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To efficiently connect a high-current-density load to a magnetically insulated vacuum transmission line, we require that the regularity of the electron flow be conserved. Regularity, i.e., Brillouin flow, can be preserved by keeping the invariants of the motion constant along the flow and requiring that small changes in these be adiabatic. It is shown that by matching the vacuum impedances of the various transmission line sections and assuring smooth transition between them, Brillouin flow of sheath electrons continues almost unperturbed to the load without losses. This idea is numerically demonstrated for a new type of diode which we name Brillouin diode.