A theory has been developed that describes relativistic cyclotron resonance masers with axially inhomogeneous external magnetic fields. The axial profiling of the magnetic field permits one to maintain the cyclotron resonance between the electromagnetic wave and decelerating electrons that is especially important for relativistic gyrotraveling wave tubes, gyroklystrons, and gyrotwystrons. Some general features of particle trapping by the wave are considered and the possibility of achieving a high electron efficiency is analyzed in a simple model of such an interaction.

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