I am not sure that there is a single 'field' (of academic study) that would focus on such a matter. The phenomenon of energy transmission without wires can be achieved in a great many ways (microwave EM, magnetic coupling, even optical (!)) and so if you could describe the power requirements (mW or MW), the challenges (underwater, through vacuum etc.) then I would be happy to discuss matters here.
Your questions has so many potential answers that I think some filtering would be necessary.
Your question confuses me, a little. Amplification *is* the process of increasing power. If the actual energy powering the remote client is being transmitted, amplification is not needed. Either the question is one of remote control (I don't think so) or it is one of power transmission and regulation.
At it's heart, these are physics questions. Energy transmission and transformation. Page Carnot, Stirling, Otto, Johnson - and particularly Tesla.