To: ALMA-FEIC group

### On the limit of necessary sampling interval for the beam scanner:

This is a follow-up to our telecon today.

Clearly lambda/2 is not always sufficient. For example, if there were a (e.g. reflected) wave coming in at +90 degrees to the axis of the scanner, that could not be distinguished from one arriving at -90 degrees. Both waves would cause a phase gradient across the plane of the scanner of 2.pi radians per wavelength, so sampling at lambda/2 we could not tell the difference between +pi or -pi per sample point. This is an aliased signal. We need to oversample enough so that we are at least a beamwidth away from receiving an aliased signal, or roughly (lambda/2)*[(N-2)/N] where N is the number of points across the map, and the FF point spread function is assumed to extend over 2 points. Very rough approximation. For N=80 that would mean sampling at about 0.487*lambda

So, although to be completely unambiguous, slightly more often than lambda/2 sampling is needed, nevertheless we were too conservative in suggesting lambda/4 in our teleconference today. Lambda/4 is not required. In "An Overview of Near-Field Antenna Measurements", IEEE Trans. Antennas & Prop. Vol 34 p.30 (1986) by A.D. Yaghjian, on p.39 he gives a formula:
ds=(lambda/2)/sqrt{1+(lambda/d)2}
where ds is the maximum allowable sampling interval, lambda the wavelength, and d the minimum distance between the test probe of the near-field scanner and the antenna under test. For lambda=10mm and d=100mm, this would be a maximum sample interval of 0.4875*lambda. For lambda=4mm it would be 0.4996*lambda.

I've ignored it here, but the angular response of the probe helps too (by attenuating signals at ±90 degs.)

On the grounds that at least a small amount of oversampling is always good, I'd suggest for our wide-angle beam measurement test that we sample at about 0.4*lambda. More often never does any harm.

Does this sound reasonable?

Cheers,

Darrel.