1.3 The perfect Single-Dish antenna

We now place an optical instrument (a mirror, lens, telescope etc.) in the beam between $\cal A$ and $\cal I$ 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 ($\vec r$) of the aperture plane ($\cal A$) and the points ($\vec u$) of the image plane ($\cal I$) by the phase transfer function $\Omega_{{\rm O}}({\vec r},{\vec u})$, so that the wavefront converges in the focal point. The receiver(R)/detector introduces an additional modulation of the amplitude $\Omega_{\rm R}({\vec r},{\vec u})$, as described below. Using this information, the field distribution in the focal plane ($\cal I$) of the telescope becomes

$${\rm E}({\vec u}) = [{\rm exp(ikR)/s}_0]{\int}_{{\cal A}}{\rm... ...O} {\Omega}_{\rm R}{\rm exp[ik(g(x,y,R) - (ux + vy)/R)] dx dy}$$ (1.6)

The phase modulation of the parabolic reflector used in a radio telescope is, fortunately,

$${\Omega}_{\rm O} = {\rm exp[-ik\,g(x,y,F)]}$$ (1.7)

(where F is the focal length of the reflector), which inserted into Eq.1.6 eliminates this term in the exponent so that

$${\rm E}({\vec u}) = {\rm [exp(ikF)/F]}{\int}_{{\cal A}}{\rm A... ...uiv}\ \ {\cal FT}[{\rm A}({\vec r}){\Omega}_{\rm R}({\vec r})]$$ (1.8)

This equation says that the field distribution E($\vec u$) in the focal plane of the telescope is the Fourier transform ($\cal FT$) of the receiver-weighted field distribution A($\vec r$) $\Omega_{\rm R}(\vec r)$ in the aperture plane. Since E($\vec u$)E $^*(\vec u) \neq \delta(\vec u - \vec u_o)$ 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 $\Theta_{\rm b} \propto \lambda$/D, with D the diameter of the reflector.

  

Figure: Phase modulation of a reflector. The plane wavefront (W) propagates to the left and the ray reflected at P(x,y) toward the focus F is shifted in phase by the amount $\Delta$.

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

$$({\rm F} - {\Delta})^{2} +({\rm x}^{2} + {\rm y}^{2}) = {\rm F}^{2}$$ (1.9)

which for small $\Delta$ becomes

$${\Delta} = - ({\rm x}^{2} + {\rm y}^{2})/{\rm F}\ {\equiv}\ - {\rm g(x,y,F)}\ {\equiv}\ {\Omega}_{\rm O}({\vec r})$$ (1.10)

which is the instrumental phase modulation function $\Omega_{\rm O}$ used above. The proof is given for a simple parabolic reflector; however, a combined telescope with main reflector and subreflector can be treated in a similar way, leading to the same result.

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( $\vec r) \equiv 1$ and $\Omega_{\rm R} \equiv 1$, then the power pattern P($\vec u$) (beam pattern) in the focal plane of the single antenna is

$${\rm P}({\vec u}) = {\rm E}({\vec u}){\rm E}^{*}({\vec u}) = ... ...xdy)}_{2}\ \ {\propto}\ \ [{\rm J}_{1}({\vec u})/{\rm u}]^{2}$$ (1.11)

where J₁ is the Bessel function of first order (see Born and Wolf, 1980). The function [J₁(u)/u]² is called Airy function, or Airy pattern. The interferometer does not simulate a continuous surface, but consists of individual aperture sections $\cal A_{1}$, $\cal A_{2}$, .... of the individual telescopes, so that its power pattern P $_{\Sigma}(\vec u$) (beam pattern) in the focal plane is

$${\rm P}_{\Sigma}({\vec u}) = {\sum}_{\rm n}{\sum}_{\rm m}{\in... ...\rm (dxdy)}_{1}{\rm (dxdy)}_{2}\ \ {\neq}\ \ {\rm P}({\vec u})$$ (1.12)

The important result of this equation is the fact that the image obtained with the interferometer is ``incomplete'', though certainly not as blurred as seen with a single telescope ( $\Theta_{\rm D} \propto \lambda$/D), but having the superior resolution of the spatial dimension (approximately the longest baseline B) of the array ( $\Theta_{\rm B} \propto \lambda$/B). For the Plateau de Bure interferometer B/D $\approx$ 300m/15m $\approx$ 20 so that $\Theta_{\rm B} \approx 1/20\ \Theta_{\rm D}$. The incompleteness sometimes requires (in particular for mm-VLBI observations which are very similar) additional information for a full image reconstruction, for instance that the object consists of several point-like sources, or a point-like source and a surrounding halo, etc. (see for instance the number of components in CLEAN).

  

Figure 1.3: The figure shows a Cassegrain telescope and the illumination of the incident plane wavefront W.

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 $\Omega_{\rm R}(\vec r)$. 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

$$\Omega_{\mathrm{R}}(\rho) = \mathrm{K} + [1 - {\rho}^2]^p$$ (1.13)

with $\rho$ the normalized radius of the circular aperture, and K and p being constants. For A $(\vec r) \equiv 1$ (i.e. an incident wavefront without structure) the diffraction integral is

$${\rm E}_{\rm T}({\vec u}) = {\int}_{\cal A}{\Omega}_{\rm R}({... ... u}){\rm E}^{*}({\vec u})\ {\equiv}\ {\rm A}_{\rm T}({\vec u})$$ (1.14)

E$_{\rm T}$ is the tapered field distribution in the focal plane, and A$_{\rm T}$ the tapered beam pattern.

  

Figure 1.4: The figure shows the calculated beam pattern of the IRAM 15-m telescope at 3 mm without taper, and for a -10 dB edge taper. The horizontal scale is in arcseconds, the vertical scale is normalized power.

Figure 1.4 shows as example a two-dimensional cut through the calculated beam pattern A$_{\rm T}$ of the IRAM 15-m telescope at $\lambda$ = 3mm, once without taper (i.e. for $\Omega_{\rm R}(\vec x) \equiv 1$), 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: $\Theta_{\rm b}$), the position of the first null ( $\Theta_{\rm fb}$), 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 (from [Minnet and Thomas 1968]). The influence of several taper forms is given in Table 1.1 (from [Christiansen and Hogbom 1969]).

  

Figure: Illustration of the tapered beam pattern for telescopes using parabolic main reflectors of different steepness, expressed by the opening angle $\theta _{o}$. IRAM 30-m and 15-m telescopes: $\theta _{o} \approx 63^{\circ }$, optical telescopes: $\theta _{o} \approx 5^{\circ }$.

 

Table 1.1: Beamwidths, side lobe levels, and maximum aperture efficiency for various parameters of the tapering function.

$\Omega_\mathrm{R}(x,y) = K + (1-\rho^2)^p$
$E({\vec u}) = \int_A \Omega_\mathrm{R}(x,y) \exp (-ik{\vec u}{\vec x}) dS$
		Beamwidth	First null	First	Maximum
		$\Theta_b$	$\Theta_{fb}$	Sidelobe	Aperture
p	K	(radian)	(radian)	(dB)	Efficiency
0	0	1.02 $\lambda/D$	1.22 $\lambda/D$	17.6	1.00
1	0	1.27 $\lambda/D$	1.62 $\lambda/D$	24.7	0.75
2	0	1.47 $\lambda/D$	2.03 $\lambda/D$	30.7	0.55
1	0.25	1.17 $\lambda/D$	1.49 $\lambda/D$	23.7	0.87
2	0.25	1.23 $\lambda/D$	1.68 $\lambda/D$	32.3	0.81
1	0.5	1.13 $\lambda/D$	1.33 $\lambda/D$	22.0	0.92
2	0.5	1.16 $\lambda/D$	1.51 $\lambda/D$	26.5	0.88

 

The complete telescope, i.e. the optics combined with the receiver, has a beam pattern A $_{\rm T}(\vec u)$ (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$_{\rm S}$ as function of wavelength. The telescope thus provides information of the form

$$I({\vec u})\ {\propto}\ {\int}_{\rm Source}{\rm A}_{\rm T}({\vec u} - {\vec u}'){\rm B}_{\rm S}({\vec u}'){\rm d}{\vec u}'$$ (1.15)

If the telescope is perfect, and we know A$_{\rm T}$, we can use the information I($\vec u$) to derive the calibrated brightness distribution B$_{\rm S}$ of the source. distribution.

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 ( $\Delta\Theta$) of the telescope should be small in comparison to the beam width. The loss in gain is small, and acceptable, if the mispointing $\Delta\Theta < 1/10\,\Theta_{\rm b}$. Since modern radio telescopes use an alt-azimuth mount, this criterion says the mispointing in azimuth ( $\Delta\Theta_{\rm az}$) and elevation ( $\Delta\Theta_{\rm el}$) direction should not exceed 1/$\sqrt{2}$ 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.

  

Figure 1.6: Protocol of pointing corrections applied in azimuth and elevation direction, and focus corrections; shown for 5 antennas during an observation which lasted 6 hours.