Holographic Measurements of the Effelsberg 100-m Radiotelescope via Phase Retrieval
by
W. Reich and E. Fürst

Kleinheubacher Berichte, 1999


Contents:
- Introduction
- Previous Surface Measurements and Adjustments
- Holograpy via Phase Retrieval
- Sampling and Sensitivity Requirements
- First test and Satellite Properties
- Holographic Measurements in 1999
- Results
- Summary and Outlook
- Aknowlegdements
- References
- Fig 1: Focussed Antenna Pattern
- Fig 2: Defocussed Antenna Pattern
- Fig 3: Surface Current Amplitudes
- Fig 4: Distribution of Surface Errors

ABSTRACT
Holographic measurements of the Effelsberg 100-m radiotelescope have been made using the beacons of the geostationary satellites ITALSAT F1 and F2 as well as EUTELSAT at 18.685 GHz and 11.698 GHz, respectively. From two antenna pattern measurements with a high dynamic range at different focus settings the surface error distribution, which is proportional to the phase distribution of the surface current, are found by the iterative Misell algorithmn. The aim of the measurements is the adjustment of the recently exchanged 692 outer wire mesh panels by perforated aluminium panels to increase the aperture efficiency of the 100-m telescope at high frequencies.

ABSTRACT
Holographische Vermessungen des Effelsberger 100-m Radioteleskops wurden mittels der auf den geostationären Satelliten ITALSAT F1 und F2 sowie EUTELSAT installierten Beacons bei 18.685 GHz bzw. 11.698 GHz durchgeführt. Aus jeweils zwei unterschiedlich fokussierten Antennendiagrammen mit hohem Dynamikbereich wird die den Oberflächenfehlern proportionale Phasenverteilung des Oberflächenstromes mittels des iterativen Misell-Algorithmus ermittelt. Ziel der Vermessung ist die Justage der kürzlich ausgetauschten 692 äusseren Drahtnetzpaneele durch perforierte Aluminiumpaneele zur Steigerung des Flächenwirkungsgrades des 100-m Teleskops bei hohen Frequenzen.

Introduction

According to Ruze,1966 the aperture efficiency $\displaystyle\eta$ of a parabolic dish is given for the case of uniformly distributed surface errors with rms-error $\displaystyle\sigma_{rms}^{}$at a wavelength $\displaystyle\lambda$ by
$\displaystyle\eta$= $\displaystyle\eta_{0}^{}$exp(-4$\displaystyle\pi$$\displaystyle\sigma_{rms}^{}$/$\displaystyle\lambda$)2.
$\eta_{0}^{}$is the maximum aperture efficiency for long wavelengths, where the surface errors can be neglected. For the Effelsberg 100-m telescope $\eta_{0}^{}$ is about 55% at wavelengths longer than 6 cm. Blocking by the four subreflector support legs and the edge taper of the illuminating feeds defines this value. At shorter wavelengths than 6 cm the measured aperture efficiencies decrease. For instance at 9 mm wavelength the aperture efficiency drops to a maximum of about 23%. The Effelsberg 100-m telescope has a homological design, but residual gravitational deformations cause increasing losses with increasing elevation distance from the position of maximum efficiency.

The surface error of a telescope results from the accuracy of the individual panels and the accuracy of their adjustment relative to the ideal surface. The surface of the Effelsberg 100-m telescope consists of 2356 paneels arranged in 17 rings, where the inner 13 rings (up to a radius of 40 m) are aluminium panels with 0.3 mm rms-accuracy. Ring 14 (radius 40 m - 42.5 m) consists of perforated aluminium panels with 0.5 mm accuracy, while the three outer rings consist of wire mesh panels with a filling factor of 20% to 30% and a rms-accuracy, which decreased over the years, to about 3 mm. During the construction phase of the telescope it was assumed that wire mesh panels for the outer rings are required to keep the wind pressure load low. Operating the telescope for meanwhile more than 25 years showed that the wind pressure effect was overestimated. Therefore, it was decided to replace the damaged outer wire mesh panels by perforated aluminium panels with 0.5 mm accuracy and a filling factor of about 65%, similar to the panels of ring 14. The panel exchange has been made in 1997 and 1998. The remaining step is the adjustment of the new panels, which requires the knowledge of their position relative to the ideal surface. For this purpose we have carried out holographic measurements using geostationary satellites.

Previous Surface Measurements and Adjustments

The first adjustment of the surface of the 100-m telescope was based on measuring the distance and angles of about 9000 targets placed at the panel corners from the vertex of the telescope by steel tape and theodolite  (Greve, 1981). This method of adjustment can only be done with the telescope in zenith position. A $\sigma_{rms}^{}$  = 0.5 mm has been measured (excluding the outer wire mesh panels). Unfortunately at lower elevations, where most of the radioastronomical observations are carried out, $\sigma_{rms}^{}$ increases (for instance $\sigma_{rms}^{}$  = 0.8 mm at 20 o elevation) and a reduced aperture efficiency had to be accepted.

The method of holographic measurement using the beacons of geostationary satellites is better suited. The elevation of these satellites is around 32 o at the site of the Effelsberg telescope, which is close to the majority of astronomical observations. Such measurements have first been made at Effelsberg by EIKONTECH LIMITED in 1982 using the OTS-2 beacon at 11.786 GHz (Godwin et al., 1986). The method of these measurements was to continuously record the amplitude and phase of the satellite signal by a 2.4 m reference antenna, while the 100-m telescope was observing a fully sampled rectangular field centered on the satellites position. Feeding the signals from both antennas into a suitable backend amplitude and phase for each pixel of the antenna pattern was obtained. A subsequent Fourier transform of the complex antenna pattern results in the amplitude and phase distribution of the complex reflector current, where the phase components are proportional to the surface error distribution.

The result of the following panel adjustment based on these measurements was a significant improvement of the aperture efficiency at high frequencies and for low elevations.A$\displaystyle\sigma_{rms}^{}$ = 0.4 mm was measured for the inner 80 m of the telescope between 10 oand 45 o elevation, which increases towards higher elevations. A repeated holographic measurement in 1989 by EIKONTECH LIMITED using the 11.451 GHz beacon signal from the satellite ECS-1 showed the success of the adjustment for the panels up to ring 13 (Godwin and Schoessow,1990). However, some significant errors remain for the panels of ring 14, which unfortunately caused large deviations of the initial settings of the outer panels exchanged in 1997 and 1998.

Holography via Phase Retrieval

If the phases of the farfield antenna pattern are not measured along with the amplitude, it is possible to retrieve the phases when two antenna pattern are available, which are observed at different focus settings.

The relation of the focused farfield pattern E and the aperture current F is given by:

F(x,y) = $\int$$\int$E($\theta$,$\varphi$) exp(-2 $\pi$i (x$\theta$+y$\varphi$)) d$\theta$d$\varphi$

commonly noted as:

F(x,y) = $\cal$JE($\theta$,$\varphi$)

A modified relation holds for the defocused case:

F(x,y) = $\cal$JE($\theta$,$\varphi$) exp( -i$\Delta$f (x 2+y 2)/f 2) ,

where $\Delta$f and f are the focal displacement and the focal length, respectively. $\theta$,$\varphi$ are the spherical coordinates of the far field, x,y the coordinates of the aperture plane.

Misell (1973) found an iterative procedure to retrieve the missing phases for the case of electron microscopy, which  Morris (1985) adapted to retrieve the missing phases from two antenna patterns | E1 | and | E2 | with different focal settings.

Briefly, the Misell-algorithmn starts with an assumend current distribution A taking the feed illumination into account and uses a random phase distribution. These are transformed into focal position 1, where the phases are combined with the measured amplitudes | E1 | . A transform into the aperture plane and into focal positon 2 results in an antenna pattern, which is to be compared with the measured amplitudes | E2 | .

[E2($\theta$,$\varphi$)]1  = $\cal$Jexp( i$\Delta$f2 (x 2+y 2)/f 2) $\cal$J - 1[| E1($\theta$,$\varphi$)| arg( $\cal$J [A(x,y) exp ( -i$\Delta$f1 (x 2+y 2)/f 2)])]

where arg F(z) = arctan ImF(z)  / Re F(z)
Replacing the calculated amplitudes [E2($\theta$,$\varphi$)]1 by the measured ones |E2| a new antenna pattern is calulated for the focal position 1:

[E1($\theta$,$\varphi$)]2  = $\cal$Jexp( i$\Delta$f1 (x 2+y 2)/f 2) $\cal$J - 1[| E2($\theta$,$\varphi$)|arg($\cal$J (J - 1[| E2($\theta$,$\varphi$)|1exp( -i$\Delta$f1 (x 2+y 2)/f 2)]))]

These forward and backward transformations are repeated until the calculated amplitudes and the measured amplitudes of the two antenna pattern agree within some limit. Then the phase distribution in the aperture plane is reconstructed.

Holography via Phase Retrieval needs substantial computing power and requires twice the observing time if compared to holographic measurements using a reference antenna. The large observing time may cause problems if temperature or wind pressure changes occur during the measurements. The computing power is less significant due to the today's powerful computers. A linear backend is needed to achive the required high dynamic range of the measurements. In comparison the hardware requirements for holography with a reference antenna are much higher.

The Phase Retrieval method has been successfuly used by Dave Morris and co-workers at the IRAM 30-m telescope, which operates at mm and submm-wavelengths. We gained very much from their experience, when applying this method to measure the surface errors of the Effelsberg 100-m telescope.

Sampling and Sensitivity Requirements

Planning of holographic measurements requires some considerations on the antenna pattern extend and its sensitivity. These requirements have been already discussed in the literature, that we just give the results. The angular resolution of a telescope of diameter D at a wavelength $\lambda$ is defined as its half-power-beam-width (HPBW), which is given by:

HPBW [ o ] = k $\lambda$ / D

k depends on the feed illumination of the telescope and varies for the 100-m telescope between 58 and 70. The minimum sampling interval is half the HPBW. For a fully sampled antenna pattern with NxN pixels a map with MxM pixels of the aperture current distribution is obtained, where M and N depend as:

M = 0.5 N k $\pi$ / 180

This means that for a spatial resolution of 1 m over the surface of the 100-m telescope the size of the fully sampled antenna pattern needs to be as large as 176 x 176 pixels (for k = 65).

The theoretical accuracy to measure the surface errors $\Delta$D depends also on the signal-to-noise ratio (SNR) of the measurements, defined as the ratio of the r.m.s.-noise and the peak amplitude of the antenna pattern:

$\Delta$D [mm] = 0.5 N / SNR 0.5

For $\Delta$D  = 0.1 mm and N = 176 a SNR of 63 db is required. Such a large dynamic range is not achievable by the standard quadratic detectors used for radioastronomical continuum observations. For this reason a backend with a linear detector was built and installed.

First Tests and Satellite Properties

First test observations were made in January and December 1996 using the beacon onboard ITALSAT F1 and F2 at 18.685 GHz. At that frequency the HPBW of the 100-m telescope is about 50''. For each experiment the appropriate coordinates have been provides by Telespacio, the ITALSAT Control Center, shortly before the observing run. The beacon signal was found to be quite stable variing less than 4% during one night and we noted a very small drift of the beacons frequency of less than 2 kHz, which is the bandwidth of the linear backend. Despite the actually provided expected satellite positions pointing corrections where found necessary to do about every 15 minutes due to the small beam of the 100-m telescope.

The maser receiver in the prime focus of the telescope was used for the observations, which can be tuned in a wide frequency range from about 18 GHz to 26 GHz. Antenna pattern of 64x64 pixels separated by 15'' were observed in a first run. We used three different focus settings (focussed and defocussed by +1 $\lambda$ and -1 $\lambda$ ) and observed four maps for each setting, two by scanning along the azimut direction and two along the elevation direction. Standard reduction techniques for radio continuum observations were applied, where orthogonal scanned maps give the most reliable combined result. The Misell-algorithmn was run with three pairs of pattern to check the resulting phase distributions. This is helpful to decide on the reliability and accuracy of the solution of the iteration process. In fact very similar results were found. The spatial resolution of about 3.5 m across the aperture does not allow to identify individual panels. The SNR of the combined antenna pattern was about 50 dB. This was sufficient to show the area of the previous outer wire mesh panels outstanding with quite large surface errors, as it was expected. For details of these observations see  Vassen (1997), where also a second observing run with 128x128 pixel maps is described. These data had a spatial resolution high enough to identify a panel de-adjusted by 3 mm. Panels, which were de-adjusted by a definite amount, are important to prove the scale accuracy of the derived surface deviations.

Unfortunately, the frequency of 18.685 GHz is close to the operation limit of the maser receiver leading to a substantial loss in sensitivity and a limited dynamic range of our results. In addition, the highly tapered feed of the maser is not ideal for the actual purpose of the holographic measurements, namely to adjust the replaced outer panels. For subsequent observations we used the 11.698 GHz beacon of EUTELSAT.

Holographic measurements in 1999

A new receiver with uncooled HEMTs was built in particular for observations of the 11.698 GHz beacon of EUTELSAT. A feed with low taper was selected to have sufficient sensitivity in the outer area of the antenna. After successful tests of the receiver and the suitability of the EUTELSAT beacon for our purpose in spring 1999, we run holographic measurements in May 1999. At 11.698 GHz a HPBW of 67'' is measured for the 100-m telescope. The beacon signal of EUTELSAT was found to be very stable in amplitude ( < 1% variation during one night), but its frequency drifted within about 7 kHz. In particular these drifts show up at sun-set or sun-rise and required to change the observing frequency accordingly, because of the 2 kHz bandwidth of the linear backend. Although actual positions were kindly provided by RegTp (Regulierungsbehörde für Telekommunikation und Post), it was necessary to do additional pointing corrections every 30 minutes. The total drift in position from the predicted coordinates was about 4' in AZ and in EL during one night.

Since we already know the error distribution across the antenna in general, it was sufficient to observe at two focus settings and we choose +0.75 $\lambda$ for the defocussed observations in addition to the focussed observations. For scale control two panels from ring 7 and 8 have been de-adjusted by +6 mm and -6 mm, respectively. The size of the antenna pattern was 140x140 pixels with a sampling of 35''. One coverage takes about 3 hours of observng time and was five times interrupted for pointing corrections. For each focus setting in total four maps were observed, which were scanned in orthogonal directions. The SNR for the combined focussed pattern was found to be 63 dB, thus we expect to identify surface deviations exceeding 0.1 mm.

Figure 1: Focussed antenna pattern at 11.698 GHz. The log-colourscale starts at -24 db below the peak intensity. Additional contours are shown at -3, -10, -20, -30, -40 and -50 dB of the peak intensity.
Click in the picture to get full resolution picture

1

Figure 2: Defocussed antenna pattern at 11.698 GHz. Contour levels are the same as in Fig. 1, while the log-colourscale starts at -20 dB below the peak intensity.
Click in the picture to get full resolution picture

  Results

We show the focused and the defoscused antenna pattern, which are combined from all measurements, in Figs. 1 and 2, respectively. Both antenna pattern show the cross like structure of enhanced sidelobes due to the four feed support legs of the 6.5 m subreflector. Linear higher order diffraction feature run roughly parallel to the arms of the cross. The large diameter diffraction ring has to be attributed to the subreflector.

In Fig. 3 and Fig. 4 we show the surface current amplitude and phase distribution resulting from the Phase Retrieval procedure. The areas which are blocked by the subreflector and the four feed support legs are blanked out. The amplitude distribution reflects the feed illumination with a decrease in sensitivity with increasing radius. The two de-adjusted panels are clearly visible in these figures. We measure values of -5.8 mm and +5.0 mm, which are slightly below the de-adjusted values of 6 mm. The reason is likely the limited spatial resolution of about 1.1 m, which causes some smoothing of the 6 mm steps at the edges of the de-adjusted panels. Also the peak deviations are not exactly centered on the panels, which indicates residual pointing effects despite our correction for the actual position of the satellite every 30 minutes. We measure a r.m.s. of the deviations for inner surface area of about 0.5 mm, which is the expected value and agrees with the previous holographic results by EIKONTECH LIMITED.

We have run the Phase Retrieval software for various sets of antenna pattern with small variations in pointing corrections and scaling of the different individual measurements, but found quite small changes in the phase distributions. Differences are below 0.4 mm for the phase distributions across the inner area. Exceptions are noted for the edges of the de-adjusted panels and in particular for the outer area with the new panels, where also the absolute surface errors are large. The uncertainty in this area is of the order of 1 mm.

Figure 3: Surface current amplitudes in arbitrary units. Maximum intensity is shown red and black is the minimum of the atan-colourscale. The superimposed grid shows the individual panels of the 100-m telescope.
Click in the picture to get full resolution picture

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Figure 4: Distribution of surface errors of the 100-m telescope in summer 1999. The linear colourscale (see wedge) covers the range from +3 mm to -5 mm. The two de-adjusted panels of ring 7 and 8 are clearly visible. Positive deviations are further away from the focus.
Click in the picture to get full resolution picture

Summary and Outlook

We have successfully measured the distribution of surface errors for the Effelsberg 100-m telescope by holography using satellite beacons and applying the Phase Retrieval method. We found quite large adjustment errors for the new outer panels, which have been mounted relative to their neighbouring panels. The existing errors along the outer edge of the ring 14 panels are transfered to ring 15 and the systematic errors increase for ring 16 and 17. A first correction based on the holographic result shown in Fig. 4 has been applied in late summer 1999. Afterwards the efficiency and beam width at 32 GHz are just marginally changed, which shows that the applied adjustment needs further improvement. New holographic measurements and a further correction of the panels setting will be made.

Acknowledgements: Many collegues contributed to this project. In particular we like to thank Dave Morris, IRAM, for providing his Phase Retrieval software package and for helpful advice during the initial stages of this project. O. Lochner  and K. Grypstra build the 11.698 GHz receiver and the linear backend. J. Neidhöfer helped to set-up some special observing procedures and P. Reich improved the reduction software. We are very grateful to Mr. A. Calfapietro (Telespacio) for providing the actual positions of ITALSAT and Mr. P. Steiner (RegTp) for those of EUTELSAT.
 

References
Greve(1981)
Greve, A., Zeitschrift für Vermessungswesen, 106, 308, 1981.
Godwin et al.(1986)
Godwin, M. P., Schoessow, E. P., and Grahl, B. H., Improvement of the Effelsberg 100-m telescope based on holographic reflector surface measurement, Astron. Astrophys., 167, 390-394, 1986.
Godwin and Schoessow(1990)
Godwin, M. P., and Schoessow, E. P., Final report on prime focus mode holographic tests on the 100 metre antenna, Report by EIKONTECH LIMITED, 1990.
D.L. Misell(1973)
Misell, D.L., A method for the solution of the phase problem in electron microscopy, Journal Phys. D: Applied Physics, 6, L6-L9, 1973.
D. Morris(1985)
Morris, D., Phase Retrieval in the Radio Holography of Reflector Antennas and Radio Telescopes, IEEE Transactions on Antennas and Propagation, AP-33, No. 7, 749-755, 1985.
Ruze(1966)
Ruze, J., Antenna tolerance theory - a review, Proc. IEEE, 54, 633-640, 1966.
Vassen(1997)
Vassen, S., Holographische Vermessung des Effelsberger 100-m Teleskops, Diplomarbeit, Universiät Bonn, 1997.
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