External microwave propulsion technology for single-stage-to-orbit space flight [pdf]
escapedynamics.com
escapedynamics.com
They all seem to peter out at 25 km or so. The goal with lasers was to pick a wavelength that the atmosphere was relatively transparent too but could be easily converted into heat through absorption on the receiving end. Microwaves would work to if you are careful about your wavelengths and don't point them at satellites.
[1] http://science.nasa.gov/science-news/science-at-nasa/1999/pr...
[2] https://youtu.be/KtH-SxqdtaA?t=1m
[3] http://www.space.com/7067-laser-propulsion-wild-idea-finally...
The breakdown field strength of gases decreases as pressure declines until a minimum of a few Torr (see Paschen's Law: https://en.wikipedia.org/wiki/Paschen%27s_law).
The power density in a beam focussed on a launch vehicle would be pretty high. Anybody know if air can sustain beam propagation at the E-field gradients that would be needed for this to work?
http://www.technologyreview.com/view/420700/microwave-powere...
Interesting. But since your power requirements are basically your specific impulse times your thrust that'll up the power they need. Still, electricity is cheaper than rockets.
Edit: compressed air taken from the atmosphere in-flight
Edit: My dad mentioned in school doing some scut work for a group designing an external combustion engine. Use the shock waves off the body to compress air, add fuel, zoom. Not sure if you could make that work with with an external heat source.
Side note: My understanding why people want to use hydrogen is it's a light particle, making it easy to accelerate it to high speeds. The higher exhaust velocity the better the ratio of payload to fuel becomes.
There are some significant drawbacks to this approach (assuming you managed to overcome the technical limitation of making everything work at the temperature and radiation involved) such as massive radiation leakage, which is why this was intended to be part of a weapon system that could drop nuclear bombs and then just spend its time flying around your enemies cities irradiating them until they shoot it down or it runs out of fuel, causing yet another nuclear disaster to deal with as it crashes.
The successful test run shows the design can work, so an external microwave heat source could do the same if you could transfer enough power.