Do these massive telescopes have a future, when we could just build smaller ones and use aperture synthesis to pretend like we have a really really big telescope?
> The thinned-array curse (sometimes, sparse-array curse) is a theorem in electromagnetic theory of antennas. It states that a transmitting antenna which is synthesized from a coherent phased array of smaller antenna apertures that are spaced apart will have a smaller minimum beam spot size, but the amount of power that is beamed into this main lobe is reduced by an exactly proportional amount, so that the total power density in the beam is constant
So you can transmit with a sparse array, but the power you can deliver with it (for a set synthetic aperture size) decreases with the degree of sparseness. No free lunch.
What's the tl;dr on the math here?
If you leave "gaps" in your array elements, that's not a problem receiving strong, periodic signals. You can physically move the elements to fill in the gaps to remove those lobes; the equivalent of an equivalent time oscilloscope or a synthetic aperture radar. The problem is you can't do that when transmitting since you only have one opportunity to send something (non-periodic).
There are other disadvantages with phased arrays. Each amplifier has to be matched in amplitude and phase. I've had to bin lots of amplifiers as some have phase compression and some phase expansion over amplitude. That distorts the pattern over the pulse.
On the receive side, each receiver has thermal noise un-correlated with the others. So for N antenna elements, you received signal-to-noise scale proportional to sqrt(N) instead of N like you have on transmit, or with a big reflector. This is why arrays suck for very weak signals. Notice NASA keeps building big dishes for their DSN instead of "cheaper" arrays.
Then there's the whole timing and phase noise issue with distributing an oscillator.
Tough you can only make a reflector with about 100 dB gain.