This device doesn't use POC but it is interesting to analyze all the same. I think the principle of operation is charging by induction.
This is as close as I can get with a schematic. The diode bridge connects to one plate of the capacitor only. The load is applied between the spark gap and ground (I think). He says there is a full wave bridge rectifier (I've just put a single diode) on the negative side of the pancake coil to ground. The magnetic field from the capacitor plate will cause a voltage drop on the pancake coil.