_{ n = 1,K}({\delta}_{ n})^{2}/{ K},\ \ {\sum}{\delta}_{ n} = 0$$ (3)

for deformations $\delta_{ n}$ measured in the normal direction of the best-fit reflector surface. The quantity $\sigma$ is the geometrical rms-value of the surface errors (Greve $\&$ Hooghoudt 1981). The rms-value $\sigma$$_{\varphi}$ of the associated wavefront deformation (including a factor 2 because of reflection) is

$${\sigma}_{\varphi} = { R}\ (4{\pi}/{\lambda}){\sigma} = { 2\... ...{\sigma} = {2\,k\,}{\sigma}_{\rm T},\ \ { k} = 2{\pi}/{\lambda}$$ (4)

with the reduction factor R = 0.8- 0.9 for a steep radio reflector of focal ratio $N = F/D \approx$ 0.3 and a -10 dB to -15 dB edge taper of the illumination. $\sigma$$_{\rm T}$ = R$\sigma$ is the tapered, or illumination weighted, rms-value of the surface errors (Greve $\&$ Hooghoudt 1981) as derived, for instance, from aperture efficiency and holography measurements. The normalized correlation function $C_{\delta}(d)$ with $C_{\delta}(0) = 1$ is

$${ C}_{\delta}({ d})\ = \left[{\sum}_{(1,2)}{\delta}({\bf r}... ..._{\rm 1,2}\right]/{\sigma}^{2} = { C}_{\delta}^{+}/{\sigma}^{2}$$ (5)

with the summation (1,2) extending over the number $N_{\rm 1,2}$ of pairs (${\bf r}$₁, ${\bf r}$₂) $[{\bf r} = (x,y)]$ of positions in the aperture (A) with separation

$${\mid} {\bf r}_{1} - {\bf r}_{2}{\mid}\ =\ { d}\ {\leq}\ { D}.$$ (6)

The correlation function of the corresponding wavefront deformations [$\delta$$_{\varphi}$] is

$${ C}_{\varphi}({ d}) = \left[{\sum}_{(1,2)}{\delta}_{\varph... ...igma}_{\varphi}^{2} = { C}_{\varphi}^{+}/{\sigma}_{\varphi}^{2}$$ (7)

with $C_{\varphi}$ $\approx$ $C_{\delta}$. The degraded beam pattern at the position ${\bf u} = (u,v)$ of the focal plane is (Scheffler 1962)

$${\cal F}({\theta})\ {\equiv}\ {\cal F}({\bf u})\ {\propto}\$$

$${\rm exp}[-({\sigma}_{\varphi})^{2}]{\int}{\int}_{A}{\rm d}S... ..._{\varphi}^{+}( d)]{\exp}[ik{\bf u}({\bf r}_{1}-{\bf r}_{2})/f]$$ (8)

with d$S_{\rm i}$ = (dxdy)_i a surface element of the aperture (A). The angular distance $\theta$ from the focal axis of the position ${\bf u} = (u,v)$ is $\theta$ = $\sqrt{{ u}^{2} + { v}^{2}}/f$ [rad], with f the effective focal length of the telescope. In order to obtain an analytic expression of Eq. (8), it has been assumed (Scheffler 1962; Ruze 1966; Shifrin 1971; see the criticism/support by Schwesinger 1972; Greve et al. 1994a) that

$${ C}_{\delta}({ d})\ {\approx}\ { C}_{\varphi}({ d})\ {\propto}\ {\exp}[ -({d/L})^{2}]$$ (9)

with L the correlation length of the deformations [$\delta$]. Under the assumption of a Gaussian correlation length distribution (Eq. (9)), the degraded beam pattern $\cal F$($\theta$) is circular symmetric and

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

$$= {\rm exp}[-({\sigma}_{\varphi})^{2}]{\cal A}_{\rm T}({\the... ... + { a}_{\rm e}\ {\rm exp}[-({\pi}{\theta}{ L}/2{\lambda})^{2}]$$ (10)

with

$${ a}_{\rm e} = ({ L/D})^{2}[1 - {\rm exp}[-({\sigma}_{\varphi})^{2}]]/{\epsilon}_{0}$$ (11)

for $\sigma$$_{\varphi}$ 1 (Scheffler 1962; Ruze 1966; Vu 1969; Baars 1973) as existing on a good quality telescope. $\epsilon$₀ is the aperture efficiency at long wavelengths. The normalization is $\cal F$($\theta$ = 0) = 1 for $\sigma$ = 0 (which implies L = 0). The width (FWHP) of the error beam is

$${\theta}_{\rm e} = 0.53{\lambda}/({ L}/2)\ \ [{\rm rad}].$$ (12)

This tolerance theory of a single small-scale error distribution [$\delta$] with correlation length L, as discussed in the basic publication of Scheffler (1962), Ruze (1966), and Robieux (1966), is verified by a large number of investigations of optical telescopes and radio telescopes (see in particular Ruze 1966; Vu 1970a).

2.4 The combination of small-scale and large-scale surface deformations

It is reasonable to assume that a paneled reflector surface may have several independent error distributions [$\delta$ _i] with different rms-values [$\sigma$ _i] and different correlation lengths [L _i]. The surface of the IRAM 30-m reflector contains independent large-scale and small-scale errors of which the characteristic correlation lengths are anticipated from the mechanical construction.

The surface of the 30-m reflector consists of 7 rings of panel frames (in total 210) with each frame holding two panels. A panel (average size $\sim$ $1 \times 2$ meter) is attached to its frame by 15 screws, arranged in 5 parallel rows with approximately 1/4 $\times$ 2 m $\approx$ 0.5 m spacing. These support screws were used to adjust the panel contours to an average precision ${\sigma}_{\rm p}$ $\approx$ 0.03 mm, as measured in the factory (Baars et al. 1987). From the geometry of the panel support and contour maps of the panel surfaces (Fig. 1) as measured in the factory, we anticipate that the residual deformations of the adjustments have a correlation length $L_{\rm p}$ of approximately 1/4 length of a panel, so that $L_{\rm p} = 0.3 - 0.5$ m and $L_{\rm p}/D$ $\approx$ 1/75. The width of the anticipated error beam is $\theta$$_{\rm e,p}$ $\approx$ 75$\theta$$_{\rm b}$ (Eqs. (1, 12)).

A panel frame (average size $\sim$ $2 \times 2$ meter) is attached to the backstructure by adjustment screws located at the four frame corners. A panel frame which is misaligned in piston and/or tilt represents a surface area of correlated deformations. The weighted distance between the centers of adjacent panel frames gives the correlation length $L_{\rm a} = 1.5 - 2.0$ m so that $L_{\rm a}/D \approx$ 1/17. The width of the anticipated error beam is $\theta$$_{\rm e,a}$ $\approx$ 17 $\theta$$_{\rm b}$(Eqs. (1, 12)).

  

Figure 1: Surface contours as measured in the factory of five randomly selected panels. The dots show the positions of the adjustment screws; the circles illustrate the size of a correlation cell of $L_{\rm p} = 0.3 - 0.5$ m diameter as the influence area of an adjustment screw. Contour levels at 0.015 mm

For two independent small-scale surface error distributions [$\delta$₁] and [$\delta$₂], with Gaussian correlation length distributions L₁ and L₂, the effective surface rms-value $\sigma$ is (see Eq. (3))

$${\sigma}^{2} = {\sum}({\delta}_{ 1,n} + {\delta}_{ 2,n})^{2}/{ K}$$

$$= {\sum}({\delta}_{ 1,n})^{2}/{\rm K} + {\sum}({\delta}_{ 2,n})^{2}/ { K} = {\sigma}_{1}^{2} + {\sigma}_{2}^{2}$$ (13)

since ${\sum}{\delta}_{ 1,n}{\delta}_{ 2,n}$ = 0. The correlation function of the combined error distribution [$\delta$] = [$\delta$₁] + [$\delta$₂] is

$${ C}_{\delta}({d}) = [{ C}_{\delta,1}^{+}({ d}) + { C}_{\delta,2}^{+}({ d})]/[{\sigma}_{1}^{2} + {\sigma}_{2}^{2}]$$

$${\propto}\ {\sum}_{ i=1,2}{\exp[-(d/L}_{ i})^{2}] /{\sum}{\sigma}_{ i}^{2}.$$ (14)

When using the correlation function of the corresponding wavefront deformation (see Eq. (7)) in Eq. (8) under the assumption that the phase rms-values $\sigma$$_{\varphi,i}$ are small so that the integral can be split into the sums of the individual contributions, i.e.

$${\cal F}({\theta})\! =\! {\exp}[-\!{\sum_{i}}({\sigma}_{\var... ...m d}S_{2}{\exp}[{\sum_{i}}{ C}_{\varphi,i}^{+} ( d)]{\exp}[...]$$

$${\approx}\ \ {\sum}_{i}{\exp}[-({\sigma}_{\varphi,i})^{2}]{\... ...m d}S_{1}{\rm d}S_{2}{\exp}[{C}_{\varphi,i}^{+}( d)]{\exp}[...]$$ (15)

then the beam pattern degraded by several independent small-scale deformations [$\delta$ _i] with Gaussian correlation length distributions [L _i] (Eq. (14)) is

$${\cal F}({\theta}) = {\exp}[-({\sigma}_{\varphi})^{2}]\ {\c... ... i}a_{{\rm e},i}{\exp}[-({\pi}{\theta}{ L}_{i}/2{\lambda})^{2}]$$ (16)

where the amplitude $a_{{\rm e},i} = ({ L}_{ i}/{ D})^{2}[1 - {\exp}[-({\sigma}_{\varphi,i})^{2}]]/{\epsilon}_{0}$ (Eq. (11)) of the error beam [i] is related only to the rms-value $\sigma$ _i and the correlation length L _i of the error distribution [$\delta$ _i]. Equivalent to Eq. (13), in Eq. (16) $\sigma$$_{\varphi}$² = $\sum$($\sigma$$_{\varphi,i}$)². For a rigorous discussion of several small-scale error distributions, the proofs of Eq. (14) and Eq. (16), and the validity of relation (15) see Shifrin (1971, chapter 5).

Following the explanation of Sect. 2.2, additional large-scale deformations cover areas of several panel frames so that their correlation length is, say, $L_{\rm l}$ $\approx$ 5 $L_{\rm a}$. The width of the anticipated error beam is $\theta$$_{\rm e,l}$ $\approx$ 5 $\theta$$_{\rm b}$ (Eqs. (1, 12)). For large-scale deformations the diffraction pattern $\cal A$$_{\rm T}$ of Eq. (16) is replaced by the corresponding low order Zernike polynomial diffraction pattern $\cal Z$$_{\rm c,T}$, for instance a comatic or astigmatic beam.