Little Luneburg Lenses - Part 1

Peter Hall, 26 October 1999

 

In my e-mail of 20 September, I mentioned a few of the advantages of using relatively small, completely spherical, Luneburg lenses as opposed to the large hemispheres which had been introduced by Graeme James and his colleagues, and which were then dominating our thinking. I also made the point that, if mechanical movement of any type is involved, it's probably easier to steer a ~5m lens and its associated integral feed structure (e.g. printed feed) than to mechanically translate, with high precision, the large feed boom structure needed for the big lenses. In this short note, I want to summarize briefly some other thoughts, especially as they relate to little Luneburg Lens (LLL) realization.

First, to build a "mid-range" SKA, it is clear than an LLL cannot be smaller than about 5 m in diameter if it is to form an effective primary beam at the lower frequencies. A 5 m lens is about a 5l aperture at 300 MHz - just adequate for effective operation. Like the small paraboloids of the SETI Institute's 1hT though, such an LLL has a chance of operating to beyond 12 GHz: the minimum scientifically useful high-frequency limit set by the Amsterdam deliberations in April this year. With suitable broadband feeds, both the reflector and the lens could eliminate the need for a separate high frequency SKA - something which is impractical for planar arrays needing to close-pack elements of widely differing sizes. Of course, the actual upper frequency limit for the LLL is limited largely by the loss in the lens dielectric.

The small lens looks to be a good trade-off because its optical beamforming effectively removes the most difficult first-level element interconnect problem of the planar arrays, while avoiding the much higher loss of the large lenses. In a related vein, the amount of (lossy) dielectric material is decreased for a given collecting area made of small lenses (radius r), since the lens aperture scales as r2, while the amount of dielectric scales as r3; apart from any loss consideration, the cost of the dielectric used in forming the (large) given area is then proportional to the radius of the constituent lenses.

Although the calculations are simple, it is worth mentioning some typical numbers for the losses of ambient temperature lenses. A loss of 0.1 dB ahead of the feed contributes an additional 7 K equivalent noise temperature; 0.5 dB loss adds 35 K; 1 dB adds 75 K. Over the mid-range SKA band, one might hope for an integrated LNA noise temperature of perhaps 35 K. It is important to appreciate therefore that even a modest 0.5 dB dielectric loss halves the sensitivity of the astronomical system. Of course, all antennas have their imperfections and, while the degradations in a reflector-based SKA may not amount to a factor of 2, the losses in ambient temperature planar array aggregation networks are probably in the same league as the 0.5 dB Luneburg realization. These numbers are critical in examining the various SKA concepts and those of us interested in Luneburg Lenses must soon produce some loss estimates, together with representative scaling relationships for lens diameter and frequency.

The feed question is of course critical to the practicality of any SKA Luneburg Lens implementation. Having raised the earlier idea of printed feeds on the (presumably hardened) surface of movable lenses, I concede the difficult of actually connecting to the feed array. While fibre optics to individual feeds are a possibility, it is unlikely that optical links with sufficient bandwidth and signal-to-noise ratio will become cheaply available on a decade timescale. Also, the actual mechanical flexibility and robustness of the connections are obviously a problem. The switchable "invisible antenna" feeds mentioned by Ray Norris share most of the opto-electronic difficulties, as well as adding the uncertainty of whether such elements can in fact be made, at least with even usably small losses. If we are looking for reliable, cheap, feeds likely to be practical in the next 10 years for an SKA of perhaps 40,000 LLLs, we are most likely looking at stationary arrays of printed feeds, perhaps augmented by a small number of high efficiency, or even cooled, translatable feeds.

Apart from the obvious practical reasons for displacing such a feed array from the surface of the lens, a little doodling with curvilinear squares will show that Luneburg Lenses do exhibit an aperture edge-brightening effect (should I call this "limb brightening" given the likely origin of Mr Luneburg's name?). The analysis by Morgan [1] derives an expression for the aperture power flow in terms of the feed pattern and a multiplicative factor which is maximum at the edge of the sphere. If the feed is placed at the surface of lens, then the edge brightening is infinite unless the feed has zero output 90° off its boresight. If the feed is set back from the sphere, the edge brightening becomes finite. The important point though is that a degree of care is necessary in illuminating the lens: my first notion of printing a relatively simple excitation element on to the sphere's surface results in a markedly inferior aperture illumination and (by the Fourier relationship) increased far-field sidelobe levels. Controlled illumination most likely demands a feed or feed surface displaced some distance from the lens.

Finally, I note that we need to consider the imaging advantages of the LLL solution - all the traditional arguments favouring large-N arrays apply. Of course, faced with the possibility of 40,000 x 5 m lenses, a first-generation SKA still needs to implement some sort of aggregration hierarchy, if only to allow a practical correlator to be built. Four hundred such lenses give the same collecting area as a 100 m diameter dish, but of course offer enormously more flexible imaging and beamforming performance. The picture below, produced by Ben Simons of VisLab, shows what such an SKA station might look like when located in the Australian desert.

 

 

Reference

1. Morgan, S.P., General Solution of the Luneburg Lens Problem, J. App. Phys., 29(9), 1958.

 

 

 

 

 

 

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