Power beaming is clearly central to space-based solar power concepts. Here I will provide a quick overview of my understanding of power beaming, the various equations involved, typical example calculations.
If power beaming were efficient and cheap, I believe space-based solar power would be quite viable even for grid power. However it’s not, and that largely has to do with the distances involved AND the fact that you need to convert energy multiple times, with losses along the way. The distances involved aren’t a complete show-stopper, since you can solve that problem just by operating at a large enough scale. However, the conversion inefficiencies (and the need to dump waste heat, etc) is not going to go away simply by operating at greater scale (although it helps).
The first equation we need is the diffraction limit. Roughly speaking, the spot size of a transmitted beam (microwave or laser) is:
Spot size = distance-to-spot * wavelength/(aperture diameter).
This is close enough for an order-of-magnitude estimate. More detailed work to follow.
But if we have a satellite out in Geosynchronous orbit (36000km altitude) transmitting power at roughly 10GHz (3cm wavelength, the shortest wavelength that still penetrates readily through the atmosphere) with an antenna 300m in diameter (NRO SIGINT/ELINT satellites are rumored to be that big, but maybe only around 100m in diameter), you’d have a spot size on the order of:
3.6E7m*3E-2m/(3E2m) = 3.6E3m or 3.6km in diameter…
…turns out that not all the energy of your beam is contained in this diameter (“Where’s that factor of 1.22,” you cry), but that’s a halfway decent start (and you’d need an infinitely wide aperture to collect all the energy in the beam…). 3.6km is obviously huge. The biggest full-aperture dish ever built is the half-way finished Chinese Arecibo clone at 500m. Still, there are ways to tweak this.
With a laser operating at 1micron, in medium-Earth-orbit (10000km) with 1 meter diameter optics needs only a:
1E7m*1E-6m/1m=10m diameter receiver to receive the vast majority of the beam’s energy. This is much, much better, obviously. You could put a 10m diameter receiver on top of a tethered airship or drone or something that allows you to transmit it to the ground without interference from clouds.
Or heck, use it to power high-altitude aircraft… but that’s a whole ‘nother blog post! (And suffice it to say, there are lots of caveats about laser transmission of energy, too.)