Wireless transmission of energy through a layer of non-conductive material
To transmit power across a gap you can do it by transformer action but the efficiency will be low because the magnetic field from the power coil doesn't 100% couple to the receive coil. Depending on the gap the coupling may be only (say) 10% and this means you have to use a lot of brute force to get the power you need on the receive coil.

A significant improvement on this is to use tuned coils operating at resonant frequency. 

I'm going to describe one I was involved with last year - it didn't achieve anything more than 10% power coupling but, the constraints on the application were enormous - this is why I'm using it as an example - if you can avoid these constraints (described below) then you should get easily better than 50% power efficiency.

It needed to transfer power to the end of a 500MW Power generator's turbine rotor and the shafts are quite large. In effect the receive coil was about 1.5 metres diameter: -



The red circular line is the coil sitting in a raised insulated groove above the metal of the rotor. The metal was expected to give poor coupling but at the frequency used, it wasn't too bad at sucking the power away (maybe 20% loss). The coil was a single turn of 1.6mm Cu. 

The stator coil coupled to about 30º of the circumference i.e. it wasn't a full turn of 1.5m diameter. It was a 4 turn coil. Each of the 4 turns used 3 litz wires of diameter 1mm with 250 strands in each litz wire, 750 strands in total. With a full circular coil, power-coupling would be better; probably something like double.

The coil-gap was about 40mm and, despite all the constraints, the arrangement could easily produce 50VRMS on the tuned rotor coil. The power needed on the rotor was about 2W maximum. Smaller gaps are going to be more effective.

The stator coil had two tuning capacitors, one in parallel and one to bring the 600kHz AC feed in. The feed-in voltage was about 30VRMS but due to the tuning it was generating about 100VRMS across the coil. Better litz wire would improve the voltage and reduce the power losses - the coil rose about 5 to 10ºC when operating under ambient conditions. This design was also constrained with a small diameter 600kHz input power-feed cable (about 3mm for screened twisted pair) and the cable was about 10 metres long - having the 600kHz generator up close to the power coil is the way to get more efficiency in power-coupling.

Also, the efficiency wasn't that great because of the 40mm gap and the need to run at temperatures approaching 100ºC.

Summary - yes, you can transmit power magnetically across a gap and having a full loop on both the primary and secondary will couple much better than the fractional coupling described above. Having a smaller gap also couples much better. The receive loop described above had a single turn because it was also coupling data back out from the turbine mounted electronics - more turns on this winding would help - try and match the input coil profile and turns with the receive profile and turns. Use of ferrite material was impractical on this particular job but they would help.