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Why don't we investigate this more? It sounds very promising. What are the primary obstacles?

Some would include that it super heats the air to gain thrust, as you get higher the air gets thinner meaning less thrust. Also there are limits to how narrow you can keep a laser beam through air and atmospheric absorption and diffusion.

All these things combine to make it more difficult to get energy to the craft as it get's further away.

Gentlemen, I think I've solved our problem of air thinness at high altitudes. We use a RAM jet like shape for the rocket to funnel the air and compress it.

The rocket could like this:

  thin air
  
  \   /
   | |
   ||
  
  compressed air
  
  laser 
    |
  _____

Would adaptive optics help? and/or Could the rocket carry its own propellant but get power from the beam? E.g., carry its own air.

I don't know what you mean when you say adaptive optics sorry, but yes it could carry it's own air, I'm not sure of the practicalities but it's possible.

Also there are hard physical limits on how finely you can converge a point of light, MinimumDiameter = (2.44wavelengthfrequency)/LenseDiameter This is called the minimum spot size.

Adaptive optics lets telescopes compensate for atmospheric distortions to get similar performance to telescopes in space. It's also used for lasers. I don't know how it works but it sounds like it would be applicable here.

Your minimum diameter formula doesn't sound too restrictive.

'Adaptive optics' generally means lenses and the like whose shape (and possibly other properties) varies during operation to meet current operational requirements.

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