Neat idea. How much energy do you lose (assuming a vacuum)?
You could imagine some kind of dish that catches the energy, but that's the extent of my knowledge -- I don't know how you'd convert it back to propulsion.
Not entirely the right question. Usually in spaceflight you are overwhelmingly concerned with efficiency, since you have to carry your fuel with you. But with laser-boosted light sails, you leave your engines at home. At reasonable distances, (100+ AU) most of the beam is wasted, but the spacecraft doesn't care about that.
Your main losses will be just from inverse-square law. Even if your ship is reflecting a laser off its rear end, you can only focus the beam so much; at some point the spot size will be bigger than the ship, and you'll be in the inverse-square regime (same surface area, 1/r^2 energy within it).
That doesn't seem fundamental, though. You could have a series of relays that catch the beam, convert it to energy, and re-beam it out.
But I'm sure the efficiency would be awful, and if you have a chain of N of these things, now you're dropping off exponentially with N. And N is linear in distance. Hm... this isn't sounding like such a great workaround anymore.
Not to mention that the incoming beam would be shoving your relay forward. Does the outgoing beam push it backwards? I don't know how that works. (Even if it does, you'd be shoved forward proportionally to the energy loss.)
Bleagh. You'd be better off sending out a series of energy pellets well in advance that a traveling ship would scoop up along the way. That must be what Pac-Man was all about...!
> That doesn't seem fundamental, though. You could have a series of relays that catch the beam, convert it to energy, and re-beam it out.
That's a lot of hardware to send out, though. With a dedicated relay you could afford very large collecting surfaces, to compensate a bit for the conversion/retransmission efficiency loss, but there's a bigger problem: you can't just put a chain of relays on a line inside a planetary system. You have to put them in orbit of the Sun (even if by proxy of an orbit around a planet/moon). This means your initial line of relays will quickly drift out of alignment, making the path through them much longer than beaming straight at a ship that's transferring between planets or out of the system. You'd have to put rings of relays at various heights above the sun to guarantee a reasonably short path, and that would take a lot of relays. And work only for a single plane - if you want relayed power at arbitrary plane, you'd have to build shells of relays - so the amount of satellites you need to deploy just squared.
You don't even need a dish to catch the energy; just use a mirror that reflects the laser. You will lose energy to particles in the laser beam, and possibly also due to redshift.
You could imagine some kind of dish that catches the energy, but that's the extent of my knowledge -- I don't know how you'd convert it back to propulsion.