Oh boy..... This is a bad and inefficient idea. The atmosphere causes, if I remember right, refraction. This will spread out the beam regardless of how tightly you have it focused. Further, you're talking of stacking energy losses! If you wanted to do the whole "put it in space to always get light!" Idea, why not put the solar panels themselves in space and beam the power back through microwaves, as others have suggested? I don't think the microwaves get scattered nearly as much, though there is still a large energy loss.
Further, bringing day to the night, even locally, is likely to have big consequences. Lights at night already disrupt many animals cycles. Turtles being a good, albeit overused, example.
Also cost wide, is this even practical? Yes you are utilizing solar panels at night when they are otherwise not in use. However, how many mirrors do you need to achieve worth while results? And how expensive is it to get them up their and working properly, hopefully for long enough to have a good ROI.
In short, this is short sighted. There are better, cheaper ways currently available. It's also not as effecient as other ideas suggested, and has a greater energy loss. Maybe once manufacturing in space becomes big, you might be able to make a better argument for an idea like this for another use, but for now? It's just not a good idea in my opinion. And I'm well younger than this guy is.
